The first table shows all symmetric positions, counts and depths. These are new results as of March 2014.
Summary of C_1 in QTM
dist             pos          modM      modM+inv mod48
---- --------------- ------------- ------------- -----
   0               1             1             1     1
   1              12             1             1    12
   2              18             3             3    18
   3             108             5             3    12
   4             411            19            14    27
   5           1,104            46            25     0
   6           3,744           163           102     0
   7          11,760           490           258     0
   8          36,731         1,582           902    11
   9         111,144         4,632         2,370    24
  10         358,138        15,054         7,908    10
  11       1,028,848        42,895        21,647    16
  12       3,266,949       136,691        70,284    21
  13       9,443,588       393,602       198,149    20
  14      29,201,318     1,218,544       618,945    38
  15      83,765,676     3,490,523     1,752,632    12
  16     268,136,523    11,177,948     5,644,771    27
  17     819,440,112    34,146,994    17,140,681     0
  18   3,083,699,868   128,516,587    64,687,074    12
  19  11,628,867,276   484,563,973   243,091,188    12
  20  41,538,350,563 1,730,948,894   869,224,590    19
  21  67,617,360,740 2,817,563,628 1,413,565,363    20
  22  37,373,063,137 1,557,455,774   783,700,221     1
  23   2,147,815,036    89,510,797    45,330,245    28
  24          78,820         3,635         3,324     4
  25              36             2             2    36
  26               3             1             1     3
---- --------------- ------------- ------------- -----
 tot 164,604,041,664 6,859,192,484 3,445,060,704     0
The remaining tables give counts for all the individual symmetry subgroups. In the tables below, the "pos" column gives all positions in the subgroup (including those that happen to have more symmetry); the "modM" and "modM+inv" columns however are for only positions with *exactly* that symmetry.
Summary of C_2_(a) in QTM
dist            pos          modM    modM+inv
---- -------------- ------------- -----------
   0              1             0           0
   1              4             0           0
   2             10             0           0
   3             36             4           2
   4            123            12           8
   5            368            46          25
   6          1,192           132          78
   7          3,856           482         254
   8         11,411         1,345         724
   9         36,296         4,536       2,318
  10        112,410        13,792       7,114
  11        335,920        41,965      21,157
  12      1,020,073       126,636      64,312
  13      3,058,828       382,235     192,220
  14      8,978,902     1,119,811     564,679
  15     26,377,392     3,296,942   1,653,623
  16     77,108,783     9,631,733   4,840,286
  17    227,640,076    28,452,180  14,264,211
  18    685,484,820    85,655,163  42,991,338
  19  2,007,223,500   250,890,643 125,744,086
  20  4,656,375,131   581,918,540 291,904,955
  21  5,304,119,504   662,991,213 332,570,156
  22  2,215,021,221   276,741,182 139,150,287
  23     75,323,248     9,413,673   4,800,651
  24          4,960           358         331
  25             12             1           1
  26              3             0           0
---- -------------- ------------- -----------
 tot 15,288,238,080 1,910,682,624 958,772,816

Summary of C_2_(b) in QTM
dist           pos        modM    modM+inv
---- ------------- ----------- -----------
   0             1           0           0
   2             2           0           0
   4             1           0           0
   7            32           8           4
   8           181          30          21
   9           312          78          41
  10           898         188         113
  11         2,560         640         341
  12         7,945       1,814       1,088
  13        29,548       7,387       3,823
  14        91,464      22,462      12,307
  15       385,910      96,454      49,328
  16     1,537,621     383,032     198,706
  17     7,225,614   1,806,031     914,400
  18    36,699,180   9,169,755   4,651,890
  19   180,982,220  45,241,940  22,772,973
  20   681,459,499 170,342,729  85,808,746
  21 1,061,220,706 265,286,959 133,536,783
  22   554,222,051 138,536,519  70,017,794
  23    24,172,938   6,040,783   3,116,475
  24           996         199         191
  26             1           0           0
---- ------------- ----------- -----------
 tot 2,548,039,680 636,937,008 321,085,024

Summary of C_2h_(a) in QTM
dist     pos    modM modM+inv
---- ------- ------- --------
   0       1       0        0
   2       2       0        0
   4       3       0        0
   6       8       2        1
   8      79      10        9
  10      96      22       19
  12     759     135      105
  14   1,712     364      304
  16   8,013   1,755    1,321
  18  34,418   7,893    5,624
  20 229,733  55,892   35,839
  22 314,539  77,374   47,539
  24     458     105      103
  26       3       0        0
---- ------- ------- --------
 tot 589,824 143,552   90,864

Summary of C_2h_(b) in QTM
dist    pos   modM modM+inv
---- ------ ------ --------
   0      1      0        0
   4      1      0        0
   8      9      4        4
  10      8      3        3
  12    107     38       38
  14    310    121      121
  15     50     25       20
  16  1,169    495      484
  17    828    413      278
  18  4,444  2,030    1,899
  19  7,896  3,947    2,633
  20 20,703 10,013    8,380
  21 35,616 17,802   12,031
  22 22,347 10,823    9,014
  23  4,762  2,381    1,926
  24     52     21       21
  26      1      0        0
---- ------ ------ --------
 tot 98,304 48,116   36,852

Summary of C_2v_(a1) in QTM
dist    pos   modM modM+inv
---- ------ ------ --------
   0      1      0        0
   2      2      0        0
   4      1      0        0
   8      1      0        0
  10      8      0        0
  12     97      8        6
  14    448     66       56
  15    156     39       27
  16  1,179    227      173
  17    988    247      183
  18  5,222  1,119      877
  19  4,744  1,186      908
  20 17,395  4,060    3,212
  21 22,660  5,665    4,318
  22  8,369  1,873    1,621
  23  4,220  1,055      964
  24     44      7        7
  26      1      0        0
---- ------ ------ --------
 tot 65,536 15,552   12,352

Summary of C_2v_(a2) in QTM
dist       pos    modM modM+inv
---- --------- ------- --------
   0         1       0        0
   2         2       0        0
   4         3       0        0
   6         8       2        1
   8        71       6        3
  10       108      23       16
  12       691     105       78
  14     2,622     587      415
  16    10,627   2,411    1,737
  18    60,046  14,253    9,436
  20   461,903 113,824   68,566
  22   642,105 159,312   94,815
  24     1,458     357      309
  26         3       0        0
---- --------- ------- --------
 tot 1,179,648 290,880  175,376

Summary of C_2v_(b) in QTM
dist    pos   modM modM+inv
---- ------ ------ --------
   0      1      0        0
   4      1      0        0
   8      5      2        2
  10     12      5        5
  12     81     25       25
  14    208     70       70
  15     44     22       18
  16  1,049    435      425
  17    662    331      231
  18  3,980  1,799    1,716
  19  6,504  3,252    2,250
  20 19,581  9,454    8,010
  21 37,002 18,501   12,423
  22 24,163 11,732    9,686
  23  4,940  2,470    1,974
  24     70     30       29
  26      1      0        0
---- ------ ------ --------
 tot 98,304 48,128   36,864

Summary of C_3 in QTM
dist       pos    modM modM+inv
---- --------- ------- --------
   0         1       0        0
   8         2       0        0
  10         4       1        1
  11        16       4        2
  12        51       9        7
  13         8       2        1
  14       254      60       41
  15       360      89       48
  16     2,004     487      309
  17     5,397   1,345      705
  18    31,131   7,732    4,431
  19   142,371  35,515   18,206
  20   678,691 169,386   90,740
  21 1,572,830 392,559  199,807
  22 1,176,979 293,578  161,531
  23   168,586  41,850   21,959
  24       451      99       94
---- --------- ------- --------
 tot 3,779,136 942,716  497,882

Summary of C_3v in QTM
dist pos modM modM+inv
---- --- ---- --------
   0   1    0        0
  12   1    0        0
  16   2    1        1
  17   3    1        1
  18   5    2        2
  19   7    3        3
  20   7    2        2
  21  10    2        2
  22   7    3        3
  23   4    2        2
  24   1    0        0
---- --- ---- --------
 tot  48   16       16

Summary of C_4 in QTM
dist     pos   modM modM+inv
---- ------- ------ --------
   0       1      0        0
   1       4      1        1
   2       6      0        0
   3       4      1        1
   4       1      0        0
   8       1      0        0
   9       8      2        1
  10      42      6        5
  11     160     40       21
  12     407     85       50
  13     820    205      125
  14     972    199      148
  15     912    228      146
  16   2,095    486      340
  17   8,252  2,063    1,225
  18  18,524  4,505    2,727
  19  26,848  6,712    3,817
  20  33,399  8,092    4,920
  21  34,476  8,619    4,868
  22  18,205  4,343    2,838
  23   2,240    560      418
  24      74     12       12
  25       4      1        1
  26       1      0        0
---- ------- ------ --------
 tot 147,456 36,160   21,664

Summary of C_4h in QTM
dist   pos modM modM+inv
---- ----- ---- --------
   0     1    0        0
   2     2    1        1
   4     1    0        0
   8     1    0        0
  10     6    3        3
  12    15    1        1
  14    76   33       33
  16    51   17       17
  18   172   78       77
  20   529  246      217
  22   659  315      250
  24    22   10        9
  26     1    0        0
---- ----- ---- --------
 tot 1,536  704      608

Summary of C_4v in QTM
dist   pos modM modM+inv
---- ----- ---- --------
   0     1    0        0
   2     2    1        1
   4     1    0        0
   8     1    0        0
  10     4    2        1
  12    35   11       10
  14    70   30       26
  16    71   27       25
  18   286  135      114
  20   399  181      163
  22   149   60       59
  24     4    1        1
  26     1    0        0
---- ----- ---- --------
 tot 1,024  448      400

Summary of C_i in QTM
dist            pos          modM    modM+inv
---- -------------- ------------- -----------
   0              1             0           0
   2              6             0           0
   4             27             1           1
   6            120             4           4
   8            863            27          23
   9            192             8           4
  10          7,620           304         182
  11          2,208            92          47
  12         75,567         3,051       1,720
  13         34,344         1,431         737
  14        739,556        30,549      16,360
  15        947,548        39,468      20,036
  16     12,058,761       501,258     258,311
  17     40,061,232     1,669,009     839,237
  18    359,580,434    14,977,366   7,553,717
  19  2,012,791,496    83,864,293  42,035,816
  20 10,624,291,329   442,645,324 222,062,307
  21 20,173,117,696   840,537,259 421,158,871
  22 11,935,588,411   497,271,348 249,789,819
  23    705,402,404    29,390,388  14,815,911
  24         14,422           526         465
  26              3             0           0
---- -------------- ------------- -----------
 tot 45,864,714,240 1,910,931,706 958,553,568

Summary of C_s_(a) in QTM
dist            pos          modM      modM+inv
---- -------------- ------------- -------------
   0              1             0             0
   2              6             0             0
   4             27             3             2
   6            120            12            10
   8            727            72            48
   9             32             4             4
  10          4,164           479           290
  11          1,008           126            65
  12         32,163         3,790         2,168
  13         16,368         2,046         1,084
  14        316,452        38,685        20,965
  15        406,704        50,827        25,984
  16      4,822,181       599,096       311,322
  17     16,168,084     2,020,845     1,020,096
  18    139,213,210    17,382,019     8,794,229
  19    782,908,280    97,861,909    49,127,729
  20  4,065,210,281   508,003,625   255,202,212
  21  8,026,900,428 1,003,353,303   503,169,354
  22  4,963,326,423   620,211,029   312,119,302
  23    346,541,944    43,316,508    21,865,028
  24         17,090         1,702         1,548
  26              3             0             0
---- -------------- ------------- -------------
 tot 18,345,885,696 2,292,846,080 1,151,661,440

Summary of C_s_(b) in QTM
dist         pos        modM   modM+inv
---- ----------- ----------- ----------
   0           1           0          0
   2           2           0          0
   4           1           0          0
   6           4           1          1
   8          57          11          6
   9          16           4          2
  10         168          36         27
  11          72          18          9
  12       1,503         320        204
  13       1,056         264        142
  14      14,754       3,481      2,042
  15      25,170       6,230      3,294
  16     209,771      51,682     28,121
  17     767,170     191,172     98,222
  18   5,101,310   1,272,105    656,218
  19  26,570,072   6,637,729  3,373,726
  20 110,516,385  27,615,011 13,993,693
  21 179,764,414  44,917,282 22,878,230
  22  96,491,977  24,109,621 12,278,007
  23   5,208,670   1,298,685    703,519
  24         706         140        137
  26           1           0          0
---- ----------- ----------- ----------
 tot 424,673,280 106,103,792 54,015,600

Summary of D_2_(edge) in QTM
dist    pos   modM modM+inv
---- ------ ------ --------
   0      1      0        0
   2      2      0        0
   4      1      0        0
   8     49     10        6
  10    130     27       23
  12    555    111       86
  14  1,234    261      204
  16  3,621    816      637
  18 12,456  2,893    2,173
  20 49,553 11,883    8,381
  22 30,615  7,217    5,196
  24     86     14       14
  26      1      0        0
---- ------ ------ --------
 tot 98,304 23,232   16,720

Summary of D_2_(face) in QTM
dist     pos   modM modM+inv
---- ------- ------ --------
   0       1      0        0
   2       6      0        0
   4      27      2        2
   6     120     10        7
   8     471     36       27
  10   1,686    138       85
  12   4,797    372      229
  14  11,472    921      563
  16  25,313  2,022    1,194
  18  77,460  6,222    3,623
  20 141,809 11,331    6,589
  22  31,707  2,301    1,432
  24      40      1        1
  26       3      0        0
---- ------- ------ --------
 tot 294,912 23,356   13,752

Summary of D_2d_(edge) in QTM
dist   pos  modM modM+inv
---- ----- ----- --------
   0     1     0        0
   4     1     0        0
   8     9     4        4
  10    12     6        6
  12    63    25       21
  14    92    41       34
  16   149    66       63
  18   456   220      191
  20 1,245   604      456
  22 1,023   497      368
  24    20     9        9
  26     1     0        0
---- ----- ----- --------
 tot 3,072 1,472    1,152

Summary of D_2d_(face) in QTM
dist pos modM modM+inv
---- --- ---- --------
   0   1    0        0
   4   1    0        0
   8   1    0        0
  10   2    1        1
  12  25    6        6
  14  66   28       25
  16  55   19       16
  18 110   47       43
  20 157   60       55
  22  87   29       28
  24   6    2        2
  26   1    0        0
---- --- ---- --------
 tot 512  192      176

Summary of D_2h_(edge) in QTM
dist   pos modM modM+inv
---- ----- ---- --------
   0     1    0        0
   4     1    0        0
   8     1    0        0
  10     2    1        1
  12    31    9        9
  14    68   29       29
  16   179   81       81
  18   382  183      183
  20   673  318      318
  22   699  335      335
  24    10    4        4
  26     1    0        0
---- ----- ---- --------
 tot 2,048  960      960

Summary of D_2h_(face) in QTM
dist    pos  modM modM+inv
---- ------ ----- --------
   0      1     0        0
   4      3     0        0
   8     39     6        6
  12    199    27       27
  14    132    17       17
  16    797   124      122
  18  2,324   379      373
  20  5,037   820      684
  22  3,743   608      442
  24     10     1        1
  26      3     0        0
---- ------ ----- --------
 tot 12,288 1,982    1,672

Summary of D_3 in QTM
dist pos modM modM+inv
---- --- ---- --------
   0   1    0        0
  12   5    2        2
  16   6    3        3
  17   1    0        0
  18  23   11        9
  19  31   15       10
  20  49   23       20
  21 132   63       43
  22 125   62       52
  23  52   26       19
  24   7    3        2
---- --- ---- --------
 tot 432  208      160

Summary of D_3d in QTM
dist pos modM modM+inv
---- --- ---- --------
   0   1    0        0
  12   1    0        0
  17   1    1        1
  18   1    1        1
  19   1    1        1
  20   3    2        2
  21   6    6        6
  22   1    1        1
  24   1    0        0
---- --- ---- --------
 tot  16   12       12

Summary of D_4 in QTM
dist pos modM modM+inv
---- --- ---- --------
   0   1    0        0
   2   2    1        1
   4   1    0        0
   8   1    0        0
  10   8    4        3
  12  43   15       14
  14  50   20       17
  16  63   23       20
  18  78   31       30
  20 177   70       63
  22  83   27       27
  24   4    1        1
  26   1    0        0
---- --- ---- --------
 tot 512  192      176

Summary of D_4h in QTM
dist pos modM modM+inv
---- --- ---- --------
   0   1    0        0
   4   1    1        1
   8   1    1        1
  12  13   12       12
  14  10   10       10
  16  17   17       17
  18  16   16       16
  20  37   36       36
  22  29   29       29
  24   2    1        1
  26   1    1        1
---- --- ---- --------
 tot 128  124      124

Summary of O_h in QTM
dist pos modM modM+inv
---- --- ---- --------
   0   1    1        1
  12   1    1        1
  20   1    1        1
  24   1    1        1
---- --- ---- --------
 tot   4    4        4

Summary of S_4 in QTM
dist     pos    modM modM+inv
---- ------- ------- --------
   0       1       0        0
   2       2       0        0
   4       1       0        0
   8      77      17       17
  10      72      13       10
  11      40      10        5
  12     397      80       62
  13     128      32       17
  14   2,982     692      473
  15     788     197      107
  16   6,665   1,611    1,022
  17  13,396   3,349    1,887
  18  42,878  10,543    6,073
  19  66,764  16,691    8,944
  20 109,157  26,825   15,621
  21 132,664  33,166   17,756
  22  58,879  14,292    8,795
  23   7,404   1,851    1,044
  24      72       7        7
  26       1       0        0
---- ------- ------- --------
 tot 442,368 109,376   61,840

Summary of S_6 in QTM
dist   pos  modM modM+inv
---- ----- ----- --------
   0     1     0        0
   8     2     1        1
  12     3     1        1
  14    14     7        6
  15     4     2        1
  16    48    23       17
  17    15     7        4
  18   173    85       62
  19   275   137       86
  20 1,077   533      394
  21 2,464 1,229      715
  22 2,521 1,256      944
  23 1,130   565      355
  24    49    24       24
---- ----- ----- --------
 tot 7,776 3,870    2,610

Summary of T in QTM
dist pos modM modM+inv
---- --- ---- --------
   0   1    0        0
  12   9    2        2
  16   2    0        0
  18   6    1        1
  20  29    5        5
  22  24    4        4
  24   1    0        0
---- --- ---- --------
 tot  72   12       12

Summary of T_h in QTM
dist pos modM modM+inv
---- --- ---- --------
   0   1    0        0
  12   1    0        0
  16   2    1        1
  18   2    1        1
  20   9    4        3
  22   8    4        3
  24   1    0        0
---- --- ---- --------
 tot  24   10        8

Summary of hard positions in QTM
       symm    pos  modM modM+inv
----------- ------ ----- --------
        O_h      1     1        1
       D_4h      6     2        2
        D_4      6     1        1
       C_4v      6     1        1
D_2h_(face)      6     1        1
D_2d_(face)     12     2        2
 D_2_(face)     12     1        1
D_2h_(edge)     24     4        4
        D_3     24     3        2
D_2d_(edge)     54     9        9
       C_4h     60    10        9
  C_2v_(a1)     84     7        7
        S_4     84     7        7
        C_4    156    13       13
 D_2_(edge)    168    14       14
        S_6    192    24       24
   C_2h_(b)    252    21       21
   C_2v_(b)    360    30       29
   C_2h_(a)  1,260   105      103
        C_3  1,584    99       94
    C_s_(b)  3,360   140      137
  C_2v_(a2)  4,284   357      309
    C_2_(b)  4,776   199      191
    C_2_(a)  8,616   359      332
        C_i 12,624   526      465
    C_s_(a) 40,848 1,702    1,548
----------- ------ ----- --------
        tot 78,859 3,638    3,327