← #30869522
Subgroups — #30869522
Order 17152.
614 conjugacy classes of subgroups.
The subgroup lattice has 250 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22569144 | C2 | 1 | 2 | yes | maximal |
| 2 | #22561187 | C2 | 1 | 2 | yes | maximal |
| 3 | #22560810 | C2 | 1 | 8 | yes | maximal |
| 4 | #22560347 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #22559954 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C67×#3160 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #16863527 | — | 2 | 2 | no | — |
| 8 | #16863527 | — | 2 | 2 | no | — |
| 9 | Q8×C67⋊C8 | — | 2 | 2 | no | — |
| 10 | Q8×C67⋊C8 | — | 2 | 2 | no | — |
| 11 | #16863423 | C22 | 1 | 4 | yes | — |
| 12 | #16863420 | C22 | 1 | 2 | yes | — |
| 13 | #16863352 | C22 | 1 | 4 | yes | — |
| 14 | C2×C4×C67⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 15 | C42×Dic67 | C22 | 1 | 1 | yes | characteristic |
| 16 | #16863161 | C22 | 1 | 4 | yes | — |
| 17 | #16863155 | C22 | 1 | 8 | yes | — |
| 18 | #16863150 | C22 | 1 | 4 | yes | — |
| 19 | #533×C67 | C22 | 1 | 2 | yes | — |
| 20 | #500×C67 | C4 | 1 | 1 | yes | characteristic, complemented |
| 21 | #500×C67 | C4 | 1 | 1 | yes | characteristic, complemented |
| 22 | #499×C67 | C4 | 1 | 4 | yes | complemented |
| 23 | #498×C67 | C4 | 1 | 2 | yes | complemented |
| 24 | #427×C67 | C22 | 1 | 4 | yes | — |
| 25 | C2×C4×C8×C67 | C22 | 1 | 1 | yes | characteristic |
| 26 | C2×#11761612 | C23 | 1 | 4 | yes | — |
| 27 | C22×C67⋊C8 | C23 | 1 | 2 | yes | — |
| 28 | C2×C4×Dic67 | — | 2 | 1 | no | — |
| 29 | C2×C4×Dic67 | — | 2 | 1 | no | — |
| 30 | C2×C4×Dic67 | C23 | 1 | 2 | yes | — |
| 31 | C4⋊(C67⋊C8) | — | 2 | 4 | no | — |
| 32 | C4⋊(C67⋊C8) | — | 2 | 4 | no | — |
| 33 | C4⋊(C67⋊C8) | — | 2 | 2 | no | — |
| 34 | C4⋊(C67⋊C8) | — | 2 | 2 | no | — |
| 35 | #14523954 | — | 2 | 4 | no | — |
| 36 | C4×C67⋊C8 | — | 2 | 4 | no | — |
| 37 | C4×C67⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 38 | C4×C67⋊C8 | D4 | 1 | 2 | yes | — |
| 39 | C4×C67⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 40 | C22×C8×C67 | C23 | 1 | 2 | yes | — |
| 41 | C4⋊Q8×C67 | C2×C4 | 1 | 2 | yes | — |
| 42 | C4⋊Q8×C67 | C2×C4 | 1 | 2 | yes | — |
| 43 | #125×C67 | C2×C4 | 1 | 4 | yes | — |
| 44 | C22⋊Q8×C67 | — | 2 | 2 | no | — |
| 45 | C22⋊Q8×C67 | — | 2 | 2 | no | — |
| 46 | C4×Q8×C67 | — | 2 | 2 | no | — |
| 47 | C4×Q8×C67 | — | 2 | 2 | no | — |
| 48 | #117×C67 | C23 | 1 | 4 | yes | — |
| 49 | C2×C42×C67 | C23 | 1 | 1 | yes | characteristic |
| 50 | #107×C67 | C2×C4 | 1 | 4 | yes | — |
| 51 | #107×C67 | C2×C4 | 1 | 4 | yes | — |
| 52 | #106×C67 | C2×C4 | 1 | 8 | yes | — |
| 53 | C4×C8×C67 | C2×C4 | 1 | 1 | yes | characteristic |
| 54 | C4×C8×C67 | C2×C4 | 1 | 2 | yes | — |
| 55 | C4×C8×C67 | C2×C4 | 1 | 1 | yes | characteristic |
| 56 | C22×Dic67 | — | 2 | 2 | no | — |
| 57 | #11761612 | — | 2 | 8 | no | — |
| 58 | C2×C67⋊C8 | — | 2 | 4 | no | — |
| 59 | C2×C67⋊C8 | — | 2 | 4 | no | — |
| 60 | C2×C67⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 61 | C2×C67⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 62 | C4×Dic67 | — | 2 | 4 | no | — |
| 63 | C4×Dic67 | — | 2 | 4 | no | — |
| 64 | C4×Dic67 | — | 4 | 2 | no | — |
| 65 | C2×Q8×C67 | — | 2 | 2 | no | — |
| 66 | C2×Q8×C67 | — | 2 | 2 | no | — |
| 67 | C22×C4×C67 | #41 | 1 | 1 | yes | characteristic |
| 68 | C22×C4×C67 | #41 | 1 | 1 | yes | characteristic |
| 69 | C22×C4×C67 | C24 | 1 | 1 | yes | characteristic |
| 70 | C2×C8×C67 | — | 2 | 4 | no | — |
| 71 | C2×C8×C67 | C22×C4 | 1 | 2 | yes | — |
| 72 | C2×C8×C67 | C22×C4 | 1 | 2 | yes | — |
| 73 | C4⋊C4×C67 | — | 2 | 4 | no | — |
| 74 | C4⋊C4×C67 | — | 2 | 4 | no | — |
| 75 | C4⋊C4×C67 | — | 2 | 2 | no | — |
| 76 | C4⋊C4×C67 | — | 2 | 2 | no | — |
| 77 | C4⋊C4×C67 | C22×C4 | 1 | 4 | yes | — |
| 78 | C4⋊C4×C67 | C22×C4 | 1 | 4 | yes | — |
| 79 | C22⋊C4×C67 | — | 2 | 4 | no | — |
| 80 | C42×C67 | — | 2 | 4 | no | — |
| 81 | C42×C67 | C22×C4 | 1 | 1 | yes | characteristic |
| 82 | C42×C67 | C22×C4 | 1 | 1 | yes | characteristic |
| 83 | C42×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | C42×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | C23×C67 | C2×#41 | 1 | 1 | yes | characteristic |
| 86 | Q8×C67 | — | 4 | 4 | no | — |
| 87 | Q8×C67 | — | 4 | 4 | no | — |
| 88 | C2×C4×C67 | — | 2 | 2 | no | — |
| 89 | C2×C4×C67 | — | 2 | 2 | no | — |
| 90 | C2×C4×C67 | — | 2 | 4 | no | — |
| 91 | C2×C4×C67 | — | 2 | 4 | no | — |
| 92 | C2×C4×C67 | C4×D4 | 1 | 1 | yes | characteristic |
| 93 | C2×C4×C67 | C4×D4 | 1 | 1 | yes | characteristic |
| 94 | C2×C4×C67 | C4×D4 | 1 | 1 | yes | characteristic |
| 95 | C2×C4×C67 | C4×D4 | 1 | 1 | yes | characteristic |
| 96 | C2×C4×C67 | C23×C4 | 1 | 1 | yes | characteristic |
| 97 | C2×C4×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 98 | C2×C4×C67 | 2−1+4 | 1 | 4 | yes | — |
| 99 | C2×Dic67 | — | 2 | 4 | no | — |
| 100 | C2×Dic67 | — | 4 | 2 | no | — |
| 101 | C2×Dic67 | — | 4 | 2 | no | — |
| 102 | C8×C67 | — | 2 | 2 | no | — |
| 103 | C8×C67 | — | 2 | 2 | no | — |
| 104 | C67⋊C8 | — | 2 | 2 | no | — |
| 105 | C67⋊C8 | — | 2 | 2 | no | — |
| 106 | C67⋊C8 | — | 4 | 4 | no | — |
| 107 | C22×C67 | — | 2 | 1 | no | — |
| 108 | C22×C67 | — | 2 | 1 | no | — |
| 109 | C22×C67 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 110 | C22×C67 | #541 | 1 | 2 | yes | — |
| 111 | C4×C67 | — | 2 | 1 | no | — |
| 112 | C4×C67 | — | 2 | 1 | no | — |
| 113 | C4×C67 | — | 4 | 4 | no | — |
| 114 | C4×C67 | — | 4 | 4 | no | — |
| 115 | C4×C67 | #443 | 1 | 2 | yes | — |
| 116 | C4×C67 | #444 | 1 | 2 | yes | — |
| 117 | C4×C67 | #519 | 1 | 2 | yes | — |
| 118 | C4×C67 | #536 | 1 | 2 | yes | — |
| 119 | Dic67 | — | 4 | 4 | no | — |
| 120 | #22256 | — | 67 | 1 | no | maximal, Sylow |
| 121 | C2×C67 | — | 2 | 2 | no | — |
| 122 | C2×C67 | #2327 | 1 | 1 | yes | characteristic |
| 123 | C2×C67 | C2×#443 | 1 | 1 | yes | characteristic |
| 124 | C2×C67 | C2×#444 | 1 | 1 | yes | characteristic |
| 125 | #3160 | D67 | 1 | 1 | yes | characteristic |
| 126 | #2960 | — | 67 | 2 | no | — |
| 127 | #1994 | — | 67 | 2 | no | — |
| 128 | #1922 | — | 67 | 8 | no | — |
| 129 | #1837 | — | 67 | 1 | no | — |
| 130 | #1744 | — | 67 | 1 | no | — |
| 131 | C67 | #22256 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 132 | #533 | C2×D67 | 1 | 2 | yes | — |
| 133 | #500 | Dic67 | 1 | 1 | yes | characteristic, complemented |
| 134 | #500 | Dic67 | 1 | 1 | yes | characteristic, complemented |
| 135 | #499 | Dic67 | 1 | 4 | yes | complemented |
| 136 | #498 | Dic67 | 1 | 2 | yes | complemented |
| 137 | #446 | — | 134 | 2 | no | — |
| 138 | #446 | — | 134 | 2 | no | — |
| 139 | C8×Q8 | — | 134 | 2 | no | — |
| 140 | C8×Q8 | — | 134 | 2 | no | — |
| 141 | #427 | C2×D67 | 1 | 4 | yes | — |
| 142 | #424 | — | 67 | 4 | no | — |
| 143 | #423 | — | 67 | 2 | no | — |
| 144 | #405 | — | 67 | 4 | no | — |
| 145 | C2×C4×C8 | — | 67 | 1 | no | — |
| 146 | C2×C4×C8 | C2×D67 | 1 | 1 | yes | characteristic |
| 147 | C43 | — | 67 | 1 | no | — |
| 148 | #341 | — | 67 | 4 | no | — |
| 149 | #339 | — | 67 | 8 | no | — |
| 150 | #336 | — | 67 | 4 | no | — |
| 151 | C2×M16 | — | 67 | 4 | no | — |
| 152 | C22×C8 | — | 67 | 2 | no | — |
| 153 | C22×C8 | C22×D67 | 1 | 2 | yes | — |
| 154 | C4⋊Q8 | C2×Dic67 | 1 | 2 | yes | — |
| 155 | C4⋊Q8 | C2×Dic67 | 1 | 2 | yes | — |
| 156 | #125 | C2×Dic67 | 1 | 4 | yes | — |
| 157 | C22⋊Q8 | — | 2 | 2 | no | — |
| 158 | C22⋊Q8 | — | 2 | 2 | no | — |
| 159 | C4×Q8 | — | 2 | 2 | no | — |
| 160 | C4×Q8 | — | 2 | 2 | no | — |
| 161 | #117 | C22×D67 | 1 | 4 | yes | — |
| 162 | C2×C42 | — | 67 | 2 | no | — |
| 163 | C2×C42 | — | 134 | 1 | no | — |
| 164 | C2×C42 | — | 134 | 1 | no | — |
| 165 | C2×C42 | C22×D67 | 1 | 1 | yes | characteristic |
| 166 | #107 | C2×Dic67 | 1 | 4 | yes | — |
| 167 | #107 | C2×Dic67 | 1 | 4 | yes | — |
| 168 | #106 | C2×Dic67 | 1 | 8 | yes | — |
| 169 | C4⋊C8 | — | 134 | 4 | no | — |
| 170 | C4⋊C8 | — | 134 | 4 | no | — |
| 171 | C4⋊C8 | — | 134 | 2 | no | — |
| 172 | C4⋊C8 | — | 134 | 2 | no | — |
| 173 | #97 | — | 134 | 4 | no | — |
| 174 | C4×C8 | — | 67 | 1 | no | — |
| 175 | C4×C8 | — | 67 | 2 | no | — |
| 176 | C4×C8 | — | 67 | 1 | no | — |
| 177 | C4×C8 | — | 134 | 4 | no | — |
| 178 | C4×C8 | C2×Dic67 | 1 | 1 | yes | characteristic |
| 179 | C4×C8 | C2×Dic67 | 1 | 2 | yes | — |
| 180 | C4×C8 | C2×Dic67 | 1 | 1 | yes | characteristic |
| 181 | C2×Q8 | — | 2 | 2 | no | — |
| 182 | C2×Q8 | — | 2 | 2 | no | — |
| 183 | C22×C4 | — | 134 | 2 | no | — |
| 184 | C22×C4 | #11761627 | 1 | 1 | yes | characteristic |
| 185 | C22×C4 | #11761627 | 1 | 1 | yes | characteristic |
| 186 | C22×C4 | C23×D67 | 1 | 1 | yes | characteristic |
| 187 | M16 | — | 134 | 8 | no | — |
| 188 | C2×C8 | — | 2 | 4 | no | — |
| 189 | C2×C8 | — | 67 | 2 | no | — |
| 190 | C2×C8 | — | 67 | 2 | no | — |
| 191 | C2×C8 | — | 134 | 4 | no | — |
| 192 | C2×C8 | — | 134 | 4 | no | — |
| 193 | C2×C8 | C22×Dic67 | 1 | 2 | yes | — |
| 194 | C2×C8 | C22×Dic67 | 1 | 2 | yes | — |
| 195 | C4⋊C4 | — | 2 | 4 | no | — |
| 196 | C4⋊C4 | — | 2 | 4 | no | — |
| 197 | C4⋊C4 | — | 2 | 2 | no | — |
| 198 | C4⋊C4 | — | 2 | 2 | no | — |
| 199 | C4⋊C4 | C22×Dic67 | 1 | 4 | yes | — |
| 200 | C4⋊C4 | C22×Dic67 | 1 | 4 | yes | — |
| 201 | C22⋊C4 | — | 2 | 4 | no | — |
| 202 | C42 | — | 2 | 4 | no | — |
| 203 | C42 | — | 268 | 2 | no | — |
| 204 | C42 | — | 134 | 4 | no | — |
| 205 | C42 | — | 134 | 4 | no | — |
| 206 | C42 | C22×Dic67 | 1 | 1 | yes | characteristic |
| 207 | C42 | C22×Dic67 | 1 | 1 | yes | characteristic |
| 208 | C42 | D4×D67 | 1 | 1 | yes | characteristic |
| 209 | C42 | D4×D67 | 1 | 1 | yes | characteristic |
| 210 | C23 | C2×#11761627 | 1 | 1 | yes | characteristic |
| 211 | Q8 | — | 4 | 4 | no | — |
| 212 | Q8 | — | 4 | 4 | no | — |
| 213 | C2×C4 | — | 2 | 2 | no | — |
| 214 | C2×C4 | — | 2 | 2 | no | — |
| 215 | C2×C4 | — | 2 | 4 | no | — |
| 216 | C2×C4 | — | 2 | 4 | no | — |
| 217 | C2×C4 | — | 268 | 2 | no | — |
| 218 | C2×C4 | — | 268 | 2 | no | — |
| 219 | C2×C4 | — | 134 | 4 | no | — |
| 220 | C2×C4 | D4×Dic67 | 1 | 1 | yes | characteristic |
| 221 | C2×C4 | D4×Dic67 | 1 | 1 | yes | characteristic |
| 222 | C2×C4 | D4×Dic67 | 1 | 1 | yes | characteristic |
| 223 | C2×C4 | D4×Dic67 | 1 | 1 | yes | characteristic |
| 224 | C2×C4 | C23×Dic67 | 1 | 1 | yes | characteristic |
| 225 | C2×C4 | C2×D4×D67 | 1 | 1 | yes | characteristic |
| 226 | C2×C4 | #14524091 | 1 | 4 | yes | — |
| 227 | C8 | — | 2 | 2 | no | — |
| 228 | C8 | — | 2 | 2 | no | — |
| 229 | C8 | — | 268 | 4 | no | — |
| 230 | C8 | — | 134 | 2 | no | — |
| 231 | C8 | — | 134 | 2 | no | — |
| 232 | C22 | — | 2 | 1 | no | — |
| 233 | C22 | — | 2 | 1 | no | — |
| 234 | C22 | C2×D4×Dic67 | 1 | 1 | yes | characteristic |
| 235 | C22 | #16864007 | 1 | 2 | yes | — |
| 236 | C4 | — | 2 | 1 | no | — |
| 237 | C4 | — | 2 | 1 | no | — |
| 238 | C4 | — | 4 | 4 | no | — |
| 239 | C4 | — | 4 | 4 | no | — |
| 240 | C4 | — | 268 | 4 | no | — |
| 241 | C4 | #16863513 | 1 | 2 | yes | — |
| 242 | C4 | #16863518 | 1 | 2 | yes | — |
| 243 | C4 | #16863874 | 1 | 2 | yes | — |
| 244 | C4 | #16863971 | 1 | 2 | yes | — |
| 245 | C2 | — | 2 | 2 | no | — |
| 246 | C2 | #22562929 | 1 | 1 | yes | characteristic |
| 247 | C2 | C2×#16863513 | 1 | 1 | yes | characteristic |
| 248 | C2 | C2×#16863518 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Maximal subgroups
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
Finite Group Databasebeta