← #30752358
Subgroups — #30752358
Order 17152.
412 conjugacy classes of subgroups.
The subgroup lattice has 330 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22571360 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22569156 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #22558900 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#16863142 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #22558104 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #22558060 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C67×#1997 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #16863953 | C22 | 1 | 1 | yes | characteristic |
| 9 | #16863799 | — | 2 | 1 | no | — |
| 10 | #16863748 | — | 2 | 1 | no | — |
| 11 | #16863731 | C4 | 1 | 4 | yes | — |
| 12 | #16863688 | — | 2 | 2 | no | — |
| 13 | #16863526 | — | 2 | 1 | no | — |
| 14 | #16863524 | — | 2 | 2 | no | — |
| 15 | C8×Dic134 | — | 2 | 1 | no | — |
| 16 | #16863449 | C22 | 1 | 1 | yes | characteristic |
| 17 | #16863424 | — | 2 | 1 | no | — |
| 18 | C2×C4⋊(C67⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×C4⋊(C67⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 20 | #16863354 | — | 2 | 1 | no | — |
| 21 | #16863144 | — | 2 | 1 | no | — |
| 22 | #16863142 | — | 2 | 1 | no | — |
| 23 | #16863142 | — | 2 | 1 | no | — |
| 24 | #16863141 | — | 2 | 1 | no | — |
| 25 | C2×C4⋊Q8×C67 | C22 | 1 | 1 | yes | characteristic |
| 26 | #423×C67 | C22 | 1 | 1 | yes | characteristic |
| 27 | C4×M16×C67 | C22 | 1 | 1 | yes | characteristic |
| 28 | #343×C67 | — | 2 | 1 | no | — |
| 29 | #343×C67 | — | 2 | 1 | no | — |
| 30 | C22×C67⋊C8 | — | 2 | 1 | no | — |
| 31 | C22×C67⋊C8 | — | 2 | 1 | no | — |
| 32 | #14524043 | D4 | 1 | 1 | yes | characteristic |
| 33 | #14524043 | D4 | 1 | 1 | yes | characteristic |
| 34 | #14524035 | — | 2 | 1 | no | — |
| 35 | #14524022 | — | 2 | 1 | no | — |
| 36 | #14524022 | C2×C4 | 1 | 2 | yes | — |
| 37 | C4×Dic134 | — | 2 | 1 | no | — |
| 38 | C4×Dic134 | D4 | 1 | 2 | yes | — |
| 39 | #14524002 | — | 2 | 1 | no | — |
| 40 | #14523976 | — | 2 | 4 | no | — |
| 41 | C4⋊(C67⋊C8) | — | 2 | 1 | no | — |
| 42 | C4⋊(C67⋊C8) | — | 2 | 1 | no | — |
| 43 | C4⋊(C67⋊C8) | — | 2 | 1 | no | — |
| 44 | C4⋊(C67⋊C8) | — | 2 | 1 | no | — |
| 45 | C4⋊(C67⋊C8) | D4 | 1 | 1 | yes | characteristic |
| 46 | C4⋊(C67⋊C8) | D4 | 1 | 1 | yes | characteristic |
| 47 | Dic67⋊C8 | — | 2 | 2 | no | — |
| 48 | Dic67⋊C8 | — | 2 | 2 | no | — |
| 49 | #14523966 | — | 2 | 4 | no | — |
| 50 | C22⋊(C67⋊C8) | C2×C4 | 1 | 2 | yes | — |
| 51 | #14523953 | — | 2 | 2 | no | — |
| 52 | C4×C67⋊C8 | — | 2 | 1 | no | — |
| 53 | C8×Dic67 | — | 2 | 2 | no | — |
| 54 | C22×Q8×C67 | — | 2 | 1 | no | — |
| 55 | C2×M16×C67 | — | 2 | 1 | no | — |
| 56 | C2×M16×C67 | D4 | 1 | 1 | yes | characteristic |
| 57 | C2×M16×C67 | D4 | 1 | 1 | yes | characteristic |
| 58 | C4⋊Q8×C67 | — | 2 | 1 | no | — |
| 59 | C4⋊Q8×C67 | — | 2 | 1 | no | — |
| 60 | C4⋊Q8×C67 | — | 2 | 1 | no | — |
| 61 | C4⋊Q8×C67 | — | 2 | 1 | no | — |
| 62 | C2×C4⋊C4×C67 | — | 2 | 1 | no | — |
| 63 | C2×C4⋊C4×C67 | C2×C4 | 1 | 2 | yes | — |
| 64 | C2×C42×C67 | C23 | 1 | 1 | yes | characteristic |
| 65 | C4⋊C8×C67 | — | 2 | 1 | no | — |
| 66 | C4⋊C8×C67 | — | 2 | 1 | no | — |
| 67 | #97×C67 | — | 2 | 1 | no | — |
| 68 | C4×C8×C67 | — | 2 | 1 | no | — |
| 69 | C2×Dic134 | — | 2 | 1 | no | — |
| 70 | C2×Dic134 | C22⋊C4 | 1 | 2 | yes | — |
| 71 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 72 | C2×C67⋊C8 | — | 2 | 1 | no | — |
| 73 | C2×C67⋊C8 | — | 2 | 1 | no | — |
| 74 | C2×C67⋊C8 | — | 4 | 1 | no | — |
| 75 | C2×C67⋊C8 | — | 4 | 1 | no | — |
| 76 | C2×C67⋊C8 | — | 4 | 1 | no | — |
| 77 | C2×C67⋊C8 | — | 8 | 1 | no | — |
| 78 | C4⋊Dic67 | — | 2 | 1 | no | — |
| 79 | C4⋊Dic67 | — | 2 | 1 | no | — |
| 80 | C4⋊Dic67 | — | 2 | 1 | no | — |
| 81 | C4⋊Dic67 | C22⋊C4 | 1 | 2 | yes | — |
| 82 | Dic67⋊C4 | — | 2 | 2 | no | — |
| 83 | Dic67⋊C4 | — | 2 | 2 | no | — |
| 84 | C22⋊Dic67 | — | 2 | 2 | no | — |
| 85 | C4×Dic67 | — | 2 | 2 | no | — |
| 86 | C2×Q8×C67 | — | 4 | 1 | no | — |
| 87 | C2×Q8×C67 | — | 4 | 1 | no | — |
| 88 | C2×Q8×C67 | — | 8 | 1 | no | — |
| 89 | C2×Q8×C67 | — | 8 | 1 | no | — |
| 90 | C22×C4×C67 | — | 2 | 2 | no | — |
| 91 | C22×C4×C67 | C22×C4 | 1 | 1 | yes | characteristic |
| 92 | C22×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 93 | C22×C4×C67 | #41 | 1 | 1 | yes | characteristic |
| 94 | M16×C67 | — | 2 | 1 | no | — |
| 95 | M16×C67 | — | 2 | 1 | no | — |
| 96 | M16×C67 | — | 2 | 1 | no | — |
| 97 | M16×C67 | — | 4 | 1 | no | — |
| 98 | M16×C67 | — | 4 | 1 | no | — |
| 99 | M16×C67 | Q16 | 1 | 2 | yes | — |
| 100 | C2×C8×C67 | — | 2 | 1 | no | — |
| 101 | C2×C8×C67 | — | 2 | 1 | no | — |
| 102 | C2×C8×C67 | — | 4 | 1 | no | — |
| 103 | C4⋊C4×C67 | — | 2 | 2 | no | — |
| 104 | C4⋊C4×C67 | — | 2 | 2 | no | — |
| 105 | C4⋊C4×C67 | — | 4 | 1 | no | — |
| 106 | C4⋊C4×C67 | — | 4 | 1 | no | — |
| 107 | C42×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 108 | C42×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 109 | C42×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | C42×C67 | #41 | 1 | 1 | yes | characteristic |
| 111 | C23×C67 | C4×D4 | 1 | 1 | yes | characteristic |
| 112 | Q8×C67 | — | 8 | 2 | no | — |
| 113 | Q8×C67 | — | 8 | 1 | no | — |
| 114 | Q8×C67 | — | 8 | 1 | no | — |
| 115 | C2×C4×C67 | — | 2 | 1 | no | — |
| 116 | C2×C4×C67 | — | 2 | 1 | no | — |
| 117 | C2×C4×C67 | — | 4 | 1 | no | — |
| 118 | C2×C4×C67 | — | 4 | 2 | no | — |
| 119 | C2×C4×C67 | — | 8 | 2 | no | — |
| 120 | C2×C4×C67 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 121 | C2×C4×C67 | C4×D4 | 1 | 1 | yes | characteristic |
| 122 | C2×C4×C67 | #120 | 1 | 1 | yes | characteristic |
| 123 | C2×C4×C67 | #121 | 1 | 1 | yes | characteristic |
| 124 | C2×C4×C67 | #121 | 1 | 1 | yes | characteristic |
| 125 | C2×C4×C67 | #123 | 1 | 1 | yes | characteristic |
| 126 | C2×C4×C67 | C2×Q16 | 1 | 1 | yes | characteristic |
| 127 | C2×C4×C67 | #135 | 1 | 1 | yes | characteristic |
| 128 | C2×C4×C67 | #137 | 1 | 1 | yes | characteristic |
| 129 | C2×C4×C67 | #137 | 1 | 1 | yes | characteristic |
| 130 | C2×Dic67 | — | 2 | 2 | no | — |
| 131 | C2×Dic67 | — | 2 | 2 | no | — |
| 132 | Dic134 | — | 4 | 1 | no | — |
| 133 | Dic134 | — | 4 | 1 | no | — |
| 134 | Dic134 | — | 4 | 2 | no | — |
| 135 | C8×C67 | — | 2 | 2 | no | — |
| 136 | C8×C67 | — | 4 | 1 | no | — |
| 137 | C8×C67 | — | 8 | 1 | no | — |
| 138 | C67⋊C8 | — | 4 | 2 | no | — |
| 139 | C67⋊C8 | — | 4 | 1 | no | — |
| 140 | C67⋊C8 | — | 4 | 1 | no | — |
| 141 | C67⋊C8 | — | 8 | 1 | no | — |
| 142 | C22×C67 | — | 2 | 1 | no | — |
| 143 | C22×C67 | — | 2 | 1 | no | — |
| 144 | C22×C67 | #386 | 1 | 1 | yes | characteristic |
| 145 | C22×C67 | C4×Q16 | 1 | 1 | yes | characteristic |
| 146 | C22×C67 | #441 | 1 | 1 | yes | characteristic |
| 147 | C4×C67 | — | 2 | 1 | no | — |
| 148 | C4×C67 | — | 2 | 1 | no | — |
| 149 | C4×C67 | — | 2 | 1 | no | — |
| 150 | C4×C67 | — | 8 | 2 | no | — |
| 151 | C4×C67 | #451 | 1 | 1 | yes | characteristic |
| 152 | C4×C67 | #452 | 1 | 1 | yes | characteristic |
| 153 | C4×C67 | #462 | 1 | 1 | yes | characteristic |
| 154 | C4×C67 | #464 | 1 | 1 | yes | characteristic |
| 155 | C4×C67 | #471 | 1 | 1 | yes | characteristic |
| 156 | C4×C67 | #472 | 1 | 1 | yes | characteristic |
| 157 | Dic67 | — | 4 | 2 | no | — |
| 158 | Dic67 | — | 4 | 2 | no | — |
| 159 | #9270 | — | 67 | 1 | no | maximal, Sylow |
| 160 | C2×C67 | — | 2 | 1 | no | — |
| 161 | C2×C67 | — | 2 | 1 | no | — |
| 162 | C2×C67 | #1878 | 1 | 1 | yes | characteristic |
| 163 | C2×C67 | #1907 | 1 | 1 | yes | characteristic |
| 164 | C2×C67 | #1915 | 1 | 1 | yes | characteristic |
| 165 | #3230 | — | 67 | 1 | no | — |
| 166 | #2962 | — | 67 | 1 | no | — |
| 167 | #1997 | D67 | 1 | 1 | yes | characteristic |
| 168 | #1550 | — | 67 | 1 | no | — |
| 169 | C2×#332 | — | 67 | 1 | no | — |
| 170 | #1301 | — | 67 | 1 | no | — |
| 171 | #1283 | — | 67 | 1 | no | — |
| 172 | C67 | #9270 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 173 | #533 | — | 67 | 1 | no | — |
| 174 | C2×C4⋊Q8 | C2×D67 | 1 | 1 | yes | characteristic |
| 175 | #500 | — | 134 | 1 | no | — |
| 176 | #487 | — | 134 | 1 | no | — |
| 177 | #484 | — | 67 | 4 | no | — |
| 178 | #477 | — | 134 | 2 | no | — |
| 179 | #446 | — | 134 | 1 | no | — |
| 180 | #446 | — | 134 | 2 | no | — |
| 181 | C8×Q8 | — | 134 | 1 | no | — |
| 182 | #431 | — | 67 | 1 | no | — |
| 183 | #424 | — | 134 | 1 | no | — |
| 184 | #423 | C2×D67 | 1 | 1 | yes | characteristic |
| 185 | C2×C4⋊C8 | — | 67 | 1 | no | — |
| 186 | C2×C4⋊C8 | — | 67 | 1 | no | — |
| 187 | #405 | — | 134 | 1 | no | — |
| 188 | C4×M16 | C2×D67 | 1 | 1 | yes | characteristic |
| 189 | #343 | — | 2 | 1 | no | — |
| 190 | #343 | — | 2 | 1 | no | — |
| 191 | #333 | — | 134 | 1 | no | — |
| 192 | #332 | — | 134 | 1 | no | — |
| 193 | #332 | — | 134 | 1 | no | — |
| 194 | #332 | — | 134 | 1 | no | — |
| 195 | C22×Q8 | — | 2 | 1 | no | — |
| 196 | C2×M16 | — | 2 | 1 | no | — |
| 197 | C2×M16 | C22⋊D67 | 1 | 1 | yes | characteristic |
| 198 | C2×M16 | C22⋊D67 | 1 | 1 | yes | characteristic |
| 199 | C22×C8 | — | 134 | 1 | no | — |
| 200 | C22×C8 | — | 134 | 1 | no | — |
| 201 | C4⋊Q8 | — | 2 | 1 | no | — |
| 202 | C4⋊Q8 | — | 2 | 1 | no | — |
| 203 | C4⋊Q8 | — | 2 | 1 | no | — |
| 204 | C4⋊Q8 | — | 2 | 1 | no | — |
| 205 | C4⋊Q8 | — | 67 | 1 | no | — |
| 206 | C4⋊Q8 | — | 67 | 1 | no | — |
| 207 | #125 | — | 134 | 1 | no | — |
| 208 | C22⋊Q8 | — | 67 | 2 | no | — |
| 209 | C22⋊Q8 | — | 134 | 1 | no | — |
| 210 | C4×Q8 | — | 67 | 2 | no | — |
| 211 | C4×Q8 | — | 134 | 1 | no | — |
| 212 | #117 | — | 134 | 1 | no | — |
| 213 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 214 | C2×C4⋊C4 | C4×D67 | 1 | 2 | yes | — |
| 215 | C2×C42 | C22×D67 | 1 | 1 | yes | characteristic |
| 216 | #107 | — | 134 | 4 | no | — |
| 217 | C4⋊C8 | — | 2 | 1 | no | — |
| 218 | C4⋊C8 | — | 2 | 1 | no | — |
| 219 | C4⋊C8 | — | 67 | 1 | no | — |
| 220 | C4⋊C8 | — | 67 | 1 | no | — |
| 221 | C4⋊C8 | — | 134 | 2 | no | — |
| 222 | C4⋊C8 | — | 134 | 2 | no | — |
| 223 | C4⋊C8 | — | 134 | 1 | no | — |
| 224 | C4⋊C8 | — | 134 | 1 | no | — |
| 225 | C4⋊C8 | — | 134 | 1 | no | — |
| 226 | C4⋊C8 | — | 134 | 1 | no | — |
| 227 | #103 | — | 134 | 4 | no | — |
| 228 | C22⋊C8 | — | 67 | 2 | no | — |
| 229 | #97 | — | 2 | 1 | no | — |
| 230 | #97 | — | 134 | 2 | no | — |
| 231 | C4×C8 | — | 2 | 1 | no | — |
| 232 | C4×C8 | — | 134 | 2 | no | — |
| 233 | C4×C8 | — | 134 | 1 | no | — |
| 234 | C2×Q8 | — | 4 | 1 | no | — |
| 235 | C2×Q8 | — | 4 | 1 | no | — |
| 236 | C2×Q8 | — | 8 | 1 | no | — |
| 237 | C2×Q8 | — | 8 | 1 | no | — |
| 238 | C2×Q8 | — | 67 | 2 | no | — |
| 239 | C2×Q8 | — | 134 | 1 | no | — |
| 240 | C22×C4 | — | 2 | 2 | no | — |
| 241 | C22×C4 | C2×C4×D67 | 1 | 1 | yes | characteristic |
| 242 | C22×C4 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 243 | C22×C4 | #11761628 | 1 | 1 | yes | characteristic |
| 244 | M16 | — | 2 | 1 | no | — |
| 245 | M16 | — | 2 | 1 | no | — |
| 246 | M16 | — | 2 | 1 | no | — |
| 247 | M16 | — | 4 | 1 | no | — |
| 248 | M16 | — | 4 | 1 | no | — |
| 249 | M16 | #11761618 | 1 | 2 | yes | — |
| 250 | C2×C8 | — | 2 | 1 | no | — |
| 251 | C2×C8 | — | 2 | 1 | no | — |
| 252 | C2×C8 | — | 4 | 1 | no | — |
| 253 | C2×C8 | — | 268 | 1 | no | — |
| 254 | C2×C8 | — | 268 | 1 | no | — |
| 255 | C2×C8 | — | 268 | 1 | no | — |
| 256 | C2×C8 | — | 536 | 1 | no | — |
| 257 | C2×C8 | — | 134 | 2 | no | — |
| 258 | C2×C8 | — | 134 | 1 | no | — |
| 259 | C2×C8 | — | 134 | 1 | no | — |
| 260 | C4⋊C4 | — | 2 | 2 | no | — |
| 261 | C4⋊C4 | — | 2 | 2 | no | — |
| 262 | C4⋊C4 | — | 4 | 1 | no | — |
| 263 | C4⋊C4 | — | 4 | 1 | no | — |
| 264 | C4⋊C4 | — | 67 | 2 | no | — |
| 265 | C4⋊C4 | — | 134 | 2 | no | — |
| 266 | C4⋊C4 | — | 134 | 1 | no | — |
| 267 | C4⋊C4 | — | 134 | 1 | no | — |
| 268 | C4⋊C4 | — | 134 | 1 | no | — |
| 269 | C4⋊C4 | — | 134 | 2 | no | — |
| 270 | C22⋊C4 | — | 134 | 2 | no | — |
| 271 | C42 | — | 134 | 2 | no | — |
| 272 | C42 | D4×D67 | 1 | 1 | yes | characteristic |
| 273 | C42 | D4×D67 | 1 | 1 | yes | characteristic |
| 274 | C42 | D4×D67 | 1 | 1 | yes | characteristic |
| 275 | C42 | #11761627 | 1 | 1 | yes | characteristic |
| 276 | C23 | C4×C22⋊D67 | 1 | 1 | yes | characteristic |
| 277 | Q8 | — | 8 | 2 | no | — |
| 278 | Q8 | — | 8 | 1 | no | — |
| 279 | Q8 | — | 8 | 1 | no | — |
| 280 | Q8 | — | 268 | 2 | no | — |
| 281 | Q8 | — | 268 | 1 | no | — |
| 282 | Q8 | — | 268 | 1 | no | — |
| 283 | C2×C4 | — | 2 | 1 | no | — |
| 284 | C2×C4 | — | 2 | 1 | no | — |
| 285 | C2×C4 | — | 4 | 1 | no | — |
| 286 | C2×C4 | — | 4 | 2 | no | — |
| 287 | C2×C4 | — | 8 | 2 | no | — |
| 288 | C2×C4 | — | 134 | 2 | no | — |
| 289 | C2×C4 | — | 134 | 2 | no | — |
| 290 | C2×C4 | C22⋊C4×D67 | 1 | 1 | yes | characteristic |
| 291 | C2×C4 | (C22⋊D67)⋊C4 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | #14524011 | 1 | 1 | yes | characteristic |
| 293 | C2×C4 | #14524014 | 1 | 1 | yes | characteristic |
| 294 | C2×C4 | #14524016 | 1 | 1 | yes | characteristic |
| 295 | C2×C4 | #14524025 | 1 | 1 | yes | characteristic |
| 296 | C2×C4 | C2×#11761618 | 1 | 1 | yes | characteristic |
| 297 | C2×C4 | #14524063 | 1 | 1 | yes | characteristic |
| 298 | C2×C4 | #14524073 | 1 | 1 | yes | characteristic |
| 299 | C2×C4 | #14524074 | 1 | 1 | yes | characteristic |
| 300 | C8 | — | 2 | 2 | no | — |
| 301 | C8 | — | 4 | 1 | no | — |
| 302 | C8 | — | 8 | 1 | no | — |
| 303 | C8 | — | 268 | 2 | no | — |
| 304 | C8 | — | 268 | 1 | no | — |
| 305 | C8 | — | 268 | 1 | no | — |
| 306 | C8 | — | 536 | 1 | no | — |
| 307 | C22 | — | 2 | 1 | no | — |
| 308 | C22 | — | 2 | 1 | no | — |
| 309 | C22 | #16863272 | 1 | 1 | yes | characteristic |
| 310 | C22 | C4×#11761618 | 1 | 1 | yes | characteristic |
| 311 | C22 | #16863501 | 1 | 1 | yes | characteristic |
| 312 | C4 | — | 2 | 1 | no | — |
| 313 | C4 | — | 2 | 1 | no | — |
| 314 | C4 | — | 2 | 1 | no | — |
| 315 | C4 | — | 8 | 2 | no | — |
| 316 | C4 | — | 268 | 2 | no | — |
| 317 | C4 | — | 268 | 2 | no | — |
| 318 | C4 | #16863551 | 1 | 1 | yes | characteristic |
| 319 | C4 | #16863556 | 1 | 1 | yes | characteristic |
| 320 | C4 | #16863603 | 1 | 1 | yes | characteristic |
| 321 | C4 | #16863615 | 1 | 1 | yes | characteristic |
| 322 | C4 | #16863658 | 1 | 1 | yes | characteristic |
| 323 | C4 | #16863665 | 1 | 1 | yes | characteristic |
| 324 | C2 | — | 2 | 1 | no | — |
| 325 | C2 | — | 2 | 1 | no | — |
| 326 | C2 | #22560553 | 1 | 1 | yes | characteristic |
| 327 | C2 | #22560714 | 1 | 1 | yes | characteristic |
| 328 | C2 | #22560758 | 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