← #30852903
Subgroups — #30852903
Order 17152.
638 conjugacy classes of subgroups.
The subgroup lattice has 352 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22572516 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #22572515 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #22559819 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #22559757 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #22559756 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #22559447 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #22559446 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #22559095 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C67×#1589 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | #16864051 | C22 | 1 | 2 | yes | — |
| 11 | #16864050 | C22 | 1 | 2 | yes | — |
| 12 | #16863801 | C22 | 1 | 1 | yes | characteristic |
| 13 | #16863801 | C22 | 1 | 1 | yes | characteristic |
| 14 | #16863747 | C22 | 1 | 2 | yes | — |
| 15 | #16863747 | C22 | 1 | 2 | yes | — |
| 16 | #16863733 | — | 2 | 2 | no | — |
| 17 | #16863732 | — | 2 | 2 | no | — |
| 18 | #16863677 | C22 | 1 | 2 | yes | — |
| 19 | #16863676 | C22 | 1 | 2 | yes | — |
| 20 | #16863642 | — | 2 | 2 | no | — |
| 21 | #16863641 | — | 2 | 2 | no | — |
| 22 | #16863602 | C22 | 1 | 2 | yes | — |
| 23 | #16863601 | C22 | 1 | 2 | yes | — |
| 24 | #16863543 | — | 2 | 2 | no | — |
| 25 | #16863538 | — | 2 | 2 | no | — |
| 26 | C4×D536 | C22 | 1 | 1 | yes | characteristic |
| 27 | #16863475 | C22 | 1 | 2 | yes | — |
| 28 | #16863398 | — | 2 | 2 | no | — |
| 29 | #16863394 | — | 2 | 2 | no | — |
| 30 | #16863149 | C22 | 1 | 1 | yes | characteristic |
| 31 | #16863143 | C22 | 1 | 2 | yes | — |
| 32 | #16863141 | C22 | 1 | 2 | yes | — |
| 33 | #16863125 | C22 | 1 | 2 | yes | — |
| 34 | C4×D8×C67 | C22 | 1 | 1 | yes | characteristic |
| 35 | #436×C67 | C22 | 1 | 2 | yes | — |
| 36 | #335×C67 | C22 | 1 | 1 | yes | characteristic |
| 37 | D4⋊C8×C67 | C22 | 1 | 2 | yes | — |
| 38 | C82×C67 | C22 | 1 | 1 | yes | characteristic |
| 39 | C2×#11761628 | — | 2 | 2 | no | — |
| 40 | C2×#11761627 | — | 2 | 2 | no | — |
| 41 | C2×#11761617 | — | 2 | 2 | no | — |
| 42 | C2×#11761616 | — | 2 | 2 | no | — |
| 43 | C2×D536 | D4 | 1 | 1 | yes | characteristic |
| 44 | C2×#11761611 | — | 2 | 2 | no | — |
| 45 | #14524043 | C23 | 1 | 2 | yes | — |
| 46 | #14524043 | C23 | 1 | 2 | yes | — |
| 47 | #14524027 | — | 4 | 2 | no | — |
| 48 | #14524025 | — | 4 | 2 | no | — |
| 49 | #14524023 | — | 2 | 2 | no | — |
| 50 | #14524022 | — | 2 | 2 | no | — |
| 51 | #14524017 | — | 2 | 2 | no | — |
| 52 | #14524014 | — | 2 | 2 | no | — |
| 53 | C4×D268 | C23 | 1 | 2 | yes | — |
| 54 | C2×C4⋊Dic67 | — | 2 | 2 | no | — |
| 55 | C4⋊C4×D67 | — | 2 | 2 | no | — |
| 56 | #14523977 | — | 2 | 2 | no | — |
| 57 | #14523977 | — | 4 | 1 | no | — |
| 58 | #14523977 | — | 4 | 1 | no | — |
| 59 | #14523977 | D4 | 1 | 1 | yes | characteristic |
| 60 | #14523976 | — | 2 | 2 | no | — |
| 61 | C4⋊(C67⋊C8) | C23 | 1 | 2 | yes | — |
| 62 | #14523968 | — | 2 | 2 | no | — |
| 63 | #14523968 | — | 2 | 2 | no | — |
| 64 | #14523968 | — | 2 | 2 | no | — |
| 65 | #14523968 | — | 2 | 2 | no | — |
| 66 | #14523968 | — | 2 | 2 | no | — |
| 67 | #14523966 | — | 2 | 2 | no | — |
| 68 | #14523965 | — | 2 | 2 | no | — |
| 69 | #14523965 | D4 | 1 | 2 | yes | — |
| 70 | #14523964 | — | 2 | 2 | no | — |
| 71 | (C2×D67)⋊C8 | — | 2 | 2 | no | — |
| 72 | C2×D8×C67 | D4 | 1 | 1 | yes | characteristic |
| 73 | C2×M16×C67 | — | 2 | 2 | no | — |
| 74 | C4×D4×C67 | C23 | 1 | 2 | yes | — |
| 75 | #107×C67 | D4 | 1 | 1 | yes | characteristic |
| 76 | C4⋊C8×C67 | C23 | 1 | 2 | yes | — |
| 77 | #102×C67 | D4 | 1 | 2 | yes | — |
| 78 | C22⋊C8×C67 | — | 2 | 2 | no | — |
| 79 | C4×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 80 | C4×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 81 | C4×C8×C67 | C23 | 1 | 1 | yes | characteristic |
| 82 | #11761628 | — | 4 | 4 | no | — |
| 83 | #11761628 | — | 8 | 2 | no | — |
| 84 | #11761627 | — | 4 | 4 | no | — |
| 85 | #11761627 | — | 8 | 2 | no | — |
| 86 | C2×Dic134 | — | 2 | 2 | no | — |
| 87 | C2×Dic134 | — | 2 | 2 | no | — |
| 88 | C2×D268 | C2×D4 | 1 | 2 | yes | — |
| 89 | C2×C22⋊D67 | — | 4 | 2 | no | — |
| 90 | C2×C22⋊D67 | — | 4 | 2 | no | — |
| 91 | C22×Dic67 | — | 4 | 2 | no | — |
| 92 | C2×C4×D67 | — | 2 | 2 | no | — |
| 93 | C2×C4×D67 | — | 4 | 2 | no | — |
| 94 | #11761617 | — | 8 | 2 | no | — |
| 95 | #11761616 | — | 8 | 2 | no | — |
| 96 | D536 | — | 4 | 1 | no | — |
| 97 | #11761611 | — | 8 | 2 | no | — |
| 98 | C2×C67⋊C8 | — | 2 | 2 | no | — |
| 99 | C4⋊Dic67 | — | 2 | 2 | no | — |
| 100 | C4⋊Dic67 | — | 2 | 2 | no | — |
| 101 | C4⋊Dic67 | — | 2 | 2 | no | — |
| 102 | C4⋊Dic67 | — | 2 | 2 | no | — |
| 103 | C4⋊Dic67 | — | 4 | 2 | no | — |
| 104 | C4⋊Dic67 | C2×D4 | 1 | 2 | yes | — |
| 105 | Dic67⋊C4 | — | 4 | 2 | no | — |
| 106 | C22⋊Dic67 | — | 4 | 2 | no | — |
| 107 | (C2×D67)⋊C4 | — | 2 | 2 | no | — |
| 108 | (C2×D67)⋊C4 | — | 4 | 2 | no | — |
| 109 | C2×D4×C67 | C2×D4 | 1 | 2 | yes | — |
| 110 | C22×C4×C67 | — | 2 | 2 | no | — |
| 111 | D8×C67 | — | 4 | 1 | no | — |
| 112 | M16×C67 | — | 8 | 2 | no | — |
| 113 | C2×C8×C67 | — | 2 | 1 | no | — |
| 114 | C2×C8×C67 | — | 2 | 2 | no | — |
| 115 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 116 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 117 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 118 | C2×C8×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 119 | C4⋊C4×C67 | C2×D4 | 1 | 2 | yes | — |
| 120 | C22⋊C4×C67 | — | 2 | 2 | no | — |
| 121 | C42×C67 | C24 | 1 | 1 | yes | characteristic |
| 122 | C23×C67 | — | 2 | 2 | no | — |
| 123 | C22×D67 | — | 2 | 2 | no | — |
| 124 | D4×C67 | — | 2 | 2 | no | — |
| 125 | D4×C67 | — | 2 | 2 | no | — |
| 126 | C2×C4×C67 | — | 2 | 2 | no | — |
| 127 | C2×C4×C67 | — | 4 | 2 | no | — |
| 128 | C2×C4×C67 | — | 4 | 2 | no | — |
| 129 | C2×C4×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 130 | C2×C4×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 131 | C2×C4×C67 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 132 | C22⋊D67 | — | 8 | 4 | no | — |
| 133 | C22⋊D67 | — | 8 | 4 | no | — |
| 134 | C2×Dic67 | — | 2 | 2 | no | — |
| 135 | C2×Dic67 | — | 4 | 2 | no | — |
| 136 | C2×Dic67 | — | 4 | 2 | no | — |
| 137 | C2×Dic67 | — | 8 | 4 | no | — |
| 138 | D268 | — | 2 | 2 | no | — |
| 139 | D268 | — | 2 | 2 | no | — |
| 140 | C4×D67 | — | 4 | 2 | no | — |
| 141 | C4×D67 | — | 4 | 2 | no | — |
| 142 | C4×D67 | — | 8 | 4 | no | — |
| 143 | Dic134 | — | 4 | 2 | no | — |
| 144 | Dic134 | — | 4 | 2 | no | — |
| 145 | Dic134 | — | 4 | 2 | no | — |
| 146 | Dic134 | — | 4 | 2 | no | — |
| 147 | C8×C67 | — | 2 | 2 | no | — |
| 148 | C8×C67 | — | 2 | 1 | no | — |
| 149 | C8×C67 | — | 2 | 1 | no | — |
| 150 | C8×C67 | — | 2 | 1 | no | — |
| 151 | C8×C67 | — | 2 | 1 | no | — |
| 152 | C8×C67 | — | 4 | 2 | no | — |
| 153 | C67⋊C8 | — | 4 | 2 | no | — |
| 154 | C22×C67 | — | 4 | 2 | no | — |
| 155 | C22×C67 | — | 4 | 2 | no | — |
| 156 | C22×C67 | — | 4 | 2 | no | — |
| 157 | C22×C67 | D42 | 1 | 1 | yes | characteristic |
| 158 | C2×D67 | — | 4 | 2 | no | — |
| 159 | C2×D67 | — | 4 | 2 | no | — |
| 160 | C2×D67 | — | 4 | 2 | no | — |
| 161 | C4×C67 | — | 2 | 1 | no | — |
| 162 | C4×C67 | — | 4 | 2 | no | — |
| 163 | C4×C67 | #575 | 1 | 1 | yes | characteristic |
| 164 | C4×C67 | #575 | 1 | 1 | yes | characteristic |
| 165 | C4×C67 | #576 | 1 | 1 | yes | characteristic |
| 166 | C4×C67 | #576 | 1 | 1 | yes | characteristic |
| 167 | Dic67 | — | 4 | 2 | no | — |
| 168 | Dic67 | — | 8 | 2 | no | — |
| 169 | Dic67 | — | 8 | 2 | no | — |
| 170 | #20304 | — | 67 | 1 | no | maximal, Sylow |
| 171 | C2×C67 | — | 8 | 2 | no | — |
| 172 | C2×C67 | #3273 | 1 | 1 | yes | characteristic |
| 173 | C2×C67 | #3273 | 1 | 1 | yes | characteristic |
| 174 | C2×C67 | #3293 | 1 | 1 | yes | characteristic |
| 175 | D67 | — | 8 | 2 | no | — |
| 176 | #3330 | — | 67 | 2 | no | — |
| 177 | #3330 | — | 67 | 2 | no | — |
| 178 | #1710 | — | 67 | 1 | no | — |
| 179 | #1699 | — | 67 | 2 | no | — |
| 180 | #1699 | — | 67 | 2 | no | — |
| 181 | #1647 | — | 67 | 2 | no | — |
| 182 | #1647 | — | 67 | 2 | no | — |
| 183 | #1589 | — | 67 | 1 | no | — |
| 184 | #1589 | D67 | 1 | 1 | yes | characteristic, complemented |
| 185 | C67 | #20304 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 186 | #547 | — | 67 | 2 | no | — |
| 187 | #547 | — | 67 | 2 | no | — |
| 188 | #500 | — | 67 | 1 | no | — |
| 189 | #500 | — | 67 | 1 | no | — |
| 190 | #487 | — | 67 | 2 | no | — |
| 191 | #487 | — | 67 | 2 | no | — |
| 192 | #484 | — | 134 | 2 | no | — |
| 193 | #484 | — | 134 | 2 | no | — |
| 194 | #474 | — | 67 | 2 | no | — |
| 195 | #474 | — | 67 | 2 | no | — |
| 196 | #468 | — | 134 | 2 | no | — |
| 197 | #468 | — | 134 | 2 | no | — |
| 198 | #461 | — | 67 | 2 | no | — |
| 199 | #461 | — | 67 | 2 | no | — |
| 200 | #449 | — | 134 | 2 | no | — |
| 201 | #449 | — | 134 | 2 | no | — |
| 202 | C4×D8 | — | 67 | 1 | no | — |
| 203 | C4×D8 | C2×D67 | 1 | 1 | yes | characteristic |
| 204 | #436 | — | 67 | 2 | no | — |
| 205 | #436 | C2×D67 | 1 | 2 | yes | — |
| 206 | #417 | — | 134 | 2 | no | — |
| 207 | #417 | — | 134 | 2 | no | — |
| 208 | #335 | — | 67 | 1 | no | — |
| 209 | #335 | C2×D67 | 1 | 1 | yes | characteristic |
| 210 | #332 | — | 67 | 2 | no | — |
| 211 | #332 | — | 67 | 2 | no | — |
| 212 | D4⋊C8 | — | 67 | 2 | no | — |
| 213 | D4⋊C8 | C2×D67 | 1 | 2 | yes | — |
| 214 | C82 | C2×D67 | 1 | 1 | yes | characteristic |
| 215 | C2×#41 | — | 134 | 2 | no | — |
| 216 | C2×#41 | — | 134 | 2 | no | — |
| 217 | C2×SD16 | — | 134 | 2 | no | — |
| 218 | C2×SD16 | — | 134 | 2 | no | — |
| 219 | C2×D8 | — | 67 | 1 | no | — |
| 220 | C2×D8 | D268 | 1 | 1 | yes | characteristic |
| 221 | C2×M16 | — | 2 | 2 | no | — |
| 222 | C2×M16 | — | 134 | 2 | no | — |
| 223 | C4⋊Q8 | — | 67 | 2 | no | — |
| 224 | C4⋊Q8 | — | 67 | 2 | no | — |
| 225 | #123 | — | 268 | 2 | no | — |
| 226 | #123 | — | 268 | 2 | no | — |
| 227 | C22⋊Q8 | — | 134 | 2 | no | — |
| 228 | C22⋊Q8 | — | 134 | 2 | no | — |
| 229 | #121 | — | 134 | 2 | no | — |
| 230 | #121 | — | 134 | 2 | no | — |
| 231 | C4×D4 | — | 67 | 2 | no | — |
| 232 | C4×D4 | C22×D67 | 1 | 2 | yes | — |
| 233 | C2×C4⋊C4 | — | 134 | 2 | no | — |
| 234 | C2×C4⋊C4 | — | 134 | 2 | no | — |
| 235 | #107 | — | 268 | 1 | no | — |
| 236 | #107 | — | 268 | 1 | no | — |
| 237 | #107 | — | 67 | 1 | no | — |
| 238 | #107 | — | 134 | 2 | no | — |
| 239 | #107 | — | 134 | 2 | no | — |
| 240 | #107 | D268 | 1 | 1 | yes | characteristic |
| 241 | C4⋊C8 | — | 67 | 2 | no | — |
| 242 | C4⋊C8 | C22×D67 | 1 | 2 | yes | — |
| 243 | #103 | — | 134 | 2 | no | — |
| 244 | #103 | — | 134 | 2 | no | — |
| 245 | #103 | — | 134 | 2 | no | — |
| 246 | #103 | — | 134 | 2 | no | — |
| 247 | #103 | — | 134 | 2 | no | — |
| 248 | #103 | — | 134 | 2 | no | — |
| 249 | #102 | — | 67 | 2 | no | — |
| 250 | #102 | — | 134 | 2 | no | — |
| 251 | #102 | — | 134 | 2 | no | — |
| 252 | #102 | D268 | 1 | 2 | yes | — |
| 253 | C22⋊C8 | — | 2 | 2 | no | — |
| 254 | C22⋊C8 | — | 134 | 2 | no | — |
| 255 | C4×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 256 | C4×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 257 | C4×C8 | C22×D67 | 1 | 1 | yes | characteristic |
| 258 | #41 | — | 268 | 4 | no | — |
| 259 | #41 | — | 268 | 4 | no | — |
| 260 | #41 | — | 536 | 2 | no | — |
| 261 | #41 | — | 536 | 2 | no | — |
| 262 | C2×Q8 | — | 134 | 2 | no | — |
| 263 | C2×Q8 | — | 134 | 2 | no | — |
| 264 | C2×D4 | — | 268 | 2 | no | — |
| 265 | C2×D4 | — | 268 | 2 | no | — |
| 266 | C2×D4 | — | 67 | 2 | no | — |
| 267 | C2×D4 | C2×D268 | 1 | 2 | yes | — |
| 268 | C22×C4 | — | 2 | 2 | no | — |
| 269 | C22×C4 | — | 268 | 2 | no | — |
| 270 | C22×C4 | — | 268 | 2 | no | — |
| 271 | C22×C4 | — | 134 | 2 | no | — |
| 272 | SD16 | — | 536 | 2 | no | — |
| 273 | SD16 | — | 536 | 2 | no | — |
| 274 | D8 | — | 4 | 1 | no | — |
| 275 | D8 | — | 268 | 1 | no | — |
| 276 | M16 | — | 8 | 2 | no | — |
| 277 | M16 | — | 536 | 2 | no | — |
| 278 | C2×C8 | — | 2 | 1 | no | — |
| 279 | C2×C8 | — | 2 | 2 | no | — |
| 280 | C2×C8 | — | 134 | 2 | no | — |
| 281 | C2×C8 | D4×D67 | 1 | 1 | yes | characteristic |
| 282 | C2×C8 | D4×D67 | 1 | 1 | yes | characteristic |
| 283 | C2×C8 | C2×D268 | 1 | 1 | yes | characteristic |
| 284 | C2×C8 | C2×D268 | 1 | 1 | yes | characteristic |
| 285 | C4⋊C4 | — | 268 | 2 | no | — |
| 286 | C4⋊C4 | — | 268 | 2 | no | — |
| 287 | C4⋊C4 | — | 67 | 2 | no | — |
| 288 | C4⋊C4 | — | 134 | 2 | no | — |
| 289 | C4⋊C4 | — | 134 | 2 | no | — |
| 290 | C4⋊C4 | — | 134 | 2 | no | — |
| 291 | C4⋊C4 | — | 134 | 2 | no | — |
| 292 | C4⋊C4 | C2×D268 | 1 | 2 | yes | — |
| 293 | C22⋊C4 | — | 2 | 2 | no | — |
| 294 | C22⋊C4 | — | 268 | 2 | no | — |
| 295 | C22⋊C4 | — | 268 | 2 | no | — |
| 296 | C22⋊C4 | — | 134 | 2 | no | — |
| 297 | C42 | C23×D67 | 1 | 1 | yes | characteristic |
| 298 | C23 | — | 2 | 2 | no | — |
| 299 | C23 | — | 134 | 2 | no | — |
| 300 | Q8 | — | 268 | 2 | no | — |
| 301 | Q8 | — | 268 | 2 | no | — |
| 302 | Q8 | — | 268 | 2 | no | — |
| 303 | Q8 | — | 268 | 2 | no | — |
| 304 | D4 | — | 2 | 2 | no | — |
| 305 | D4 | — | 2 | 2 | no | — |
| 306 | D4 | — | 536 | 4 | no | — |
| 307 | D4 | — | 536 | 4 | no | — |
| 308 | D4 | — | 134 | 2 | no | — |
| 309 | D4 | — | 134 | 2 | no | — |
| 310 | C2×C4 | — | 2 | 2 | no | — |
| 311 | C2×C4 | — | 4 | 2 | no | — |
| 312 | C2×C4 | — | 4 | 2 | no | — |
| 313 | C2×C4 | — | 268 | 2 | no | — |
| 314 | C2×C4 | — | 268 | 2 | no | — |
| 315 | C2×C4 | — | 268 | 2 | no | — |
| 316 | C2×C4 | — | 268 | 2 | no | — |
| 317 | C2×C4 | — | 536 | 4 | no | — |
| 318 | C2×C4 | — | 536 | 4 | no | — |
| 319 | C2×C4 | — | 134 | 2 | no | — |
| 320 | C2×C4 | C2×D4×D67 | 1 | 1 | yes | characteristic |
| 321 | C2×C4 | C22×D268 | 1 | 1 | yes | characteristic |
| 322 | C2×C4 | #14524089 | 1 | 1 | yes | characteristic |
| 323 | C8 | — | 2 | 2 | no | — |
| 324 | C8 | — | 2 | 1 | no | — |
| 325 | C8 | — | 2 | 1 | no | — |
| 326 | C8 | — | 2 | 1 | no | — |
| 327 | C8 | — | 2 | 1 | no | — |
| 328 | C8 | — | 4 | 2 | no | — |
| 329 | C8 | — | 268 | 2 | no | — |
| 330 | C22 | — | 4 | 2 | no | — |
| 331 | C22 | — | 4 | 2 | no | — |
| 332 | C22 | — | 4 | 2 | no | — |
| 333 | C22 | — | 268 | 2 | no | — |
| 334 | C22 | — | 268 | 2 | no | — |
| 335 | C22 | — | 268 | 2 | no | — |
| 336 | C22 | D4×D268 | 1 | 1 | yes | characteristic |
| 337 | C4 | — | 2 | 1 | no | — |
| 338 | C4 | — | 4 | 2 | no | — |
| 339 | C4 | — | 268 | 2 | no | — |
| 340 | C4 | — | 536 | 2 | no | — |
| 341 | C4 | — | 536 | 2 | no | — |
| 342 | C4 | #16864193 | 1 | 1 | yes | characteristic |
| 343 | C4 | #16864196 | 1 | 1 | yes | characteristic |
| 344 | C4 | #16864197 | 1 | 1 | yes | characteristic |
| 345 | C4 | #16864201 | 1 | 1 | yes | characteristic |
| 346 | C2 | — | 8 | 2 | no | — |
| 347 | C2 | — | 536 | 2 | no | — |
| 348 | C2 | #22571819 | 1 | 1 | yes | characteristic |
| 349 | C2 | #22571821 | 1 | 1 | yes | characteristic |
| 350 | C2 | #22572014 | 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