← #32155337
Subgroups — #32155337
Order 18176.
462 conjugacy classes of subgroups.
The subgroup lattice has 318 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22601232 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #22590620 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #22590616 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #22590501 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #22590497 | C2 | 1 | 2 | yes | maximal |
| 6 | #22590489 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #22590477 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | Q8×C71⋊C16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C4×#14556072 | C2 | 1 | 2 | yes | maximal |
| 10 | C4×#14556072 | C2 | 1 | 1 | yes | maximal, characteristic |
| 11 | C71×#3380 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #17986727 | — | 2 | 1 | no | — |
| 13 | C2×#14556072 | C22 | 1 | 1 | yes | characteristic |
| 14 | #17986404 | C22 | 1 | 1 | yes | characteristic |
| 15 | #17986401 | C22 | 1 | 2 | yes | — |
| 16 | #17986307 | — | 2 | 2 | no | — |
| 17 | #17986285 | — | 2 | 1 | no | — |
| 18 | C4×Dic284 | C22 | 1 | 1 | yes | characteristic |
| 19 | C4×Dic284 | C22 | 1 | 2 | yes | — |
| 20 | #17985826 | C22 | 1 | 2 | yes | — |
| 21 | #17985826 | C22 | 1 | 1 | yes | characteristic |
| 22 | #17985826 | C22 | 1 | 2 | yes | — |
| 23 | #17985824 | C22 | 1 | 2 | yes | — |
| 24 | #17985824 | C22 | 1 | 2 | yes | — |
| 25 | C4⋊(C71⋊C16) | C22 | 1 | 1 | yes | characteristic |
| 26 | C4⋊(C71⋊C16) | C22 | 1 | 2 | yes | — |
| 27 | #17985802 | C22 | 1 | 2 | yes | — |
| 28 | #17985802 | C22 | 1 | 1 | yes | characteristic |
| 29 | #17985800 | C22 | 1 | 2 | yes | — |
| 30 | #17985800 | C22 | 1 | 1 | yes | characteristic |
| 31 | C4×C71⋊C16 | C22 | 1 | 2 | yes | — |
| 32 | C4×C71⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 33 | #554×C71 | — | 2 | 1 | no | — |
| 34 | #500×C71 | C22 | 1 | 1 | yes | characteristic |
| 35 | #499×C71 | C22 | 1 | 2 | yes | — |
| 36 | #478×C71 | — | 2 | 2 | no | — |
| 37 | #476×C71 | — | 2 | 1 | no | — |
| 38 | C8×Q8×C71 | C22 | 1 | 1 | yes | characteristic |
| 39 | C4×D8×C71 | C22 | 1 | 1 | yes | characteristic |
| 40 | C4×D8×C71 | C22 | 1 | 2 | yes | — |
| 41 | C2×Dic284 | C23 | 1 | 1 | yes | characteristic |
| 42 | #14556128 | — | 2 | 1 | no | — |
| 43 | #14556126 | — | 4 | 1 | no | — |
| 44 | #14556120 | — | 2 | 2 | no | — |
| 45 | #14556118 | — | 4 | 2 | no | — |
| 46 | C4×Dic142 | — | 2 | 2 | no | — |
| 47 | C4×Dic142 | — | 2 | 1 | no | — |
| 48 | #14556093 | — | 4 | 1 | no | — |
| 49 | #14556072 | Q8 | 1 | 4 | yes | complemented |
| 50 | C2×C71⋊C16 | C23 | 1 | 2 | yes | — |
| 51 | C2×C71⋊C16 | C23 | 1 | 1 | yes | characteristic |
| 52 | C2×C71⋊C16 | C23 | 1 | 1 | yes | characteristic |
| 53 | #14556062 | — | 2 | 1 | no | — |
| 54 | #14556062 | C23 | 1 | 1 | yes | characteristic |
| 55 | #14556062 | C23 | 1 | 2 | yes | — |
| 56 | #14556060 | — | 2 | 2 | no | — |
| 57 | #14556053 | — | 2 | 1 | no | — |
| 58 | #14556053 | — | 2 | 2 | no | — |
| 59 | C2×D8×C71 | C23 | 1 | 1 | yes | characteristic |
| 60 | C4⋊Q8×C71 | — | 2 | 1 | no | — |
| 61 | #125×C71 | — | 2 | 2 | no | — |
| 62 | C22⋊Q8×C71 | — | 4 | 1 | no | — |
| 63 | C22⋊Q8×C71 | — | 4 | 2 | no | — |
| 64 | C4×Q8×C71 | D4 | 1 | 1 | yes | characteristic |
| 65 | C4×D4×C71 | — | 2 | 1 | no | — |
| 66 | C4×D4×C71 | — | 2 | 2 | no | — |
| 67 | C2×C4⋊C4×C71 | — | 4 | 1 | no | — |
| 68 | #107×C71 | — | 2 | 1 | no | — |
| 69 | #107×C71 | C23 | 1 | 1 | yes | characteristic |
| 70 | #107×C71 | C23 | 1 | 2 | yes | — |
| 71 | #106×C71 | — | 2 | 2 | no | — |
| 72 | C4⋊C8×C71 | D4 | 1 | 2 | yes | — |
| 73 | C4⋊C8×C71 | D4 | 1 | 1 | yes | characteristic |
| 74 | #102×C71 | — | 2 | 2 | no | — |
| 75 | #102×C71 | — | 2 | 1 | no | — |
| 76 | C4×C8×C71 | C23 | 1 | 2 | yes | — |
| 77 | C4×C8×C71 | C23 | 1 | 1 | yes | characteristic |
| 78 | C2×Dic142 | — | 2 | 1 | no | — |
| 79 | C2×Dic142 | — | 4 | 1 | no | — |
| 80 | Dic284 | — | 2 | 1 | no | — |
| 81 | Dic284 | C2×Q8 | 1 | 2 | yes | — |
| 82 | C4⋊Dic71 | — | 2 | 1 | no | — |
| 83 | C4⋊Dic71 | — | 2 | 2 | no | — |
| 84 | C4⋊Dic71 | — | 4 | 1 | no | — |
| 85 | Dic71⋊C4 | — | 4 | 2 | no | — |
| 86 | Dic71⋊C4 | — | 4 | 1 | no | — |
| 87 | Dic71⋊C4 | — | 4 | 2 | no | — |
| 88 | Dic71⋊C4 | — | 4 | 2 | no | — |
| 89 | C4×Dic71 | — | 4 | 2 | no | — |
| 90 | C4×Dic71 | — | 4 | 1 | no | — |
| 91 | C71⋊C16 | — | 2 | 2 | no | — |
| 92 | C71⋊C16 | — | 2 | 1 | no | — |
| 93 | C71⋊C16 | C2×Q8 | 1 | 2 | yes | — |
| 94 | C2×Q8×C71 | — | 4 | 1 | no | — |
| 95 | C2×Q8×C71 | D8 | 1 | 1 | yes | characteristic |
| 96 | C2×D4×C71 | — | 2 | 1 | no | — |
| 97 | C22×C4×C71 | — | 4 | 1 | no | — |
| 98 | C22×C4×C71 | — | 4 | 2 | no | — |
| 99 | D8×C71 | — | 2 | 1 | no | — |
| 100 | D8×C71 | C2×Q8 | 1 | 2 | yes | — |
| 101 | C2×C8×C71 | C2×D4 | 1 | 2 | yes | — |
| 102 | C2×C8×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 103 | C2×C8×C71 | C24 | 1 | 1 | yes | characteristic |
| 104 | C4⋊C4×C71 | — | 2 | 1 | no | — |
| 105 | C4⋊C4×C71 | — | 2 | 2 | no | — |
| 106 | C4⋊C4×C71 | — | 4 | 2 | no | — |
| 107 | C4⋊C4×C71 | — | 4 | 1 | no | — |
| 108 | C4⋊C4×C71 | — | 4 | 2 | no | — |
| 109 | C4⋊C4×C71 | D8 | 1 | 1 | yes | characteristic |
| 110 | C4⋊C4×C71 | D8 | 1 | 2 | yes | — |
| 111 | C22⋊C4×C71 | — | 4 | 1 | no | — |
| 112 | C22⋊C4×C71 | — | 4 | 2 | no | — |
| 113 | C42×C71 | C2×D4 | 1 | 2 | yes | — |
| 114 | C42×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 115 | C23×C71 | — | 4 | 1 | no | — |
| 116 | Q8×C71 | — | 2 | 2 | no | — |
| 117 | Q8×C71 | — | 16 | 1 | no | — |
| 118 | D4×C71 | — | 2 | 2 | no | — |
| 119 | D4×C71 | — | 4 | 1 | no | — |
| 120 | C2×C4×C71 | — | 4 | 1 | no | — |
| 121 | C2×C4×C71 | — | 4 | 2 | no | — |
| 122 | C2×C4×C71 | — | 8 | 2 | no | — |
| 123 | C2×C4×C71 | — | 16 | 2 | no | — |
| 124 | C2×C4×C71 | C2×D8 | 1 | 2 | yes | — |
| 125 | C2×C4×C71 | C2×D8 | 1 | 1 | yes | characteristic |
| 126 | C2×C4×C71 | C2×D8 | 1 | 2 | yes | — |
| 127 | C2×C4×C71 | C2×D8 | 1 | 1 | yes | characteristic |
| 128 | C2×C4×C71 | C22×D4 | 1 | 1 | yes | characteristic |
| 129 | C2×Dic71 | — | 4 | 1 | no | — |
| 130 | C2×Dic71 | — | 4 | 1 | no | — |
| 131 | C2×Dic71 | — | 4 | 2 | no | — |
| 132 | Dic142 | — | 2 | 2 | no | — |
| 133 | Dic142 | — | 4 | 1 | no | — |
| 134 | Dic142 | — | 8 | 2 | no | — |
| 135 | C8×C71 | — | 2 | 2 | no | — |
| 136 | C8×C71 | — | 2 | 1 | no | — |
| 137 | C8×C71 | C22×Q8 | 1 | 1 | yes | characteristic |
| 138 | C8×C71 | 2−1+4 | 1 | 1 | yes | characteristic |
| 139 | C22×C71 | — | 4 | 2 | no | — |
| 140 | C22×C71 | — | 8 | 1 | no | — |
| 141 | C22×C71 | — | 8 | 1 | no | — |
| 142 | C22×C71 | C22×D8 | 1 | 1 | yes | characteristic |
| 143 | C4×C71 | — | 2 | 2 | no | — |
| 144 | C4×C71 | — | 2 | 2 | no | — |
| 145 | C4×C71 | — | 2 | 1 | no | — |
| 146 | C4×C71 | — | 8 | 1 | no | — |
| 147 | C4×C71 | — | 8 | 2 | no | — |
| 148 | C4×C71 | #508 | 1 | 2 | yes | — |
| 149 | C4×C71 | D4×Q8 | 1 | 1 | yes | characteristic |
| 150 | C4×C71 | #578 | 1 | 1 | yes | characteristic |
| 151 | Dic71 | — | 4 | 2 | no | — |
| 152 | Dic71 | — | 8 | 2 | no | — |
| 153 | Dic71 | — | 8 | 1 | no | — |
| 154 | #33948 | — | 71 | 1 | no | maximal, Sylow |
| 155 | C2×C71 | — | 8 | 2 | no | — |
| 156 | C2×C71 | Q8×D8 | 1 | 1 | yes | characteristic |
| 157 | C2×C71 | C2×#508 | 1 | 1 | yes | characteristic |
| 158 | C2×C71 | #3416 | 1 | 1 | yes | characteristic |
| 159 | #3383 | — | 71 | 1 | no | — |
| 160 | #3380 | D71 | 1 | 1 | yes | characteristic |
| 161 | #2254 | — | 71 | 1 | no | — |
| 162 | #2253 | — | 71 | 2 | no | — |
| 163 | #2229 | — | 71 | 2 | no | — |
| 164 | #2228 | — | 71 | 2 | no | — |
| 165 | #2227 | — | 71 | 1 | no | — |
| 166 | #2224 | — | 71 | 1 | no | — |
| 167 | Q8×C16 | — | 71 | 1 | no | — |
| 168 | C4×SD32 | — | 71 | 2 | no | — |
| 169 | C4×SD32 | — | 71 | 1 | no | — |
| 170 | C71 | #33948 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 171 | #557 | — | 142 | 1 | no | — |
| 172 | #554 | — | 2 | 1 | no | — |
| 173 | C2×SD32 | — | 71 | 1 | no | — |
| 174 | #500 | — | 71 | 1 | no | — |
| 175 | #500 | C2×D71 | 1 | 1 | yes | characteristic |
| 176 | #499 | — | 71 | 2 | no | — |
| 177 | #499 | C2×D71 | 1 | 2 | yes | — |
| 178 | #479 | — | 142 | 2 | no | — |
| 179 | #478 | — | 2 | 2 | no | — |
| 180 | #476 | — | 2 | 1 | no | — |
| 181 | #475 | — | 142 | 1 | no | — |
| 182 | C8×Q8 | C2×D71 | 1 | 1 | yes | characteristic |
| 183 | C4×Q16 | — | 71 | 1 | no | — |
| 184 | C4×Q16 | — | 71 | 2 | no | — |
| 185 | C4×D8 | C2×D71 | 1 | 1 | yes | characteristic |
| 186 | C4×D8 | C2×D71 | 1 | 2 | yes | — |
| 187 | #367 | — | 71 | 1 | no | — |
| 188 | #367 | — | 71 | 2 | no | — |
| 189 | #367 | — | 71 | 2 | no | — |
| 190 | #366 | — | 71 | 2 | no | — |
| 191 | #366 | — | 71 | 2 | no | — |
| 192 | C4⋊C16 | — | 71 | 1 | no | — |
| 193 | C4⋊C16 | — | 71 | 2 | no | — |
| 194 | #358 | — | 71 | 2 | no | — |
| 195 | #358 | — | 71 | 1 | no | — |
| 196 | #357 | — | 71 | 2 | no | — |
| 197 | #357 | — | 71 | 1 | no | — |
| 198 | C4×C16 | — | 71 | 2 | no | — |
| 199 | C4×C16 | — | 71 | 1 | no | — |
| 200 | C2×Q16 | — | 71 | 1 | no | — |
| 201 | C2×D8 | C22×D71 | 1 | 1 | yes | characteristic |
| 202 | C4⋊Q8 | — | 2 | 1 | no | — |
| 203 | C4⋊Q8 | — | 284 | 1 | no | — |
| 204 | C4⋊Q8 | — | 142 | 1 | no | — |
| 205 | #125 | — | 2 | 2 | no | — |
| 206 | #125 | — | 284 | 2 | no | — |
| 207 | #125 | — | 142 | 2 | no | — |
| 208 | C22⋊Q8 | — | 4 | 1 | no | — |
| 209 | C22⋊Q8 | — | 4 | 2 | no | — |
| 210 | C4×Q8 | — | 284 | 1 | no | — |
| 211 | C4×Q8 | — | 142 | 1 | no | — |
| 212 | C4×Q8 | — | 142 | 2 | no | — |
| 213 | C4×Q8 | C22⋊D71 | 1 | 1 | yes | characteristic |
| 214 | C4×D4 | — | 2 | 1 | no | — |
| 215 | C4×D4 | — | 2 | 2 | no | — |
| 216 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 217 | SD32 | — | 71 | 4 | no | — |
| 218 | C2×C16 | — | 71 | 1 | no | — |
| 219 | C2×C16 | — | 71 | 2 | no | — |
| 220 | C2×C16 | — | 71 | 1 | no | — |
| 221 | #107 | — | 2 | 1 | no | — |
| 222 | #107 | — | 71 | 1 | no | — |
| 223 | #107 | — | 71 | 2 | no | — |
| 224 | #107 | — | 142 | 1 | no | — |
| 225 | #107 | C22×D71 | 1 | 1 | yes | characteristic |
| 226 | #107 | C22×D71 | 1 | 2 | yes | — |
| 227 | #106 | — | 2 | 2 | no | — |
| 228 | #106 | — | 142 | 2 | no | — |
| 229 | C4⋊C8 | C22⋊D71 | 1 | 1 | yes | characteristic |
| 230 | C4⋊C8 | C22⋊D71 | 1 | 2 | yes | — |
| 231 | #103 | — | 142 | 2 | no | — |
| 232 | #103 | — | 142 | 1 | no | — |
| 233 | #102 | — | 2 | 2 | no | — |
| 234 | #102 | — | 2 | 1 | no | — |
| 235 | C4×C8 | C22×D71 | 1 | 2 | yes | — |
| 236 | C4×C8 | C22×D71 | 1 | 1 | yes | characteristic |
| 237 | C2×Q8 | — | 4 | 1 | no | — |
| 238 | C2×Q8 | — | 284 | 1 | no | — |
| 239 | C2×Q8 | — | 142 | 1 | no | — |
| 240 | C2×Q8 | #11765850 | 1 | 1 | yes | characteristic |
| 241 | C2×D4 | — | 2 | 1 | no | — |
| 242 | C22×C4 | — | 4 | 1 | no | — |
| 243 | C22×C4 | — | 4 | 2 | no | — |
| 244 | Q16 | — | 71 | 2 | no | — |
| 245 | Q16 | — | 142 | 1 | no | — |
| 246 | D8 | — | 2 | 1 | no | — |
| 247 | D8 | Q8×D71 | 1 | 2 | yes | — |
| 248 | C2×C8 | C2×C22⋊D71 | 1 | 2 | yes | — |
| 249 | C2×C8 | C2×C22⋊D71 | 1 | 1 | yes | characteristic |
| 250 | C2×C8 | C23×D71 | 1 | 1 | yes | characteristic |
| 251 | C4⋊C4 | — | 2 | 1 | no | — |
| 252 | C4⋊C4 | — | 2 | 2 | no | — |
| 253 | C4⋊C4 | — | 4 | 2 | no | — |
| 254 | C4⋊C4 | — | 4 | 1 | no | — |
| 255 | C4⋊C4 | — | 4 | 2 | no | — |
| 256 | C4⋊C4 | — | 284 | 2 | no | — |
| 257 | C4⋊C4 | — | 284 | 1 | no | — |
| 258 | C4⋊C4 | — | 284 | 2 | no | — |
| 259 | C4⋊C4 | — | 284 | 2 | no | — |
| 260 | C4⋊C4 | — | 284 | 1 | no | — |
| 261 | C4⋊C4 | — | 142 | 1 | no | — |
| 262 | C4⋊C4 | — | 142 | 2 | no | — |
| 263 | C4⋊C4 | #11765850 | 1 | 1 | yes | characteristic |
| 264 | C4⋊C4 | #11765850 | 1 | 2 | yes | — |
| 265 | C22⋊C4 | — | 4 | 1 | no | — |
| 266 | C22⋊C4 | — | 4 | 2 | no | — |
| 267 | C42 | — | 284 | 2 | no | — |
| 268 | C42 | — | 284 | 1 | no | — |
| 269 | C42 | C2×C22⋊D71 | 1 | 2 | yes | — |
| 270 | C42 | C2×C22⋊D71 | 1 | 1 | yes | characteristic |
| 271 | C16 | — | 71 | 2 | no | — |
| 272 | C16 | — | 142 | 2 | no | — |
| 273 | C16 | — | 142 | 1 | no | — |
| 274 | C23 | — | 4 | 1 | no | — |
| 275 | Q8 | — | 2 | 2 | no | — |
| 276 | Q8 | — | 16 | 1 | no | — |
| 277 | Q8 | — | 284 | 1 | no | — |
| 278 | Q8 | — | 568 | 2 | no | — |
| 279 | Q8 | — | 142 | 2 | no | — |
| 280 | D4 | — | 2 | 2 | no | — |
| 281 | D4 | — | 4 | 1 | no | — |
| 282 | C2×C4 | — | 4 | 1 | no | — |
| 283 | C2×C4 | — | 4 | 2 | no | — |
| 284 | C2×C4 | — | 8 | 2 | no | — |
| 285 | C2×C4 | — | 16 | 2 | no | — |
| 286 | C2×C4 | — | 284 | 1 | no | — |
| 287 | C2×C4 | — | 284 | 1 | no | — |
| 288 | C2×C4 | — | 284 | 2 | no | — |
| 289 | C2×C4 | C2×#11765850 | 1 | 2 | yes | — |
| 290 | C2×C4 | C2×#11765850 | 1 | 2 | yes | — |
| 291 | C2×C4 | C2×#11765850 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | C2×#11765850 | 1 | 1 | yes | characteristic |
| 293 | C2×C4 | C22×C22⋊D71 | 1 | 1 | yes | characteristic |
| 294 | C8 | — | 2 | 2 | no | — |
| 295 | C8 | — | 2 | 1 | no | — |
| 296 | C8 | C2×Q8×D71 | 1 | 1 | yes | characteristic |
| 297 | C8 | #14556176 | 1 | 1 | yes | characteristic |
| 298 | C22 | — | 4 | 2 | no | — |
| 299 | C22 | — | 8 | 1 | no | — |
| 300 | C22 | — | 8 | 1 | no | — |
| 301 | C22 | C22×#11765850 | 1 | 1 | yes | characteristic |
| 302 | C4 | — | 2 | 2 | no | — |
| 303 | C4 | — | 2 | 1 | no | — |
| 304 | C4 | — | 2 | 2 | no | — |
| 305 | C4 | — | 8 | 2 | no | — |
| 306 | C4 | — | 8 | 1 | no | — |
| 307 | C4 | — | 284 | 2 | no | — |
| 308 | C4 | — | 568 | 1 | no | — |
| 309 | C4 | — | 568 | 2 | no | — |
| 310 | C4 | #17986429 | 1 | 2 | yes | — |
| 311 | C4 | Q8×C22⋊D71 | 1 | 1 | yes | characteristic |
| 312 | C4 | #17986814 | 1 | 1 | yes | characteristic |
| 313 | C2 | — | 8 | 2 | no | — |
| 314 | C2 | Q8×#11765850 | 1 | 1 | yes | characteristic |
| 315 | C2 | C2×#17986429 | 1 | 1 | yes | characteristic |
| 316 | C2 | #22601487 | 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