← #32155447
Subgroups — #32155447
Order 18176.
508 conjugacy classes of subgroups.
The subgroup lattice has 320 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22601325 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | #22590598 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #22590591 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #22590400 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #22590399 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #22590383 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C16×Dic142 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C4×Dic568 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | Q32×Dic71 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 10 | D16×Dic71 | C2 | 1 | 1 | yes | maximal, characteristic |
| 11 | C71×#2249 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #17986718 | — | 2 | 2 | no | — |
| 13 | C2×Dic568 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#14556074 | C22 | 1 | 2 | yes | — |
| 15 | C2×#14556072 | C22 | 1 | 2 | yes | — |
| 16 | #17986382 | C22 | 1 | 2 | yes | — |
| 17 | #17986377 | C22 | 1 | 2 | yes | — |
| 18 | #17986219 | — | 2 | 2 | no | — |
| 19 | #17986218 | — | 2 | 2 | no | — |
| 20 | #17986202 | — | 2 | 2 | no | — |
| 21 | C8×Dic142 | C22 | 1 | 1 | yes | characteristic |
| 22 | C4×Dic284 | C22 | 1 | 2 | yes | — |
| 23 | Q16×Dic71 | C22 | 1 | 2 | yes | — |
| 24 | D8×Dic71 | C22 | 1 | 2 | yes | — |
| 25 | #17985825 | C22 | 1 | 1 | yes | characteristic |
| 26 | C4⋊(C71⋊C16) | C22 | 1 | 1 | yes | characteristic |
| 27 | Dic71⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 28 | Dic71⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 29 | #17985804 | C22 | 1 | 2 | yes | — |
| 30 | #17985803 | C22 | 1 | 2 | yes | — |
| 31 | #17985800 | C22 | 1 | 2 | yes | — |
| 32 | C16×Dic71 | C22 | 1 | 1 | yes | characteristic |
| 33 | C16×Dic71 | C22 | 1 | 1 | yes | characteristic |
| 34 | C2×Q32×C71 | C22 | 1 | 1 | yes | characteristic |
| 35 | C2×SD32×C71 | C22 | 1 | 2 | yes | — |
| 36 | C2×D16×C71 | C22 | 1 | 1 | yes | characteristic |
| 37 | #495×C71 | C22 | 1 | 2 | yes | — |
| 38 | C4×C16×C71 | C22 | 1 | 1 | yes | characteristic |
| 39 | C2×Dic284 | C23 | 1 | 2 | yes | — |
| 40 | C2×#11765855 | — | 2 | 2 | no | — |
| 41 | C2×#11765853 | — | 2 | 2 | no | — |
| 42 | #14556126 | — | 2 | 2 | no | — |
| 43 | #14556121 | — | 4 | 2 | no | — |
| 44 | #14556121 | — | 4 | 2 | no | — |
| 45 | #14556115 | — | 2 | 2 | no | — |
| 46 | #14556105 | — | 4 | 2 | no | — |
| 47 | Q8×Dic71 | — | 2 | 2 | no | — |
| 48 | C4×Dic142 | — | 2 | 2 | no | — |
| 49 | C4×Dic142 | D4 | 1 | 1 | yes | characteristic |
| 50 | D4×Dic71 | — | 2 | 2 | no | — |
| 51 | #14556084 | — | 4 | 2 | no | — |
| 52 | Dic568 | — | 2 | 2 | no | — |
| 53 | #14556074 | — | 4 | 2 | no | — |
| 54 | #14556072 | — | 4 | 2 | no | — |
| 55 | C2×C71⋊C16 | C23 | 1 | 1 | yes | characteristic |
| 56 | C2×C71⋊C16 | C23 | 1 | 1 | yes | characteristic |
| 57 | #14556062 | C23 | 1 | 2 | yes | — |
| 58 | C4⋊(C71⋊C8) | D4 | 1 | 1 | yes | characteristic |
| 59 | Dic71⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 60 | Dic71⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 61 | #14556053 | — | 2 | 2 | no | — |
| 62 | #14556052 | — | 2 | 2 | no | — |
| 63 | #14556049 | — | 2 | 2 | no | — |
| 64 | C8×Dic71 | C23 | 1 | 1 | yes | characteristic |
| 65 | C8×Dic71 | C23 | 1 | 1 | yes | characteristic |
| 66 | C2×Q16×C71 | C23 | 1 | 2 | yes | — |
| 67 | C2×SD16×C71 | — | 2 | 2 | no | — |
| 68 | C2×D8×C71 | C23 | 1 | 2 | yes | — |
| 69 | #124×C71 | — | 2 | 2 | no | — |
| 70 | Q32×C71 | — | 2 | 2 | no | — |
| 71 | SD32×C71 | — | 4 | 2 | no | — |
| 72 | D16×C71 | — | 2 | 2 | no | — |
| 73 | C2×C16×C71 | C23 | 1 | 1 | yes | characteristic |
| 74 | C2×C16×C71 | C23 | 1 | 1 | yes | characteristic |
| 75 | C4×C8×C71 | C23 | 1 | 1 | yes | characteristic |
| 76 | C2×Dic142 | — | 2 | 2 | no | — |
| 77 | C2×Dic142 | D8 | 1 | 1 | yes | characteristic |
| 78 | C22×Dic71 | — | 4 | 2 | no | — |
| 79 | Dic284 | — | 2 | 2 | no | — |
| 80 | Dic284 | — | 2 | 2 | no | — |
| 81 | #11765855 | — | 8 | 2 | no | — |
| 82 | #11765853 | — | 8 | 2 | no | — |
| 83 | C2×C71⋊C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 84 | C2×C71⋊C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 85 | C4⋊Dic71 | — | 2 | 2 | no | — |
| 86 | C4⋊Dic71 | — | 4 | 2 | no | — |
| 87 | C4⋊Dic71 | D8 | 1 | 1 | yes | characteristic |
| 88 | Dic71⋊C4 | — | 4 | 2 | no | — |
| 89 | Dic71⋊C4 | — | 4 | 2 | no | — |
| 90 | Dic71⋊C4 | — | 4 | 2 | no | — |
| 91 | Dic71⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 92 | Dic71⋊C4 | D8 | 1 | 1 | yes | characteristic |
| 93 | C22⋊Dic71 | — | 4 | 2 | no | — |
| 94 | C22⋊Dic71 | — | 4 | 2 | no | — |
| 95 | C22⋊Dic71 | — | 4 | 2 | no | — |
| 96 | C4×Dic71 | — | 4 | 2 | no | — |
| 97 | C4×Dic71 | — | 4 | 2 | no | — |
| 98 | C4×Dic71 | C2×D4 | 1 | 1 | yes | characteristic |
| 99 | C4×Dic71 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C71⋊C16 | — | 2 | 1 | no | — |
| 101 | C71⋊C16 | — | 2 | 1 | no | — |
| 102 | C2×Q8×C71 | — | 2 | 2 | no | — |
| 103 | C2×D4×C71 | — | 2 | 2 | no | — |
| 104 | Q16×C71 | — | 2 | 2 | no | — |
| 105 | Q16×C71 | — | 2 | 2 | no | — |
| 106 | SD16×C71 | — | 8 | 2 | no | — |
| 107 | D8×C71 | — | 2 | 2 | no | — |
| 108 | D8×C71 | — | 2 | 2 | no | — |
| 109 | C2×C8×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 110 | C2×C8×C71 | C24 | 1 | 1 | yes | characteristic |
| 111 | C22⋊C4×C71 | — | 4 | 2 | no | — |
| 112 | C22⋊C4×C71 | — | 4 | 2 | no | — |
| 113 | C42×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 114 | C16×C71 | — | 2 | 1 | no | — |
| 115 | C16×C71 | #41 | 1 | 2 | yes | — |
| 116 | C23×C71 | — | 4 | 2 | no | — |
| 117 | Q8×C71 | — | 4 | 2 | no | — |
| 118 | Q8×C71 | — | 4 | 2 | no | — |
| 119 | D4×C71 | — | 4 | 2 | no | — |
| 120 | D4×C71 | — | 4 | 2 | no | — |
| 121 | C2×C4×C71 | — | 4 | 2 | no | — |
| 122 | C2×C4×C71 | C2×D8 | 1 | 1 | yes | characteristic |
| 123 | C2×C4×C71 | C2×D8 | 1 | 1 | yes | characteristic |
| 124 | C2×C4×C71 | C22×D4 | 1 | 1 | yes | characteristic |
| 125 | C2×Dic71 | — | 4 | 2 | no | — |
| 126 | C2×Dic71 | — | 4 | 2 | no | — |
| 127 | C2×Dic71 | — | 8 | 4 | no | — |
| 128 | C2×Dic71 | C2×D8 | 1 | 1 | yes | characteristic |
| 129 | C2×Dic71 | C2×D8 | 1 | 1 | yes | characteristic |
| 130 | C2×Dic71 | C2×D8 | 1 | 1 | yes | characteristic |
| 131 | C2×Dic71 | C2×D8 | 1 | 1 | yes | characteristic |
| 132 | Dic142 | — | 2 | 2 | no | — |
| 133 | Dic142 | — | 4 | 2 | no | — |
| 134 | Dic142 | — | 4 | 2 | no | — |
| 135 | C8×C71 | — | 2 | 1 | no | — |
| 136 | C8×C71 | C2×#41 | 1 | 1 | yes | characteristic |
| 137 | C8×C71 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 138 | C71⋊C8 | — | 2 | 1 | no | — |
| 139 | C71⋊C8 | — | 2 | 1 | no | — |
| 140 | C22×C71 | — | 8 | 2 | no | — |
| 141 | C22×C71 | — | 8 | 2 | no | — |
| 142 | C22×C71 | — | 8 | 2 | no | — |
| 143 | C22×C71 | C22×D8 | 1 | 1 | yes | characteristic |
| 144 | C4×C71 | — | 2 | 1 | no | — |
| 145 | C4×C71 | — | 2 | 1 | no | — |
| 146 | C4×C71 | — | 8 | 2 | no | — |
| 147 | C4×C71 | #550 | 1 | 1 | yes | characteristic |
| 148 | C4×C71 | #576 | 1 | 1 | yes | characteristic |
| 149 | Dic71 | — | 2 | 1 | no | — |
| 150 | Dic71 | — | 2 | 1 | no | — |
| 151 | Dic71 | — | 2 | 1 | no | — |
| 152 | Dic71 | — | 8 | 2 | no | — |
| 153 | Dic71 | — | 8 | 2 | no | — |
| 154 | Dic71 | #508 | 1 | 2 | yes | — |
| 155 | #33960 | — | 71 | 1 | no | maximal, Sylow |
| 156 | C2×C71 | — | 16 | 2 | no | — |
| 157 | C2×C71 | #3391 | 1 | 1 | yes | characteristic |
| 158 | C2×C71 | C2×#508 | 1 | 1 | yes | characteristic |
| 159 | C2×C71 | #3417 | 1 | 1 | yes | characteristic |
| 160 | #3393 | — | 71 | 2 | no | — |
| 161 | #2249 | — | 71 | 1 | no | — |
| 162 | #2249 | D71 | 1 | 1 | yes | characteristic |
| 163 | #2247 | — | 71 | 1 | no | — |
| 164 | #2211 | — | 71 | 2 | no | — |
| 165 | #2211 | — | 71 | 2 | no | — |
| 166 | #2208 | — | 71 | 2 | no | — |
| 167 | Q8×C16 | — | 71 | 1 | no | — |
| 168 | C4×Q32 | — | 71 | 1 | no | — |
| 169 | C4×Q32 | — | 71 | 1 | no | — |
| 170 | C4×D16 | — | 71 | 1 | no | — |
| 171 | C71 | #33960 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 172 | #556 | — | 142 | 2 | no | — |
| 173 | C2×Q32 | — | 71 | 2 | no | — |
| 174 | C2×Q32 | — | 71 | 1 | no | — |
| 175 | C2×Q32 | C2×D71 | 1 | 1 | yes | characteristic |
| 176 | C2×SD32 | — | 71 | 2 | no | — |
| 177 | C2×SD32 | C2×D71 | 1 | 2 | yes | — |
| 178 | C2×D16 | C2×D71 | 1 | 1 | yes | characteristic |
| 179 | #495 | — | 71 | 2 | no | — |
| 180 | #495 | C2×D71 | 1 | 2 | yes | — |
| 181 | #494 | — | 71 | 2 | no | — |
| 182 | #464 | — | 142 | 2 | no | — |
| 183 | #464 | — | 142 | 2 | no | — |
| 184 | #461 | — | 142 | 2 | no | — |
| 185 | C8×Q8 | — | 71 | 1 | no | — |
| 186 | C4×Q16 | — | 71 | 2 | no | — |
| 187 | C4×Q16 | — | 71 | 2 | no | — |
| 188 | C4×D8 | — | 71 | 2 | no | — |
| 189 | #366 | — | 71 | 1 | no | — |
| 190 | C4⋊C16 | — | 71 | 1 | no | — |
| 191 | C4⋊C16 | — | 71 | 1 | no | — |
| 192 | C4⋊C16 | — | 71 | 1 | no | — |
| 193 | #358 | — | 71 | 2 | no | — |
| 194 | #358 | — | 71 | 2 | no | — |
| 195 | #357 | — | 71 | 2 | no | — |
| 196 | C4×C16 | — | 71 | 1 | no | — |
| 197 | C4×C16 | — | 71 | 1 | no | — |
| 198 | C4×C16 | C2×D71 | 1 | 1 | yes | characteristic |
| 199 | C2×Q16 | — | 71 | 2 | no | — |
| 200 | C2×Q16 | — | 142 | 2 | no | — |
| 201 | C2×Q16 | C22×D71 | 1 | 2 | yes | — |
| 202 | C2×SD16 | — | 2 | 2 | no | — |
| 203 | C2×SD16 | — | 142 | 2 | no | — |
| 204 | C2×D8 | C22×D71 | 1 | 2 | yes | — |
| 205 | C4⋊Q8 | — | 142 | 2 | no | — |
| 206 | #126 | — | 284 | 2 | no | — |
| 207 | #126 | — | 284 | 2 | no | — |
| 208 | #124 | — | 2 | 2 | no | — |
| 209 | #124 | — | 142 | 2 | no | — |
| 210 | C22⋊Q8 | — | 284 | 2 | no | — |
| 211 | C4×Q8 | — | 71 | 1 | no | — |
| 212 | C4×Q8 | — | 142 | 2 | no | — |
| 213 | C4×Q8 | — | 142 | 2 | no | — |
| 214 | C4×D4 | — | 142 | 2 | no | — |
| 215 | #117 | — | 284 | 2 | no | — |
| 216 | Q32 | — | 2 | 2 | no | — |
| 217 | Q32 | — | 284 | 2 | no | — |
| 218 | Q32 | — | 142 | 2 | no | — |
| 219 | SD32 | — | 4 | 2 | no | — |
| 220 | SD32 | — | 284 | 2 | no | — |
| 221 | D16 | — | 2 | 2 | no | — |
| 222 | C2×C16 | — | 71 | 1 | no | — |
| 223 | C2×C16 | — | 71 | 1 | no | — |
| 224 | C2×C16 | C22×D71 | 1 | 1 | yes | characteristic |
| 225 | C2×C16 | C22×D71 | 1 | 1 | yes | characteristic |
| 226 | #107 | — | 71 | 2 | no | — |
| 227 | C4⋊C8 | — | 71 | 1 | no | — |
| 228 | C4⋊C8 | — | 71 | 1 | no | — |
| 229 | C4⋊C8 | — | 71 | 1 | no | — |
| 230 | #103 | — | 142 | 2 | no | — |
| 231 | #103 | — | 142 | 2 | no | — |
| 232 | #102 | — | 142 | 2 | no | — |
| 233 | C4×C8 | — | 71 | 1 | no | — |
| 234 | C4×C8 | — | 71 | 1 | no | — |
| 235 | C4×C8 | C22×D71 | 1 | 1 | yes | characteristic |
| 236 | C2×Q8 | — | 2 | 2 | no | — |
| 237 | C2×Q8 | — | 71 | 1 | no | — |
| 238 | C2×Q8 | — | 142 | 2 | no | — |
| 239 | C2×D4 | — | 2 | 2 | no | — |
| 240 | C22×C4 | — | 284 | 2 | no | — |
| 241 | Q16 | — | 2 | 2 | no | — |
| 242 | Q16 | — | 2 | 2 | no | — |
| 243 | Q16 | — | 568 | 2 | no | — |
| 244 | Q16 | — | 142 | 2 | no | — |
| 245 | Q16 | — | 142 | 2 | no | — |
| 246 | SD16 | — | 8 | 2 | no | — |
| 247 | SD16 | — | 568 | 2 | no | — |
| 248 | D8 | — | 2 | 2 | no | — |
| 249 | D8 | — | 2 | 2 | no | — |
| 250 | C2×C8 | — | 71 | 1 | no | — |
| 251 | C2×C8 | — | 71 | 1 | no | — |
| 252 | C2×C8 | D4×D71 | 1 | 1 | yes | characteristic |
| 253 | C2×C8 | C23×D71 | 1 | 1 | yes | characteristic |
| 254 | C4⋊C4 | — | 284 | 2 | no | — |
| 255 | C4⋊C4 | — | 284 | 2 | no | — |
| 256 | C4⋊C4 | — | 284 | 2 | no | — |
| 257 | C4⋊C4 | — | 284 | 2 | no | — |
| 258 | C4⋊C4 | — | 71 | 1 | no | — |
| 259 | C4⋊C4 | — | 71 | 1 | no | — |
| 260 | C4⋊C4 | — | 71 | 1 | no | — |
| 261 | C4⋊C4 | — | 142 | 2 | no | — |
| 262 | C22⋊C4 | — | 4 | 2 | no | — |
| 263 | C22⋊C4 | — | 4 | 2 | no | — |
| 264 | C22⋊C4 | — | 284 | 2 | no | — |
| 265 | C22⋊C4 | — | 284 | 2 | no | — |
| 266 | C22⋊C4 | — | 284 | 2 | no | — |
| 267 | C42 | — | 284 | 2 | no | — |
| 268 | C42 | — | 284 | 2 | no | — |
| 269 | C42 | — | 71 | 1 | no | — |
| 270 | C42 | — | 71 | 1 | no | — |
| 271 | C42 | D4×D71 | 1 | 1 | yes | characteristic |
| 272 | C16 | — | 2 | 1 | no | — |
| 273 | C16 | — | 142 | 1 | no | — |
| 274 | C16 | — | 142 | 1 | no | — |
| 275 | C16 | #11765864 | 1 | 2 | yes | — |
| 276 | C23 | — | 4 | 2 | no | — |
| 277 | Q8 | — | 4 | 2 | no | — |
| 278 | Q8 | — | 4 | 2 | no | — |
| 279 | Q8 | — | 284 | 2 | no | — |
| 280 | Q8 | — | 284 | 2 | no | — |
| 281 | Q8 | — | 142 | 2 | no | — |
| 282 | D4 | — | 4 | 2 | no | — |
| 283 | D4 | — | 4 | 2 | no | — |
| 284 | C2×C4 | — | 4 | 2 | no | — |
| 285 | C2×C4 | — | 284 | 2 | no | — |
| 286 | C2×C4 | — | 284 | 2 | no | — |
| 287 | C2×C4 | — | 568 | 4 | no | — |
| 288 | C2×C4 | — | 71 | 1 | no | — |
| 289 | C2×C4 | — | 71 | 1 | no | — |
| 290 | C2×C4 | — | 71 | 1 | no | — |
| 291 | C2×C4 | — | 71 | 1 | no | — |
| 292 | C2×C4 | D8×D71 | 1 | 1 | yes | characteristic |
| 293 | C2×C4 | D8×D71 | 1 | 1 | yes | characteristic |
| 294 | C2×C4 | C2×D4×D71 | 1 | 1 | yes | characteristic |
| 295 | C8 | — | 2 | 1 | no | — |
| 296 | C8 | — | 142 | 1 | no | — |
| 297 | C8 | — | 142 | 1 | no | — |
| 298 | C8 | C2×#11765864 | 1 | 1 | yes | characteristic |
| 299 | C8 | #14556173 | 1 | 1 | yes | characteristic |
| 300 | C22 | — | 8 | 2 | no | — |
| 301 | C22 | — | 8 | 2 | no | — |
| 302 | C22 | — | 8 | 2 | no | — |
| 303 | C22 | C2×D8×D71 | 1 | 1 | yes | characteristic |
| 304 | C4 | — | 2 | 1 | no | — |
| 305 | C4 | — | 2 | 1 | no | — |
| 306 | C4 | — | 8 | 2 | no | — |
| 307 | C4 | — | 568 | 2 | no | — |
| 308 | C4 | — | 568 | 2 | no | — |
| 309 | C4 | — | 71 | 2 | no | — |
| 310 | C4 | — | 142 | 1 | no | — |
| 311 | C4 | — | 142 | 1 | no | — |
| 312 | C4 | — | 142 | 1 | no | — |
| 313 | C4 | #17986669 | 1 | 1 | yes | characteristic |
| 314 | C4 | #17986800 | 1 | 1 | yes | characteristic |
| 315 | C2 | — | 16 | 2 | no | — |
| 316 | C2 | #22601313 | 1 | 1 | yes | characteristic |
| 317 | C2 | #508×D71 | 1 | 1 | yes | characteristic |
| 318 | C2 | #22601490 | 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