← #32006307
Subgroups — #32006307
Order 18176.
654 conjugacy classes of subgroups.
The subgroup lattice has 334 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #22598852 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22597279 | C2 | 1 | 2 | yes | maximal |
| 3 | #22589319 | C2 | 1 | 2 | yes | maximal |
| 4 | #22589093 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #22589085 | C2 | 1 | 2 | yes | maximal |
| 6 | #22588510 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #22588227 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #22588226 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #22588218 | C2 | 1 | 2 | yes | maximal |
| 10 | #22588028 | C2 | 1 | 1 | yes | maximal, characteristic |
| 11 | C71×#1946 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #17986552 | C22 | 1 | 2 | yes | — |
| 13 | #17986403 | C4 | 1 | 2 | yes | complemented |
| 14 | #17986400 | C4 | 1 | 2 | yes | complemented |
| 15 | #17986399 | C4 | 1 | 2 | yes | complemented |
| 16 | #17986396 | C4 | 1 | 2 | yes | complemented |
| 17 | #17986128 | — | 2 | 4 | no | — |
| 18 | #17986126 | — | 2 | 4 | no | — |
| 19 | C2×#14556064 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×#14556063 | C22 | 1 | 2 | yes | — |
| 21 | #17986037 | C22 | 1 | 1 | yes | characteristic |
| 22 | #17986035 | C22 | 1 | 2 | yes | — |
| 23 | #17986025 | C22 | 1 | 2 | yes | — |
| 24 | #17986022 | C22 | 1 | 2 | yes | — |
| 25 | #17985957 | C22 | 1 | 2 | yes | — |
| 26 | #17985955 | C22 | 1 | 2 | yes | — |
| 27 | C2×#14556038 | C22 | 1 | 1 | yes | characteristic |
| 28 | C2×#14556038 | C22 | 1 | 1 | yes | characteristic |
| 29 | C2×C4×C71⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 30 | C2×C8×Dic71 | C22 | 1 | 1 | yes | characteristic |
| 31 | C2×C8×Dic71 | C22 | 1 | 1 | yes | characteristic |
| 32 | #17985845 | C22 | 1 | 1 | yes | characteristic |
| 33 | #17985760 | C22 | 1 | 2 | yes | — |
| 34 | #17985759 | C22 | 1 | 2 | yes | — |
| 35 | #17985758 | C22 | 1 | 2 | yes | — |
| 36 | C2×#108×C71 | C22 | 1 | 1 | yes | characteristic |
| 37 | #427×C71 | C22 | 1 | 1 | yes | characteristic |
| 38 | #424×C71 | C22 | 1 | 2 | yes | — |
| 39 | C2×#97×C71 | C22 | 1 | 1 | yes | characteristic |
| 40 | #339×C71 | C22 | 1 | 2 | yes | — |
| 41 | C2×#11765849 | C23 | 1 | 2 | yes | — |
| 42 | C22×C71⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 43 | C22×C71⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 44 | #14556127 | C2×C4 | 1 | 4 | yes | — |
| 45 | #14556119 | C2×C4 | 1 | 4 | yes | — |
| 46 | #14556105 | — | 2 | 4 | no | — |
| 47 | C4×Dic142 | — | 2 | 4 | no | — |
| 48 | #14556087 | C23 | 1 | 2 | yes | — |
| 49 | C2×C4×Dic71 | C23 | 1 | 1 | yes | characteristic |
| 50 | C2×C4×Dic71 | C23 | 1 | 1 | yes | characteristic |
| 51 | #14556064 | — | 2 | 2 | no | — |
| 52 | #14556063 | — | 2 | 4 | no | — |
| 53 | #14556062 | C2×C4 | 1 | 2 | yes | — |
| 54 | #14556061 | C2×C4 | 1 | 4 | yes | — |
| 55 | #14556060 | C2×C4 | 1 | 2 | yes | — |
| 56 | #14556059 | C2×C4 | 1 | 4 | yes | — |
| 57 | Dic71⋊C8 | — | 2 | 4 | no | — |
| 58 | Dic71⋊C8 | — | 2 | 4 | no | — |
| 59 | #14556039 | — | 2 | 2 | no | — |
| 60 | #14556038 | — | 2 | 2 | no | — |
| 61 | #14556038 | D4 | 1 | 4 | yes | — |
| 62 | #14556038 | Q8 | 1 | 4 | yes | complemented |
| 63 | C4×C71⋊C8 | — | 2 | 2 | no | — |
| 64 | C4×C71⋊C8 | Q8 | 1 | 4 | yes | complemented |
| 65 | C8×Dic71 | — | 2 | 2 | no | — |
| 66 | C8×Dic71 | — | 2 | 2 | no | — |
| 67 | C8×Dic71 | C2×C4 | 1 | 1 | yes | characteristic |
| 68 | C8×Dic71 | C2×C4 | 1 | 1 | yes | characteristic |
| 69 | C8×Dic71 | C2×C4 | 1 | 1 | yes | characteristic |
| 70 | C8×Dic71 | C2×C4 | 1 | 1 | yes | characteristic |
| 71 | C71⋊#95 | — | 2 | 2 | no | — |
| 72 | C2×M16×C71 | C23 | 1 | 2 | yes | — |
| 73 | C22×C8×C71 | C23 | 1 | 1 | yes | characteristic |
| 74 | C22×C8×C71 | C23 | 1 | 1 | yes | characteristic |
| 75 | #117×C71 | C23 | 1 | 2 | yes | — |
| 76 | C2×C42×C71 | C23 | 1 | 1 | yes | characteristic |
| 77 | #108×C71 | — | 2 | 2 | no | — |
| 78 | #107×C71 | C2×C4 | 1 | 2 | yes | — |
| 79 | #106×C71 | C2×C4 | 1 | 2 | yes | — |
| 80 | C4⋊C8×C71 | — | 2 | 4 | no | — |
| 81 | #97×C71 | — | 2 | 2 | no | — |
| 82 | C2×Dic142 | — | 2 | 4 | no | — |
| 83 | C22×Dic71 | — | 2 | 1 | no | — |
| 84 | C22×Dic71 | #41 | 1 | 2 | yes | — |
| 85 | #11765849 | — | 2 | 2 | no | — |
| 86 | #11765849 | — | 2 | 2 | no | — |
| 87 | C2×C71⋊C8 | — | 2 | 2 | no | — |
| 88 | C2×C71⋊C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 89 | C2×C71⋊C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 90 | C2×C71⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 91 | C2×C71⋊C8 | C2×Q8 | 1 | 2 | yes | — |
| 92 | C2×C71⋊C8 | C2×Q8 | 1 | 2 | yes | — |
| 93 | C2×C71⋊C8 | C2×Q8 | 1 | 2 | yes | — |
| 94 | C2×C71⋊C8 | #41 | 1 | 2 | yes | — |
| 95 | C4⋊Dic71 | — | 2 | 4 | no | — |
| 96 | C4⋊Dic71 | C22×C4 | 1 | 4 | yes | — |
| 97 | Dic71⋊C4 | — | 2 | 4 | no | — |
| 98 | Dic71⋊C4 | — | 2 | 4 | no | — |
| 99 | C22⋊Dic71 | — | 2 | 2 | no | — |
| 100 | C4×Dic71 | — | 2 | 2 | no | — |
| 101 | C4×Dic71 | — | 2 | 1 | no | — |
| 102 | C4×Dic71 | C22×C4 | 1 | 1 | yes | characteristic |
| 103 | C4×Dic71 | C22×C4 | 1 | 1 | yes | characteristic |
| 104 | C4×Dic71 | C2×D4 | 1 | 2 | yes | — |
| 105 | C4×Dic71 | C2×Q8 | 1 | 2 | yes | — |
| 106 | C22×C4×C71 | — | 2 | 1 | no | — |
| 107 | C22×C4×C71 | C24 | 1 | 1 | yes | characteristic |
| 108 | M16×C71 | — | 2 | 2 | no | — |
| 109 | M16×C71 | — | 2 | 2 | no | — |
| 110 | C2×C8×C71 | — | 2 | 1 | no | — |
| 111 | C2×C8×C71 | — | 2 | 1 | no | — |
| 112 | C2×C8×C71 | — | 2 | 2 | no | — |
| 113 | C2×C8×C71 | C22×C4 | 1 | 1 | yes | characteristic |
| 114 | C2×C8×C71 | C22×C4 | 1 | 1 | yes | characteristic |
| 115 | C2×C8×C71 | C2×D4 | 1 | 2 | yes | — |
| 116 | C2×C8×C71 | C2×Q8 | 1 | 2 | yes | — |
| 117 | C2×C8×C71 | #41 | 1 | 2 | yes | — |
| 118 | C4⋊C4×C71 | C22×C4 | 1 | 4 | yes | — |
| 119 | C22⋊C4×C71 | — | 2 | 2 | no | — |
| 120 | C42×C71 | — | 2 | 2 | no | — |
| 121 | C42×C71 | — | 2 | 1 | no | — |
| 122 | C42×C71 | C2×Q8 | 1 | 2 | yes | — |
| 123 | C23×C71 | C2×#41 | 1 | 1 | yes | characteristic |
| 124 | C2×C4×C71 | — | 2 | 4 | no | — |
| 125 | C2×C4×C71 | — | 2 | 2 | no | — |
| 126 | C2×C4×C71 | — | 4 | 2 | no | — |
| 127 | C2×C4×C71 | C23×C4 | 1 | 1 | yes | characteristic |
| 128 | C2×C4×C71 | C22×D4 | 1 | 1 | yes | characteristic |
| 129 | C2×C4×C71 | C22×Q8 | 1 | 1 | yes | characteristic |
| 130 | C2×C4×C71 | C22×Q8 | 1 | 1 | yes | characteristic |
| 131 | C2×C4×C71 | C2×#41 | 1 | 1 | yes | characteristic |
| 132 | C2×C4×C71 | C2×#41 | 1 | 1 | yes | characteristic |
| 133 | C2×Dic71 | — | 2 | 4 | no | — |
| 134 | C2×Dic71 | — | 2 | 2 | no | — |
| 135 | C2×Dic71 | — | 4 | 2 | no | — |
| 136 | C2×Dic71 | — | 4 | 2 | no | — |
| 137 | C2×Dic71 | C4×D4 | 1 | 2 | yes | — |
| 138 | C2×Dic71 | C4×D4 | 1 | 2 | yes | — |
| 139 | Dic142 | — | 8 | 4 | no | — |
| 140 | C8×C71 | — | 2 | 2 | no | — |
| 141 | C8×C71 | — | 4 | 2 | no | — |
| 142 | C8×C71 | C4×Q8 | 1 | 2 | yes | — |
| 143 | C8×C71 | C4×Q8 | 1 | 2 | yes | — |
| 144 | C71⋊C8 | — | 4 | 2 | no | — |
| 145 | C71⋊C8 | C4×Q8 | 1 | 2 | yes | — |
| 146 | C71⋊C8 | C4×Q8 | 1 | 2 | yes | — |
| 147 | C71⋊C8 | C4⋊Q8 | 1 | 4 | yes | — |
| 148 | C22×C71 | — | 2 | 2 | no | — |
| 149 | C22×C71 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 150 | C22×C71 | #533 | 1 | 1 | yes | characteristic |
| 151 | C22×C71 | #533 | 1 | 1 | yes | characteristic |
| 152 | C4×C71 | — | 4 | 2 | no | — |
| 153 | C4×C71 | — | 4 | 4 | no | — |
| 154 | C4×C71 | C2×C4×Q8 | 1 | 1 | yes | characteristic |
| 155 | C4×C71 | C2×C4×Q8 | 1 | 1 | yes | characteristic |
| 156 | C4×C71 | C2×C4⋊Q8 | 1 | 1 | yes | characteristic |
| 157 | C4×C71 | #532 | 1 | 1 | yes | characteristic |
| 158 | Dic71 | — | 2 | 2 | no | — |
| 159 | Dic71 | — | 2 | 2 | no | — |
| 160 | Dic71 | — | 4 | 2 | no | — |
| 161 | Dic71 | — | 4 | 4 | no | — |
| 162 | #21733 | — | 71 | 1 | no | maximal, Sylow |
| 163 | C2×C71 | — | 2 | 1 | no | — |
| 164 | C2×C71 | — | 2 | 1 | no | — |
| 165 | C2×C71 | C4×C4⋊Q8 | 1 | 1 | yes | characteristic |
| 166 | C2×C71 | #2957 | 1 | 2 | yes | — |
| 167 | #3160 | — | 71 | 1 | no | — |
| 168 | #2961 | — | 71 | 2 | no | — |
| 169 | #1995 | — | 71 | 2 | no | — |
| 170 | #1946 | — | 71 | 1 | no | — |
| 171 | #1946 | D71 | 1 | 1 | yes | characteristic |
| 172 | #1945 | — | 71 | 2 | no | — |
| 173 | #1848 | — | 71 | 1 | no | — |
| 174 | #1773 | — | 71 | 1 | no | — |
| 175 | #1773 | — | 71 | 1 | no | — |
| 176 | #1772 | — | 71 | 2 | no | — |
| 177 | #1726 | — | 71 | 1 | no | — |
| 178 | C71 | #21733 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 179 | #533 | — | 71 | 2 | no | — |
| 180 | #500 | — | 71 | 2 | no | — |
| 181 | #499 | — | 71 | 2 | no | — |
| 182 | #499 | — | 71 | 2 | no | — |
| 183 | #498 | — | 71 | 2 | no | — |
| 184 | #446 | — | 142 | 4 | no | — |
| 185 | #446 | — | 142 | 4 | no | — |
| 186 | C2×#108 | — | 71 | 2 | no | — |
| 187 | C2×#108 | — | 71 | 1 | no | — |
| 188 | C2×#108 | C2×D71 | 1 | 1 | yes | characteristic |
| 189 | #427 | — | 71 | 1 | no | — |
| 190 | #427 | — | 71 | 2 | no | — |
| 191 | #427 | C2×D71 | 1 | 1 | yes | characteristic |
| 192 | #424 | — | 71 | 2 | no | — |
| 193 | #424 | C2×D71 | 1 | 2 | yes | — |
| 194 | #423 | — | 71 | 2 | no | — |
| 195 | #405 | — | 71 | 2 | no | — |
| 196 | #405 | — | 71 | 2 | no | — |
| 197 | C2×#97 | — | 71 | 1 | no | — |
| 198 | C2×#97 | — | 71 | 1 | no | — |
| 199 | C2×#97 | C2×D71 | 1 | 1 | yes | characteristic |
| 200 | C2×C4×C8 | — | 71 | 1 | no | — |
| 201 | C2×C4×C8 | — | 71 | 1 | no | — |
| 202 | C2×C4×C8 | — | 71 | 1 | no | — |
| 203 | #376 | — | 71 | 1 | no | — |
| 204 | #339 | — | 71 | 2 | no | — |
| 205 | #339 | — | 71 | 2 | no | — |
| 206 | #339 | — | 71 | 2 | no | — |
| 207 | #339 | C2×D71 | 1 | 2 | yes | — |
| 208 | C2×M16 | — | 71 | 2 | no | — |
| 209 | C2×M16 | C22×D71 | 1 | 2 | yes | — |
| 210 | C22×C8 | — | 71 | 1 | no | — |
| 211 | C22×C8 | — | 71 | 1 | no | — |
| 212 | C22×C8 | C22×D71 | 1 | 1 | yes | characteristic |
| 213 | C22×C8 | C22×D71 | 1 | 1 | yes | characteristic |
| 214 | C4⋊Q8 | — | 71 | 4 | no | — |
| 215 | #125 | — | 71 | 4 | no | — |
| 216 | C22⋊Q8 | — | 142 | 4 | no | — |
| 217 | C4×Q8 | — | 142 | 4 | no | — |
| 218 | #117 | — | 71 | 2 | no | — |
| 219 | #117 | C22×D71 | 1 | 2 | yes | — |
| 220 | C2×C42 | — | 71 | 1 | no | — |
| 221 | C2×C42 | — | 71 | 1 | no | — |
| 222 | C2×C42 | C22×D71 | 1 | 1 | yes | characteristic |
| 223 | #108 | — | 2 | 2 | no | — |
| 224 | #108 | — | 142 | 4 | no | — |
| 225 | #108 | — | 142 | 2 | no | — |
| 226 | #107 | — | 71 | 2 | no | — |
| 227 | #107 | — | 71 | 4 | no | — |
| 228 | #107 | C4×D71 | 1 | 2 | yes | — |
| 229 | #106 | — | 71 | 2 | no | — |
| 230 | #106 | — | 71 | 4 | no | — |
| 231 | #106 | C4×D71 | 1 | 2 | yes | — |
| 232 | C4⋊C8 | — | 2 | 4 | no | — |
| 233 | C4⋊C8 | — | 142 | 4 | no | — |
| 234 | C4⋊C8 | — | 142 | 4 | no | — |
| 235 | #97 | — | 2 | 2 | no | — |
| 236 | #97 | — | 71 | 4 | no | — |
| 237 | #97 | — | 71 | 4 | no | — |
| 238 | #97 | — | 142 | 2 | no | — |
| 239 | #97 | — | 142 | 2 | no | — |
| 240 | C4×C8 | — | 71 | 1 | no | — |
| 241 | C4×C8 | — | 71 | 1 | no | — |
| 242 | C4×C8 | — | 71 | 1 | no | — |
| 243 | C4×C8 | — | 71 | 1 | no | — |
| 244 | C4×C8 | — | 71 | 4 | no | — |
| 245 | C4×C8 | — | 142 | 2 | no | — |
| 246 | C4×C8 | — | 142 | 2 | no | — |
| 247 | C4×C8 | — | 142 | 2 | no | — |
| 248 | #95 | — | 142 | 2 | no | — |
| 249 | C2×Q8 | — | 142 | 4 | no | — |
| 250 | C22×C4 | — | 2 | 1 | no | — |
| 251 | C22×C4 | — | 71 | 2 | no | — |
| 252 | C22×C4 | — | 142 | 1 | no | — |
| 253 | C22×C4 | C23×D71 | 1 | 1 | yes | characteristic |
| 254 | M16 | — | 2 | 2 | no | — |
| 255 | M16 | — | 2 | 2 | no | — |
| 256 | M16 | — | 142 | 2 | no | — |
| 257 | M16 | — | 142 | 2 | no | — |
| 258 | C2×C8 | — | 2 | 1 | no | — |
| 259 | C2×C8 | — | 2 | 1 | no | — |
| 260 | C2×C8 | — | 2 | 2 | no | — |
| 261 | C2×C8 | — | 71 | 1 | no | — |
| 262 | C2×C8 | — | 71 | 1 | no | — |
| 263 | C2×C8 | — | 71 | 2 | no | — |
| 264 | C2×C8 | — | 71 | 2 | no | — |
| 265 | C2×C8 | — | 71 | 2 | no | — |
| 266 | C2×C8 | — | 71 | 2 | no | — |
| 267 | C2×C8 | — | 71 | 2 | no | — |
| 268 | C2×C8 | — | 142 | 2 | no | — |
| 269 | C2×C8 | C2×C4×D71 | 1 | 1 | yes | characteristic |
| 270 | C2×C8 | C2×C4×D71 | 1 | 1 | yes | characteristic |
| 271 | C2×C8 | D4×D71 | 1 | 2 | yes | — |
| 272 | C2×C8 | Q8×D71 | 1 | 2 | yes | — |
| 273 | C2×C8 | #11765864 | 1 | 2 | yes | — |
| 274 | C4⋊C4 | — | 71 | 4 | no | — |
| 275 | C4⋊C4 | — | 142 | 4 | no | — |
| 276 | C4⋊C4 | — | 142 | 4 | no | — |
| 277 | C4⋊C4 | — | 142 | 4 | no | — |
| 278 | C4⋊C4 | C2×C4×D71 | 1 | 4 | yes | — |
| 279 | C22⋊C4 | — | 2 | 2 | no | — |
| 280 | C22⋊C4 | — | 142 | 2 | no | — |
| 281 | C42 | — | 2 | 2 | no | — |
| 282 | C42 | — | 2 | 1 | no | — |
| 283 | C42 | — | 71 | 1 | no | — |
| 284 | C42 | — | 71 | 1 | no | — |
| 285 | C42 | — | 71 | 2 | no | — |
| 286 | C42 | — | 71 | 2 | no | — |
| 287 | C42 | — | 142 | 2 | no | — |
| 288 | C42 | — | 142 | 1 | no | — |
| 289 | C42 | Q8×D71 | 1 | 2 | yes | — |
| 290 | C23 | #41×D71 | 1 | 1 | yes | characteristic |
| 291 | Q8 | — | 568 | 4 | no | — |
| 292 | C2×C4 | — | 2 | 4 | no | — |
| 293 | C2×C4 | — | 2 | 2 | no | — |
| 294 | C2×C4 | — | 4 | 2 | no | — |
| 295 | C2×C4 | — | 284 | 2 | no | — |
| 296 | C2×C4 | — | 284 | 2 | no | — |
| 297 | C2×C4 | — | 71 | 2 | no | — |
| 298 | C2×C4 | — | 71 | 2 | no | — |
| 299 | C2×C4 | — | 142 | 4 | no | — |
| 300 | C2×C4 | — | 142 | 2 | no | — |
| 301 | C2×C4 | C22×C4×D71 | 1 | 1 | yes | characteristic |
| 302 | C2×C4 | C2×D4×D71 | 1 | 1 | yes | characteristic |
| 303 | C2×C4 | C2×Q8×D71 | 1 | 1 | yes | characteristic |
| 304 | C2×C4 | C2×Q8×D71 | 1 | 1 | yes | characteristic |
| 305 | C2×C4 | #41×D71 | 1 | 1 | yes | characteristic |
| 306 | C2×C4 | C2×#11765864 | 1 | 1 | yes | characteristic |
| 307 | C8 | — | 2 | 2 | no | — |
| 308 | C8 | — | 4 | 2 | no | — |
| 309 | C8 | — | 284 | 2 | no | — |
| 310 | C8 | — | 71 | 2 | no | — |
| 311 | C8 | — | 71 | 2 | no | — |
| 312 | C8 | — | 71 | 4 | no | — |
| 313 | C8 | #14556093 | 1 | 2 | yes | — |
| 314 | C8 | #14556093 | 1 | 2 | yes | — |
| 315 | C22 | — | 2 | 2 | no | — |
| 316 | C22 | C4×D4×D71 | 1 | 1 | yes | characteristic |
| 317 | C22 | #17986551 | 1 | 1 | yes | characteristic |
| 318 | C22 | #17986555 | 1 | 1 | yes | characteristic |
| 319 | C4 | — | 4 | 2 | no | — |
| 320 | C4 | — | 4 | 4 | no | — |
| 321 | C4 | — | 284 | 2 | no | — |
| 322 | C4 | — | 284 | 4 | no | — |
| 323 | C4 | — | 142 | 2 | no | — |
| 324 | C4 | — | 142 | 2 | no | — |
| 325 | C4 | C4×Q8×D71 | 1 | 1 | yes | characteristic |
| 326 | C4 | C2×#14556093 | 1 | 1 | yes | characteristic |
| 327 | C4 | C4⋊Q8×D71 | 1 | 1 | yes | characteristic |
| 328 | C4 | #17986544 | 1 | 1 | yes | characteristic |
| 329 | C2 | — | 2 | 1 | no | — |
| 330 | C2 | — | 2 | 1 | no | — |
| 331 | C2 | #22590911 | 1 | 1 | yes | characteristic |
| 332 | C2 | #22597254 | 1 | 2 | yes | — |
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