← #32015944
Subgroups — #32015944
Order 18176.
846 conjugacy classes of subgroups.
The subgroup lattice has 324 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#17986117 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #22597225 | C2 | 1 | 2 | yes | maximal |
| 3 | #22596937 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #22589120 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #22588574 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #22588504 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #22588401 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | C71×#1772 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | C2×#14556150 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#14556134 | C22 | 1 | 2 | yes | — |
| 11 | #17986479 | C22 | 1 | 4 | yes | — |
| 12 | #17986117 | C4 | 1 | 8 | yes | complemented |
| 13 | #17986112 | — | 2 | 2 | no | — |
| 14 | #17986107 | — | 2 | 2 | no | — |
| 15 | C4×#11765854 | — | 2 | 4 | no | — |
| 16 | C2×#14556064 | C22 | 1 | 1 | yes | characteristic |
| 17 | #17986042 | C22 | 1 | 4 | yes | — |
| 18 | #17986038 | — | 2 | 4 | no | — |
| 19 | #17986036 | — | 2 | 2 | no | — |
| 20 | C2×#14556062 | C22 | 1 | 2 | yes | — |
| 21 | #107×D71 | — | 2 | 2 | no | — |
| 22 | C2×#14556060 | C22 | 1 | 2 | yes | — |
| 23 | C2×#14556056 | C22 | 1 | 2 | yes | — |
| 24 | #17986001 | C22 | 1 | 4 | yes | — |
| 25 | #17985865 | C22 | 1 | 2 | yes | — |
| 26 | #17985767 | C22 | 1 | 4 | yes | — |
| 27 | C2×#108×C71 | C22 | 1 | 1 | yes | characteristic |
| 28 | #427×C71 | C22 | 1 | 1 | yes | characteristic |
| 29 | #405×C71 | C22 | 1 | 2 | yes | — |
| 30 | C2×C4×C8×C71 | C22 | 1 | 1 | yes | characteristic |
| 31 | #339×C71 | C22 | 1 | 2 | yes | — |
| 32 | C2×#11765865 | C23 | 1 | 2 | yes | — |
| 33 | #14556150 | C2×C4 | 1 | 4 | yes | — |
| 34 | #14556150 | Q8 | 1 | 4 | yes | — |
| 35 | C2×Dic284 | D4 | 1 | 1 | yes | characteristic |
| 36 | C2×#11765854 | D4 | 1 | 2 | yes | — |
| 37 | C2×D568 | D4 | 1 | 1 | yes | characteristic |
| 38 | #14556134 | — | 2 | 4 | no | — |
| 39 | #14556134 | C2×C4 | 1 | 8 | yes | complemented |
| 40 | C2×#11765849 | C23 | 1 | 2 | yes | — |
| 41 | C2×#11765848 | — | 2 | 2 | no | — |
| 42 | C2×C8×D71 | — | 2 | 2 | no | — |
| 43 | C4×Dic142 | — | 2 | 4 | no | — |
| 44 | C4×D284 | — | 2 | 4 | no | — |
| 45 | (C22⋊D71)⋊C4 | — | 4 | 4 | no | — |
| 46 | #14556086 | — | 2 | 4 | no | — |
| 47 | C2×C4⋊Dic71 | C23 | 1 | 4 | yes | — |
| 48 | C4⋊C4×D71 | — | 2 | 4 | no | — |
| 49 | #14556064 | C2×C4 | 1 | 4 | yes | complemented |
| 50 | #14556062 | — | 2 | 2 | no | — |
| 51 | #14556062 | — | 2 | 2 | no | — |
| 52 | #14556061 | — | 2 | 4 | no | — |
| 53 | #14556060 | — | 2 | 2 | no | — |
| 54 | #14556060 | — | 2 | 2 | no | — |
| 55 | #14556059 | — | 2 | 4 | no | — |
| 56 | #14556056 | C2×C4 | 1 | 8 | yes | complemented |
| 57 | #14556053 | — | 2 | 4 | no | — |
| 58 | #14556050 | — | 2 | 4 | no | — |
| 59 | C71⋊#95 | — | 4 | 2 | no | — |
| 60 | C2×M16×C71 | C23 | 1 | 2 | yes | — |
| 61 | C22×C8×C71 | C23 | 1 | 1 | yes | characteristic |
| 62 | C22×C8×C71 | C23 | 1 | 1 | yes | characteristic |
| 63 | #117×C71 | C23 | 1 | 2 | yes | — |
| 64 | C2×C42×C71 | C23 | 1 | 1 | yes | characteristic |
| 65 | #108×C71 | — | 2 | 2 | no | — |
| 66 | #107×C71 | D4 | 1 | 2 | yes | — |
| 67 | #106×C71 | D4 | 1 | 2 | yes | — |
| 68 | #97×C71 | — | 2 | 2 | no | — |
| 69 | C4×C8×C71 | — | 2 | 2 | no | — |
| 70 | C4×C8×C71 | C2×C4 | 1 | 4 | yes | complemented |
| 71 | #11765865 | — | 2 | 2 | no | — |
| 72 | #11765865 | — | 2 | 2 | no | — |
| 73 | #11765865 | C22×C4 | 1 | 4 | yes | — |
| 74 | #11765865 | C2×Q8 | 1 | 4 | yes | — |
| 75 | C2×Dic142 | C2×D4 | 1 | 2 | yes | — |
| 76 | C2×D284 | C2×D4 | 1 | 2 | yes | — |
| 77 | C2×C22⋊D71 | — | 2 | 2 | no | — |
| 78 | C22×Dic71 | — | 2 | 4 | no | — |
| 79 | C2×C4×D71 | — | 2 | 2 | no | — |
| 80 | C2×C4×D71 | — | 4 | 4 | no | — |
| 81 | Dic284 | C4⋊C4 | 1 | 4 | yes | — |
| 82 | #11765854 | C4⋊C4 | 1 | 8 | yes | — |
| 83 | D568 | C4⋊C4 | 1 | 4 | yes | — |
| 84 | #11765849 | — | 2 | 2 | no | — |
| 85 | #11765849 | C22×C4 | 1 | 4 | yes | — |
| 86 | #11765848 | — | 2 | 8 | no | — |
| 87 | C2×C71⋊C8 | — | 2 | 2 | no | — |
| 88 | C8×D71 | — | 2 | 8 | no | — |
| 89 | C4⋊Dic71 | — | 2 | 4 | no | — |
| 90 | C4⋊Dic71 | — | 2 | 4 | no | — |
| 91 | Dic71⋊C4 | — | 4 | 4 | no | — |
| 92 | (C2×D71)⋊C4 | — | 4 | 4 | no | — |
| 93 | C4×Dic71 | — | 4 | 4 | no | — |
| 94 | C22×C4×C71 | — | 2 | 1 | no | — |
| 95 | C22×C4×C71 | C24 | 1 | 1 | yes | characteristic |
| 96 | M16×C71 | — | 2 | 2 | no | — |
| 97 | M16×C71 | — | 2 | 2 | no | — |
| 98 | C2×C8×C71 | — | 2 | 2 | no | — |
| 99 | C2×C8×C71 | C22×C4 | 1 | 2 | yes | — |
| 100 | C2×C8×C71 | C22×C4 | 1 | 2 | yes | — |
| 101 | C2×C8×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 102 | C2×C8×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 103 | C2×C8×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 104 | C2×C8×C71 | C2×D4 | 1 | 1 | yes | characteristic |
| 105 | C2×C8×C71 | C2×Q8 | 1 | 2 | yes | — |
| 106 | C2×C8×C71 | #41 | 1 | 2 | yes | — |
| 107 | C4⋊C4×C71 | C2×D4 | 1 | 4 | yes | — |
| 108 | C22⋊C4×C71 | — | 2 | 2 | no | — |
| 109 | C42×C71 | — | 2 | 2 | no | — |
| 110 | C42×C71 | — | 2 | 1 | no | — |
| 111 | C42×C71 | C22×C4 | 1 | 2 | yes | — |
| 112 | C23×C71 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 113 | C22×D71 | — | 2 | 2 | no | — |
| 114 | C2×C4×C71 | — | 2 | 2 | no | — |
| 115 | C2×C4×C71 | — | 2 | 4 | no | — |
| 116 | C2×C4×C71 | — | 4 | 2 | no | — |
| 117 | C2×C4×C71 | C23×C4 | 1 | 1 | yes | characteristic |
| 118 | C2×C4×C71 | C22×D4 | 1 | 1 | yes | characteristic |
| 119 | C2×C4×C71 | C22×D4 | 1 | 1 | yes | characteristic |
| 120 | C2×C4×C71 | C22×Q8 | 1 | 1 | yes | characteristic |
| 121 | C2×C4×C71 | C2×#41 | 1 | 1 | yes | characteristic |
| 122 | C2×C4×C71 | 2−1+4 | 1 | 1 | yes | characteristic |
| 123 | C22⋊D71 | — | 2 | 4 | no | — |
| 124 | C22⋊D71 | — | 2 | 4 | no | — |
| 125 | C2×Dic71 | — | 2 | 2 | no | — |
| 126 | C2×Dic71 | — | 4 | 4 | no | — |
| 127 | C2×Dic71 | — | 8 | 4 | no | — |
| 128 | D284 | — | 2 | 2 | no | — |
| 129 | D284 | C2×C4⋊C4 | 1 | 4 | yes | — |
| 130 | C4×D71 | — | 2 | 4 | no | — |
| 131 | C4×D71 | — | 2 | 4 | no | — |
| 132 | C4×D71 | — | 16 | 4 | no | — |
| 133 | Dic142 | — | 2 | 2 | no | — |
| 134 | Dic142 | C2×C4⋊C4 | 1 | 4 | yes | — |
| 135 | C8×C71 | — | 4 | 2 | no | — |
| 136 | C8×C71 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 137 | C8×C71 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 138 | C8×C71 | C4×D4 | 1 | 2 | yes | — |
| 139 | C8×C71 | C4×D4 | 1 | 2 | yes | — |
| 140 | C71⋊C8 | — | 2 | 4 | no | — |
| 141 | C22×C71 | — | 2 | 2 | no | — |
| 142 | C22×C71 | #520 | 1 | 1 | yes | characteristic |
| 143 | C22×C71 | D42 | 1 | 1 | yes | characteristic |
| 144 | C22×C71 | #554 | 1 | 1 | yes | characteristic |
| 145 | C2×D71 | — | 2 | 4 | no | — |
| 146 | C2×D71 | — | 4 | 2 | no | — |
| 147 | C2×D71 | — | 4 | 2 | no | — |
| 148 | C4×C71 | — | 4 | 2 | no | — |
| 149 | C4×C71 | — | 4 | 4 | no | — |
| 150 | C4×C71 | C22×C4⋊C4 | 1 | 1 | yes | characteristic |
| 151 | C4×C71 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 152 | C4×C71 | #547 | 1 | 1 | yes | characteristic |
| 153 | C4×C71 | D4×Q8 | 1 | 1 | yes | characteristic |
| 154 | Dic71 | — | 2 | 4 | no | — |
| 155 | Dic71 | — | 8 | 4 | no | — |
| 156 | #22691 | — | 71 | 1 | no | maximal, Sylow |
| 157 | C2×C71 | — | 2 | 1 | no | — |
| 158 | C2×C71 | — | 2 | 1 | no | — |
| 159 | C2×C71 | D4×C4⋊C4 | 1 | 1 | yes | characteristic |
| 160 | C2×C71 | #3290 | 1 | 2 | yes | — |
| 161 | D71 | — | 4 | 4 | no | — |
| 162 | C2×#443 | — | 71 | 1 | no | — |
| 163 | #2952 | — | 71 | 2 | no | — |
| 164 | #2913 | — | 71 | 2 | no | — |
| 165 | #1951 | — | 71 | 2 | no | — |
| 166 | #1860 | — | 71 | 4 | no | — |
| 167 | #1846 | — | 71 | 1 | no | — |
| 168 | #1816 | — | 71 | 2 | no | — |
| 169 | #1772 | D71 | 1 | 1 | yes | characteristic, complemented |
| 170 | C71 | #22691 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 171 | C2×#135 | — | 71 | 1 | no | — |
| 172 | C2×#131 | — | 71 | 2 | no | — |
| 173 | #520 | — | 71 | 4 | no | — |
| 174 | #443 | — | 71 | 8 | no | — |
| 175 | #442 | — | 142 | 2 | no | — |
| 176 | #441 | — | 142 | 2 | no | — |
| 177 | C4×SD16 | — | 142 | 4 | no | — |
| 178 | C2×#108 | — | 71 | 1 | no | — |
| 179 | C2×#108 | C2×D71 | 1 | 1 | yes | characteristic |
| 180 | #428 | — | 71 | 4 | no | — |
| 181 | #428 | — | 142 | 4 | no | — |
| 182 | #427 | — | 142 | 2 | no | — |
| 183 | #427 | C2×D71 | 1 | 1 | yes | characteristic |
| 184 | C2×#107 | — | 71 | 2 | no | — |
| 185 | C2×#107 | — | 142 | 2 | no | — |
| 186 | C2×#106 | — | 71 | 2 | no | — |
| 187 | C2×#104 | — | 71 | 2 | no | — |
| 188 | #417 | — | 71 | 4 | no | — |
| 189 | #405 | C2×D71 | 1 | 2 | yes | — |
| 190 | C2×C4×C8 | C2×D71 | 1 | 1 | yes | characteristic |
| 191 | #382 | — | 71 | 2 | no | — |
| 192 | #343 | — | 71 | 4 | no | — |
| 193 | #339 | C2×D71 | 1 | 2 | yes | — |
| 194 | C2×#41 | — | 71 | 2 | no | — |
| 195 | #135 | — | 71 | 4 | no | — |
| 196 | #135 | — | 71 | 4 | no | — |
| 197 | C2×Q16 | — | 71 | 1 | no | — |
| 198 | C2×SD16 | — | 71 | 2 | no | — |
| 199 | C2×D8 | — | 71 | 1 | no | — |
| 200 | #131 | — | 71 | 8 | no | — |
| 201 | #131 | — | 142 | 4 | no | — |
| 202 | C2×M16 | — | 71 | 2 | no | — |
| 203 | C2×M16 | — | 142 | 2 | no | — |
| 204 | C2×M16 | C22×D71 | 1 | 2 | yes | — |
| 205 | C22×C8 | — | 142 | 2 | no | — |
| 206 | C22×C8 | C22×D71 | 1 | 1 | yes | characteristic |
| 207 | C22×C8 | C22×D71 | 1 | 1 | yes | characteristic |
| 208 | C4×Q8 | — | 142 | 4 | no | — |
| 209 | C4×D4 | — | 284 | 4 | no | — |
| 210 | C4×D4 | — | 142 | 4 | no | — |
| 211 | #117 | — | 142 | 4 | no | — |
| 212 | #117 | C22×D71 | 1 | 2 | yes | — |
| 213 | C2×C4⋊C4 | — | 71 | 4 | no | — |
| 214 | C2×C4⋊C4 | — | 142 | 4 | no | — |
| 215 | C2×C42 | C22×D71 | 1 | 1 | yes | characteristic |
| 216 | #108 | — | 2 | 2 | no | — |
| 217 | #108 | — | 71 | 4 | no | — |
| 218 | #107 | — | 142 | 2 | no | — |
| 219 | #107 | — | 142 | 2 | no | — |
| 220 | #107 | — | 142 | 4 | no | — |
| 221 | #107 | D284 | 1 | 2 | yes | — |
| 222 | #106 | — | 142 | 2 | no | — |
| 223 | #106 | — | 142 | 2 | no | — |
| 224 | #106 | — | 142 | 4 | no | — |
| 225 | #106 | D284 | 1 | 2 | yes | — |
| 226 | #104 | — | 71 | 8 | no | — |
| 227 | #103 | — | 142 | 4 | no | — |
| 228 | #102 | — | 142 | 4 | no | — |
| 229 | #97 | — | 2 | 2 | no | — |
| 230 | C4×C8 | — | 2 | 2 | no | — |
| 231 | C4×C8 | C4×D71 | 1 | 4 | yes | complemented |
| 232 | #95 | — | 284 | 2 | no | — |
| 233 | #41 | — | 71 | 4 | no | — |
| 234 | #41 | — | 71 | 4 | no | — |
| 235 | #41 | — | 142 | 2 | no | — |
| 236 | #41 | — | 142 | 2 | no | — |
| 237 | C2×Q8 | — | 71 | 2 | no | — |
| 238 | C2×D4 | — | 71 | 2 | no | — |
| 239 | C2×D4 | — | 142 | 2 | no | — |
| 240 | C22×C4 | — | 2 | 1 | no | — |
| 241 | C22×C4 | — | 284 | 4 | no | — |
| 242 | C22×C4 | — | 142 | 4 | no | — |
| 243 | C22×C4 | — | 142 | 2 | no | — |
| 244 | C22×C4 | C23×D71 | 1 | 1 | yes | characteristic |
| 245 | Q16 | — | 71 | 4 | no | — |
| 246 | SD16 | — | 71 | 8 | no | — |
| 247 | D8 | — | 71 | 4 | no | — |
| 248 | M16 | — | 2 | 2 | no | — |
| 249 | M16 | — | 2 | 2 | no | — |
| 250 | M16 | — | 71 | 4 | no | — |
| 251 | M16 | — | 142 | 2 | no | — |
| 252 | M16 | — | 142 | 8 | no | — |
| 253 | C2×C8 | — | 2 | 2 | no | — |
| 254 | C2×C8 | — | 142 | 2 | no | — |
| 255 | C2×C8 | — | 142 | 8 | no | — |
| 256 | C2×C8 | C2×C4×D71 | 1 | 2 | yes | — |
| 257 | C2×C8 | C2×C4×D71 | 1 | 2 | yes | — |
| 258 | C2×C8 | D4×D71 | 1 | 1 | yes | characteristic |
| 259 | C2×C8 | D4×D71 | 1 | 1 | yes | characteristic |
| 260 | C2×C8 | C2×D284 | 1 | 1 | yes | characteristic |
| 261 | C2×C8 | C2×D284 | 1 | 1 | yes | characteristic |
| 262 | C2×C8 | Q8×D71 | 1 | 2 | yes | — |
| 263 | C2×C8 | #11765865 | 1 | 2 | yes | — |
| 264 | C4⋊C4 | — | 284 | 4 | no | — |
| 265 | C4⋊C4 | — | 142 | 4 | no | — |
| 266 | C4⋊C4 | — | 142 | 4 | no | — |
| 267 | C4⋊C4 | C2×D284 | 1 | 4 | yes | — |
| 268 | C22⋊C4 | — | 2 | 2 | no | — |
| 269 | C22⋊C4 | — | 284 | 4 | no | — |
| 270 | C42 | — | 2 | 2 | no | — |
| 271 | C42 | — | 2 | 1 | no | — |
| 272 | C42 | — | 284 | 4 | no | — |
| 273 | C42 | C2×C4×D71 | 1 | 2 | yes | — |
| 274 | C23 | — | 142 | 2 | no | — |
| 275 | C23 | #14556174 | 1 | 1 | yes | characteristic |
| 276 | Q8 | — | 71 | 4 | no | — |
| 277 | Q8 | — | 142 | 2 | no | — |
| 278 | D4 | — | 71 | 4 | no | — |
| 279 | D4 | — | 142 | 4 | no | — |
| 280 | D4 | — | 142 | 4 | no | — |
| 281 | D4 | — | 142 | 2 | no | — |
| 282 | C2×C4 | — | 2 | 4 | no | — |
| 283 | C2×C4 | — | 2 | 2 | no | — |
| 284 | C2×C4 | — | 4 | 2 | no | — |
| 285 | C2×C4 | — | 284 | 4 | no | — |
| 286 | C2×C4 | — | 568 | 4 | no | — |
| 287 | C2×C4 | — | 1136 | 4 | no | — |
| 288 | C2×C4 | — | 142 | 2 | no | — |
| 289 | C2×C4 | — | 142 | 4 | no | — |
| 290 | C2×C4 | — | 142 | 4 | no | — |
| 291 | C2×C4 | C22×C4×D71 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | C2×D4×D71 | 1 | 1 | yes | characteristic |
| 293 | C2×C4 | C22×D284 | 1 | 1 | yes | characteristic |
| 294 | C2×C4 | C2×Q8×D71 | 1 | 1 | yes | characteristic |
| 295 | C2×C4 | C2×#11765865 | 1 | 1 | yes | characteristic |
| 296 | C2×C4 | #14556175 | 1 | 1 | yes | characteristic |
| 297 | C8 | — | 4 | 2 | no | — |
| 298 | C8 | — | 142 | 4 | no | — |
| 299 | C8 | C4⋊C4×D71 | 1 | 2 | yes | — |
| 300 | C8 | C4⋊C4×D71 | 1 | 2 | yes | — |
| 301 | C8 | C4×D284 | 1 | 2 | yes | — |
| 302 | C8 | C4×D284 | 1 | 2 | yes | — |
| 303 | C22 | — | 2 | 2 | no | — |
| 304 | C22 | — | 284 | 2 | no | — |
| 305 | C22 | — | 284 | 2 | no | — |
| 306 | C22 | — | 142 | 4 | no | — |
| 307 | C22 | #17986479 | 1 | 1 | yes | characteristic |
| 308 | C22 | D4×D284 | 1 | 1 | yes | characteristic |
| 309 | C22 | #17986709 | 1 | 1 | yes | characteristic |
| 310 | C4 | — | 4 | 2 | no | — |
| 311 | C4 | — | 4 | 4 | no | — |
| 312 | C4 | — | 568 | 4 | no | — |
| 313 | C4 | — | 142 | 4 | no | — |
| 314 | C4 | C2×C4⋊C4×D71 | 1 | 1 | yes | characteristic |
| 315 | C4 | C2×C4×D284 | 1 | 1 | yes | characteristic |
| 316 | C4 | #17986654 | 1 | 1 | yes | characteristic |
| 317 | C4 | Q8×D284 | 1 | 1 | yes | characteristic |
| 318 | C2 | — | 2 | 1 | no | — |
| 319 | C2 | — | 2 | 1 | no | — |
| 320 | C2 | — | 284 | 4 | no | — |
| 321 | C2 | C4⋊C4×D284 | 1 | 1 | yes | characteristic |
| 322 | C2 | #22600128 | 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