← #22573172
Subgroups — #22573172
Order 8576.
320 conjugacy classes of subgroups.
The subgroup lattice has 300 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #16864124 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #16863806 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #16863802 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #16863795 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #16863703 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #16863703 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #16863700 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #16863700 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #16863691 | C2 | 1 | 1 | yes | maximal, characteristic |
| 10 | #16863688 | C2 | 1 | 1 | yes | maximal, characteristic |
| 11 | #16863527 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #16863501 | C2 | 1 | 1 | yes | maximal, characteristic |
| 13 | #16863501 | C2 | 1 | 1 | yes | maximal, characteristic |
| 14 | C4×#11761618 | C2 | 1 | 1 | yes | maximal, characteristic |
| 15 | #557×C67 | C2 | 1 | 1 | yes | maximal, characteristic |
| 16 | C2×#11761618 | C22 | 1 | 1 | yes | characteristic |
| 17 | #14524043 | C22 | 1 | 1 | yes | characteristic |
| 18 | #14524041 | — | 2 | 1 | no | — |
| 19 | #14524035 | C22 | 1 | 1 | yes | characteristic |
| 20 | #14524035 | C22 | 1 | 1 | yes | characteristic |
| 21 | #14524033 | — | 2 | 1 | no | — |
| 22 | #14524033 | — | 2 | 1 | no | — |
| 23 | C4×Dic134 | C22 | 1 | 1 | yes | characteristic |
| 24 | C4×Dic134 | C22 | 1 | 1 | yes | characteristic |
| 25 | C4×Dic134 | C22 | 1 | 1 | yes | characteristic |
| 26 | #14524008 | — | 2 | 1 | no | — |
| 27 | #14523976 | C22 | 1 | 1 | yes | characteristic |
| 28 | #14523976 | C22 | 1 | 1 | yes | characteristic |
| 29 | #14523976 | C22 | 1 | 1 | yes | characteristic |
| 30 | #14523976 | C22 | 1 | 1 | yes | characteristic |
| 31 | #14523976 | C22 | 1 | 1 | yes | characteristic |
| 32 | #14523974 | C22 | 1 | 1 | yes | characteristic |
| 33 | #14523974 | C22 | 1 | 1 | yes | characteristic |
| 34 | #14523974 | C22 | 1 | 1 | yes | characteristic |
| 35 | #14523974 | C22 | 1 | 1 | yes | characteristic |
| 36 | C4⋊(C67⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 37 | C4⋊(C67⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 38 | C4⋊(C67⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 39 | #14523967 | C22 | 1 | 1 | yes | characteristic |
| 40 | #14523967 | C22 | 1 | 1 | yes | characteristic |
| 41 | #14523967 | C22 | 1 | 1 | yes | characteristic |
| 42 | #14523966 | C22 | 1 | 1 | yes | characteristic |
| 43 | #14523966 | C22 | 1 | 1 | yes | characteristic |
| 44 | #14523966 | C22 | 1 | 1 | yes | characteristic |
| 45 | #14523954 | C22 | 1 | 1 | yes | characteristic |
| 46 | #14523954 | C22 | 1 | 1 | yes | characteristic |
| 47 | C4×C67⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 48 | C4⋊Q8×C67 | — | 2 | 1 | no | — |
| 49 | C4⋊Q8×C67 | C22 | 1 | 1 | yes | characteristic |
| 50 | #125×C67 | — | 2 | 1 | no | — |
| 51 | #125×C67 | — | 2 | 1 | no | — |
| 52 | #125×C67 | C22 | 1 | 1 | yes | characteristic |
| 53 | #125×C67 | C22 | 1 | 1 | yes | characteristic |
| 54 | C4×Q8×C67 | — | 2 | 1 | no | — |
| 55 | C4×Q8×C67 | C22 | 1 | 1 | yes | characteristic |
| 56 | C4×Q8×C67 | C22 | 1 | 1 | yes | characteristic |
| 57 | C4×Q8×C67 | C22 | 1 | 1 | yes | characteristic |
| 58 | C4×Q8×C67 | C22 | 1 | 1 | yes | characteristic |
| 59 | C2×Dic134 | — | 2 | 1 | no | — |
| 60 | C2×Dic134 | C23 | 1 | 1 | yes | characteristic |
| 61 | #11761618 | Q8 | 1 | 4 | yes | complemented |
| 62 | C2×C67⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 63 | C2×C67⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 64 | C2×C67⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 65 | C2×C67⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 66 | C4⋊Dic67 | — | 2 | 1 | no | — |
| 67 | C4⋊Dic67 | C23 | 1 | 1 | yes | characteristic |
| 68 | C4⋊Dic67 | C23 | 1 | 1 | yes | characteristic |
| 69 | C4⋊Dic67 | C23 | 1 | 1 | yes | characteristic |
| 70 | Dic67⋊C4 | — | 2 | 1 | no | — |
| 71 | Dic67⋊C4 | — | 2 | 1 | no | — |
| 72 | Dic67⋊C4 | — | 2 | 1 | no | — |
| 73 | Dic67⋊C4 | — | 2 | 1 | no | — |
| 74 | Dic67⋊C4 | — | 2 | 1 | no | — |
| 75 | Dic67⋊C4 | — | 2 | 1 | no | — |
| 76 | Dic67⋊C4 | — | 2 | 1 | no | — |
| 77 | C4×Dic67 | — | 2 | 1 | no | — |
| 78 | C4×Dic67 | — | 2 | 1 | no | — |
| 79 | C4×Dic67 | — | 2 | 1 | no | — |
| 80 | C2×Q8×C67 | — | 2 | 1 | no | — |
| 81 | C2×Q8×C67 | D4 | 1 | 1 | yes | characteristic |
| 82 | C2×Q8×C67 | C23 | 1 | 1 | yes | characteristic |
| 83 | C4⋊C4×C67 | — | 2 | 1 | no | — |
| 84 | C4⋊C4×C67 | — | 2 | 1 | no | — |
| 85 | C4⋊C4×C67 | — | 2 | 1 | no | — |
| 86 | C4⋊C4×C67 | — | 2 | 1 | no | — |
| 87 | C4⋊C4×C67 | — | 2 | 1 | no | — |
| 88 | C4⋊C4×C67 | — | 2 | 1 | no | — |
| 89 | C4⋊C4×C67 | — | 2 | 1 | no | — |
| 90 | C4⋊C4×C67 | — | 2 | 1 | no | — |
| 91 | C4⋊C4×C67 | D4 | 1 | 1 | yes | characteristic |
| 92 | C4⋊C4×C67 | D4 | 1 | 1 | yes | characteristic |
| 93 | C4⋊C4×C67 | D4 | 1 | 1 | yes | characteristic |
| 94 | C4⋊C4×C67 | C23 | 1 | 1 | yes | characteristic |
| 95 | C4⋊C4×C67 | C23 | 1 | 1 | yes | characteristic |
| 96 | C4⋊C4×C67 | C23 | 1 | 1 | yes | characteristic |
| 97 | C42×C67 | — | 2 | 1 | no | — |
| 98 | C42×C67 | — | 2 | 1 | no | — |
| 99 | C42×C67 | — | 2 | 1 | no | — |
| 100 | C42×C67 | C23 | 1 | 1 | yes | characteristic |
| 101 | C42×C67 | C23 | 1 | 1 | yes | characteristic |
| 102 | C42×C67 | C23 | 1 | 1 | yes | characteristic |
| 103 | Q8×C67 | — | 2 | 1 | no | — |
| 104 | Q8×C67 | — | 4 | 1 | no | — |
| 105 | Q8×C67 | — | 4 | 2 | no | — |
| 106 | Q8×C67 | C2×Q8 | 1 | 2 | yes | — |
| 107 | C2×C4×C67 | — | 2 | 1 | no | — |
| 108 | C2×C4×C67 | — | 2 | 1 | no | — |
| 109 | C2×C4×C67 | — | 2 | 1 | no | — |
| 110 | C2×C4×C67 | — | 2 | 1 | no | — |
| 111 | C2×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 112 | C2×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 113 | C2×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 114 | C2×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 115 | C2×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 116 | C2×C4×C67 | C2×D4 | 1 | 1 | yes | characteristic |
| 117 | C2×C4×C67 | C24 | 1 | 1 | yes | characteristic |
| 118 | C2×Dic67 | — | 2 | 1 | no | — |
| 119 | C2×Dic67 | — | 2 | 1 | no | — |
| 120 | C2×Dic67 | — | 2 | 1 | no | — |
| 121 | C2×Dic67 | — | 2 | 1 | no | — |
| 122 | Dic134 | — | 2 | 1 | no | — |
| 123 | Dic134 | — | 4 | 2 | no | — |
| 124 | Dic134 | C2×Q8 | 1 | 2 | yes | — |
| 125 | C67⋊C8 | — | 2 | 1 | no | — |
| 126 | C67⋊C8 | — | 2 | 1 | no | — |
| 127 | C67⋊C8 | — | 2 | 1 | no | — |
| 128 | C67⋊C8 | C2×Q8 | 1 | 2 | yes | — |
| 129 | C22×C67 | C22×D4 | 1 | 1 | yes | characteristic |
| 130 | C4×C67 | — | 2 | 2 | no | — |
| 131 | C4×C67 | — | 2 | 1 | no | — |
| 132 | C4×C67 | — | 2 | 1 | no | — |
| 133 | C4×C67 | — | 2 | 1 | no | — |
| 134 | C4×C67 | — | 2 | 1 | no | — |
| 135 | C4×C67 | — | 2 | 1 | no | — |
| 136 | C4×C67 | — | 2 | 1 | no | — |
| 137 | C4×C67 | — | 4 | 1 | no | — |
| 138 | C4×C67 | — | 4 | 1 | no | — |
| 139 | C4×C67 | — | 4 | 1 | no | — |
| 140 | C4×C67 | C22×Q8 | 1 | 1 | yes | characteristic |
| 141 | C4×C67 | 2−1+4 | 1 | 1 | yes | characteristic |
| 142 | Dic67 | — | 2 | 2 | no | — |
| 143 | Dic67 | — | 4 | 1 | no | — |
| 144 | Dic67 | — | 4 | 1 | no | — |
| 145 | Dic67 | — | 4 | 1 | no | — |
| 146 | C2×C67 | D4×Q8 | 1 | 1 | yes | characteristic |
| 147 | C2×C67 | #575 | 1 | 1 | yes | characteristic |
| 148 | C2×C67 | #578 | 1 | 1 | yes | characteristic |
| 149 | #3390 | — | 67 | 1 | no | maximal, Sylow |
| 150 | C67 | #3390 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 151 | #557 | — | 67 | 1 | no | — |
| 152 | #557 | D67 | 1 | 1 | yes | characteristic |
| 153 | #501 | — | 67 | 1 | no | — |
| 154 | #501 | — | 67 | 1 | no | — |
| 155 | #499 | — | 67 | 1 | no | — |
| 156 | #479 | — | 67 | 1 | no | — |
| 157 | #479 | — | 67 | 1 | no | — |
| 158 | #479 | — | 67 | 1 | no | — |
| 159 | #479 | — | 67 | 1 | no | — |
| 160 | #477 | — | 67 | 1 | no | — |
| 161 | #477 | — | 67 | 1 | no | — |
| 162 | #446 | — | 67 | 1 | no | — |
| 163 | #441 | — | 67 | 1 | no | — |
| 164 | #441 | — | 67 | 1 | no | — |
| 165 | C4×Q16 | — | 67 | 1 | no | — |
| 166 | C2×Q16 | — | 67 | 1 | no | — |
| 167 | C4⋊Q8 | — | 2 | 1 | no | — |
| 168 | C4⋊Q8 | — | 67 | 1 | no | — |
| 169 | C4⋊Q8 | — | 134 | 1 | no | — |
| 170 | C4⋊Q8 | C2×D67 | 1 | 1 | yes | characteristic |
| 171 | #125 | — | 2 | 1 | no | — |
| 172 | #125 | — | 2 | 1 | no | — |
| 173 | #125 | — | 67 | 1 | no | — |
| 174 | #125 | — | 67 | 1 | no | — |
| 175 | #125 | — | 134 | 1 | no | — |
| 176 | #125 | — | 134 | 1 | no | — |
| 177 | #125 | C2×D67 | 1 | 1 | yes | characteristic |
| 178 | #125 | C2×D67 | 1 | 1 | yes | characteristic |
| 179 | C4×Q8 | — | 2 | 1 | no | — |
| 180 | C4×Q8 | — | 67 | 1 | no | — |
| 181 | C4×Q8 | — | 67 | 1 | no | — |
| 182 | C4×Q8 | — | 67 | 1 | no | — |
| 183 | C4×Q8 | — | 134 | 1 | no | — |
| 184 | C4×Q8 | C2×D67 | 1 | 1 | yes | characteristic |
| 185 | C4×Q8 | C2×D67 | 1 | 1 | yes | characteristic |
| 186 | C4×Q8 | C2×D67 | 1 | 1 | yes | characteristic |
| 187 | C4×Q8 | C2×D67 | 1 | 1 | yes | characteristic |
| 188 | #107 | — | 67 | 1 | no | — |
| 189 | #107 | — | 67 | 1 | no | — |
| 190 | #107 | — | 67 | 1 | no | — |
| 191 | #107 | — | 67 | 1 | no | — |
| 192 | #107 | — | 67 | 1 | no | — |
| 193 | #106 | — | 67 | 1 | no | — |
| 194 | #106 | — | 67 | 1 | no | — |
| 195 | #106 | — | 67 | 1 | no | — |
| 196 | #106 | — | 67 | 1 | no | — |
| 197 | C4⋊C8 | — | 67 | 1 | no | — |
| 198 | C4⋊C8 | — | 67 | 1 | no | — |
| 199 | C4⋊C8 | — | 67 | 1 | no | — |
| 200 | #103 | — | 67 | 1 | no | — |
| 201 | #103 | — | 67 | 1 | no | — |
| 202 | #103 | — | 67 | 1 | no | — |
| 203 | #103 | — | 67 | 1 | no | — |
| 204 | #103 | — | 67 | 1 | no | — |
| 205 | #103 | — | 67 | 1 | no | — |
| 206 | #97 | — | 67 | 1 | no | — |
| 207 | #97 | — | 67 | 1 | no | — |
| 208 | C4×C8 | — | 67 | 1 | no | — |
| 209 | C2×Q8 | — | 2 | 1 | no | — |
| 210 | C2×Q8 | — | 67 | 1 | no | — |
| 211 | C2×Q8 | — | 134 | 1 | no | — |
| 212 | C2×Q8 | C22⋊D67 | 1 | 1 | yes | characteristic |
| 213 | C2×Q8 | C22×D67 | 1 | 1 | yes | characteristic |
| 214 | Q16 | — | 67 | 4 | no | — |
| 215 | C2×C8 | — | 67 | 1 | no | — |
| 216 | C2×C8 | — | 67 | 1 | no | — |
| 217 | C2×C8 | — | 67 | 1 | no | — |
| 218 | C2×C8 | — | 67 | 1 | no | — |
| 219 | C4⋊C4 | — | 2 | 1 | no | — |
| 220 | C4⋊C4 | — | 2 | 1 | no | — |
| 221 | C4⋊C4 | — | 2 | 1 | no | — |
| 222 | C4⋊C4 | — | 2 | 1 | no | — |
| 223 | C4⋊C4 | — | 2 | 1 | no | — |
| 224 | C4⋊C4 | — | 2 | 1 | no | — |
| 225 | C4⋊C4 | — | 2 | 1 | no | — |
| 226 | C4⋊C4 | — | 2 | 1 | no | — |
| 227 | C4⋊C4 | — | 67 | 1 | no | — |
| 228 | C4⋊C4 | — | 67 | 1 | no | — |
| 229 | C4⋊C4 | — | 67 | 1 | no | — |
| 230 | C4⋊C4 | — | 134 | 1 | no | — |
| 231 | C4⋊C4 | — | 134 | 1 | no | — |
| 232 | C4⋊C4 | — | 134 | 1 | no | — |
| 233 | C4⋊C4 | — | 134 | 1 | no | — |
| 234 | C4⋊C4 | — | 134 | 1 | no | — |
| 235 | C4⋊C4 | — | 134 | 1 | no | — |
| 236 | C4⋊C4 | — | 134 | 1 | no | — |
| 237 | C4⋊C4 | — | 134 | 1 | no | — |
| 238 | C4⋊C4 | C22⋊D67 | 1 | 1 | yes | characteristic |
| 239 | C4⋊C4 | C22⋊D67 | 1 | 1 | yes | characteristic |
| 240 | C4⋊C4 | C22⋊D67 | 1 | 1 | yes | characteristic |
| 241 | C4⋊C4 | C22×D67 | 1 | 1 | yes | characteristic |
| 242 | C4⋊C4 | C22×D67 | 1 | 1 | yes | characteristic |
| 243 | C4⋊C4 | C22×D67 | 1 | 1 | yes | characteristic |
| 244 | C42 | — | 2 | 1 | no | — |
| 245 | C42 | — | 2 | 1 | no | — |
| 246 | C42 | — | 2 | 1 | no | — |
| 247 | C42 | — | 134 | 1 | no | — |
| 248 | C42 | — | 134 | 1 | no | — |
| 249 | C42 | — | 134 | 1 | no | — |
| 250 | C42 | C22×D67 | 1 | 1 | yes | characteristic |
| 251 | C42 | C22×D67 | 1 | 1 | yes | characteristic |
| 252 | C42 | C22×D67 | 1 | 1 | yes | characteristic |
| 253 | Q8 | — | 2 | 1 | no | — |
| 254 | Q8 | — | 4 | 1 | no | — |
| 255 | Q8 | — | 4 | 2 | no | — |
| 256 | Q8 | — | 268 | 2 | no | — |
| 257 | Q8 | — | 67 | 2 | no | — |
| 258 | Q8 | — | 134 | 1 | no | — |
| 259 | Q8 | Q8×D67 | 1 | 2 | yes | — |
| 260 | C2×C4 | — | 2 | 1 | no | — |
| 261 | C2×C4 | — | 2 | 1 | no | — |
| 262 | C2×C4 | — | 2 | 1 | no | — |
| 263 | C2×C4 | — | 2 | 1 | no | — |
| 264 | C2×C4 | — | 134 | 1 | no | — |
| 265 | C2×C4 | — | 134 | 1 | no | — |
| 266 | C2×C4 | — | 134 | 1 | no | — |
| 267 | C2×C4 | — | 134 | 1 | no | — |
| 268 | C2×C4 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 269 | C2×C4 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 270 | C2×C4 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 271 | C2×C4 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 272 | C2×C4 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 273 | C2×C4 | C2×C22⋊D67 | 1 | 1 | yes | characteristic |
| 274 | C2×C4 | C23×D67 | 1 | 1 | yes | characteristic |
| 275 | C8 | — | 67 | 2 | no | — |
| 276 | C8 | — | 134 | 1 | no | — |
| 277 | C8 | — | 134 | 1 | no | — |
| 278 | C8 | — | 134 | 1 | no | — |
| 279 | C22 | C22×C22⋊D67 | 1 | 1 | yes | characteristic |
| 280 | C4 | — | 2 | 2 | no | — |
| 281 | C4 | — | 2 | 1 | no | — |
| 282 | C4 | — | 2 | 1 | no | — |
| 283 | C4 | — | 2 | 1 | no | — |
| 284 | C4 | — | 2 | 1 | no | — |
| 285 | C4 | — | 2 | 1 | no | — |
| 286 | C4 | — | 2 | 1 | no | — |
| 287 | C4 | — | 4 | 1 | no | — |
| 288 | C4 | — | 4 | 1 | no | — |
| 289 | C4 | — | 4 | 1 | no | — |
| 290 | C4 | — | 268 | 1 | no | — |
| 291 | C4 | — | 268 | 1 | no | — |
| 292 | C4 | — | 268 | 1 | no | — |
| 293 | C4 | — | 134 | 2 | no | — |
| 294 | C4 | C2×Q8×D67 | 1 | 1 | yes | characteristic |
| 295 | C4 | #14524091 | 1 | 1 | yes | characteristic |
| 296 | C2 | Q8×C22⋊D67 | 1 | 1 | yes | characteristic |
| 297 | C2 | #16864194 | 1 | 1 | yes | characteristic |
| 298 | C2 | #16864211 | 1 | 1 | yes | characteristic |
Complements
Splittings G = N ⋊ H with N normal; H ≅ G/N.
Finite Group Databasebeta