← #13250309
Subgroups — #13250309
Order 1792.
424 conjugacy classes of subgroups.
The subgroup lattice has 328 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #11740667 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11735953 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C7×#1997 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#89341 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #11728263 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | #11725520 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | #11725474 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×C7×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 9 | #90161 | C22 | 1 | 1 | yes | characteristic |
| 10 | C7×#423 | C22 | 1 | 1 | yes | characteristic |
| 11 | C4×C7×M16 | C22 | 1 | 1 | yes | characteristic |
| 12 | #90000 | C4 | 1 | 4 | yes | — |
| 13 | #89894 | — | 2 | 1 | no | — |
| 14 | #89891 | — | 2 | 1 | no | — |
| 15 | #89864 | — | 2 | 1 | no | — |
| 16 | #89862 | — | 2 | 2 | no | — |
| 17 | #89853 | — | 2 | 1 | no | — |
| 18 | #89697 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×C4⋊(C7⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×C4⋊(C7⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 21 | #89473 | — | 2 | 2 | no | — |
| 22 | #89451 | — | 2 | 1 | no | — |
| 23 | C8×Dic14 | — | 2 | 1 | no | — |
| 24 | C7×#343 | — | 2 | 1 | no | — |
| 25 | C7×#343 | — | 2 | 1 | no | — |
| 26 | #89341 | — | 2 | 1 | no | — |
| 27 | #89341 | — | 2 | 1 | no | — |
| 28 | #89281 | — | 2 | 1 | no | — |
| 29 | #89276 | — | 2 | 1 | no | — |
| 30 | #9268 | — | 7 | 1 | no | maximal, Sylow |
| 31 | C22×C7×Q8 | C2×C4 | 1 | 2 | yes | — |
| 32 | C2×C7×M16 | — | 2 | 1 | no | — |
| 33 | C2×C7×M16 | D4 | 1 | 1 | yes | characteristic |
| 34 | C2×C7×M16 | D4 | 1 | 1 | yes | characteristic |
| 35 | C7×C4⋊Q8 | — | 2 | 1 | no | — |
| 36 | C7×C4⋊Q8 | — | 2 | 1 | no | — |
| 37 | C7×C4⋊Q8 | — | 2 | 1 | no | — |
| 38 | C7×C4⋊Q8 | — | 2 | 1 | no | — |
| 39 | C2×C7×C4⋊C4 | — | 2 | 1 | no | — |
| 40 | C2×C7×C4⋊C4 | — | 2 | 1 | no | — |
| 41 | C2×C42×C7 | C23 | 1 | 1 | yes | characteristic |
| 42 | C2×#1144 | — | 2 | 4 | no | — |
| 43 | #6489 | — | 2 | 1 | no | — |
| 44 | #6487 | — | 2 | 1 | no | — |
| 45 | #6487 | C2×C4 | 1 | 2 | yes | — |
| 46 | C22×C7⋊C8 | — | 2 | 1 | no | — |
| 47 | C22×C7⋊C8 | — | 2 | 1 | no | — |
| 48 | #6433 | — | 2 | 1 | no | — |
| 49 | #6432 | D4 | 1 | 1 | yes | characteristic |
| 50 | #6432 | D4 | 1 | 1 | yes | characteristic |
| 51 | C4×Dic14 | — | 2 | 1 | no | — |
| 52 | C4×Dic14 | D4 | 1 | 2 | yes | — |
| 53 | C7×C4⋊C8 | — | 2 | 1 | no | — |
| 54 | C7×C4⋊C8 | — | 2 | 1 | no | — |
| 55 | C7×#97 | — | 2 | 1 | no | — |
| 56 | C4×C7×C8 | — | 2 | 1 | no | — |
| 57 | #6409 | — | 2 | 4 | no | — |
| 58 | C22⋊(C7⋊C8) | C2×C4 | 1 | 2 | yes | — |
| 59 | #6389 | — | 2 | 2 | no | — |
| 60 | Dic7⋊C8 | — | 2 | 2 | no | — |
| 61 | Dic7⋊C8 | — | 2 | 2 | no | — |
| 62 | C8×Dic7 | — | 2 | 2 | no | — |
| 63 | C4⋊(C7⋊C8) | — | 2 | 1 | no | — |
| 64 | C4⋊(C7⋊C8) | — | 2 | 1 | no | — |
| 65 | C4⋊(C7⋊C8) | — | 2 | 1 | no | — |
| 66 | C4⋊(C7⋊C8) | — | 2 | 1 | no | — |
| 67 | C4⋊(C7⋊C8) | D4 | 1 | 1 | yes | characteristic |
| 68 | C4⋊(C7⋊C8) | D4 | 1 | 1 | yes | characteristic |
| 69 | C4×C7⋊C8 | — | 2 | 1 | no | — |
| 70 | #3209 | — | 7 | 1 | no | — |
| 71 | #2962 | — | 7 | 1 | no | — |
| 72 | #1997 | D7 | 1 | 1 | yes | characteristic |
| 73 | #1550 | — | 7 | 1 | no | — |
| 74 | C2×#333 | — | 7 | 1 | no | — |
| 75 | #1301 | — | 7 | 1 | no | — |
| 76 | #1283 | — | 7 | 1 | no | — |
| 77 | C2×C7×Q8 | — | 2 | 2 | no | — |
| 78 | C2×C7×Q8 | — | 2 | 2 | no | — |
| 79 | C2×C7×Q8 | — | 8 | 2 | no | — |
| 80 | C22×C4×C7 | — | 2 | 2 | no | — |
| 81 | C22×C4×C7 | C22×C4 | 1 | 1 | yes | characteristic |
| 82 | C22×C4×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 83 | C22×C4×C7 | #41 | 1 | 1 | yes | characteristic |
| 84 | C2×Dic14 | — | 2 | 1 | no | — |
| 85 | C2×Dic14 | C22⋊C4 | 1 | 2 | yes | — |
| 86 | C7×M16 | — | 2 | 1 | no | — |
| 87 | C7×M16 | — | 2 | 1 | no | — |
| 88 | C7×M16 | — | 2 | 1 | no | — |
| 89 | C7×M16 | — | 4 | 1 | no | — |
| 90 | C7×M16 | — | 4 | 1 | no | — |
| 91 | C7×M16 | Q16 | 1 | 2 | yes | complemented |
| 92 | C2×C7×C8 | — | 2 | 1 | no | — |
| 93 | C2×C7×C8 | — | 2 | 1 | no | — |
| 94 | C2×C7×C8 | — | 4 | 1 | no | — |
| 95 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 96 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 97 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 98 | C7×C4⋊C4 | — | 4 | 1 | no | — |
| 99 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 100 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 101 | C42×C7 | C2×D4 | 1 | 1 | yes | characteristic |
| 102 | C42×C7 | #41 | 1 | 1 | yes | characteristic |
| 103 | C22⋊Dic7 | — | 2 | 2 | no | — |
| 104 | #1144 | — | 8 | 4 | no | — |
| 105 | C4⋊Dic7 | — | 2 | 1 | no | — |
| 106 | C4⋊Dic7 | — | 2 | 1 | no | — |
| 107 | C4⋊Dic7 | — | 2 | 1 | no | — |
| 108 | C4⋊Dic7 | C22⋊C4 | 1 | 2 | yes | — |
| 109 | Dic7⋊C4 | — | 2 | 2 | no | — |
| 110 | Dic7⋊C4 | — | 2 | 2 | no | — |
| 111 | C4×Dic7 | — | 2 | 2 | no | — |
| 112 | C2×C7⋊C8 | — | 2 | 2 | no | — |
| 113 | C2×C7⋊C8 | — | 2 | 1 | no | — |
| 114 | C2×C7⋊C8 | — | 2 | 1 | no | — |
| 115 | C2×C7⋊C8 | — | 4 | 1 | no | — |
| 116 | C2×C7⋊C8 | — | 4 | 1 | no | — |
| 117 | C2×C7⋊C8 | — | 4 | 1 | no | — |
| 118 | C2×C7⋊C8 | — | 8 | 1 | no | — |
| 119 | #533 | — | 7 | 1 | no | — |
| 120 | C2×C4⋊Q8 | C2×D7 | 1 | 1 | yes | characteristic |
| 121 | #494 | — | 14 | 1 | no | — |
| 122 | #487 | — | 14 | 1 | no | — |
| 123 | #462 | — | 14 | 2 | no | — |
| 124 | #451 | — | 7 | 4 | no | — |
| 125 | #446 | — | 14 | 1 | no | — |
| 126 | #446 | — | 14 | 2 | no | — |
| 127 | C8×Q8 | — | 14 | 1 | no | — |
| 128 | #431 | — | 7 | 1 | no | — |
| 129 | #424 | — | 14 | 1 | no | — |
| 130 | #423 | C2×D7 | 1 | 1 | yes | characteristic |
| 131 | C2×C4⋊C8 | — | 7 | 1 | no | — |
| 132 | C2×C4⋊C8 | — | 7 | 1 | no | — |
| 133 | #405 | — | 14 | 1 | no | — |
| 134 | C4×M16 | C2×D7 | 1 | 1 | yes | characteristic |
| 135 | #343 | — | 2 | 1 | no | — |
| 136 | #343 | — | 2 | 1 | no | — |
| 137 | #333 | — | 14 | 1 | no | — |
| 138 | #333 | — | 14 | 1 | no | — |
| 139 | #333 | — | 14 | 1 | no | — |
| 140 | #332 | — | 14 | 1 | no | — |
| 141 | C23×C7 | C4×D4 | 1 | 1 | yes | characteristic |
| 142 | C7×Q8 | — | 4 | 4 | no | — |
| 143 | C7×Q8 | — | 8 | 2 | no | — |
| 144 | C2×C4×C7 | — | 2 | 1 | no | — |
| 145 | C2×C4×C7 | — | 2 | 1 | no | — |
| 146 | C2×C4×C7 | — | 4 | 1 | no | — |
| 147 | C2×C4×C7 | — | 4 | 2 | no | — |
| 148 | C2×C4×C7 | — | 8 | 2 | no | — |
| 149 | C2×C4×C7 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 150 | C2×C4×C7 | C4×D4 | 1 | 1 | yes | characteristic |
| 151 | C2×C4×C7 | #120 | 1 | 1 | yes | characteristic |
| 152 | C2×C4×C7 | #121 | 1 | 1 | yes | characteristic |
| 153 | C2×C4×C7 | #121 | 1 | 1 | yes | characteristic |
| 154 | C2×C4×C7 | #123 | 1 | 1 | yes | characteristic |
| 155 | C2×C4×C7 | C2×Q16 | 1 | 1 | yes | characteristic |
| 156 | C2×C4×C7 | #135 | 1 | 1 | yes | characteristic |
| 157 | C2×C4×C7 | #137 | 1 | 1 | yes | characteristic |
| 158 | C2×C4×C7 | #137 | 1 | 1 | yes | characteristic |
| 159 | C2×Dic7 | — | 2 | 2 | no | — |
| 160 | C2×Dic7 | — | 2 | 2 | no | — |
| 161 | Dic14 | — | 4 | 1 | no | — |
| 162 | Dic14 | — | 4 | 1 | no | — |
| 163 | Dic14 | — | 4 | 2 | no | — |
| 164 | C7×C8 | — | 2 | 2 | no | — |
| 165 | C7×C8 | — | 4 | 1 | no | — |
| 166 | C7×C8 | — | 8 | 1 | no | — |
| 167 | C7⋊C8 | — | 4 | 2 | no | — |
| 168 | C7⋊C8 | — | 4 | 1 | no | — |
| 169 | C7⋊C8 | — | 4 | 1 | no | — |
| 170 | C7⋊C8 | — | 8 | 1 | no | — |
| 171 | C22×Q8 | C4×D7 | 1 | 2 | yes | — |
| 172 | C2×Q16 | — | 14 | 4 | no | — |
| 173 | C2×M16 | — | 2 | 1 | no | — |
| 174 | C2×M16 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 175 | C2×M16 | C22⋊D7 | 1 | 1 | yes | characteristic |
| 176 | C22×C8 | — | 14 | 1 | no | — |
| 177 | C22×C8 | — | 14 | 1 | no | — |
| 178 | C4⋊Q8 | — | 2 | 1 | no | — |
| 179 | C4⋊Q8 | — | 2 | 1 | no | — |
| 180 | C4⋊Q8 | — | 2 | 1 | no | — |
| 181 | C4⋊Q8 | — | 2 | 1 | no | — |
| 182 | C4⋊Q8 | — | 7 | 1 | no | — |
| 183 | C4⋊Q8 | — | 7 | 1 | no | — |
| 184 | #125 | — | 14 | 1 | no | — |
| 185 | C22⋊Q8 | — | 7 | 2 | no | — |
| 186 | C22⋊Q8 | — | 14 | 1 | no | — |
| 187 | C4×Q8 | — | 7 | 2 | no | — |
| 188 | C4×Q8 | — | 14 | 1 | no | — |
| 189 | #117 | — | 14 | 1 | no | — |
| 190 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 191 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 192 | C2×C42 | C22×D7 | 1 | 1 | yes | characteristic |
| 193 | C4⋊C8 | — | 2 | 1 | no | — |
| 194 | C4⋊C8 | — | 2 | 1 | no | — |
| 195 | C4⋊C8 | — | 7 | 1 | no | — |
| 196 | C4⋊C8 | — | 7 | 1 | no | — |
| 197 | C4⋊C8 | — | 14 | 2 | no | — |
| 198 | C4⋊C8 | — | 14 | 2 | no | — |
| 199 | C4⋊C8 | — | 14 | 1 | no | — |
| 200 | C4⋊C8 | — | 14 | 1 | no | — |
| 201 | C4⋊C8 | — | 14 | 1 | no | — |
| 202 | C4⋊C8 | — | 14 | 1 | no | — |
| 203 | #103 | — | 14 | 4 | no | — |
| 204 | C22⋊C8 | — | 7 | 2 | no | — |
| 205 | #97 | — | 2 | 1 | no | — |
| 206 | #97 | — | 14 | 2 | no | — |
| 207 | C4×C8 | — | 2 | 1 | no | — |
| 208 | C4×C8 | — | 14 | 2 | no | — |
| 209 | C4×C8 | — | 14 | 1 | no | — |
| 210 | C22×C7 | — | 2 | 1 | no | — |
| 211 | C22×C7 | — | 2 | 1 | no | — |
| 212 | C22×C7 | #386 | 1 | 1 | yes | characteristic |
| 213 | C22×C7 | C4×Q16 | 1 | 1 | yes | characteristic |
| 214 | C22×C7 | #441 | 1 | 1 | yes | characteristic |
| 215 | C4×C7 | — | 2 | 1 | no | — |
| 216 | C4×C7 | — | 2 | 1 | no | — |
| 217 | C4×C7 | — | 2 | 1 | no | — |
| 218 | C4×C7 | — | 8 | 2 | no | — |
| 219 | C4×C7 | #451 | 1 | 1 | yes | characteristic |
| 220 | C4×C7 | #452 | 1 | 1 | yes | characteristic |
| 221 | C4×C7 | #462 | 1 | 1 | yes | characteristic |
| 222 | C4×C7 | #464 | 1 | 1 | yes | characteristic |
| 223 | C4×C7 | #471 | 1 | 1 | yes | characteristic |
| 224 | C4×C7 | #473 | 1 | 1 | yes | characteristic |
| 225 | Dic7 | — | 4 | 2 | no | — |
| 226 | Dic7 | — | 4 | 2 | no | — |
| 227 | C2×Q8 | — | 2 | 2 | no | — |
| 228 | C2×Q8 | — | 2 | 2 | no | — |
| 229 | C2×Q8 | — | 7 | 2 | no | — |
| 230 | C2×Q8 | — | 8 | 2 | no | — |
| 231 | C2×Q8 | — | 14 | 1 | no | — |
| 232 | C22×C4 | — | 2 | 2 | no | — |
| 233 | C22×C4 | C2×C4×D7 | 1 | 1 | yes | characteristic |
| 234 | C22×C4 | #1157 | 1 | 1 | yes | characteristic |
| 235 | C22×C4 | C2×C22⋊D7 | 1 | 1 | yes | characteristic |
| 236 | Q16 | — | 56 | 4 | no | — |
| 237 | M16 | — | 2 | 1 | no | — |
| 238 | M16 | — | 2 | 1 | no | — |
| 239 | M16 | — | 2 | 1 | no | — |
| 240 | M16 | — | 4 | 1 | no | — |
| 241 | M16 | — | 4 | 1 | no | — |
| 242 | M16 | #1144 | 1 | 2 | yes | complemented |
| 243 | C2×C8 | — | 2 | 1 | no | — |
| 244 | C2×C8 | — | 2 | 1 | no | — |
| 245 | C2×C8 | — | 4 | 1 | no | — |
| 246 | C2×C8 | — | 14 | 2 | no | — |
| 247 | C2×C8 | — | 14 | 1 | no | — |
| 248 | C2×C8 | — | 14 | 1 | no | — |
| 249 | C2×C8 | — | 28 | 1 | no | — |
| 250 | C2×C8 | — | 28 | 1 | no | — |
| 251 | C2×C8 | — | 28 | 1 | no | — |
| 252 | C2×C8 | — | 56 | 1 | no | — |
| 253 | C4⋊C4 | — | 4 | 1 | no | — |
| 254 | C4⋊C4 | — | 4 | 1 | no | — |
| 255 | C4⋊C4 | — | 4 | 1 | no | — |
| 256 | C4⋊C4 | — | 4 | 1 | no | — |
| 257 | C4⋊C4 | — | 7 | 2 | no | — |
| 258 | C4⋊C4 | — | 14 | 2 | no | — |
| 259 | C4⋊C4 | — | 14 | 2 | no | — |
| 260 | C4⋊C4 | — | 14 | 1 | no | — |
| 261 | C4⋊C4 | — | 14 | 1 | no | — |
| 262 | C4⋊C4 | — | 14 | 1 | no | — |
| 263 | C22⋊C4 | — | 14 | 2 | no | — |
| 264 | C42 | — | 14 | 2 | no | — |
| 265 | C42 | D4×D7 | 1 | 1 | yes | characteristic |
| 266 | C42 | D4×D7 | 1 | 1 | yes | characteristic |
| 267 | C42 | D4×D7 | 1 | 1 | yes | characteristic |
| 268 | C42 | #1159 | 1 | 1 | yes | characteristic |
| 269 | C2×C7 | — | 2 | 1 | no | — |
| 270 | C2×C7 | — | 2 | 1 | no | — |
| 271 | C2×C7 | #1878 | 1 | 1 | yes | characteristic |
| 272 | C2×C7 | #1908 | 1 | 1 | yes | characteristic |
| 273 | C2×C7 | #1914 | 1 | 1 | yes | characteristic |
| 274 | C23 | C4×C22⋊D7 | 1 | 1 | yes | characteristic |
| 275 | Q8 | — | 4 | 4 | no | — |
| 276 | Q8 | — | 8 | 2 | no | — |
| 277 | Q8 | — | 28 | 2 | no | — |
| 278 | Q8 | — | 28 | 1 | no | — |
| 279 | Q8 | — | 28 | 1 | no | — |
| 280 | C2×C4 | — | 2 | 1 | no | — |
| 281 | C2×C4 | — | 2 | 1 | no | — |
| 282 | C2×C4 | — | 4 | 1 | no | — |
| 283 | C2×C4 | — | 4 | 2 | no | — |
| 284 | C2×C4 | — | 8 | 2 | no | — |
| 285 | C2×C4 | — | 14 | 2 | no | — |
| 286 | C2×C4 | — | 14 | 2 | no | — |
| 287 | C2×C4 | D7×C22⋊C4 | 1 | 1 | yes | characteristic |
| 288 | C2×C4 | (C22⋊D7)⋊C4 | 1 | 1 | yes | characteristic |
| 289 | C2×C4 | #6446 | 1 | 1 | yes | characteristic |
| 290 | C2×C4 | #6447 | 1 | 1 | yes | characteristic |
| 291 | C2×C4 | #6500 | 1 | 1 | yes | characteristic |
| 292 | C2×C4 | #6501 | 1 | 1 | yes | characteristic |
| 293 | C2×C4 | #6505 | 1 | 1 | yes | characteristic |
| 294 | C2×C4 | C2×#1144 | 1 | 1 | yes | characteristic |
| 295 | C2×C4 | #6513 | 1 | 1 | yes | characteristic |
| 296 | C2×C4 | #6514 | 1 | 1 | yes | characteristic |
| 297 | C8 | — | 2 | 2 | no | — |
| 298 | C8 | — | 4 | 1 | no | — |
| 299 | C8 | — | 8 | 1 | no | — |
| 300 | C8 | — | 28 | 2 | no | — |
| 301 | C8 | — | 28 | 1 | no | — |
| 302 | C8 | — | 28 | 1 | no | — |
| 303 | C8 | — | 56 | 1 | no | — |
| 304 | C7 | #9268 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 305 | C22 | — | 2 | 1 | no | — |
| 306 | C22 | — | 2 | 1 | no | — |
| 307 | C22 | #89728 | 1 | 1 | yes | characteristic |
| 308 | C22 | C4×#1144 | 1 | 1 | yes | characteristic |
| 309 | C22 | #89802 | 1 | 1 | yes | characteristic |
| 310 | C4 | — | 2 | 1 | no | — |
| 311 | C4 | — | 2 | 1 | no | — |
| 312 | C4 | — | 2 | 1 | no | — |
| 313 | C4 | — | 8 | 2 | no | — |
| 314 | C4 | — | 28 | 2 | no | — |
| 315 | C4 | — | 28 | 2 | no | — |
| 316 | C4 | #89526 | 1 | 1 | yes | characteristic |
| 317 | C4 | #89527 | 1 | 1 | yes | characteristic |
| 318 | C4 | #89819 | 1 | 1 | yes | characteristic |
| 319 | C4 | #89822 | 1 | 1 | yes | characteristic |
| 320 | C4 | #89826 | 1 | 1 | yes | characteristic |
| 321 | C4 | #89862 | 1 | 1 | yes | characteristic |
| 322 | C2 | — | 2 | 1 | no | — |
| 323 | C2 | — | 2 | 1 | no | — |
| 324 | C2 | #11727444 | 1 | 1 | yes | characteristic |
| 325 | C2 | #11727843 | 1 | 1 | yes | characteristic |
| 326 | C2 | #11727877 | 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