← #10782892
Subgroups — #10782892
Order 768.
706 conjugacy classes of subgroups.
The subgroup lattice has 336 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C3×#3170 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #83344 | C2 | 1 | 2 | yes | maximal |
| 3 | #83339 | C2 | 1 | 2 | yes | maximal |
| 4 | #78475 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #71362 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #71360 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #71298 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #71296 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #71214 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #71212 | C2 | 1 | 2 | yes | maximal, complemented |
| 11 | C2×C8×C3⋊C8 | C2 | 1 | 1 | yes | maximal, characteristic |
| 12 | #19604 | — | 3 | 1 | no | maximal, Sylow |
| 13 | C3×#533 | C22 | 1 | 2 | yes | — |
| 14 | #5470 | C22 | 1 | 2 | yes | — |
| 15 | C3×#500 | C22 | 1 | 1 | yes | characteristic |
| 16 | C3×#494 | C22 | 1 | 1 | yes | characteristic |
| 17 | C3×#487 | C22 | 1 | 2 | yes | — |
| 18 | C3×#467 | — | 2 | 2 | no | — |
| 19 | C3×C4×Q16 | — | 2 | 2 | no | — |
| 20 | C2×C3×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 21 | #5116 | — | 2 | 2 | no | — |
| 22 | #5093 | C22 | 1 | 2 | yes | — |
| 23 | #5092 | C22 | 1 | 2 | yes | — |
| 24 | #5080 | C22 | 1 | 2 | yes | — |
| 25 | #5080 | C22 | 1 | 2 | yes | — |
| 26 | #5079 | C22 | 1 | 2 | yes | — |
| 27 | #5078 | C22 | 1 | 2 | yes | — |
| 28 | #5078 | C22 | 1 | 2 | yes | — |
| 29 | #5078 | C22 | 1 | 2 | yes | — |
| 30 | #5065 | — | 2 | 4 | no | — |
| 31 | #5051 | — | 2 | 4 | no | — |
| 32 | C4×#216 | — | 2 | 4 | no | — |
| 33 | #4998 | — | 2 | 4 | no | — |
| 34 | C2×C4×C3⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 35 | C2×C4×C3⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 36 | #4699 | C22 | 1 | 1 | yes | characteristic |
| 37 | C4×Dic12 | — | 2 | 2 | no | — |
| 38 | #4682 | C22 | 1 | 1 | yes | characteristic |
| 39 | #4681 | C22 | 1 | 2 | yes | — |
| 40 | C8×C3⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 41 | C8×C3⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 42 | C8×C3⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 43 | C8×C3⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 44 | #3170 | — | 3 | 1 | no | — |
| 45 | #3170 | — | 3 | 2 | no | — |
| 46 | #3170 | D3 | 1 | 1 | yes | characteristic |
| 47 | #3160 | — | 3 | 2 | no | — |
| 48 | #1713 | — | 3 | 1 | no | — |
| 49 | #1713 | — | 3 | 1 | no | — |
| 50 | #1710 | — | 3 | 2 | no | — |
| 51 | #1707 | — | 3 | 1 | no | — |
| 52 | #1707 | — | 3 | 1 | no | — |
| 53 | #1707 | — | 3 | 2 | no | — |
| 54 | C2×C82 | — | 3 | 1 | no | — |
| 55 | C2×C3×Q16 | — | 2 | 2 | no | — |
| 56 | C22×C3×C8 | D4 | 1 | 2 | yes | — |
| 57 | C3×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 58 | C3×C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 59 | C3×#125 | — | 2 | 2 | no | — |
| 60 | C3×C22⋊Q8 | — | 2 | 2 | no | — |
| 61 | C3×C22⋊Q8 | — | 2 | 2 | no | — |
| 62 | C3×C4×Q8 | — | 2 | 2 | no | — |
| 63 | C3×C4×Q8 | — | 2 | 2 | no | — |
| 64 | C3×#117 | — | 2 | 2 | no | — |
| 65 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 66 | C2×#216 | — | 2 | 4 | no | — |
| 67 | #926 | — | 2 | 2 | no | — |
| 68 | #924 | — | 2 | 2 | no | — |
| 69 | #924 | — | 2 | 2 | no | — |
| 70 | C22×C3⋊C8 | D4 | 1 | 2 | yes | — |
| 71 | C22×C3⋊C8 | D4 | 1 | 2 | yes | — |
| 72 | C2×Dic12 | — | 2 | 2 | no | — |
| 73 | #870 | — | 2 | 2 | no | — |
| 74 | #869 | C23 | 1 | 2 | yes | — |
| 75 | #869 | C23 | 1 | 2 | yes | — |
| 76 | C4×Dic6 | — | 2 | 2 | no | — |
| 77 | C4×Dic6 | — | 2 | 2 | no | — |
| 78 | C3×#107 | — | 2 | 2 | no | — |
| 79 | C3×#103 | — | 2 | 4 | no | — |
| 80 | C3×C4×C8 | D4 | 1 | 2 | yes | — |
| 81 | C3×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 82 | C3×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 83 | #835 | — | 2 | 4 | no | — |
| 84 | #818 | — | 2 | 2 | no | — |
| 85 | #816 | — | 2 | 4 | no | — |
| 86 | #810 | — | 2 | 4 | no | — |
| 87 | #808 | — | 2 | 4 | no | — |
| 88 | #808 | — | 2 | 4 | no | — |
| 89 | #807 | — | 2 | 4 | no | — |
| 90 | #807 | — | 2 | 4 | no | — |
| 91 | #807 | — | 2 | 4 | no | — |
| 92 | C4×C3⋊C8 | D4 | 1 | 2 | yes | — |
| 93 | C4×C3⋊C8 | D4 | 1 | 2 | yes | — |
| 94 | C4×C3⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 95 | C4×C3⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 96 | C4×C3⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 97 | C4×C3⋊C8 | C23 | 1 | 1 | yes | characteristic |
| 98 | #533 | — | 3 | 2 | no | — |
| 99 | #533 | C2×D3 | 1 | 2 | yes | — |
| 100 | #500 | — | 3 | 2 | no | — |
| 101 | #500 | — | 3 | 2 | no | — |
| 102 | #500 | — | 3 | 2 | no | — |
| 103 | #500 | — | 3 | 1 | no | — |
| 104 | #500 | C2×D3 | 1 | 1 | yes | characteristic |
| 105 | #499 | — | 6 | 4 | no | — |
| 106 | #498 | — | 3 | 2 | no | — |
| 107 | #498 | — | 3 | 2 | no | — |
| 108 | #494 | — | 3 | 2 | no | — |
| 109 | #494 | — | 3 | 1 | no | — |
| 110 | #494 | C2×D3 | 1 | 1 | yes | characteristic |
| 111 | #487 | — | 3 | 2 | no | — |
| 112 | #487 | — | 3 | 2 | no | — |
| 113 | #487 | — | 3 | 2 | no | — |
| 114 | #487 | C2×D3 | 1 | 2 | yes | — |
| 115 | #467 | — | 2 | 2 | no | — |
| 116 | #467 | — | 6 | 4 | no | — |
| 117 | #467 | — | 6 | 2 | no | — |
| 118 | C4×Q16 | — | 2 | 2 | no | — |
| 119 | C4×Q16 | — | 6 | 4 | no | — |
| 120 | C4×Q16 | — | 6 | 2 | no | — |
| 121 | #427 | — | 6 | 4 | no | — |
| 122 | C2×C4×C8 | — | 3 | 1 | no | — |
| 123 | C2×C4×C8 | — | 3 | 1 | no | — |
| 124 | C2×C4×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 125 | C82 | — | 3 | 1 | no | — |
| 126 | C82 | — | 3 | 1 | no | — |
| 127 | C82 | — | 3 | 1 | no | — |
| 128 | C82 | — | 3 | 1 | no | — |
| 129 | C2×C3×Q8 | — | 2 | 2 | no | — |
| 130 | C2×C3×Q8 | — | 2 | 2 | no | — |
| 131 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 132 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 133 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 134 | C2×Dic6 | — | 2 | 2 | no | — |
| 135 | C2×Dic6 | — | 2 | 2 | no | — |
| 136 | C3×Q16 | — | 4 | 4 | no | — |
| 137 | C2×C3×C8 | — | 2 | 4 | no | — |
| 138 | C2×C3×C8 | C2×D4 | 1 | 2 | yes | — |
| 139 | C2×C3×C8 | C2×D4 | 1 | 2 | yes | — |
| 140 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 141 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 142 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 143 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 144 | C3×C4⋊C4 | — | 4 | 2 | no | — |
| 145 | C3×C4⋊C4 | — | 4 | 2 | no | — |
| 146 | C3×C22⋊C4 | — | 4 | 2 | no | — |
| 147 | C3×C42 | — | 4 | 2 | no | — |
| 148 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 149 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 150 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 151 | C3×C42 | C24 | 1 | 1 | yes | characteristic |
| 152 | #217 | — | 4 | 2 | no | — |
| 153 | #216 | — | 4 | 8 | no | — |
| 154 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 155 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 156 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 157 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 158 | Dic3⋊C4 | — | 4 | 2 | no | — |
| 159 | Dic3⋊C4 | — | 4 | 2 | no | — |
| 160 | C4×Dic3 | — | 4 | 2 | no | — |
| 161 | C2×C3⋊C8 | — | 2 | 4 | no | — |
| 162 | C2×C3⋊C8 | Q16 | 1 | 8 | yes | complemented |
| 163 | C2×C3⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 164 | C2×C3⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 165 | C2×C3⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 166 | C2×C3⋊C8 | C2×D4 | 1 | 2 | yes | — |
| 167 | Dic12 | — | 4 | 4 | no | — |
| 168 | C2×Q16 | — | 2 | 2 | no | — |
| 169 | C2×Q16 | — | 6 | 4 | no | — |
| 170 | C2×Q16 | — | 6 | 2 | no | — |
| 171 | C22×C8 | — | 3 | 2 | no | — |
| 172 | C22×C8 | — | 3 | 2 | no | — |
| 173 | C22×C8 | #67 | 1 | 2 | yes | — |
| 174 | C4⋊Q8 | — | 3 | 2 | no | — |
| 175 | C4⋊Q8 | — | 3 | 2 | no | — |
| 176 | C4⋊Q8 | C22×D3 | 1 | 2 | yes | — |
| 177 | C4⋊Q8 | C22×D3 | 1 | 2 | yes | — |
| 178 | #125 | — | 2 | 2 | no | — |
| 179 | #125 | — | 6 | 2 | no | — |
| 180 | C22⋊Q8 | — | 2 | 2 | no | — |
| 181 | C22⋊Q8 | — | 2 | 2 | no | — |
| 182 | C22⋊Q8 | — | 6 | 2 | no | — |
| 183 | C22⋊Q8 | — | 6 | 2 | no | — |
| 184 | C4×Q8 | — | 2 | 2 | no | — |
| 185 | C4×Q8 | — | 2 | 2 | no | — |
| 186 | C4×Q8 | — | 6 | 2 | no | — |
| 187 | C4×Q8 | — | 6 | 2 | no | — |
| 188 | #117 | — | 2 | 2 | no | — |
| 189 | #117 | — | 6 | 2 | no | — |
| 190 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 191 | #107 | — | 2 | 2 | no | — |
| 192 | #107 | — | 6 | 4 | no | — |
| 193 | #107 | — | 6 | 4 | no | — |
| 194 | #107 | — | 6 | 4 | no | — |
| 195 | #107 | — | 6 | 2 | no | — |
| 196 | #106 | — | 6 | 4 | no | — |
| 197 | #106 | — | 6 | 4 | no | — |
| 198 | #103 | — | 2 | 4 | no | — |
| 199 | #103 | — | 6 | 4 | no | — |
| 200 | #103 | — | 6 | 4 | no | — |
| 201 | #103 | — | 6 | 4 | no | — |
| 202 | C4×C8 | — | 3 | 2 | no | — |
| 203 | C4×C8 | — | 3 | 2 | no | — |
| 204 | C4×C8 | — | 3 | 1 | no | — |
| 205 | C4×C8 | — | 3 | 1 | no | — |
| 206 | C4×C8 | — | 3 | 1 | no | — |
| 207 | C4×C8 | — | 3 | 1 | no | — |
| 208 | C4×C8 | #67 | 1 | 2 | yes | — |
| 209 | C4×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 210 | C4×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 211 | C23×C3 | #127 | 1 | 1 | yes | characteristic |
| 212 | C3×Q8 | — | 4 | 4 | no | — |
| 213 | C3×Q8 | — | 4 | 4 | no | — |
| 214 | C2×C3×C4 | — | 2 | 2 | no | — |
| 215 | C2×C3×C4 | — | 2 | 2 | no | — |
| 216 | C2×C3×C4 | — | 4 | 2 | no | — |
| 217 | C2×C3×C4 | — | 4 | 2 | no | — |
| 218 | C2×C3×C4 | #127 | 1 | 1 | yes | characteristic |
| 219 | C2×C3×C4 | #127 | 1 | 1 | yes | characteristic |
| 220 | C2×C3×C4 | #127 | 1 | 1 | yes | characteristic |
| 221 | C2×C3×C4 | C2×Q16 | 1 | 4 | yes | — |
| 222 | C2×C3×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 223 | C2×C3×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 224 | C2×C3×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 225 | C2×Dic3 | — | 4 | 2 | no | — |
| 226 | C2×Dic3 | — | 4 | 2 | no | — |
| 227 | Dic6 | — | 4 | 4 | no | — |
| 228 | Dic6 | — | 4 | 4 | no | — |
| 229 | C3×C8 | — | 2 | 2 | no | — |
| 230 | C3×C8 | #135 | 1 | 4 | yes | — |
| 231 | C3⋊C8 | — | 2 | 2 | no | — |
| 232 | C3⋊C8 | C2×Q16 | 1 | 4 | yes | — |
| 233 | C3⋊C8 | C2×Q16 | 1 | 4 | yes | — |
| 234 | C3⋊C8 | #135 | 1 | 4 | yes | — |
| 235 | C2×Q8 | — | 2 | 2 | no | — |
| 236 | C2×Q8 | — | 2 | 2 | no | — |
| 237 | C2×Q8 | — | 6 | 2 | no | — |
| 238 | C2×Q8 | — | 6 | 2 | no | — |
| 239 | C22×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 240 | C22×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 241 | C22×C4 | C2×#67 | 1 | 1 | yes | characteristic |
| 242 | Q16 | — | 4 | 4 | no | — |
| 243 | Q16 | — | 12 | 8 | no | — |
| 244 | Q16 | — | 12 | 4 | no | — |
| 245 | C2×C8 | — | 2 | 4 | no | — |
| 246 | C2×C8 | — | 3 | 8 | no | — |
| 247 | C2×C8 | — | 3 | 2 | no | — |
| 248 | C2×C8 | — | 3 | 2 | no | — |
| 249 | C2×C8 | — | 3 | 2 | no | — |
| 250 | C2×C8 | — | 3 | 2 | no | — |
| 251 | C2×C8 | — | 6 | 4 | no | — |
| 252 | C2×C8 | C2×#67 | 1 | 2 | yes | — |
| 253 | C2×C8 | C2×#67 | 1 | 2 | yes | — |
| 254 | C4⋊C4 | — | 2 | 2 | no | — |
| 255 | C4⋊C4 | — | 2 | 2 | no | — |
| 256 | C4⋊C4 | — | 2 | 2 | no | — |
| 257 | C4⋊C4 | — | 2 | 2 | no | — |
| 258 | C4⋊C4 | — | 4 | 2 | no | — |
| 259 | C4⋊C4 | — | 4 | 2 | no | — |
| 260 | C4⋊C4 | — | 6 | 2 | no | — |
| 261 | C4⋊C4 | — | 6 | 2 | no | — |
| 262 | C4⋊C4 | — | 6 | 2 | no | — |
| 263 | C4⋊C4 | — | 6 | 2 | no | — |
| 264 | C4⋊C4 | — | 12 | 2 | no | — |
| 265 | C4⋊C4 | — | 12 | 2 | no | — |
| 266 | C22⋊C4 | — | 4 | 2 | no | — |
| 267 | C22⋊C4 | — | 12 | 2 | no | — |
| 268 | C42 | — | 4 | 2 | no | — |
| 269 | C42 | — | 12 | 2 | no | — |
| 270 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 271 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 272 | C42 | C2×#67 | 1 | 1 | yes | characteristic |
| 273 | C42 | C23×D3 | 1 | 1 | yes | characteristic |
| 274 | C22×C3 | — | 2 | 1 | no | — |
| 275 | C22×C3 | — | 2 | 1 | no | — |
| 276 | C22×C3 | #494 | 1 | 2 | yes | — |
| 277 | C22×C3 | C2×#127 | 1 | 1 | yes | characteristic |
| 278 | C3×C4 | — | 2 | 1 | no | — |
| 279 | C3×C4 | — | 2 | 1 | no | — |
| 280 | C3×C4 | — | 8 | 2 | no | — |
| 281 | C3×C4 | — | 8 | 2 | no | — |
| 282 | C3×C4 | #494 | 1 | 2 | yes | — |
| 283 | C3×C4 | C22×Q16 | 1 | 2 | yes | — |
| 284 | C3×C4 | C2×#135 | 1 | 2 | yes | — |
| 285 | C3×C4 | C2×#135 | 1 | 2 | yes | — |
| 286 | Dic3 | — | 8 | 2 | no | — |
| 287 | Dic3 | — | 8 | 2 | no | — |
| 288 | C23 | #940 | 1 | 1 | yes | characteristic |
| 289 | Q8 | — | 4 | 4 | no | — |
| 290 | Q8 | — | 4 | 4 | no | — |
| 291 | Q8 | — | 12 | 4 | no | — |
| 292 | Q8 | — | 12 | 4 | no | — |
| 293 | C2×C4 | — | 2 | 2 | no | — |
| 294 | C2×C4 | — | 2 | 2 | no | — |
| 295 | C2×C4 | — | 4 | 2 | no | — |
| 296 | C2×C4 | — | 4 | 2 | no | — |
| 297 | C2×C4 | — | 12 | 2 | no | — |
| 298 | C2×C4 | — | 12 | 2 | no | — |
| 299 | C2×C4 | D3×Q16 | 1 | 4 | yes | — |
| 300 | C2×C4 | #940 | 1 | 1 | yes | characteristic |
| 301 | C2×C4 | #940 | 1 | 1 | yes | characteristic |
| 302 | C2×C4 | #940 | 1 | 1 | yes | characteristic |
| 303 | C2×C4 | C2×D3×D4 | 1 | 1 | yes | characteristic |
| 304 | C2×C4 | C2×D3×D4 | 1 | 1 | yes | characteristic |
| 305 | C2×C4 | C22×#67 | 1 | 1 | yes | characteristic |
| 306 | C8 | — | 2 | 2 | no | — |
| 307 | C8 | — | 3 | 4 | no | — |
| 308 | C8 | — | 3 | 4 | no | — |
| 309 | C8 | — | 3 | 4 | no | — |
| 310 | C8 | — | 6 | 2 | no | — |
| 311 | C8 | #950 | 1 | 4 | yes | — |
| 312 | C2×C3 | — | 2 | 2 | no | — |
| 313 | C2×C3 | C2×#494 | 1 | 1 | yes | characteristic |
| 314 | C2×C3 | #3147 | 1 | 1 | yes | characteristic |
| 315 | C2×C3 | #3147 | 1 | 1 | yes | characteristic |
| 316 | C22 | — | 2 | 1 | no | — |
| 317 | C22 | — | 2 | 1 | no | — |
| 318 | C22 | #5183 | 1 | 2 | yes | — |
| 319 | C22 | C2×#940 | 1 | 1 | yes | characteristic |
| 320 | C4 | — | 2 | 1 | no | — |
| 321 | C4 | — | 2 | 1 | no | — |
| 322 | C4 | — | 8 | 2 | no | — |
| 323 | C4 | — | 8 | 2 | no | — |
| 324 | C4 | — | 24 | 2 | no | — |
| 325 | C4 | — | 24 | 2 | no | — |
| 326 | C4 | #5183 | 1 | 2 | yes | — |
| 327 | C4 | C2×D3×Q16 | 1 | 2 | yes | — |
| 328 | C4 | D3×#135 | 1 | 2 | yes | — |
| 329 | C4 | C2×#950 | 1 | 2 | yes | — |
| 330 | C3 | #19604 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 331 | C2 | — | 2 | 2 | no | — |
| 332 | C2 | #83285 | 1 | 1 | yes | characteristic |
| 333 | C2 | C2×#5183 | 1 | 1 | yes | characteristic |
| 334 | C2 | #83903 | 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