← #10948741
Subgroups — #10948741
Order 768.
518 conjugacy classes of subgroups.
The subgroup lattice has 320 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | #78448 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C3×#346 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #71407 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C16×#67 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #69744 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #69075 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #69074 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #69072 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #69068 | C2 | 1 | 2 | yes | maximal, complemented |
| 10 | #33482 | — | 3 | 1 | no | maximal, Sylow |
| 11 | #5479 | C4 | 1 | 1 | yes | characteristic |
| 12 | C22×C3×C16 | C22 | 1 | 2 | yes | — |
| 13 | C2×C3×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 14 | #5111 | C4 | 1 | 2 | yes | — |
| 15 | C8×#67 | C22 | 1 | 2 | yes | — |
| 16 | #4926 | C4 | 1 | 1 | yes | characteristic |
| 17 | C2×#798 | C22 | 1 | 2 | yes | — |
| 18 | C2×D3×C16 | C22 | 1 | 2 | yes | — |
| 19 | #4690 | C4 | 1 | 2 | yes | — |
| 20 | #4689 | C22 | 1 | 2 | yes | — |
| 21 | #4688 | C4 | 1 | 1 | yes | characteristic |
| 22 | C8×D12 | C22 | 1 | 1 | yes | characteristic |
| 23 | #4679 | C4 | 1 | 1 | yes | characteristic |
| 24 | C8×Dic6 | C22 | 1 | 1 | yes | characteristic |
| 25 | C3×#346 | C22 | 1 | 2 | yes | — |
| 26 | C3×#346 | C22 | 1 | 1 | yes | characteristic |
| 27 | C3×#346 | C22 | 1 | 1 | yes | characteristic |
| 28 | #4549 | C22 | 1 | 1 | yes | characteristic |
| 29 | #4549 | C22 | 1 | 1 | yes | characteristic |
| 30 | (C2×D3)⋊C16 | C22 | 1 | 2 | yes | — |
| 31 | (C2×D3)⋊C16 | C22 | 1 | 2 | yes | — |
| 32 | #4502 | C22 | 1 | 2 | yes | — |
| 33 | Dic3⋊C16 | C22 | 1 | 2 | yes | — |
| 34 | Dic3⋊C16 | C22 | 1 | 2 | yes | — |
| 35 | Dic3×C16 | C22 | 1 | 2 | yes | — |
| 36 | C4⋊(C3⋊C16) | C22 | 1 | 2 | yes | — |
| 37 | C4×C3⋊C16 | C22 | 1 | 2 | yes | — |
| 38 | #2966 | — | 3 | 1 | no | — |
| 39 | #2183 | — | 3 | 2 | no | — |
| 40 | #2168 | — | 3 | 2 | no | — |
| 41 | #2168 | — | 3 | 2 | no | — |
| 42 | D4×C16 | — | 3 | 2 | no | — |
| 43 | #2164 | — | 3 | 2 | no | — |
| 44 | #2162 | — | 3 | 1 | no | — |
| 45 | #2108 | — | 3 | 2 | no | — |
| 46 | C2×#346 | D3 | 1 | 1 | yes | characteristic, complemented |
| 47 | C22×C3×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 48 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 49 | C2×C3×C42 | C2×C4 | 1 | 1 | yes | characteristic |
| 50 | #930 | C8 | 1 | 1 | yes | characteristic |
| 51 | #929 | C8 | 1 | 2 | yes | — |
| 52 | C4×#67 | C2×C4 | 1 | 2 | yes | — |
| 53 | #924 | C8 | 1 | 1 | yes | characteristic |
| 54 | C2×ΓL(1,𝔽25) | C2×C4 | 1 | 2 | yes | — |
| 55 | C2×D3×C8 | C23 | 1 | 2 | yes | — |
| 56 | #876 | C8 | 1 | 2 | yes | — |
| 57 | #875 | C8 | 1 | 1 | yes | characteristic |
| 58 | C4×D12 | C2×C4 | 1 | 1 | yes | characteristic |
| 59 | #872 | C2×C4 | 1 | 2 | yes | — |
| 60 | #870 | C8 | 1 | 1 | yes | characteristic |
| 61 | C4×Dic6 | C2×C4 | 1 | 1 | yes | characteristic |
| 62 | C2×C3×C16 | — | 2 | 4 | no | — |
| 63 | C2×C3×C16 | C23 | 1 | 2 | yes | — |
| 64 | C2×C3×C16 | C23 | 1 | 2 | yes | — |
| 65 | C3×C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 66 | C3×C4×C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 67 | C3×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 68 | C3×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 69 | #830 | C2×C4 | 1 | 2 | yes | — |
| 70 | (C2×D3)⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 71 | (C2×D3)⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 72 | #815 | C2×C4 | 1 | 2 | yes | — |
| 73 | Dic3⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 74 | Dic3⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 75 | C8×Dic3 | C23 | 1 | 2 | yes | — |
| 76 | C2×C3⋊C16 | C23 | 1 | 2 | yes | — |
| 77 | C2×C3⋊C16 | C23 | 1 | 2 | yes | — |
| 78 | C4⋊(C3⋊C8) | C2×C4 | 1 | 1 | yes | characteristic |
| 79 | C4⋊(C3⋊C8) | C2×C4 | 1 | 1 | yes | characteristic |
| 80 | #803 | C2×C4 | 1 | 2 | yes | — |
| 81 | #798 | — | 2 | 4 | no | — |
| 82 | D3×C16 | — | 2 | 4 | no | — |
| 83 | #529 | — | 3 | 1 | no | — |
| 84 | C2×M32 | — | 3 | 2 | no | — |
| 85 | C22×C16 | — | 3 | 2 | no | — |
| 86 | C22×C16 | C2×D3 | 1 | 2 | yes | — |
| 87 | #446 | — | 3 | 1 | no | — |
| 88 | C8×Q8 | — | 3 | 1 | no | — |
| 89 | #436 | — | 3 | 1 | no | — |
| 90 | #435 | — | 3 | 2 | no | — |
| 91 | C8×D4 | — | 3 | 2 | no | — |
| 92 | C8×D4 | — | 3 | 1 | no | — |
| 93 | #433 | — | 3 | 2 | no | — |
| 94 | #432 | — | 3 | 1 | no | — |
| 95 | #405 | — | 3 | 2 | no | — |
| 96 | C2×C4×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 97 | C4⋊C16 | — | 3 | 2 | no | — |
| 98 | C4⋊C16 | — | 3 | 2 | no | — |
| 99 | C4⋊C16 | — | 3 | 2 | no | — |
| 100 | C22⋊C16 | — | 3 | 1 | no | — |
| 101 | C22⋊C16 | — | 3 | 1 | no | — |
| 102 | C22⋊C16 | — | 3 | 2 | no | — |
| 103 | C22⋊C16 | — | 3 | 2 | no | — |
| 104 | #346 | — | 3 | 2 | no | — |
| 105 | #346 | C2×D3 | 1 | 1 | yes | characteristic |
| 106 | #346 | C2×D3 | 1 | 1 | yes | characteristic |
| 107 | #346 | C2×D3 | 1 | 2 | yes | — |
| 108 | C4×C16 | — | 3 | 2 | no | — |
| 109 | C4×C16 | — | 3 | 2 | no | — |
| 110 | C22×C3×C4 | C2×C8 | 1 | 1 | yes | characteristic |
| 111 | C22×C3×C4 | C2×C8 | 1 | 1 | yes | characteristic |
| 112 | C22×C3×C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 113 | C2×#67 | C2×C8 | 1 | 2 | yes | — |
| 114 | C2×D12 | C2×C8 | 1 | 1 | yes | characteristic |
| 115 | C2×C4×D3 | C22×C4 | 1 | 2 | yes | — |
| 116 | C2×Dic6 | C2×C8 | 1 | 1 | yes | characteristic |
| 117 | C2×C3×C8 | — | 2 | 2 | no | — |
| 118 | C2×C3×C8 | — | 2 | 2 | no | — |
| 119 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 120 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 121 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 122 | C2×C3×C8 | C24 | 1 | 1 | yes | characteristic |
| 123 | C3×C42 | C2×C8 | 1 | 1 | yes | characteristic |
| 124 | C3×C42 | C2×C8 | 1 | 1 | yes | characteristic |
| 125 | C3×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 126 | C3×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 127 | #217 | C2×C8 | 1 | 2 | yes | — |
| 128 | (C2×D3)⋊C4 | C2×C8 | 1 | 2 | yes | — |
| 129 | (C2×D3)⋊C4 | C2×C8 | 1 | 2 | yes | — |
| 130 | (C2×D3)⋊C4 | C2×C8 | 1 | 2 | yes | — |
| 131 | (C2×D3)⋊C4 | C2×C8 | 1 | 2 | yes | — |
| 132 | C4⋊Dic3 | C2×C8 | 1 | 1 | yes | characteristic |
| 133 | C4⋊Dic3 | C2×C8 | 1 | 1 | yes | characteristic |
| 134 | Dic3⋊C4 | C2×C8 | 1 | 2 | yes | — |
| 135 | Dic3⋊C4 | C2×C8 | 1 | 2 | yes | — |
| 136 | Dic3⋊C4 | C2×C8 | 1 | 2 | yes | — |
| 137 | Dic3⋊C4 | C2×C8 | 1 | 2 | yes | — |
| 138 | C4×Dic3 | C22×C4 | 1 | 2 | yes | — |
| 139 | C2×C3⋊C8 | C22×C4 | 1 | 2 | yes | — |
| 140 | C2×C3⋊C8 | C22×C4 | 1 | 2 | yes | — |
| 141 | ΓL(1,𝔽25) | — | 2 | 4 | no | — |
| 142 | D3×C8 | — | 2 | 2 | no | — |
| 143 | D3×C8 | — | 2 | 2 | no | — |
| 144 | C3×C16 | — | 2 | 2 | no | — |
| 145 | C3×C16 | #41 | 1 | 4 | yes | — |
| 146 | C3⋊C16 | — | 2 | 2 | no | — |
| 147 | C3⋊C16 | — | 2 | 2 | no | — |
| 148 | C2×M16 | — | 3 | 2 | no | — |
| 149 | C22×C8 | — | 3 | 2 | no | — |
| 150 | C22×C8 | C4×D3 | 1 | 1 | yes | characteristic |
| 151 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 152 | #126 | — | 3 | 2 | no | — |
| 153 | #125 | — | 3 | 1 | no | — |
| 154 | #124 | — | 3 | 1 | no | — |
| 155 | #123 | — | 3 | 2 | no | — |
| 156 | C22⋊Q8 | — | 3 | 1 | no | — |
| 157 | #121 | — | 3 | 1 | no | — |
| 158 | C4×Q8 | — | 3 | 1 | no | — |
| 159 | C4×D4 | — | 3 | 2 | no | — |
| 160 | C4×D4 | — | 3 | 1 | no | — |
| 161 | #117 | — | 3 | 2 | no | — |
| 162 | C2×C42 | C4×D3 | 1 | 1 | yes | characteristic |
| 163 | M32 | — | 6 | 4 | no | — |
| 164 | C2×C16 | — | 2 | 4 | no | — |
| 165 | C2×C16 | — | 3 | 2 | no | — |
| 166 | C2×C16 | — | 3 | 2 | no | — |
| 167 | C2×C16 | — | 6 | 4 | no | — |
| 168 | C2×C16 | C22×D3 | 1 | 2 | yes | — |
| 169 | C2×C16 | C22×D3 | 1 | 2 | yes | — |
| 170 | C4⋊C8 | — | 3 | 2 | no | — |
| 171 | C4⋊C8 | — | 3 | 2 | no | — |
| 172 | C4⋊C8 | — | 3 | 1 | no | — |
| 173 | C4⋊C8 | — | 3 | 1 | no | — |
| 174 | C22⋊C8 | — | 3 | 2 | no | — |
| 175 | C22⋊C8 | — | 3 | 2 | no | — |
| 176 | C22⋊C8 | — | 3 | 2 | no | — |
| 177 | #97 | — | 3 | 2 | no | — |
| 178 | #97 | — | 3 | 2 | no | — |
| 179 | C4×C8 | — | 3 | 2 | no | — |
| 180 | C4×C8 | C4×D3 | 1 | 1 | yes | characteristic |
| 181 | C4×C8 | C4×D3 | 1 | 1 | yes | characteristic |
| 182 | C4×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 183 | C4×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 184 | C23×C3 | C22×C8 | 1 | 1 | yes | characteristic |
| 185 | C22×D3 | C22×C8 | 1 | 2 | yes | — |
| 186 | C2×C3×C4 | — | 2 | 2 | no | — |
| 187 | C2×C3×C4 | — | 2 | 1 | no | — |
| 188 | C2×C3×C4 | — | 2 | 1 | no | — |
| 189 | C2×C3×C4 | — | 4 | 1 | no | — |
| 190 | C2×C3×C4 | C22×C8 | 1 | 1 | yes | characteristic |
| 191 | C2×C3×C4 | C22×C8 | 1 | 1 | yes | characteristic |
| 192 | C2×C3×C4 | C22×C8 | 1 | 1 | yes | characteristic |
| 193 | C2×C3×C4 | C22×C8 | 1 | 1 | yes | characteristic |
| 194 | C2×C3×C4 | C22×C8 | 1 | 1 | yes | characteristic |
| 195 | C2×C3×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 196 | #67 | — | 2 | 4 | no | — |
| 197 | C2×Dic3 | C22×C8 | 1 | 2 | yes | — |
| 198 | C2×Dic3 | C22×C8 | 1 | 2 | yes | — |
| 199 | C2×Dic3 | C22×C8 | 1 | 2 | yes | — |
| 200 | D12 | — | 4 | 1 | no | — |
| 201 | C4×D3 | — | 2 | 2 | no | — |
| 202 | C4×D3 | — | 2 | 2 | no | — |
| 203 | Dic6 | — | 4 | 1 | no | — |
| 204 | C3×C8 | — | 2 | 1 | no | — |
| 205 | C3×C8 | — | 2 | 1 | no | — |
| 206 | C3×C8 | #131 | 1 | 2 | yes | — |
| 207 | C3×C8 | C2×#41 | 1 | 2 | yes | — |
| 208 | C3⋊C8 | — | 2 | 2 | no | — |
| 209 | C3⋊C8 | — | 2 | 2 | no | — |
| 210 | C2×Q8 | — | 3 | 1 | no | — |
| 211 | C2×D4 | — | 3 | 2 | no | — |
| 212 | C2×D4 | — | 3 | 1 | no | — |
| 213 | C22×C4 | — | 3 | 2 | no | — |
| 214 | C22×C4 | D3×C8 | 1 | 1 | yes | characteristic |
| 215 | C22×C4 | D3×C8 | 1 | 1 | yes | characteristic |
| 216 | C22×C4 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 217 | M16 | — | 6 | 4 | no | — |
| 218 | C2×C8 | — | 2 | 2 | no | — |
| 219 | C2×C8 | — | 2 | 2 | no | — |
| 220 | C2×C8 | — | 3 | 2 | no | — |
| 221 | C2×C8 | — | 3 | 2 | no | — |
| 222 | C2×C8 | — | 6 | 2 | no | — |
| 223 | C2×C8 | — | 6 | 2 | no | — |
| 224 | C2×C8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 225 | C2×C8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 226 | C2×C8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 227 | C2×C8 | C23×D3 | 1 | 1 | yes | characteristic |
| 228 | C4⋊C4 | — | 3 | 2 | no | — |
| 229 | C4⋊C4 | — | 3 | 2 | no | — |
| 230 | C4⋊C4 | — | 3 | 1 | no | — |
| 231 | C4⋊C4 | — | 3 | 2 | no | — |
| 232 | C4⋊C4 | — | 3 | 1 | no | — |
| 233 | C4⋊C4 | — | 3 | 2 | no | — |
| 234 | C22⋊C4 | — | 3 | 2 | no | — |
| 235 | C22⋊C4 | — | 3 | 2 | no | — |
| 236 | C22⋊C4 | — | 3 | 2 | no | — |
| 237 | C22⋊C4 | — | 3 | 2 | no | — |
| 238 | C22⋊C4 | — | 3 | 2 | no | — |
| 239 | C42 | — | 3 | 2 | no | — |
| 240 | C42 | D3×C8 | 1 | 1 | yes | characteristic |
| 241 | C42 | D3×C8 | 1 | 1 | yes | characteristic |
| 242 | C42 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 243 | C42 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 244 | C16 | — | 2 | 2 | no | — |
| 245 | C16 | — | 6 | 2 | no | — |
| 246 | C16 | — | 6 | 2 | no | — |
| 247 | C16 | #235 | 1 | 4 | yes | — |
| 248 | C22×C3 | — | 2 | 1 | no | — |
| 249 | C22×C3 | — | 2 | 1 | no | — |
| 250 | C22×C3 | #504 | 1 | 2 | yes | — |
| 251 | C22×C3 | C23×C8 | 1 | 1 | yes | characteristic |
| 252 | C2×D3 | — | 2 | 2 | no | — |
| 253 | C2×D3 | — | 2 | 2 | no | — |
| 254 | C2×D3 | — | 2 | 2 | no | — |
| 255 | C3×C4 | — | 2 | 1 | no | — |
| 256 | C3×C4 | — | 2 | 1 | no | — |
| 257 | C3×C4 | — | 2 | 1 | no | — |
| 258 | C3×C4 | — | 2 | 1 | no | — |
| 259 | C3×C4 | — | 2 | 1 | no | — |
| 260 | C3×C4 | C4×#41 | 1 | 1 | yes | characteristic |
| 261 | C3×C4 | C2×#131 | 1 | 1 | yes | characteristic |
| 262 | Dic3 | — | 2 | 2 | no | — |
| 263 | Dic3 | — | 2 | 2 | no | — |
| 264 | Dic3 | — | 2 | 2 | no | — |
| 265 | C23 | — | 3 | 2 | no | — |
| 266 | C23 | C2×D3×C8 | 1 | 1 | yes | characteristic |
| 267 | Q8 | — | 12 | 1 | no | — |
| 268 | D4 | — | 6 | 4 | no | — |
| 269 | D4 | — | 12 | 1 | no | — |
| 270 | C2×C4 | — | 2 | 2 | no | — |
| 271 | C2×C4 | — | 2 | 1 | no | — |
| 272 | C2×C4 | — | 2 | 1 | no | — |
| 273 | C2×C4 | — | 3 | 2 | no | — |
| 274 | C2×C4 | — | 3 | 2 | no | — |
| 275 | C2×C4 | — | 3 | 2 | no | — |
| 276 | C2×C4 | — | 4 | 1 | no | — |
| 277 | C2×C4 | — | 6 | 2 | no | — |
| 278 | C2×C4 | — | 6 | 2 | no | — |
| 279 | C2×C4 | C2×D3×C8 | 1 | 1 | yes | characteristic |
| 280 | C2×C4 | C2×D3×C8 | 1 | 1 | yes | characteristic |
| 281 | C2×C4 | C2×D3×C8 | 1 | 1 | yes | characteristic |
| 282 | C2×C4 | C2×D3×C8 | 1 | 1 | yes | characteristic |
| 283 | C2×C4 | C2×D3×C8 | 1 | 1 | yes | characteristic |
| 284 | C2×C4 | C22×C4×D3 | 1 | 1 | yes | characteristic |
| 285 | C8 | — | 2 | 1 | no | — |
| 286 | C8 | — | 2 | 1 | no | — |
| 287 | C8 | — | 6 | 2 | no | — |
| 288 | C8 | — | 6 | 2 | no | — |
| 289 | C8 | #901 | 1 | 2 | yes | — |
| 290 | C8 | C2×#235 | 1 | 2 | yes | — |
| 291 | C2×C3 | — | 2 | 2 | no | — |
| 292 | C2×C3 | C8×#41 | 1 | 1 | yes | characteristic |
| 293 | C2×C3 | C2×#504 | 1 | 1 | yes | characteristic |
| 294 | C2×C3 | #3407 | 1 | 1 | yes | characteristic |
| 295 | D3 | — | 4 | 2 | no | — |
| 296 | C22 | — | 2 | 1 | no | — |
| 297 | C22 | — | 2 | 1 | no | — |
| 298 | C22 | — | 6 | 2 | no | — |
| 299 | C22 | — | 6 | 2 | no | — |
| 300 | C22 | — | 6 | 2 | no | — |
| 301 | C22 | #4907 | 1 | 2 | yes | — |
| 302 | C22 | C22×D3×C8 | 1 | 1 | yes | characteristic |
| 303 | C4 | — | 2 | 1 | no | — |
| 304 | C4 | — | 2 | 1 | no | — |
| 305 | C4 | — | 2 | 1 | no | — |
| 306 | C4 | — | 2 | 1 | no | — |
| 307 | C4 | — | 2 | 1 | no | — |
| 308 | C4 | — | 6 | 2 | no | — |
| 309 | C4 | — | 6 | 2 | no | — |
| 310 | C4 | — | 6 | 2 | no | — |
| 311 | C4 | C4×#235 | 1 | 1 | yes | characteristic |
| 312 | C4 | C2×#901 | 1 | 1 | yes | characteristic |
| 313 | C3 | #33482 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 314 | C2 | — | 2 | 2 | no | — |
| 315 | C2 | — | 12 | 2 | no | — |
| 316 | C2 | C8×#235 | 1 | 1 | yes | characteristic |
| 317 | C2 | C2×#4907 | 1 | 1 | yes | characteristic |
| 318 | C2 | #82007 | 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