← #10948283
Subgroups — #10948283
Order 768.
582 conjugacy classes of subgroups.
The subgroup lattice has 298 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | Dic3×#131 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C3×#2113 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #72004 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #71421 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | Dic3×M32 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #71401 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C2×#4502 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #69094 | C2 | 1 | 4 | yes | maximal, complemented |
| 9 | #69093 | C2 | 1 | 4 | yes | maximal, complemented |
| 10 | #33421 | — | 3 | 1 | no | maximal, Sylow |
| 11 | C2×C3×#131 | C22 | 1 | 1 | yes | characteristic |
| 12 | Dic3×#41 | C4 | 1 | 1 | yes | characteristic |
| 13 | C2×#948 | C4 | 1 | 1 | yes | characteristic |
| 14 | C2×C3×M32 | C22 | 1 | 1 | yes | characteristic |
| 15 | C22×C3×C16 | C22 | 1 | 1 | yes | characteristic |
| 16 | #5122 | C4 | 1 | 2 | yes | — |
| 17 | #5122 | C4 | 1 | 1 | yes | characteristic |
| 18 | Dic3×M16 | C4 | 1 | 1 | yes | characteristic |
| 19 | Dic3×M16 | C4 | 1 | 2 | yes | — |
| 20 | #5101 | C22 | 1 | 2 | yes | — |
| 21 | #5101 | C22 | 1 | 1 | yes | characteristic |
| 22 | C2×C8×Dic3 | C22 | 1 | 2 | yes | — |
| 23 | C2×C8×Dic3 | C22 | 1 | 1 | yes | characteristic |
| 24 | C2×#812 | C22 | 1 | 1 | yes | characteristic |
| 25 | C22×C3⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 26 | C3×C22⋊C16 | C22 | 1 | 4 | yes | — |
| 27 | #4549 | C22 | 1 | 4 | yes | — |
| 28 | #4502 | C22 | 1 | 2 | yes | — |
| 29 | #4502 | C22 | 1 | 2 | yes | — |
| 30 | #4502 | C22 | 1 | 2 | yes | — |
| 31 | Dic3⋊C16 | C22 | 1 | 4 | yes | — |
| 32 | Dic3⋊C16 | C22 | 1 | 4 | yes | — |
| 33 | Dic3×C16 | C22 | 1 | 2 | yes | — |
| 34 | C4×#131 | — | 3 | 1 | no | — |
| 35 | #2164 | — | 3 | 4 | no | — |
| 36 | #2163 | — | 3 | 4 | no | — |
| 37 | #2151 | — | 3 | 1 | no | — |
| 38 | #2151 | — | 3 | 1 | no | — |
| 39 | #2113 | — | 3 | 1 | no | — |
| 40 | #2113 | D3 | 1 | 1 | yes | characteristic |
| 41 | C4×M32 | — | 3 | 1 | no | — |
| 42 | C2×#346 | — | 3 | 1 | no | — |
| 43 | C2×C3×#41 | C2×C4 | 1 | 1 | yes | characteristic |
| 44 | C3×#131 | — | 2 | 2 | no | — |
| 45 | C3×#131 | — | 2 | 2 | no | — |
| 46 | C2×C3×M16 | C2×C4 | 1 | 1 | yes | characteristic |
| 47 | C2×C3×M16 | C2×C4 | 1 | 2 | yes | — |
| 48 | C22×C3×C8 | C23 | 1 | 2 | yes | — |
| 49 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 50 | #948 | — | 2 | 4 | no | — |
| 51 | Q8×Dic3 | C8 | 1 | 2 | yes | — |
| 52 | D4×Dic3 | C8 | 1 | 4 | yes | — |
| 53 | D4×Dic3 | C8 | 1 | 2 | yes | — |
| 54 | #926 | C2×C4 | 1 | 2 | yes | — |
| 55 | #926 | C2×C4 | 1 | 1 | yes | characteristic |
| 56 | C2×C4×Dic3 | C2×C4 | 1 | 2 | yes | — |
| 57 | C2×C4×Dic3 | C2×C4 | 1 | 1 | yes | characteristic |
| 58 | C2×#208 | C2×C4 | 1 | 1 | yes | characteristic |
| 59 | C2×#208 | C2×C4 | 1 | 2 | yes | — |
| 60 | C22×C3⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 61 | C22×C3⋊C8 | C2×C4 | 1 | 1 | yes | characteristic |
| 62 | C3×M32 | — | 2 | 2 | no | — |
| 63 | C2×C3×C16 | — | 2 | 2 | no | — |
| 64 | C2×C3×C16 | C23 | 1 | 2 | yes | — |
| 65 | C2×C3×C16 | C23 | 1 | 2 | yes | — |
| 66 | #815 | C2×C4 | 1 | 2 | yes | — |
| 67 | #815 | C2×C4 | 1 | 2 | yes | — |
| 68 | #815 | C2×C4 | 1 | 2 | yes | — |
| 69 | C8×Dic3 | C2×C4 | 1 | 2 | yes | — |
| 70 | C8×Dic3 | C2×C4 | 1 | 1 | yes | characteristic |
| 71 | C8×Dic3 | C2×C4 | 1 | 2 | yes | — |
| 72 | C8×Dic3 | C2×C4 | 1 | 1 | yes | characteristic |
| 73 | C8×Dic3 | C23 | 1 | 1 | yes | characteristic |
| 74 | C8×Dic3 | C23 | 1 | 2 | yes | — |
| 75 | C8×Dic3 | C23 | 1 | 1 | yes | characteristic |
| 76 | #812 | — | 2 | 2 | no | — |
| 77 | C2×C3⋊C16 | — | 2 | 2 | no | — |
| 78 | C2×C3⋊C16 | C23 | 1 | 2 | yes | — |
| 79 | C2×C3⋊C16 | C23 | 1 | 2 | yes | — |
| 80 | C2×#131 | — | 3 | 1 | no | — |
| 81 | C2×#131 | C2×D3 | 1 | 1 | yes | characteristic |
| 82 | C4×#41 | — | 3 | 1 | no | — |
| 83 | C2×M32 | — | 3 | 1 | no | — |
| 84 | C2×M32 | C2×D3 | 1 | 1 | yes | characteristic |
| 85 | C22×C16 | — | 3 | 1 | no | — |
| 86 | C22×C16 | C2×D3 | 1 | 1 | yes | characteristic |
| 87 | #405 | — | 3 | 1 | no | — |
| 88 | #405 | — | 3 | 2 | no | — |
| 89 | #405 | — | 3 | 1 | no | — |
| 90 | #405 | — | 3 | 2 | no | — |
| 91 | C4×M16 | — | 3 | 2 | no | — |
| 92 | C4×M16 | — | 3 | 1 | no | — |
| 93 | C2×C4×C8 | — | 3 | 1 | no | — |
| 94 | C2×C4×C8 | — | 3 | 2 | no | — |
| 95 | C4⋊C16 | — | 3 | 4 | no | — |
| 96 | C4⋊C16 | — | 3 | 4 | no | — |
| 97 | C22⋊C16 | — | 3 | 4 | no | — |
| 98 | C22⋊C16 | C2×D3 | 1 | 4 | yes | — |
| 99 | #346 | — | 3 | 2 | no | — |
| 100 | #346 | — | 3 | 2 | no | — |
| 101 | #346 | — | 3 | 2 | no | — |
| 102 | C4×C16 | — | 3 | 2 | no | — |
| 103 | C3×#41 | — | 2 | 2 | no | — |
| 104 | C3×#41 | — | 2 | 2 | no | — |
| 105 | C2×C3×Q8 | C2×C8 | 1 | 1 | yes | characteristic |
| 106 | C2×C3×D4 | C2×C8 | 1 | 2 | yes | — |
| 107 | C2×C3×D4 | C2×C8 | 1 | 1 | yes | characteristic |
| 108 | C22×C3×C4 | C22×C4 | 1 | 2 | yes | — |
| 109 | C22×C3×C4 | C22×C4 | 1 | 1 | yes | characteristic |
| 110 | C22×Dic3 | C2×C8 | 1 | 4 | yes | — |
| 111 | C22×Dic3 | C2×C8 | 1 | 2 | yes | — |
| 112 | C3×M16 | — | 2 | 2 | no | — |
| 113 | C3×M16 | — | 2 | 4 | no | — |
| 114 | C2×C3×C8 | — | 2 | 2 | no | — |
| 115 | C2×C3×C8 | — | 2 | 2 | no | — |
| 116 | C2×C3×C8 | C22×C4 | 1 | 2 | yes | — |
| 117 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 118 | C2×C3×C8 | #41 | 1 | 2 | yes | — |
| 119 | C2×C3×C8 | #41 | 1 | 2 | yes | — |
| 120 | C2×C3×C8 | C24 | 1 | 1 | yes | characteristic |
| 121 | #217 | C2×C8 | 1 | 4 | yes | — |
| 122 | #217 | C2×C8 | 1 | 2 | yes | — |
| 123 | C4⋊Dic3 | C2×C8 | 1 | 4 | yes | — |
| 124 | C4⋊Dic3 | C2×C8 | 1 | 2 | yes | — |
| 125 | C4×Dic3 | C2×C8 | 1 | 4 | yes | — |
| 126 | C4×Dic3 | C2×C8 | 1 | 2 | yes | — |
| 127 | C4×Dic3 | C22×C4 | 1 | 1 | yes | characteristic |
| 128 | C4×Dic3 | C22×C4 | 1 | 1 | yes | characteristic |
| 129 | C4×Dic3 | C22×C4 | 1 | 2 | yes | — |
| 130 | #208 | — | 2 | 2 | no | — |
| 131 | #208 | — | 2 | 4 | no | — |
| 132 | C2×C3⋊C8 | — | 2 | 4 | no | — |
| 133 | C2×C3⋊C8 | M16 | 1 | 4 | yes | — |
| 134 | C2×C3⋊C8 | C22×C4 | 1 | 2 | yes | — |
| 135 | C2×C3⋊C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 136 | C2×C3⋊C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 137 | C3×C16 | — | 2 | 2 | no | — |
| 138 | C3×C16 | — | 2 | 2 | no | — |
| 139 | C3⋊C16 | — | 2 | 2 | no | — |
| 140 | C3⋊C16 | — | 2 | 2 | no | — |
| 141 | C2×#41 | C4×D3 | 1 | 1 | yes | characteristic |
| 142 | #131 | — | 2 | 2 | no | — |
| 143 | #131 | — | 2 | 2 | no | — |
| 144 | #131 | — | 6 | 4 | no | — |
| 145 | C2×M16 | — | 3 | 2 | no | — |
| 146 | C2×M16 | — | 3 | 1 | no | — |
| 147 | C2×M16 | C4×D3 | 1 | 1 | yes | characteristic |
| 148 | C2×M16 | C4×D3 | 1 | 2 | yes | — |
| 149 | C22×C8 | — | 3 | 1 | no | — |
| 150 | C22×C8 | — | 3 | 2 | no | — |
| 151 | C22×C8 | C22×D3 | 1 | 2 | yes | — |
| 152 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 153 | C4×Q8 | — | 3 | 2 | no | — |
| 154 | C4×D4 | — | 3 | 4 | no | — |
| 155 | C4×D4 | — | 3 | 2 | no | — |
| 156 | #117 | — | 3 | 2 | no | — |
| 157 | #117 | — | 3 | 1 | no | — |
| 158 | C2×C42 | — | 3 | 2 | no | — |
| 159 | C2×C42 | — | 3 | 1 | no | — |
| 160 | M32 | — | 2 | 2 | no | — |
| 161 | M32 | — | 6 | 2 | no | — |
| 162 | C2×C16 | — | 2 | 2 | no | — |
| 163 | C2×C16 | — | 3 | 2 | no | — |
| 164 | C2×C16 | — | 3 | 2 | no | — |
| 165 | C2×C16 | — | 6 | 2 | no | — |
| 166 | C2×C16 | C22×D3 | 1 | 2 | yes | — |
| 167 | C2×C16 | C22×D3 | 1 | 2 | yes | — |
| 168 | #97 | — | 3 | 2 | no | — |
| 169 | #97 | — | 3 | 2 | no | — |
| 170 | #97 | — | 3 | 2 | no | — |
| 171 | C4×C8 | — | 3 | 1 | no | — |
| 172 | C4×C8 | — | 3 | 2 | no | — |
| 173 | C4×C8 | — | 3 | 2 | no | — |
| 174 | C4×C8 | — | 3 | 1 | no | — |
| 175 | C4×C8 | — | 3 | 1 | no | — |
| 176 | C4×C8 | — | 3 | 1 | no | — |
| 177 | C4×C8 | — | 3 | 2 | no | — |
| 178 | C23×C3 | C22×C8 | 1 | 2 | yes | — |
| 179 | C23×C3 | C22×C8 | 1 | 1 | yes | characteristic |
| 180 | C3×Q8 | — | 2 | 2 | no | — |
| 181 | C3×D4 | — | 2 | 4 | no | — |
| 182 | C3×D4 | — | 2 | 2 | no | — |
| 183 | C2×C3×C4 | — | 2 | 2 | no | — |
| 184 | C2×C3×C4 | — | 2 | 2 | no | — |
| 185 | C2×C3×C4 | #117 | 1 | 2 | yes | — |
| 186 | C2×C3×C4 | C22×C8 | 1 | 2 | yes | — |
| 187 | C2×C3×C4 | C22×C8 | 1 | 1 | yes | characteristic |
| 188 | C2×C3×C4 | C2×M16 | 1 | 2 | yes | — |
| 189 | C2×C3×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 190 | C2×Dic3 | — | 4 | 2 | no | — |
| 191 | C2×Dic3 | — | 4 | 4 | no | — |
| 192 | C2×Dic3 | C22×C8 | 1 | 2 | yes | — |
| 193 | C2×Dic3 | C22×C8 | 1 | 4 | yes | — |
| 194 | C2×Dic3 | C22×C8 | 1 | 2 | yes | — |
| 195 | C3×C8 | — | 2 | 2 | no | — |
| 196 | C3×C8 | #117 | 1 | 2 | yes | — |
| 197 | C3×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 198 | C3×C8 | C2×#41 | 1 | 1 | yes | characteristic |
| 199 | C3⋊C8 | — | 2 | 2 | no | — |
| 200 | C3⋊C8 | C2×M16 | 1 | 2 | yes | — |
| 201 | C3⋊C8 | C2×M16 | 1 | 2 | yes | — |
| 202 | #41 | — | 2 | 2 | no | — |
| 203 | #41 | — | 2 | 2 | no | — |
| 204 | C2×Q8 | D3×C8 | 1 | 1 | yes | characteristic |
| 205 | C2×D4 | D3×C8 | 1 | 2 | yes | — |
| 206 | C2×D4 | D3×C8 | 1 | 1 | yes | characteristic |
| 207 | C22×C4 | — | 3 | 4 | no | — |
| 208 | C22×C4 | — | 3 | 2 | no | — |
| 209 | C22×C4 | C2×C4×D3 | 1 | 2 | yes | — |
| 210 | C22×C4 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 211 | M16 | — | 2 | 2 | no | — |
| 212 | M16 | — | 2 | 4 | no | — |
| 213 | M16 | — | 6 | 4 | no | — |
| 214 | M16 | — | 6 | 2 | no | — |
| 215 | C2×C8 | — | 2 | 2 | no | — |
| 216 | C2×C8 | — | 2 | 2 | no | — |
| 217 | C2×C8 | — | 3 | 4 | no | — |
| 218 | C2×C8 | — | 3 | 1 | no | — |
| 219 | C2×C8 | — | 3 | 1 | no | — |
| 220 | C2×C8 | — | 3 | 2 | no | — |
| 221 | C2×C8 | — | 6 | 4 | no | — |
| 222 | C2×C8 | C2×C4×D3 | 1 | 2 | yes | — |
| 223 | C2×C8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 224 | C2×C8 | #237 | 1 | 2 | yes | — |
| 225 | C2×C8 | #237 | 1 | 2 | yes | — |
| 226 | C2×C8 | C23×D3 | 1 | 1 | yes | characteristic |
| 227 | C4⋊C4 | — | 3 | 4 | no | — |
| 228 | C4⋊C4 | — | 3 | 2 | no | — |
| 229 | C22⋊C4 | — | 3 | 4 | no | — |
| 230 | C22⋊C4 | — | 3 | 2 | no | — |
| 231 | C42 | — | 3 | 4 | no | — |
| 232 | C42 | — | 3 | 2 | no | — |
| 233 | C42 | — | 3 | 1 | no | — |
| 234 | C42 | — | 3 | 1 | no | — |
| 235 | C42 | — | 3 | 2 | no | — |
| 236 | C16 | — | 2 | 2 | no | — |
| 237 | C16 | — | 2 | 2 | no | — |
| 238 | C16 | — | 6 | 2 | no | — |
| 239 | C16 | — | 6 | 2 | no | — |
| 240 | C22×C3 | — | 2 | 2 | no | — |
| 241 | C22×C3 | — | 2 | 4 | no | — |
| 242 | C22×C3 | — | 2 | 2 | no | — |
| 243 | C22×C3 | #431 | 1 | 2 | yes | — |
| 244 | C22×C3 | C23×C8 | 1 | 1 | yes | characteristic |
| 245 | C3×C4 | — | 2 | 2 | no | — |
| 246 | C3×C4 | #431 | 1 | 2 | yes | — |
| 247 | C3×C4 | C2×#117 | 1 | 1 | yes | characteristic |
| 248 | C3×C4 | C22×M16 | 1 | 1 | yes | characteristic |
| 249 | Dic3 | — | 2 | 2 | no | — |
| 250 | Dic3 | — | 2 | 2 | no | — |
| 251 | Dic3 | — | 2 | 4 | no | — |
| 252 | C23 | C2×D3×C8 | 1 | 2 | yes | — |
| 253 | C23 | C2×D3×C8 | 1 | 1 | yes | characteristic |
| 254 | Q8 | — | 2 | 2 | no | — |
| 255 | D4 | — | 2 | 4 | no | — |
| 256 | D4 | — | 2 | 2 | no | — |
| 257 | C2×C4 | — | 2 | 2 | no | — |
| 258 | C2×C4 | — | 2 | 2 | no | — |
| 259 | C2×C4 | — | 3 | 2 | no | — |
| 260 | C2×C4 | — | 3 | 4 | no | — |
| 261 | C2×C4 | — | 3 | 2 | no | — |
| 262 | C2×C4 | — | 12 | 2 | no | — |
| 263 | C2×C4 | — | 12 | 4 | no | — |
| 264 | C2×C4 | #877 | 1 | 2 | yes | — |
| 265 | C2×C4 | C2×D3×C8 | 1 | 2 | yes | — |
| 266 | C2×C4 | C2×D3×C8 | 1 | 1 | yes | characteristic |
| 267 | C2×C4 | D3×M16 | 1 | 2 | yes | — |
| 268 | C2×C4 | C22×C4×D3 | 1 | 1 | yes | characteristic |
| 269 | C8 | — | 2 | 2 | no | — |
| 270 | C8 | — | 3 | 2 | no | — |
| 271 | C8 | — | 3 | 2 | no | — |
| 272 | C8 | — | 6 | 2 | no | — |
| 273 | C8 | #877 | 1 | 2 | yes | — |
| 274 | C8 | C2×#237 | 1 | 1 | yes | characteristic |
| 275 | C8 | C2×#237 | 1 | 1 | yes | characteristic |
| 276 | C2×C3 | — | 2 | 2 | no | — |
| 277 | C2×C3 | — | 4 | 2 | no | — |
| 278 | C2×C3 | C2×#431 | 1 | 1 | yes | characteristic |
| 279 | C2×C3 | #3407 | 1 | 2 | yes | — |
| 280 | C22 | — | 2 | 2 | no | — |
| 281 | C22 | — | 2 | 4 | no | — |
| 282 | C22 | — | 2 | 2 | no | — |
| 283 | C22 | #4718 | 1 | 2 | yes | — |
| 284 | C22 | C22×D3×C8 | 1 | 1 | yes | characteristic |
| 285 | C4 | — | 2 | 2 | no | — |
| 286 | C4 | — | 6 | 2 | no | — |
| 287 | C4 | — | 6 | 2 | no | — |
| 288 | C4 | — | 6 | 4 | no | — |
| 289 | C4 | #4718 | 1 | 2 | yes | — |
| 290 | C4 | C2×#877 | 1 | 1 | yes | characteristic |
| 291 | C4 | C2×D3×M16 | 1 | 1 | yes | characteristic |
| 292 | C3 | #33421 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 293 | C2 | — | 2 | 2 | no | — |
| 294 | C2 | — | 4 | 2 | no | — |
| 295 | C2 | C2×#4718 | 1 | 1 | yes | characteristic |
| 296 | C2 | #82012 | 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