← #10694405
Subgroups — #10694405
Order 768.
470 conjugacy classes of subgroups.
The subgroup lattice has 314 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C2×#5080 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C3×#332 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #71778 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #70465 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #70459 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C2×#4481 | C2 | 1 | 1 | yes | maximal, characteristic |
| 7 | C2×#4480 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | #9729 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C2×C3×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×#869 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×C3×C4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×C3×C4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#835 | — | 2 | 2 | no | — |
| 14 | C2×C8×Dic3 | — | 2 | 1 | no | — |
| 15 | #5080 | — | 2 | 2 | no | — |
| 16 | #5080 | C4 | 1 | 4 | yes | complemented |
| 17 | C2×#808 | — | 2 | 1 | no | — |
| 18 | C2×#808 | — | 2 | 1 | no | — |
| 19 | C4×C4⋊Dic3 | C22 | 1 | 1 | yes | characteristic |
| 20 | C2×C4⋊(C3⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×C4×C3⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 22 | C3×#332 | — | 2 | 2 | no | — |
| 23 | #4551 | — | 2 | 2 | no | — |
| 24 | #4550 | — | 2 | 1 | no | — |
| 25 | #4481 | — | 2 | 2 | no | — |
| 26 | #4480 | — | 2 | 2 | no | — |
| 27 | C2×#498 | — | 3 | 1 | no | — |
| 28 | #1987 | — | 3 | 1 | no | — |
| 29 | #1981 | — | 3 | 1 | no | — |
| 30 | C8×C4⋊C4 | — | 3 | 1 | no | — |
| 31 | C2×#334 | — | 3 | 1 | no | — |
| 32 | C2×#333 | — | 3 | 1 | no | — |
| 33 | C2×#332 | D3 | 1 | 1 | yes | characteristic |
| 34 | C22×C3×Q8 | — | 2 | 1 | no | — |
| 35 | C22×Dic6 | — | 2 | 1 | no | — |
| 36 | C22×C3×C8 | — | 2 | 1 | no | — |
| 37 | C22×C3×C8 | — | 2 | 1 | no | — |
| 38 | C3×C4⋊Q8 | — | 2 | 1 | no | — |
| 39 | C3×C4⋊Q8 | — | 2 | 1 | no | — |
| 40 | C3×C4⋊Q8 | — | 2 | 1 | no | — |
| 41 | C3×C4⋊Q8 | C2×C4 | 1 | 2 | yes | — |
| 42 | C2×C3×C4⋊C4 | — | 2 | 1 | no | — |
| 43 | C2×C3×C4⋊C4 | — | 2 | 1 | no | — |
| 44 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 45 | C2×C4⋊Dic3 | — | 2 | 1 | no | — |
| 46 | C2×C4⋊Dic3 | — | 2 | 1 | no | — |
| 47 | C2×C4⋊Dic3 | D4 | 1 | 2 | yes | — |
| 48 | C2×C4×Dic3 | — | 2 | 1 | no | — |
| 49 | C22×C3⋊C8 | — | 2 | 1 | no | — |
| 50 | C22×C3⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 51 | C22×C3⋊C8 | D4 | 1 | 1 | yes | characteristic |
| 52 | #869 | — | 2 | 1 | no | — |
| 53 | #869 | — | 2 | 1 | no | — |
| 54 | #869 | — | 2 | 1 | no | — |
| 55 | #869 | C2×C4 | 1 | 2 | yes | — |
| 56 | C3×C4⋊C8 | — | 2 | 1 | no | — |
| 57 | C3×C4⋊C8 | — | 2 | 1 | no | — |
| 58 | C3×C4⋊C8 | — | 2 | 1 | no | — |
| 59 | C3×C4⋊C8 | D4 | 1 | 2 | yes | — |
| 60 | #835 | — | 4 | 4 | no | — |
| 61 | C3⋊#95 | — | 2 | 1 | no | — |
| 62 | C8×Dic3 | — | 4 | 2 | no | — |
| 63 | #808 | — | 2 | 4 | no | — |
| 64 | #808 | — | 2 | 4 | no | — |
| 65 | C4⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 66 | C4⋊(C3⋊C8) | — | 2 | 1 | no | — |
| 67 | C4×C3⋊C8 | C2×C4 | 1 | 2 | yes | — |
| 68 | C4×C3⋊C8 | D4 | 1 | 2 | yes | — |
| 69 | C2×C4⋊Q8 | — | 3 | 1 | no | — |
| 70 | C2×C4⋊Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 71 | #498 | — | 3 | 4 | no | — |
| 72 | #498 | — | 6 | 2 | no | — |
| 73 | C2×#106 | — | 6 | 1 | no | — |
| 74 | C2×#106 | — | 6 | 1 | no | — |
| 75 | C2×C4⋊C8 | — | 3 | 1 | no | — |
| 76 | C2×C4⋊C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 77 | C2×C4⋊C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 78 | C2×#103 | — | 6 | 2 | no | — |
| 79 | C2×C4×C8 | — | 3 | 1 | no | — |
| 80 | C2×C4×C8 | — | 6 | 1 | no | — |
| 81 | C4×C4⋊C4 | — | 3 | 1 | no | — |
| 82 | #340 | — | 6 | 2 | no | — |
| 83 | #336 | — | 6 | 1 | no | — |
| 84 | #334 | — | 6 | 2 | no | — |
| 85 | #333 | — | 6 | 2 | no | — |
| 86 | #332 | — | 2 | 2 | no | — |
| 87 | C2×C3×Q8 | — | 2 | 2 | no | — |
| 88 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 89 | C2×C3×Q8 | — | 4 | 2 | no | — |
| 90 | C2×C3×Q8 | — | 4 | 2 | no | — |
| 91 | C22×C3×C4 | — | 4 | 1 | no | — |
| 92 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 93 | C22×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 94 | C22×C3×C4 | #41 | 1 | 1 | yes | characteristic |
| 95 | C22×Dic3 | — | 2 | 2 | no | — |
| 96 | C22×Dic3 | — | 4 | 1 | no | — |
| 97 | C2×Dic6 | — | 2 | 2 | no | — |
| 98 | C2×Dic6 | — | 4 | 1 | no | — |
| 99 | C2×Dic6 | — | 8 | 2 | no | — |
| 100 | C2×C3×C8 | — | 2 | 2 | no | — |
| 101 | C2×C3×C8 | — | 4 | 1 | no | — |
| 102 | C2×C3×C8 | — | 4 | 1 | no | — |
| 103 | C2×C3×C8 | — | 4 | 1 | no | — |
| 104 | C2×C3×C8 | — | 4 | 2 | no | — |
| 105 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 106 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 107 | C3×C4⋊C4 | — | 4 | 1 | no | — |
| 108 | C3×C4⋊C4 | — | 4 | 1 | no | — |
| 109 | C3×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 110 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 111 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 112 | C3×C42 | #41 | 1 | 1 | yes | characteristic |
| 113 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 114 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 115 | C4⋊Dic3 | — | 2 | 2 | no | — |
| 116 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 117 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 118 | C4⋊Dic3 | SD16 | 1 | 4 | yes | — |
| 119 | C4×Dic3 | — | 4 | 1 | no | — |
| 120 | C4×Dic3 | — | 4 | 1 | no | — |
| 121 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 122 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 123 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 124 | C2×C3⋊C8 | — | 4 | 2 | no | — |
| 125 | C2×C3⋊C8 | C22⋊C4 | 1 | 2 | yes | — |
| 126 | C2×C3⋊C8 | C22⋊C4 | 1 | 2 | yes | — |
| 127 | C2×C3⋊C8 | SD16 | 1 | 2 | yes | — |
| 128 | C2×C3⋊C8 | Q16 | 1 | 2 | yes | — |
| 129 | C22×Q8 | — | 2 | 1 | no | — |
| 130 | C22×Q8 | — | 6 | 1 | no | — |
| 131 | C22×C8 | — | 2 | 1 | no | — |
| 132 | C22×C8 | — | 2 | 1 | no | — |
| 133 | C22×C8 | — | 3 | 1 | no | — |
| 134 | C22×C8 | — | 3 | 1 | no | — |
| 135 | C22×C8 | — | 6 | 1 | no | — |
| 136 | C4⋊Q8 | — | 2 | 1 | no | — |
| 137 | C4⋊Q8 | — | 2 | 1 | no | — |
| 138 | C4⋊Q8 | — | 2 | 1 | no | — |
| 139 | C4⋊Q8 | — | 3 | 2 | no | — |
| 140 | C4⋊Q8 | — | 6 | 1 | no | — |
| 141 | C4⋊Q8 | — | 6 | 1 | no | — |
| 142 | C4⋊Q8 | — | 6 | 1 | no | — |
| 143 | C4⋊Q8 | C4×D3 | 1 | 2 | yes | — |
| 144 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 145 | C2×C4⋊C4 | — | 2 | 1 | no | — |
| 146 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 147 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 148 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 149 | C2×C42 | — | 6 | 1 | no | — |
| 150 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 151 | #106 | — | 6 | 4 | no | — |
| 152 | #106 | — | 6 | 4 | no | — |
| 153 | C4⋊C8 | — | 2 | 1 | no | — |
| 154 | C4⋊C8 | — | 2 | 1 | no | — |
| 155 | C4⋊C8 | — | 2 | 1 | no | — |
| 156 | C4⋊C8 | — | 6 | 1 | no | — |
| 157 | C4⋊C8 | — | 6 | 1 | no | — |
| 158 | C4⋊C8 | #67 | 1 | 2 | yes | — |
| 159 | #103 | — | 12 | 4 | no | — |
| 160 | C4×C8 | — | 3 | 2 | no | — |
| 161 | C4×C8 | — | 3 | 2 | no | — |
| 162 | C4×C8 | — | 12 | 2 | no | — |
| 163 | #95 | — | 6 | 1 | no | — |
| 164 | C23×C3 | #121 | 1 | 1 | yes | characteristic |
| 165 | C3×Q8 | — | 4 | 4 | no | — |
| 166 | C3×Q8 | — | 8 | 1 | no | — |
| 167 | C3×Q8 | — | 8 | 1 | no | — |
| 168 | C2×C3×C4 | — | 2 | 2 | no | — |
| 169 | C2×C3×C4 | — | 4 | 2 | no | — |
| 170 | C2×C3×C4 | — | 8 | 2 | no | — |
| 171 | C2×C3×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 172 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 173 | C2×C3×C4 | C4×D4 | 1 | 1 | yes | characteristic |
| 174 | C2×C3×C4 | #121 | 1 | 1 | yes | characteristic |
| 175 | C2×C3×C4 | #124 | 1 | 1 | yes | characteristic |
| 176 | C2×C3×C4 | #127 | 1 | 1 | yes | characteristic |
| 177 | C2×C3×C4 | C2×SD16 | 1 | 1 | yes | characteristic |
| 178 | C2×C3×C4 | C2×SD16 | 1 | 2 | yes | — |
| 179 | C2×C3×C4 | C2×Q16 | 1 | 1 | yes | characteristic |
| 180 | C2×C3×C4 | #135 | 1 | 2 | yes | — |
| 181 | C2×C3×C4 | #137 | 1 | 1 | yes | characteristic |
| 182 | C2×C3×C4 | #137 | 1 | 1 | yes | characteristic |
| 183 | C2×Dic3 | — | 2 | 4 | no | — |
| 184 | C2×Dic3 | — | 4 | 2 | no | — |
| 185 | C2×Dic3 | — | 4 | 2 | no | — |
| 186 | C2×Dic3 | — | 4 | 2 | no | — |
| 187 | C2×Dic3 | — | 8 | 2 | no | — |
| 188 | Dic6 | — | 8 | 2 | no | — |
| 189 | Dic6 | — | 8 | 2 | no | — |
| 190 | C3×C8 | — | 4 | 2 | no | — |
| 191 | C3×C8 | — | 8 | 1 | no | — |
| 192 | C3⋊C8 | — | 2 | 2 | no | — |
| 193 | C3⋊C8 | — | 8 | 1 | no | — |
| 194 | C3⋊C8 | #103 | 1 | 4 | yes | complemented |
| 195 | C2×Q8 | — | 2 | 2 | no | — |
| 196 | C2×Q8 | — | 4 | 1 | no | — |
| 197 | C2×Q8 | — | 4 | 2 | no | — |
| 198 | C2×Q8 | — | 4 | 2 | no | — |
| 199 | C2×Q8 | — | 6 | 2 | no | — |
| 200 | C2×Q8 | — | 12 | 1 | no | — |
| 201 | C2×Q8 | — | 24 | 2 | no | — |
| 202 | C22×C4 | — | 4 | 1 | no | — |
| 203 | C22×C4 | — | 6 | 2 | no | — |
| 204 | C22×C4 | — | 12 | 1 | no | — |
| 205 | C22×C4 | #235 | 1 | 1 | yes | characteristic |
| 206 | C22×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 207 | C22×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 208 | C2×C8 | — | 2 | 2 | no | — |
| 209 | C2×C8 | — | 3 | 2 | no | — |
| 210 | C2×C8 | — | 3 | 2 | no | — |
| 211 | C2×C8 | — | 3 | 2 | no | — |
| 212 | C2×C8 | — | 3 | 2 | no | — |
| 213 | C2×C8 | — | 4 | 1 | no | — |
| 214 | C2×C8 | — | 4 | 1 | no | — |
| 215 | C2×C8 | — | 4 | 2 | no | — |
| 216 | C2×C8 | — | 4 | 1 | no | — |
| 217 | C2×C8 | — | 6 | 1 | no | — |
| 218 | C2×C8 | — | 6 | 1 | no | — |
| 219 | C2×C8 | — | 12 | 2 | no | — |
| 220 | C2×C8 | — | 12 | 1 | no | — |
| 221 | C4⋊C4 | — | 2 | 2 | no | — |
| 222 | C4⋊C4 | — | 2 | 2 | no | — |
| 223 | C4⋊C4 | — | 3 | 4 | no | — |
| 224 | C4⋊C4 | — | 4 | 1 | no | — |
| 225 | C4⋊C4 | — | 4 | 1 | no | — |
| 226 | C4⋊C4 | — | 6 | 2 | no | — |
| 227 | C4⋊C4 | — | 6 | 2 | no | — |
| 228 | C4⋊C4 | — | 6 | 2 | no | — |
| 229 | C4⋊C4 | — | 12 | 1 | no | — |
| 230 | C4⋊C4 | — | 12 | 1 | no | — |
| 231 | C42 | — | 12 | 1 | no | — |
| 232 | C42 | — | 12 | 1 | no | — |
| 233 | C42 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 234 | C42 | D3×D4 | 1 | 1 | yes | characteristic |
| 235 | C42 | #237 | 1 | 1 | yes | characteristic |
| 236 | C42 | C2×#67 | 1 | 1 | yes | characteristic |
| 237 | C22×C3 | #390 | 1 | 1 | yes | characteristic |
| 238 | C22×C3 | #461 | 1 | 1 | yes | characteristic |
| 239 | C22×C3 | #462 | 1 | 1 | yes | characteristic |
| 240 | C22×C3 | #465 | 1 | 2 | yes | — |
| 241 | C22×C3 | #471 | 1 | 2 | yes | — |
| 242 | C3×C4 | — | 2 | 1 | no | — |
| 243 | C3×C4 | — | 2 | 1 | no | — |
| 244 | C3×C4 | — | 8 | 2 | no | — |
| 245 | C3×C4 | C2×#103 | 1 | 1 | yes | characteristic |
| 246 | C3×C4 | #419 | 1 | 1 | yes | characteristic |
| 247 | C3×C4 | C4×SD16 | 1 | 2 | yes | — |
| 248 | C3×C4 | #461 | 1 | 1 | yes | characteristic |
| 249 | C3×C4 | #462 | 1 | 1 | yes | characteristic |
| 250 | C3×C4 | #492 | 1 | 1 | yes | characteristic |
| 251 | C3×C4 | #495 | 1 | 1 | yes | characteristic |
| 252 | Dic3 | — | 4 | 4 | no | — |
| 253 | Dic3 | — | 8 | 2 | no | — |
| 254 | C23 | #884 | 1 | 1 | yes | characteristic |
| 255 | Q8 | — | 4 | 4 | no | — |
| 256 | Q8 | — | 8 | 1 | no | — |
| 257 | Q8 | — | 8 | 1 | no | — |
| 258 | Q8 | — | 24 | 2 | no | — |
| 259 | Q8 | — | 24 | 2 | no | — |
| 260 | C2×C4 | — | 2 | 2 | no | — |
| 261 | C2×C4 | — | 4 | 2 | no | — |
| 262 | C2×C4 | — | 6 | 4 | no | — |
| 263 | C2×C4 | — | 8 | 2 | no | — |
| 264 | C2×C4 | — | 12 | 2 | no | — |
| 265 | C2×C4 | — | 12 | 2 | no | — |
| 266 | C2×C4 | — | 12 | 2 | no | — |
| 267 | C2×C4 | — | 24 | 2 | no | — |
| 268 | C2×C4 | D3×C22⋊C4 | 1 | 1 | yes | characteristic |
| 269 | C2×C4 | #67⋊C4 | 1 | 1 | yes | characteristic |
| 270 | C2×C4 | #885 | 1 | 1 | yes | characteristic |
| 271 | C2×C4 | #912 | 1 | 2 | yes | — |
| 272 | C2×C4 | D3×SD16 | 1 | 1 | yes | characteristic |
| 273 | C2×C4 | D3×SD16 | 1 | 2 | yes | — |
| 274 | C2×C4 | #915 | 1 | 1 | yes | characteristic |
| 275 | C2×C4 | D3×Q16 | 1 | 1 | yes | characteristic |
| 276 | C2×C4 | #918 | 1 | 1 | yes | characteristic |
| 277 | C2×C4 | C4×#67 | 1 | 1 | yes | characteristic |
| 278 | C2×C4 | #939 | 1 | 1 | yes | characteristic |
| 279 | C2×C4 | #940 | 1 | 1 | yes | characteristic |
| 280 | C8 | — | 3 | 4 | no | — |
| 281 | C8 | — | 4 | 2 | no | — |
| 282 | C8 | — | 6 | 2 | no | — |
| 283 | C8 | — | 8 | 1 | no | — |
| 284 | C8 | — | 24 | 1 | no | — |
| 285 | C2×C3 | #1671 | 1 | 2 | yes | — |
| 286 | C2×C3 | #1672 | 1 | 2 | yes | — |
| 287 | C2×C3 | #1968 | 1 | 1 | yes | characteristic |
| 288 | C2×C3 | #1977 | 1 | 1 | yes | characteristic |
| 289 | C2×C3 | #1981 | 1 | 1 | yes | characteristic |
| 290 | C22 | #4754 | 1 | 2 | yes | — |
| 291 | C22 | #4778 | 1 | 2 | yes | — |
| 292 | C22 | #4795 | 1 | 1 | yes | characteristic |
| 293 | C22 | #4818 | 1 | 1 | yes | characteristic |
| 294 | C22 | #4957 | 1 | 1 | yes | characteristic |
| 295 | C4 | — | 2 | 1 | no | — |
| 296 | C4 | — | 2 | 1 | no | — |
| 297 | C4 | — | 8 | 2 | no | — |
| 298 | C4 | — | 12 | 4 | no | — |
| 299 | C4 | — | 24 | 2 | no | — |
| 300 | C4 | #4758 | 1 | 2 | yes | — |
| 301 | C4 | D3×#103 | 1 | 1 | yes | characteristic |
| 302 | C4 | #4802 | 1 | 1 | yes | characteristic |
| 303 | C4 | #5155 | 1 | 1 | yes | characteristic |
| 304 | C4 | #5162 | 1 | 1 | yes | characteristic |
| 305 | C4 | #5175 | 1 | 1 | yes | characteristic |
| 306 | C4 | #5182 | 1 | 1 | yes | characteristic |
| 307 | C3 | #9729 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 308 | C2 | #68442 | 1 | 2 | yes | — |
| 309 | C2 | #68466 | 1 | 2 | yes | — |
| 310 | C2 | #70107 | 1 | 1 | yes | characteristic |
| 311 | C2 | #70189 | 1 | 1 | yes | characteristic |
| 312 | C2 | #70229 | 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