← #10799577
Subgroups — #10799577
Order 768.
770 conjugacy classes of subgroups.
The subgroup lattice has 286 nodes — too large to display as a diagram.
Conjugacy classes of subgroups
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|
| 1 | C4×#904 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #78455 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C3×C42×C8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #69795 | C2 | 1 | 2 | yes | maximal |
| 5 | #69794 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #69793 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | C4×#819 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C4×#805 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #21292 | — | 3 | 1 | no | maximal, Sylow |
| 10 | C2×#904 | C22 | 1 | 1 | yes | characteristic |
| 11 | C4×#235 | C22 | 1 | 2 | yes | — |
| 12 | C2×C3×C4×C8 | C22 | 1 | 2 | yes | — |
| 13 | C2×C3×C4×C8 | C22 | 1 | 2 | yes | — |
| 14 | C2×C3×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×C3×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 16 | C3×C43 | C22 | 1 | 1 | yes | characteristic |
| 17 | #5113 | C22 | 1 | 2 | yes | — |
| 18 | #5110 | C22 | 1 | 4 | yes | — |
| 19 | C2×#819 | C22 | 1 | 2 | yes | — |
| 20 | #5106 | C22 | 1 | 1 | yes | characteristic |
| 21 | #5102 | C22 | 1 | 4 | yes | — |
| 22 | C2×#805 | C22 | 1 | 4 | yes | — |
| 23 | C4×#208 | C22 | 1 | 2 | yes | — |
| 24 | C4×Dic12 | C4 | 1 | 2 | yes | complemented |
| 25 | C4×D24 | C4 | 1 | 2 | yes | complemented |
| 26 | C4×#204 | C4 | 1 | 4 | yes | complemented |
| 27 | #4690 | — | 2 | 4 | no | — |
| 28 | #4685 | — | 2 | 4 | no | — |
| 29 | #4552 | C22 | 1 | 2 | yes | — |
| 30 | #4523 | C22 | 1 | 4 | yes | — |
| 31 | C4×#135 | — | 3 | 1 | no | — |
| 32 | #2922 | — | 3 | 2 | no | — |
| 33 | #1974 | — | 3 | 2 | no | — |
| 34 | #1945 | — | 3 | 2 | no | — |
| 35 | #1937 | — | 3 | 4 | no | — |
| 36 | C4×#108 | — | 3 | 1 | no | — |
| 37 | C4×#104 | — | 3 | 2 | no | — |
| 38 | C42×C8 | D3 | 1 | 1 | yes | characteristic, complemented |
| 39 | C2×#235 | C23 | 1 | 2 | yes | — |
| 40 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 41 | C22×C3×C8 | C23 | 1 | 2 | yes | — |
| 42 | C22×C3×C8 | C23 | 1 | 1 | yes | characteristic |
| 43 | C2×C3×C42 | — | 2 | 2 | no | — |
| 44 | C2×C3×C42 | C23 | 1 | 2 | yes | — |
| 45 | C2×C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 46 | C4×#67 | — | 2 | 4 | no | — |
| 47 | #926 | C23 | 1 | 2 | yes | — |
| 48 | C2×#208 | C23 | 1 | 4 | yes | — |
| 49 | C2×Dic12 | C2×C4 | 1 | 1 | yes | characteristic |
| 50 | #904 | — | 2 | 4 | no | — |
| 51 | C2×D24 | C2×C4 | 1 | 1 | yes | characteristic |
| 52 | C2×#204 | C2×C4 | 1 | 2 | yes | — |
| 53 | C4×D12 | C2×C4 | 1 | 4 | yes | — |
| 54 | #872 | — | 2 | 2 | no | — |
| 55 | C42×D3 | — | 2 | 2 | no | — |
| 56 | C4×Dic6 | C2×C4 | 1 | 4 | yes | — |
| 57 | C3×C4×C8 | — | 2 | 4 | no | — |
| 58 | C3×C4×C8 | — | 2 | 4 | no | — |
| 59 | C3×C4×C8 | C2×C4 | 1 | 2 | yes | — |
| 60 | C3×C4×C8 | C2×C4 | 1 | 2 | yes | — |
| 61 | C3×C4×C8 | D4 | 1 | 2 | yes | — |
| 62 | C3×C4×C8 | D4 | 1 | 2 | yes | — |
| 63 | #821 | C2×C4 | 1 | 4 | yes | — |
| 64 | (C2×D3)⋊C8 | — | 2 | 8 | no | — |
| 65 | #819 | — | 2 | 4 | no | — |
| 66 | #818 | C2×C4 | 1 | 2 | yes | — |
| 67 | #817 | C2×C4 | 1 | 2 | yes | — |
| 68 | #816 | C2×C4 | 1 | 4 | yes | — |
| 69 | Dic3⋊C8 | — | 2 | 8 | no | — |
| 70 | #805 | — | 2 | 8 | no | — |
| 71 | #803 | — | 2 | 2 | no | — |
| 72 | C4×C3⋊C8 | — | 2 | 2 | no | — |
| 73 | C2×#135 | — | 3 | 1 | no | — |
| 74 | C4×#41 | — | 3 | 2 | no | — |
| 75 | C4×Q16 | — | 3 | 2 | no | — |
| 76 | C4×SD16 | — | 3 | 4 | no | — |
| 77 | C4×D8 | — | 3 | 2 | no | — |
| 78 | #433 | — | 6 | 4 | no | — |
| 79 | #431 | — | 6 | 4 | no | — |
| 80 | C2×#108 | — | 3 | 2 | no | — |
| 81 | #427 | — | 3 | 1 | no | — |
| 82 | #424 | — | 3 | 4 | no | — |
| 83 | C2×#104 | — | 3 | 4 | no | — |
| 84 | #416 | — | 3 | 2 | no | — |
| 85 | #408 | — | 3 | 4 | no | — |
| 86 | C4×M16 | — | 3 | 2 | no | — |
| 87 | C2×C4×C8 | C2×D3 | 1 | 2 | yes | — |
| 88 | C2×C4×C8 | C2×D3 | 1 | 2 | yes | — |
| 89 | C2×C4×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 90 | C2×C4×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 91 | C43 | C2×D3 | 1 | 1 | yes | characteristic |
| 92 | #341 | — | 3 | 2 | no | — |
| 93 | #339 | — | 3 | 4 | no | — |
| 94 | C22×C3×C4 | — | 2 | 2 | no | — |
| 95 | C22×C3×C4 | #41 | 1 | 2 | yes | — |
| 96 | C22×C3×C4 | C24 | 1 | 1 | yes | characteristic |
| 97 | C2×#67 | — | 2 | 2 | no | — |
| 98 | #235 | — | 2 | 4 | no | — |
| 99 | #235 | — | 2 | 4 | no | — |
| 100 | C2×D12 | C22×C4 | 1 | 2 | yes | — |
| 101 | C2×C4×D3 | — | 2 | 4 | no | — |
| 102 | C2×C4×D3 | — | 2 | 2 | no | — |
| 103 | C2×Dic6 | C22×C4 | 1 | 2 | yes | — |
| 104 | C2×C3×C8 | — | 2 | 2 | no | — |
| 105 | C2×C3×C8 | — | 2 | 2 | no | — |
| 106 | C2×C3×C8 | — | 2 | 2 | no | — |
| 107 | C2×C3×C8 | C22×C4 | 1 | 2 | yes | — |
| 108 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 109 | C2×C3×C8 | C22×C4 | 1 | 1 | yes | characteristic |
| 110 | C2×C3×C8 | C2×D4 | 1 | 4 | yes | — |
| 111 | C2×C3×C8 | #41 | 1 | 4 | yes | — |
| 112 | C3×C42 | — | 2 | 8 | no | — |
| 113 | C3×C42 | — | 2 | 2 | no | — |
| 114 | C3×C42 | C22×C4 | 1 | 2 | yes | — |
| 115 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 116 | C3×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 117 | C3×C42 | #41 | 1 | 4 | yes | — |
| 118 | #217 | — | 2 | 2 | no | — |
| 119 | (C2×D3)⋊C4 | — | 2 | 4 | no | — |
| 120 | C4⋊Dic3 | C22×C4 | 1 | 4 | yes | — |
| 121 | Dic3⋊C4 | — | 2 | 4 | no | — |
| 122 | C4×Dic3 | — | 2 | 4 | no | — |
| 123 | C4×Dic3 | — | 2 | 2 | no | — |
| 124 | #208 | — | 2 | 4 | no | — |
| 125 | #208 | — | 2 | 4 | no | — |
| 126 | C2×C3⋊C8 | — | 2 | 4 | no | — |
| 127 | Dic12 | — | 2 | 2 | no | — |
| 128 | D24 | — | 2 | 2 | no | — |
| 129 | #204 | — | 2 | 4 | no | — |
| 130 | C2×#41 | — | 3 | 2 | no | — |
| 131 | #135 | — | 6 | 4 | no | — |
| 132 | C2×Q16 | — | 3 | 1 | no | — |
| 133 | C2×SD16 | — | 3 | 2 | no | — |
| 134 | C2×D8 | — | 3 | 1 | no | — |
| 135 | C2×M16 | — | 3 | 4 | no | — |
| 136 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 137 | C22×C8 | C22×D3 | 1 | 2 | yes | — |
| 138 | C22×C8 | C22×D3 | 1 | 1 | yes | characteristic |
| 139 | C4×Q8 | — | 3 | 4 | no | — |
| 140 | C4×D4 | — | 3 | 4 | no | — |
| 141 | C4×D4 | — | 6 | 4 | no | — |
| 142 | #117 | — | 3 | 2 | no | — |
| 143 | #117 | — | 6 | 2 | no | — |
| 144 | C2×C42 | — | 2 | 2 | no | — |
| 145 | C2×C42 | — | 6 | 2 | no | — |
| 146 | C2×C42 | C22×D3 | 1 | 2 | yes | — |
| 147 | C2×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 148 | #108 | — | 6 | 4 | no | — |
| 149 | #107 | — | 3 | 2 | no | — |
| 150 | #106 | — | 3 | 2 | no | — |
| 151 | C4⋊C8 | — | 6 | 8 | no | — |
| 152 | #104 | — | 6 | 8 | no | — |
| 153 | #103 | — | 3 | 4 | no | — |
| 154 | #102 | — | 3 | 4 | no | — |
| 155 | C22⋊C8 | — | 6 | 8 | no | — |
| 156 | #97 | — | 6 | 2 | no | — |
| 157 | C4×C8 | — | 2 | 4 | no | — |
| 158 | C4×C8 | — | 2 | 4 | no | — |
| 159 | C4×C8 | — | 6 | 2 | no | — |
| 160 | C4×C8 | C4×D3 | 1 | 2 | yes | — |
| 161 | C4×C8 | C4×D3 | 1 | 2 | yes | — |
| 162 | C4×C8 | D12 | 1 | 2 | yes | — |
| 163 | C4×C8 | D12 | 1 | 2 | yes | — |
| 164 | C23×C3 | C2×#41 | 1 | 1 | yes | characteristic |
| 165 | C22×D3 | — | 2 | 2 | no | — |
| 166 | C2×C3×C4 | — | 2 | 8 | no | — |
| 167 | C2×C3×C4 | — | 2 | 4 | no | — |
| 168 | C2×C3×C4 | — | 2 | 4 | no | — |
| 169 | C2×C3×C4 | C4×D4 | 1 | 2 | yes | — |
| 170 | C2×C3×C4 | C4×D4 | 1 | 2 | yes | — |
| 171 | C2×C3×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 172 | C2×C3×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 173 | C2×C3×C4 | C2×#41 | 1 | 2 | yes | — |
| 174 | C2×C3×C4 | C2×#41 | 1 | 2 | yes | — |
| 175 | #67 | — | 4 | 4 | no | — |
| 176 | C2×Dic3 | — | 2 | 2 | no | — |
| 177 | C2×Dic3 | — | 2 | 4 | no | — |
| 178 | D12 | — | 2 | 2 | no | — |
| 179 | D12 | — | 2 | 2 | no | — |
| 180 | C4×D3 | — | 4 | 4 | no | — |
| 181 | C4×D3 | — | 4 | 8 | no | — |
| 182 | Dic6 | — | 2 | 2 | no | — |
| 183 | Dic6 | — | 2 | 2 | no | — |
| 184 | C3×C8 | — | 2 | 2 | no | — |
| 185 | C3×C8 | — | 2 | 4 | no | — |
| 186 | C3×C8 | #117 | 1 | 2 | yes | — |
| 187 | C3×C8 | #117 | 1 | 2 | yes | — |
| 188 | C3⋊C8 | — | 4 | 4 | no | — |
| 189 | #41 | — | 6 | 4 | no | — |
| 190 | #41 | — | 6 | 4 | no | — |
| 191 | C2×Q8 | — | 3 | 2 | no | — |
| 192 | C2×D4 | — | 3 | 2 | no | — |
| 193 | C2×D4 | — | 6 | 2 | no | — |
| 194 | C22×C4 | — | 2 | 2 | no | — |
| 195 | C22×C4 | — | 6 | 4 | no | — |
| 196 | C22×C4 | — | 6 | 2 | no | — |
| 197 | C22×C4 | #235 | 1 | 2 | yes | — |
| 198 | C22×C4 | C23×D3 | 1 | 1 | yes | characteristic |
| 199 | Q16 | — | 6 | 2 | no | — |
| 200 | SD16 | — | 6 | 4 | no | — |
| 201 | D8 | — | 6 | 2 | no | — |
| 202 | M16 | — | 6 | 4 | no | — |
| 203 | M16 | — | 6 | 4 | no | — |
| 204 | C2×C8 | — | 2 | 2 | no | — |
| 205 | C2×C8 | — | 2 | 2 | no | — |
| 206 | C2×C8 | — | 2 | 2 | no | — |
| 207 | C2×C8 | — | 6 | 4 | no | — |
| 208 | C2×C8 | C2×C4×D3 | 1 | 2 | yes | — |
| 209 | C2×C8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 210 | C2×C8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 211 | C2×C8 | C2×D12 | 1 | 4 | yes | — |
| 212 | C2×C8 | #235 | 1 | 4 | yes | — |
| 213 | C4⋊C4 | — | 3 | 4 | no | — |
| 214 | C4⋊C4 | — | 6 | 4 | no | — |
| 215 | C22⋊C4 | — | 6 | 2 | no | — |
| 216 | C22⋊C4 | — | 6 | 4 | no | — |
| 217 | C42 | — | 2 | 8 | no | — |
| 218 | C42 | — | 2 | 2 | no | — |
| 219 | C42 | — | 6 | 4 | no | — |
| 220 | C42 | — | 6 | 2 | no | — |
| 221 | C42 | C2×C4×D3 | 1 | 2 | yes | — |
| 222 | C42 | C2×D12 | 1 | 1 | yes | characteristic |
| 223 | C42 | C2×D12 | 1 | 1 | yes | characteristic |
| 224 | C42 | #235 | 1 | 4 | yes | — |
| 225 | C22×C3 | — | 2 | 2 | no | — |
| 226 | C22×C3 | C2×C4×D4 | 1 | 1 | yes | characteristic |
| 227 | C22×C3 | #529 | 1 | 2 | yes | — |
| 228 | C2×D3 | — | 4 | 4 | no | — |
| 229 | C2×D3 | — | 4 | 2 | no | — |
| 230 | C3×C4 | — | 2 | 8 | no | — |
| 231 | C3×C4 | — | 2 | 2 | no | — |
| 232 | C3×C4 | #443 | 1 | 4 | yes | — |
| 233 | C3×C4 | C2×#117 | 1 | 1 | yes | characteristic |
| 234 | C3×C4 | C4×#41 | 1 | 1 | yes | characteristic |
| 235 | C3×C4 | #532 | 1 | 2 | yes | — |
| 236 | Dic3 | — | 4 | 4 | no | — |
| 237 | Dic3 | — | 4 | 2 | no | — |
| 238 | C23 | — | 6 | 2 | no | — |
| 239 | C23 | C2×#235 | 1 | 1 | yes | characteristic |
| 240 | Q8 | — | 6 | 2 | no | — |
| 241 | Q8 | — | 6 | 2 | no | — |
| 242 | D4 | — | 6 | 2 | no | — |
| 243 | D4 | — | 6 | 2 | no | — |
| 244 | D4 | — | 12 | 4 | no | — |
| 245 | C2×C4 | — | 2 | 8 | no | — |
| 246 | C2×C4 | — | 2 | 4 | no | — |
| 247 | C2×C4 | — | 2 | 4 | no | — |
| 248 | C2×C4 | — | 6 | 4 | no | — |
| 249 | C2×C4 | — | 6 | 2 | no | — |
| 250 | C2×C4 | — | 12 | 8 | no | — |
| 251 | C2×C4 | — | 12 | 4 | no | — |
| 252 | C2×C4 | C4×D12 | 1 | 2 | yes | — |
| 253 | C2×C4 | C4×D12 | 1 | 2 | yes | — |
| 254 | C2×C4 | C22×C4×D3 | 1 | 1 | yes | characteristic |
| 255 | C2×C4 | C22×D12 | 1 | 1 | yes | characteristic |
| 256 | C2×C4 | C2×#235 | 1 | 2 | yes | — |
| 257 | C2×C4 | C2×#235 | 1 | 2 | yes | — |
| 258 | C8 | — | 2 | 2 | no | — |
| 259 | C8 | — | 2 | 4 | no | — |
| 260 | C8 | — | 12 | 4 | no | — |
| 261 | C8 | #872 | 1 | 2 | yes | — |
| 262 | C8 | #872 | 1 | 2 | yes | — |
| 263 | C2×C3 | — | 2 | 2 | no | — |
| 264 | C2×C3 | #2293 | 1 | 1 | yes | characteristic |
| 265 | C2×C3 | C2×#443 | 1 | 2 | yes | — |
| 266 | D3 | — | 8 | 2 | no | — |
| 267 | C22 | — | 2 | 2 | no | — |
| 268 | C22 | — | 12 | 2 | no | — |
| 269 | C22 | — | 12 | 4 | no | — |
| 270 | C22 | C2×C4×D12 | 1 | 1 | yes | characteristic |
| 271 | C22 | #5479 | 1 | 2 | yes | — |
| 272 | C4 | — | 2 | 8 | no | — |
| 273 | C4 | — | 2 | 2 | no | — |
| 274 | C4 | — | 12 | 2 | no | — |
| 275 | C4 | — | 12 | 4 | no | — |
| 276 | C4 | #4700 | 1 | 4 | yes | — |
| 277 | C4 | C2×#872 | 1 | 1 | yes | characteristic |
| 278 | C4 | C4×#235 | 1 | 1 | yes | characteristic |
| 279 | C4 | #5477 | 1 | 2 | yes | — |
| 280 | C3 | #21292 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 281 | C2 | — | 2 | 2 | no | — |
| 282 | C2 | — | 24 | 2 | no | — |
| 283 | C2 | #73299 | 1 | 1 | yes | characteristic |
| 284 | C2 | C2×#4700 | 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