Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
5610 conjugacy classes of subgroups; only the 835 normal subgroups are shown.
This group has 5610 conjugacy classes of subgroups — too many to draw. Only its 835 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#5493 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | C22×#882 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C3×#2273 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #73597 | C2 | 1 | 12 | yes | maximal, complemented |
| 5 | #73587 | C2 | 1 | 12 | yes | maximal, complemented |
| 6 | #35390 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C25×D3 | C22 | 1 | 1 | yes | characteristic |
| 8 | C23×#67 | C22 | 1 | 3 | yes | — |
| 9 | C23×D12 | C22 | 1 | 6 | yes | complemented |
| 10 | C22×C3×C22⋊C4 | C22 | 1 | 3 | yes | — |
| 11 | C2×#930 | C22 | 1 | 12 | yes | complemented |
| 12 | C22×(C2×D3)⋊C4 | C22 | 1 | 6 | yes | complemented |
| 13 | #5493 | — | 2 | 24 | no | — |
| 14 | C2×#886 | C22 | 1 | 12 | yes | complemented |
| 15 | C2×#882 | — | 2 | 24 | no | — |
| 16 | C2×#882 | C22 | 1 | 24 | yes | complemented |
| 17 | C2×#882 | C22 | 1 | 12 | yes | complemented |
| 18 | C2×#874 | C22 | 1 | 8 | yes | complemented |
| 19 | C3×C4×C22⋊C4 | C22 | 1 | 12 | yes | complemented |
| 20 | #4939 | C22 | 1 | 24 | yes | complemented |
| 21 | #4677 | C22 | 1 | 8 | yes | complemented |
| 22 | #4671 | C22 | 1 | 24 | yes | complemented |
| 23 | C2×#534 | — | 3 | 3 | no | — |
| 24 | C22×#120 | — | 3 | 3 | no | — |
| 25 | #2539 | — | 3 | 12 | no | — |
| 26 | #2433 | — | 3 | 12 | no | — |
| 27 | #2273 | D3 | 1 | 1 | yes | characteristic, complemented |
| 28 | C25×C3 | C23 | 1 | 1 | yes | characteristic |
| 29 | C24×D3 | — | 2 | 12 | no | — |
| 30 | C24×D3 | — | 4 | 8 | no | — |
| 31 | C24×D3 | C23 | 1 | 6 | yes | — |
| 32 | C24×D3 | C23 | 1 | 3 | yes | — |
| 33 | C23×C3×C4 | C23 | 1 | 6 | yes | — |
| 34 | C22×#67 | — | 2 | 12 | no | — |
| 35 | C22×#67 | — | 2 | 24 | no | — |
| 36 | C22×#67 | C23 | 1 | 12 | yes | — |
| 37 | C23×Dic3 | C23 | 1 | 3 | yes | — |
| 38 | C22×D12 | — | 2 | 24 | no | — |
| 39 | C22×D12 | — | 4 | 24 | no | — |
| 40 | C22×D12 | C23 | 1 | 24 | yes | — |
| 41 | C22×D12 | C23 | 1 | 12 | yes | complemented |
| 42 | C2×C3×C22⋊C4 | D4 | 1 | 24 | yes | complemented |
| 43 | C2×C3×C22⋊C4 | C23 | 1 | 12 | yes | — |
| 44 | C2×C3×C42 | C23 | 1 | 8 | yes | complemented |
| 45 | #930 | — | 2 | 48 | no | — |
| 46 | C2×(C2×D3)⋊C4 | — | 2 | 24 | no | — |
| 47 | C2×(C2×D3)⋊C4 | C23 | 1 | 24 | yes | — |
| 48 | C2×(C2×D3)⋊C4 | C23 | 1 | 24 | yes | complemented |
| 49 | C2×C4⋊Dic3 | C23 | 1 | 12 | yes | complemented |
| 50 | #886 | — | 2 | 48 | no | — |
| 51 | #882 | — | 2 | 96 | no | — |
| 52 | #882 | — | 2 | 48 | no | — |
| 53 | #874 | — | 4 | 16 | no | — |
| 54 | C3×#95 | C23 | 1 | 8 | yes | complemented |
| 55 | C26 | — | 3 | 1 | no | — |
| 56 | C23×D4 | — | 3 | 3 | no | — |
| 57 | C23×D4 | — | 3 | 6 | no | — |
| 58 | #534 | — | 6 | 24 | no | — |
| 59 | C2×#127 | — | 3 | 8 | no | — |
| 60 | C2×#123 | — | 3 | 12 | no | — |
| 61 | C2×#121 | — | 3 | 12 | no | — |
| 62 | C2×#120 | — | 3 | 24 | no | — |
| 63 | C2×#120 | — | 3 | 12 | no | — |
| 64 | C2×#120 | — | 6 | 24 | no | — |
| 65 | C22×C22⋊C4 | — | 3 | 6 | no | — |
| 66 | C22×C22⋊C4 | C2×D3 | 1 | 3 | yes | — |
| 67 | #399 | — | 3 | 8 | no | — |
| 68 | #392 | — | 3 | 24 | no | — |
| 69 | #390 | — | 3 | 24 | no | — |
| 70 | C4×C22⋊C4 | C2×D3 | 1 | 12 | yes | complemented |
| 71 | C24×C3 | — | 4 | 4 | no | — |
| 72 | C24×C3 | C2×D4 | 1 | 12 | yes | — |
| 73 | C24×C3 | C24 | 1 | 3 | yes | — |
| 74 | C23×D3 | — | 2 | 12 | no | — |
| 75 | C23×D3 | — | 2 | 24 | no | — |
| 76 | C23×D3 | — | 2 | 24 | no | — |
| 77 | C23×D3 | — | 2 | 6 | no | — |
| 78 | C23×D3 | — | 2 | 12 | no | — |
| 79 | C23×D3 | — | 4 | 12 | no | — |
| 80 | C23×D3 | — | 4 | 12 | no | — |
| 81 | C23×D3 | — | 4 | 96 | no | — |
| 82 | C23×D3 | — | 4 | 16 | no | — |
| 83 | C23×D3 | C24 | 1 | 6 | yes | — |
| 84 | C23×D3 | C24 | 1 | 4 | yes | — |
| 85 | C22×C3×C4 | — | 4 | 12 | no | — |
| 86 | C22×C3×C4 | C2×D4 | 1 | 24 | yes | complemented |
| 87 | C22×C3×C4 | C24 | 1 | 12 | yes | — |
| 88 | C2×#67 | — | 2 | 24 | no | — |
| 89 | C2×#67 | — | 2 | 24 | no | — |
| 90 | C2×#67 | — | 2 | 24 | no | — |
| 91 | C2×#67 | — | 4 | 48 | no | — |
| 92 | C22×Dic3 | — | 2 | 12 | no | — |
| 93 | C22×Dic3 | — | 2 | 6 | no | — |
| 94 | C22×Dic3 | C24 | 1 | 6 | yes | — |
| 95 | C2×D12 | — | 2 | 48 | no | — |
| 96 | C2×D12 | — | 2 | 48 | no | — |
| 97 | C2×D12 | — | 2 | 24 | no | — |
| 98 | C2×D12 | — | 4 | 24 | no | — |
| 99 | C2×D12 | — | 4 | 96 | no | — |
| 100 | C2×D12 | — | 4 | 24 | no | — |
| 101 | C3×C22⋊C4 | C2×D4 | 1 | 48 | yes | complemented |
| 102 | C3×C42 | — | 4 | 8 | no | — |
| 103 | (C2×D3)⋊C4 | — | 2 | 48 | no | — |
| 104 | (C2×D3)⋊C4 | — | 2 | 48 | no | — |
| 105 | C4⋊Dic3 | — | 2 | 24 | no | — |
| 106 | C25 | — | 3 | 6 | no | — |
| 107 | C25 | — | 3 | 3 | no | — |
| 108 | C25 | — | 6 | 12 | no | — |
| 109 | C25 | — | 12 | 8 | no | — |
| 110 | C25 | C22×D3 | 1 | 1 | yes | characteristic |
| 111 | C22×D4 | — | 3 | 12 | no | — |
| 112 | C22×D4 | — | 3 | 24 | no | — |
| 113 | C22×D4 | — | 3 | 12 | no | — |
| 114 | C22×D4 | — | 6 | 12 | no | — |
| 115 | C22×D4 | — | 6 | 24 | no | — |
| 116 | C22×D4 | — | 6 | 24 | no | — |
| 117 | C22×D4 | — | 12 | 24 | no | — |
| 118 | C23×C4 | — | 3 | 3 | no | — |
| 119 | C23×C4 | C22×D3 | 1 | 6 | yes | — |
| 120 | #127 | — | 12 | 16 | no | — |
| 121 | #123 | — | 6 | 48 | no | — |
| 122 | #121 | — | 6 | 48 | no | — |
| 123 | #120 | — | 6 | 96 | no | — |
| 124 | #120 | — | 6 | 48 | no | — |
| 125 | C2×C4⋊C4 | — | 3 | 12 | no | — |
| 126 | C2×C22⋊C4 | — | 3 | 24 | no | — |
| 127 | C2×C22⋊C4 | — | 3 | 24 | no | — |
| 128 | C2×C22⋊C4 | — | 6 | 24 | no | — |
| 129 | C2×C22⋊C4 | D12 | 1 | 24 | yes | complemented |
| 130 | C2×C22⋊C4 | C22×D3 | 1 | 12 | yes | — |
| 131 | C2×C42 | C22×D3 | 1 | 8 | yes | complemented |
| 132 | #95 | C22×D3 | 1 | 8 | yes | complemented |
| 133 | C23×C3 | — | 2 | 12 | no | — |
| 134 | C23×C3 | — | 4 | 24 | no | — |
| 135 | C23×C3 | #127 | 1 | 16 | yes | complemented |
| 136 | C23×C3 | C22×D4 | 1 | 12 | yes | — |
| 137 | C23×C3 | C22×D4 | 1 | 6 | yes | — |
| 138 | C23×C3 | C25 | 1 | 1 | yes | characteristic |
| 139 | C22×D3 | — | 2 | 48 | no | — |
| 140 | C22×D3 | — | 2 | 12 | no | — |
| 141 | C22×D3 | — | 2 | 12 | no | — |
| 142 | C22×D3 | — | 2 | 12 | no | — |
| 143 | C22×D3 | — | 2 | 6 | no | — |
| 144 | C22×D3 | — | 2 | 24 | no | — |
| 145 | C22×D3 | — | 4 | 96 | no | — |
| 146 | C22×D3 | — | 4 | 24 | no | — |
| 147 | C22×D3 | — | 4 | 128 | no | — |
| 148 | C22×D3 | — | 4 | 24 | no | — |
| 149 | C22×D3 | — | 4 | 24 | no | — |
| 150 | C22×D3 | — | 4 | 12 | no | — |
| 151 | C22×D3 | — | 8 | 24 | no | — |
| 152 | C22×D3 | H(5,𝔽2) | 1 | 8 | yes | — |
| 153 | C2×C3×C4 | — | 2 | 24 | no | — |
| 154 | C2×C3×C4 | — | 4 | 24 | no | — |
| 155 | C2×C3×C4 | C22×D4 | 1 | 24 | yes | — |
| 156 | #67 | — | 4 | 48 | no | — |
| 157 | C2×Dic3 | — | 2 | 12 | no | — |
| 158 | C2×Dic3 | — | 2 | 6 | no | — |
| 159 | C2×Dic3 | — | 4 | 12 | no | — |
| 160 | D12 | — | 4 | 96 | no | — |
| 161 | D12 | — | 8 | 24 | no | — |
| 162 | C24 | — | 3 | 4 | no | — |
| 163 | C24 | — | 3 | 6 | no | — |
| 164 | C24 | — | 4 | 4 | no | — |
| 165 | C24 | — | 6 | 24 | no | — |
| 166 | C24 | — | 6 | 12 | no | — |
| 167 | C24 | — | 6 | 24 | no | — |
| 168 | C24 | — | 6 | 6 | no | — |
| 169 | C24 | — | 6 | 12 | no | — |
| 170 | C24 | — | 12 | 12 | no | — |
| 171 | C24 | — | 12 | 12 | no | — |
| 172 | C24 | — | 12 | 96 | no | — |
| 173 | C24 | — | 12 | 16 | no | — |
| 174 | C24 | C2×D12 | 1 | 12 | yes | — |
| 175 | C24 | C23×D3 | 1 | 3 | yes | — |
| 176 | C2×D4 | — | 6 | 48 | no | — |
| 177 | C2×D4 | — | 6 | 24 | no | — |
| 178 | C2×D4 | — | 6 | 24 | no | — |
| 179 | C2×D4 | — | 6 | 24 | no | — |
| 180 | C2×D4 | — | 6 | 48 | no | — |
| 181 | C2×D4 | — | 6 | 24 | no | — |
| 182 | C2×D4 | — | 12 | 24 | no | — |
| 183 | C2×D4 | — | 12 | 24 | no | — |
| 184 | C2×D4 | — | 12 | 96 | no | — |
| 185 | C2×D4 | — | 12 | 48 | no | — |
| 186 | C22×C4 | — | 3 | 6 | no | — |
| 187 | C22×C4 | — | 4 | 12 | no | — |
| 188 | C22×C4 | — | 6 | 6 | no | — |
| 189 | C22×C4 | — | 6 | 12 | no | — |
| 190 | C22×C4 | C2×D12 | 1 | 24 | yes | complemented |
| 191 | C22×C4 | C23×D3 | 1 | 12 | yes | — |
| 192 | C4⋊C4 | — | 6 | 24 | no | — |
| 193 | C22⋊C4 | — | 6 | 48 | no | — |
| 194 | C22⋊C4 | — | 6 | 48 | no | — |
| 195 | C22⋊C4 | C2×D12 | 1 | 48 | yes | complemented |
| 196 | C42 | — | 4 | 8 | no | — |
| 197 | C22×C3 | — | 2 | 12 | no | — |
| 198 | C22×C3 | — | 2 | 24 | no | — |
| 199 | C22×C3 | — | 4 | 16 | no | — |
| 200 | C22×C3 | C2×#127 | 1 | 12 | yes | — |
| 201 | C22×C3 | C23×D4 | 1 | 3 | yes | — |
| 202 | C22×C3 | C2×H(5,𝔽2) | 1 | 4 | yes | — |
| 203 | C2×D3 | — | 2 | 24 | no | — |
| 204 | C2×D3 | — | 4 | 96 | no | — |
| 205 | C2×D3 | — | 4 | 4 | no | — |
| 206 | C2×D3 | — | 4 | 12 | no | — |
| 207 | C2×D3 | — | 4 | 6 | no | — |
| 208 | C2×D3 | — | 4 | 12 | no | — |
| 209 | C2×D3 | — | 4 | 24 | no | — |
| 210 | C2×D3 | — | 8 | 24 | no | — |
| 211 | C3×C4 | — | 2 | 24 | no | — |
| 212 | Dic3 | — | 4 | 6 | no | — |
| 213 | C23 | — | 2 | 12 | no | — |
| 214 | C23 | — | 3 | 8 | no | — |
| 215 | C23 | — | 4 | 24 | no | — |
| 216 | C23 | — | 6 | 48 | no | — |
| 217 | C23 | — | 6 | 12 | no | — |
| 218 | C23 | — | 6 | 12 | no | — |
| 219 | C23 | — | 6 | 12 | no | — |
| 220 | C23 | — | 6 | 6 | no | — |
| 221 | C23 | — | 6 | 24 | no | — |
| 222 | C23 | — | 12 | 96 | no | — |
| 223 | C23 | — | 12 | 24 | no | — |
| 224 | C23 | — | 12 | 24 | no | — |
| 225 | C23 | — | 12 | 128 | no | — |
| 226 | C23 | — | 12 | 24 | no | — |
| 227 | C23 | — | 12 | 12 | no | — |
| 228 | C23 | — | 24 | 24 | no | — |
| 229 | C23 | #874 | 1 | 16 | yes | complemented |
| 230 | C23 | C22×D12 | 1 | 12 | yes | — |
| 231 | C23 | C22×D12 | 1 | 6 | yes | — |
| 232 | C23 | C24×D3 | 1 | 1 | yes | characteristic |
| 233 | D4 | — | 12 | 96 | no | — |
| 234 | D4 | — | 12 | 48 | no | — |
| 235 | D4 | — | 24 | 24 | no | — |
| 236 | C2×C4 | — | 2 | 24 | no | — |
| 237 | C2×C4 | — | 4 | 24 | no | — |
| 238 | C2×C4 | — | 6 | 6 | no | — |
| 239 | C2×C4 | — | 6 | 12 | no | — |
| 240 | C2×C4 | — | 12 | 12 | no | — |
| 241 | C2×C4 | C22×D12 | 1 | 24 | yes | — |
| 242 | C2×C3 | — | 2 | 12 | no | — |
| 243 | C2×C3 | C22×#127 | 1 | 1 | yes | characteristic |
| 244 | C2×C3 | #3484 | 1 | 6 | yes | — |
| 245 | D3 | — | 4 | 8 | no | — |
| 246 | D3 | — | 8 | 6 | no | — |
| 247 | C22 | — | 2 | 12 | no | — |
| 248 | C22 | — | 2 | 24 | no | — |
| 249 | C22 | — | 4 | 16 | no | — |
| 250 | C22 | — | 6 | 24 | no | — |
| 251 | C22 | — | 12 | 96 | no | — |
| 252 | C22 | — | 12 | 12 | no | — |
| 253 | C22 | — | 12 | 4 | no | — |
| 254 | C22 | — | 12 | 12 | no | — |
| 255 | C22 | — | 12 | 24 | no | — |
| 256 | C22 | — | 12 | 6 | no | — |
| 257 | C22 | — | 24 | 24 | no | — |
| 258 | C22 | C2×#874 | 1 | 12 | yes | — |
| 259 | C22 | C23×D12 | 1 | 3 | yes | — |
| 260 | C22 | D3×H(5,𝔽2) | 1 | 4 | yes | — |
| 261 | C4 | — | 2 | 24 | no | — |
| 262 | C4 | — | 12 | 6 | no | — |
| 263 | C3 | #35390 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 264 | C2 | — | 2 | 12 | no | — |
| 265 | C2 | — | 12 | 8 | no | — |
| 266 | C2 | — | 24 | 6 | no | — |
| 267 | C2 | C22×#874 | 1 | 1 | yes | characteristic |
| 268 | C2 | #85867 | 1 | 6 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C24×D3 |
| commutator, socle, cosocle | C23×C3 | C25 |
| Fitting | C3×#2273 | C2 |
| hypercenter | #2273 | D3 |
| hypocenter | C3 | #35390 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #2273 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#5493 | C2 | semidirect |
| 2. C22×#882 | C2 | semidirect |
| 3. C3×#2273 | C2 | semidirect |
| 4. #73597 | C2 | semidirect |
| 5. #73587 | C2 | semidirect |
| 9. C23×D12 | C22 | semidirect |
| 11. C2×#930 | C22 | semidirect |
| 12. C22×(C2×D3)⋊C4 | C22 | semidirect |
| 14. C2×#886 | C22 | semidirect |
| 16. C2×#882 | C22 | semidirect |
| 17. C2×#882 | C22 | semidirect |
| 18. C2×#874 | C22 | semidirect |
| 19. C3×C4×C22⋊C4 | C22 | semidirect |
| 20. #4939 | C22 | semidirect |
| 21. #4677 | C22 | semidirect |
| 22. #4671 | C22 | semidirect |
| 27. #2273 | D3 | semidirect |
| 41. C22×D12 | C23 | semidirect |
| 42. C2×C3×C22⋊C4 | D4 | semidirect |
| 44. C2×C3×C42 | C23 | semidirect |
| 48. C2×(C2×D3)⋊C4 | C23 | semidirect |
| 49. C2×C4⋊Dic3 | C23 | semidirect |
| 54. C3×#95 | C23 | semidirect |
| 70. C4×C22⋊C4 | C2×D3 | semidirect |
| 86. C22×C3×C4 | C2×D4 | semidirect |
| 101. C3×C22⋊C4 | C2×D4 | semidirect |
| 129. C2×C22⋊C4 | D12 | semidirect |
| 131. C2×C42 | C22×D3 | semidirect |
| 132. #95 | C22×D3 | semidirect |
| 135. C23×C3 | #127 | semidirect |
| 190. C22×C4 | C2×D12 | semidirect |
| 195. C22⋊C4 | C2×D12 | semidirect |
| 229. C23 | #874 | semidirect |
| 263. C3 | #35390 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#5493 | 2 |
| C22×#882 | 2 |
| C3×#2273 | 2 |
| #73597 | 2 |
| #73587 | 2 |
| #35390 | 3 |