Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
44 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | Dic96 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | D192 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C3×C128 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | SD256 | — | 3 | 1 | no | maximal, Sylow |
| 5 | Dic48 | — | 2 | 1 | no | — |
| 6 | D96 | — | 2 | 1 | no | — |
| 7 | C3×C64 | C22 | 1 | 1 | yes | characteristic |
| 8 | Q128 | — | 3 | 1 | no | — |
| 9 | D64 | — | 3 | 1 | no | — |
| 10 | C128 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | Dic24 | — | 4 | 1 | no | — |
| 12 | D48 | — | 4 | 1 | no | — |
| 13 | C3×C32 | D4 | 1 | 1 | yes | characteristic |
| 14 | Q64 | — | 6 | 1 | no | — |
| 15 | D32 | — | 6 | 1 | no | — |
| 16 | C64 | C2×D3 | 1 | 1 | yes | characteristic |
| 17 | Dic12 | — | 8 | 1 | no | — |
| 18 | D24 | — | 8 | 1 | no | — |
| 19 | C3×C16 | D8 | 1 | 1 | yes | characteristic |
| 20 | Q32 | — | 12 | 1 | no | — |
| 21 | D16 | — | 12 | 1 | no | — |
| 22 | C32 | D12 | 1 | 1 | yes | characteristic |
| 23 | D12 | — | 16 | 1 | no | — |
| 24 | Dic6 | — | 16 | 1 | no | — |
| 25 | C3×C8 | D16 | 1 | 1 | yes | characteristic |
| 26 | Q16 | — | 24 | 1 | no | — |
| 27 | D8 | — | 24 | 1 | no | — |
| 28 | C16 | D24 | 1 | 1 | yes | characteristic |
| 29 | C2×D3 | — | 32 | 1 | no | — |
| 30 | C3×C4 | D32 | 1 | 1 | yes | characteristic |
| 31 | Dic3 | — | 32 | 1 | no | — |
| 32 | Q8 | — | 48 | 1 | no | — |
| 33 | D4 | — | 48 | 1 | no | — |
| 34 | C8 | D48 | 1 | 1 | yes | characteristic |
| 35 | C2×C3 | D64 | 1 | 1 | yes | characteristic |
| 36 | D3 | — | 64 | 1 | no | — |
| 37 | C22 | — | 96 | 1 | no | — |
| 38 | C4 | — | 96 | 1 | no | — |
| 39 | C4 | D96 | 1 | 1 | yes | characteristic |
| 40 | C3 | SD256 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 41 | C2 | — | 192 | 1 | no | — |
| 42 | C2 | D192 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D192 |
| commutator, cosocle | C3×C64 | C22 |
| Frattini | C64 | C2×D3 |
| Fitting | C3×C128 | C2 |
| socle | C2×C3 | D64 |
| prime radical | D192 | C2 |
| hypercenter | C128 | D3 |
| hypocenter | C3 | SD256 |
| LCS | G ▷ C3×C64 ▷ C3×C32 ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 ◁ C128 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Dic96 | C2 | semidirect |
| 3. C3×C128 | C2 | semidirect |
| 10. C128 | D3 | semidirect |
| 40. C3 | SD256 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Dic96 | 2 |
| D192 | 2 |
| C3×C128 | 2 |
| SD256 | 3 |