Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
4886 conjugacy classes of subgroups; only the 907 normal subgroups are shown.
This group has 4886 conjugacy classes of subgroups — too many to draw. Only its 907 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C24 | C23×D3 |
| commutator, cosocle | C23×C3 | C25 |
| Frattini | C23 | C24×D3 |
| Fitting | C22×C3×#95 | C2 |
| socle | C24×C3 | C24 |
| hypercenter | C22×#95 | D3 |
| hypocenter | C3 | C2×#2656 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C22×#95 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#930 | C2 | semidirect |
| 2. C22×C3×#95 | C2 | semidirect |
| 3. #73381 | C2 | direct (N × H) |
| 4. C2×#4677 | C2 | semidirect |
| 5. C2×#4671 | C2 | semidirect |
| 11. C2×#930 | C22 | semidirect |
| 15. C2×C3×#95 | C22 | semidirect |
| 16. #4677 | C22 | semidirect |
| 17. #4671 | C22 | semidirect |
| 22. C22×#95 | D3 | semidirect |
| 26. C23×C3×C4 | D4 | semidirect |
| 51. C2×#95 | C2×D3 | semidirect |
| 65. C22×C3×C4 | C2×D4 | semidirect |
| 85. C23×C4 | D12 | semidirect |
| 138. C22×C4 | C2×D12 | semidirect |
| 183. C2×C3 | #2656 | semidirect |
| 204. C3 | C2×#2656 | semidirect |
| 207. C2 | #73381 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#930 | 2 |
| C22×C3×#95 | 2 |
| #73381 | 2 |
| C2×#4677 | 2 |
| C2×#4671 | 2 |
| C2×#2656 | 3 |