Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
706 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 706 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D32 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C22×C32×C4 | C22 |
| hypercenter | C22×C4 | D32 |
| hypocenter | C32 | #535 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#235 | C2 | semidirect |
| 2. C22×C3×D12 | C2 | semidirect |
| 3. #64135 | C2 | semidirect |
| 4. #63976 | C2 | semidirect |
| 5. #63914 | C2 | semidirect |
| 6. #63908 | C2 | semidirect |
| 7. #63863 | C2 | semidirect |
| 17. C2×C3×D12 | C22 | semidirect |
| 18. C2×C3×C4×D3 | C22 | semidirect |
| 19. C2×C3×Dic6 | C22 | semidirect |
| 20. C2×#625 | C22 | semidirect |
| 22. C4⋊(C3⋊Dic3) | C22 | semidirect |
| 23. C4×C3⋊Dic3 | C22 | semidirect |
| 24. (C2×D3)⋊Dic3 | C22 | semidirect |
| 27. C2×#235 | D3 | semidirect |
| 28. C22×D12 | D3 | semidirect |
| 45. C2×C32×C4 | D4 | semidirect |
| 78. C2×D12 | C2×D3 | semidirect |
| 79. C2×C4×D3 | C2×D3 | semidirect |
| 80. C2×Dic6 | C2×D3 | semidirect |
| 132. C2×C3×C4 | #67 | semidirect |
| 133. C2×C3×C4 | #67 | semidirect |
| 200. C32 | #535 | semidirect |
| 215. C2×C4 | #625 | semidirect |
| 244. C3 | #5805 | semidirect |
| 245. C3 | #5830 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#235 | 2 |
| C22×C3×D12 | 2 |
| #64135 | 2 |
| #63976 | 2 |
| #63914 | 2 |
| #63908 | 2 |
| #63863 | 2 |
| #5830 | 3 |
| #5805 | 3 |