Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
615 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 615 conjugacy classes of subgroups — too many to draw. Only its 226 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C3 | C23×D3 |
| commutator, cosocle | C22×C3 | C24×C3 |
| Frattini | C22 | C23×C3×D3 |
| Fitting | C32×C4×Q8 | C2 |
| socle | C22×C32 | C24 |
| hypercenter | C3×C4×Q8 | D3 |
| hypocenter | C3 | C3×#548 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#239 | C2 | semidirect |
| 2. C2×C3×D3×Q8 | C2 | semidirect |
| 3. C32×C4×Q8 | C2 | semidirect |
| 4. C3×#897 | C2 | semidirect |
| 5. C3×#895 | C2 | semidirect |
| 6. C3×#875 | C2 | semidirect |
| 7. C3×C4×D12 | C2 | semidirect |
| 9. #5576 | C3 | direct (N × H) |
| 26. C2×#239 | C2×C3 | semidirect |
| 27. C2×D3×Q8 | C2×C3 | semidirect |
| 32. C3×C4×Q8 | D3 | semidirect |
| 33. C3×C4×Q8 | C2×C3 | semidirect |
| 35. #897 | C2×C3 | semidirect |
| 36. #895 | C2×C3 | semidirect |
| 37. #875 | C2×C3 | semidirect |
| 38. C4×D12 | C2×C3 | semidirect |
| 41. C32×Q8 | D4 | semidirect |
| 90. C2×C3×D3 | #41 | semidirect |
| 101. C4×Q8 | C3×D3 | semidirect |
| 109. C3×Q8 | D12 | semidirect |
| 110. C3×Q8 | C3×D4 | semidirect |
| 166. C2×D3 | C3×#41 | semidirect |
| 180. C32 | #548 | semidirect |
| 184. Q8 | C3×D12 | semidirect |
| 220. C3 | #5576 | direct (N × H) |
| 221. C3 | C3×#548 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#239 | 2 |
| C2×C3×D3×Q8 | 2 |
| C32×C4×Q8 | 2 |
| C3×#897 | 2 |
| C3×#895 | 2 |
| C3×#875 | 2 |
| C3×C4×D12 | 2 |
| C3×#548 | 3 |
| #5576 | 3 |