Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
659 conjugacy classes of subgroups; only the 230 normal subgroups are shown.
This group has 659 conjugacy classes of subgroups — too many to draw. Only its 230 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×#550 |
| 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. C32×C4×Q8 | C2 | semidirect |
| 3. C3×#895 | C2 | semidirect |
| 4. C3×#874 | C2 | semidirect |
| 5. C3×C4×D12 | C2 | semidirect |
| 7. #5577 | C3 | direct (N × H) |
| 19. C2×#239 | C2×C3 | semidirect |
| 23. C3×C4×Q8 | D3 | semidirect |
| 24. C3×C4×Q8 | C2×C3 | semidirect |
| 26. #895 | C2×C3 | semidirect |
| 27. #874 | C2×C3 | semidirect |
| 28. C4×D12 | C2×C3 | semidirect |
| 30. C32×Q8 | D4 | semidirect |
| 69. C32×C4 | #41 | semidirect |
| 74. C4×Q8 | C3×D3 | semidirect |
| 79. C3×Q8 | D12 | semidirect |
| 80. C3×Q8 | C3×D4 | semidirect |
| 130. C3×C4 | #239 | semidirect |
| 132. C3×C4 | C3×#41 | semidirect |
| 134. C32 | #550 | semidirect |
| 136. Q8 | C3×D12 | semidirect |
| 163. C4 | C3×#239 | semidirect |
| 165. C3 | #5577 | direct (N × H) |
| 166. C3 | C3×#550 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#239 | 2 |
| C32×C4×Q8 | 2 |
| C3×#895 | 2 |
| C3×#874 | 2 |
| C3×C4×D12 | 2 |
| C3×#550 | 3 |
| #5577 | 3 |