Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
648 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 648 conjugacy classes of subgroups — too many to draw. Only its 183 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 | C23×C3⋊D3 |
| commutator, socle, cosocle | C22×C32 | C24 |
| Fitting | C32×#117 | C2 |
| prime radical | #64139 | C2 |
| hypercenter | #117 | C3⋊D3 |
| hypocenter | C32 | #542 |
| LCS | G ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ #117 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×#117 | C2 | semidirect |
| 3. #64133 | C2 | semidirect |
| 4. #64107 | C2 | semidirect |
| 5. #64105 | C2 | semidirect |
| 6. #64094 | C2 | semidirect |
| 7. #64092 | C2 | semidirect |
| 8. #64084 | C2 | semidirect |
| 9. C4×C4⋊(C3⋊D3) | C2 | semidirect |
| 10. #64078 | C2 | semidirect |
| 11. C4×#634 | C2 | semidirect |
| 36. C3×#117 | D3 | semidirect |
| 114. #117 | C3⋊D3 | semidirect |
| 176. C32 | #542 | semidirect |
| 215. C3 | #5534 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#117 | 2 |
| #64139 | 2 |
| #64133 | 2 |
| #64107 | 2 |
| #64105 | 2 |
| #64094 | 2 |
| #64092 | 2 |
| #64084 | 2 |
| C4×C4⋊(C3⋊D3) | 2 |
| #64078 | 2 |
| C4×#634 | 2 |
| #5534 | 3 |