Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
794 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 794 conjugacy classes of subgroups — too many to draw. Only its 200 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 | C2×C32 | C23×C4 |
| Fitting | C24×C32 | C22 |
| socle, cosocle | C22×C32 | C24 |
| prime radical | C22×#638 | C2 |
| hypercenter | C24 | D32 |
| hypocenter | C32 | #518 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×#217 | C2 | semidirect |
| 3. #63974 | C2 | semidirect |
| 4. #63957 | C2 | semidirect |
| 5. #63952 | C2 | semidirect |
| 9. C2×#638 | C4 | semidirect |
| 12. C22×C3×Dic3 | C22 | semidirect |
| 13. C2×#624 | C22 | semidirect |
| 15. C3×#217 | C22 | semidirect |
| 16. #3724 | C22 | semidirect |
| 17. #3722 | C22 | semidirect |
| 18. Dic32 | C22 | semidirect |
| 22. C2×#217 | D3 | semidirect |
| 28. C23×C32 | C2×C4 | semidirect |
| 32. C22×C3⋊D3 | C2×C4 | semidirect |
| 34. #638 | C2×C4 | semidirect |
| 35. C2×C3⋊Dic3 | C2×C4 | semidirect |
| 53. C22×Dic3 | C2×D3 | semidirect |
| 58. #217 | C2×D3 | semidirect |
| 83. C23×C3 | C4×D3 | semidirect |
| 146. C32 | #518 | semidirect |
| 151. C23 | #624 | semidirect |
| 183. C3 | #5486 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#638 | 2 |
| C2×C3×#217 | 2 |
| #63974 | 2 |
| #63957 | 2 |
| #63952 | 2 |
| #5486 | 3 |