Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
363 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 363 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×D9 |
| commutator, cosocle | C22×C9 | C24 |
| Frattini, socle | C22×C3 | C23×D3 |
| Fitting | C9×C22⋊Q8 | C2 |
| prime radical | C23×D9 | C22 |
| hypercenter | C22⋊Q8 | D9 |
| hypocenter | C9 | #543 |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×C22⋊Q8 | C2 | semidirect |
| 2. #63508 | C2 | semidirect |
| 4. #63486 | C2 | semidirect |
| 5. #63459 | C2 | semidirect |
| 6. #63458 | C2 | semidirect |
| 7. #63458 | C2 | semidirect |
| 9. #63440 | C2 | semidirect |
| 12. C2×Q8×C9 | C22 | semidirect |
| 17. C2×Dic18 | C22 | semidirect |
| 18. C2×Dic18 | C22 | semidirect |
| 19. C9×C4⋊C4 | C22 | semidirect |
| 20. C9×C4⋊C4 | C22 | semidirect |
| 27. C4⋊Dic9 | C22 | semidirect |
| 28. C4⋊Dic9 | C22 | semidirect |
| 29. Dic9⋊C4 | C22 | semidirect |
| 30. Dic9⋊C4 | C22 | semidirect |
| 31. Dic9⋊C4 | C22 | semidirect |
| 44. C22×D9 | Q8 | semidirect |
| 103. C22⋊Q8 | D9 | semidirect |
| 133. C2×Q8 | C2×D9 | semidirect |
| 143. C4⋊C4 | C2×D9 | semidirect |
| 144. C4⋊C4 | C2×D9 | semidirect |
| 166. C9 | #543 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×C22⋊Q8 | 2 |
| #63508 | 2 |
| C2×(C2×D9)⋊C4 | 2 |
| #63486 | 2 |
| #63459 | 2 |
| #63458 | 2 |
| C22⋊C4×D9 | 2 |
| #63440 | 2 |
| #5634 | 3 |