Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
390 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 111 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×#120 | C2 |
| hypercenter | #120 | D9 |
| hypocenter | C9 | H(3,𝔽4) |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ #120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×#120 | C2 | semidirect |
| 2. #63515 | C2 | semidirect |
| 3. #63499 | C2 | semidirect |
| 4. #63498 | C2 | semidirect |
| 5. #63447 | C2 | semidirect |
| 7. C24×C9 | C22 | semidirect |
| 8. C23×D9 | C22 | semidirect |
| 11. C2×Dic18 | C22 | semidirect |
| 12. C9×C22⋊C4 | C22 | semidirect |
| 13. #3676 | C22 | semidirect |
| 14. #3676 | C22 | semidirect |
| 15. (C2×D9)⋊C4 | C22 | semidirect |
| 16. C4×Dic9 | C22 | semidirect |
| 56. #120 | D9 | semidirect |
| 72. C24 | C2×D9 | semidirect |
| 79. C22⋊C4 | C2×D9 | semidirect |
| 90. C9 | H(3,𝔽4) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×#120 | 2 |
| #63515 | 2 |
| #63499 | 2 |
| #63498 | 2 |
| #63447 | 2 |
| #5594 | 3 |