Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
402 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 135 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#611 |
| commutator, cosocle | C4×C9 | C24 |
| Frattini | C3×C4 | C23×D3 |
| Fitting | C9×H(5,𝔽2) | C2 |
| socle | C2×C3 | C22×#67 |
| hypercenter | H(5,𝔽2) | D9 |
| hypocenter | C9 | #576 |
| LCS | G ▷ C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ H(5,𝔽2) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×H(5,𝔽2) | C2 | semidirect |
| 2. #63715 | C2 | semidirect |
| 3. #63513 | C2 | semidirect |
| 4. #63512 | C2 | semidirect |
| 5. #63510 | C2 | semidirect |
| 6. #63495 | C2 | semidirect |
| 7. C2×#3673 | C2 | semidirect |
| 11. C9×#41 | C22 | semidirect |
| 13. C2×D4×C9 | C22 | semidirect |
| 16. #3697 | C22 | semidirect |
| 17. C2×D36 | C22 | semidirect |
| 18. #3675 | C22 | semidirect |
| 19. #3674 | C22 | semidirect |
| 20. #3673 | C22 | semidirect |
| 21. #3673 | C22 | semidirect |
| 22. #3672 | C22 | semidirect |
| 23. #3667 | C22 | semidirect |
| 24. C2×C9⋊C8 | C22 | semidirect |
| 76. H(5,𝔽2) | D9 | semidirect |
| 107. #41 | C2×D9 | semidirect |
| 111. C2×D4 | C2×D9 | semidirect |
| 128. C9 | #576 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×H(5,𝔽2) | 2 |
| #63715 | 2 |
| #63513 | 2 |
| #63512 | 2 |
| #63510 | 2 |
| #63495 | 2 |
| C2×#3673 | 2 |
| #5835 | 3 |