Finite Group DatabasebetaFinite Group DatabasebetaOrder 3072.
At least 7491 conjugacy classes of subgroups; only the 537 normal subgroups are shown.
This group has at least 7491 conjugacy classes of subgroups — too many to draw. Only its 537 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #564 | C24×C3 |
| Frattini, socle | C22 | C24×C22⋊A4 |
| Fitting | #72272519 | C3 |
| LCS | G ▷ #564 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72272519 | C3 | semidirect |
| 2. #72272506 | C2 | semidirect |
| 3. #72272503 | C2 | semidirect |
| 4. #63036⋊C3 | C22 | semidirect |
| 5. #62979⋊C3 | C22 | semidirect |
| 6. #62976⋊C3 | C22 | semidirect |
| 7. #62840⋊C3 | C22 | semidirect |
| 8. #10586977 | C2×C3 | semidirect |
| 9. #10586964 | C2×C3 | semidirect |
| 10. #3567⋊C3 | C23 | semidirect |
| 11. #3556⋊C3 | C23 | semidirect |
| 12. #63036 | C22×C3 | semidirect |
| 13. #62979 | C22×C3 | semidirect |
| 14. #62976 | C22×C3 | semidirect |
| 15. #62840 | C22×C3 | semidirect |
| 17. #564⋊C3 | C24 | semidirect |
| 18. #3567 | C23×C3 | semidirect |
| 19. #3556 | C23×C3 | semidirect |
| 26. #564 | C24×C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72272519 | 3 |
| #72272506 | 2 |
| #72272503 | 2 |
| C22×#560⋊C3 | 4 |