Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
261 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 261 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | #560⋊C3 |
| commutator, hypocenter | C42 | C22×C3×C9 |
| Frattini, socle | C22×C3 | C22×C3×A4 |
| Fitting | C9×#560 | C3 |
| prime radical | C3×#560⋊C3 | C3 |
| cosocle | C3×C42 | C22×C32 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×#127⋊C3 | C2 | semidirect |
| 2. C9×#126⋊C3 | C2 | semidirect |
| 4. C9×#560 | C3 | semidirect |
| 5. #560⋊C9 | C3 | semidirect |
| 7. C9×C42⋊C3 | C22 | semidirect |
| 10. C9×#127 | C2×C3 | semidirect |
| 11. C9×#126 | C2×C3 | semidirect |
| 15. #127⋊C9 | C2×C3 | semidirect |
| 16. #126⋊C9 | C2×C3 | semidirect |
| 20. #560⋊C3 | C9 | direct (N × H) |
| 33. C42×C9 | C22×C3 | semidirect |
| 34. C42⋊C9 | C22×C3 | semidirect |
| 42. #127⋊C3 | C2×C9 | semidirect |
| 43. #126⋊C3 | C2×C9 | semidirect |
| 58. #560 | C3×C9 | semidirect |
| 72. C42⋊C3 | C22×C9 | semidirect |
| 86. #127 | C2×C3×C9 | semidirect |
| 87. #126 | C2×C3×C9 | semidirect |
| 121. C42 | C22×C3×C9 | semidirect |
| 135. C9 | #560⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×#127⋊C3 | 2 |
| C9×#126⋊C3 | 2 |
| C3×#560⋊C3 | 3 |
| C9×#560 | 3 |
| #560⋊C9 | 3 |
| C22×C9×A4 | 4 |