Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
261 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 261 conjugacy classes of subgroups — too many to draw. Only its 30 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 | #857⋊C2 |
| commutator, hypocenter | C42⋊C3 | C22×C9 |
| Frattini, socle | C22×C3 | C2×C3×S4 |
| Fitting | C9×#127 | D3 |
| prime radical | C3×#857⋊C2 | C3 |
| cosocle | C3×C42⋊C3 | C22×C3 |
| LCS | G ▷ C42⋊C3 | |
| 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×#857 | C2 | semidirect |
| 6. C9×C42⋊C3 | C22 | semidirect |
| 11. C9×#127 | D3 | semidirect |
| 17. #857⋊C2 | C9 | direct (N × H) |
| 31. C42×C9 | C2×D3 | semidirect |
| 39. #127⋊C3 | C2×C9 | semidirect |
| 40. #857 | C2×C9 | semidirect |
| 78. C42⋊C3 | C22×C9 | semidirect |
| 93. #127 | D3×C9 | semidirect |
| 133. C42 | C2×D3×C9 | semidirect |
| 148. C9 | #857⋊C2 | 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×#857 | 2 |
| C3×#857⋊C2 | 3 |
| C9×#453 | 3 |
| C2×C9×S4 | 4 |