Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
1069 conjugacy classes of subgroups; only the 262 normal subgroups are shown.
This group has 1069 conjugacy classes of subgroups — too many to draw. Only its 262 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #4169 |
| commutator, cosocle, hypocenter | C3×C9 | C24×C3 |
| Frattini | C32 | C23×C3×D3 |
| Fitting | C23×C32⋊C9 | C2 |
| socle | C23×C32 | C3×D3 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C32⋊C9 | C2 | semidirect |
| 2. C22×#4169 | C2 | direct (N × H) |
| 4. C23×C3⋊D9 | C3 | semidirect |
| 5. C22×C32⋊C9 | C22 | semidirect |
| 6. C2×#4169 | C22 | direct (N × H) |
| 10. C23×C3×C9 | C2×C3 | semidirect |
| 11. C22×C3⋊D9 | C2×C3 | semidirect |
| 12. C2×C32⋊C9 | C23 | semidirect |
| 13. #4169 | C23 | direct (N × H) |
| 22. C22×C3×C9 | C22×C3 | semidirect |
| 23. C2×C3⋊D9 | C22×C3 | semidirect |
| 24. C32⋊C9 | C24 | semidirect |
| 28. C23×C32 | D9 | semidirect |
| 40. C2×C3×C9 | C23×C3 | semidirect |
| 41. C3⋊D9 | C23×C3 | semidirect |
| 49. C22×C32 | C2×D9 | semidirect |
| 60. C3×C9 | C24×C3 | semidirect |
| 73. C2×C32 | C22×D9 | semidirect |
| 95. C32 | C23×D9 | semidirect |
| 100. C23 | #4169 | direct (N × H) |
| 111. C22 | C2×#4169 | direct (N × H) |
| 118. C2 | C22×#4169 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C32⋊C9 | 2 |
| C22×#4169 | 2 |
| C23×C3×C3⋊D3 | 3 |
| C23×C3⋊D9 | 3 |