Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
469 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 469 conjugacy classes of subgroups — too many to draw. Only its 182 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, hypocenter | C3×C9 | C22×C3×C4 |
| Frattini | C2×C32 | C22×C3×D3 |
| Fitting | C23×C32⋊C9 | C2 |
| socle | C23×C32 | C3×D3 |
| prime radical | C23×C33 | D3 |
| cosocle | C2×C3×C9 | C23×C3 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#66316 | C2 | direct (N × H) |
| 4. C22×C3⋊Dic9 | C3 | semidirect |
| 5. C22×C32⋊C9 | C4 | semidirect |
| 7. #66316 | C22 | direct (N × H) |
| 12. C2×C3⋊Dic9 | C2×C3 | semidirect |
| 13. C2×C32⋊C9 | C2×C4 | semidirect |
| 25. C22×C3×C9 | C3×C4 | semidirect |
| 27. C3⋊Dic9 | C22×C3 | semidirect |
| 28. C32⋊C9 | C22×C4 | semidirect |
| 45. C2×C3×C9 | C2×C3×C4 | semidirect |
| 57. C22×C32 | Dic9 | semidirect |
| 72. C3×C9 | C22×C3×C4 | semidirect |
| 88. C2×C32 | C2×Dic9 | semidirect |
| 114. C32 | C22×Dic9 | semidirect |
| 130. C22 | #66316 | direct (N × H) |
| 138. C2 | C2×#66316 | 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 |
| C2×#66316 | 2 |
| C22×C3×C3⋊Dic3 | 3 |
| C22×C3⋊Dic9 | 3 |