Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
378 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 139 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 | #4184 |
| commutator, hypocenter | #731 | C22×C4 |
| Frattini | C2×C32 | C22×C3⋊D3 |
| Fitting | C23×#731 | C2 |
| socle | C23×C3 | #272 |
| prime radical | C23×H(3,𝔽3) | D3 |
| cosocle | C2×#731 | C23 |
| LCS | G ▷ #731 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#66331 | C2 | direct (N × H) |
| 6. C22×#731 | C4 | semidirect |
| 8. #66331 | C22 | direct (N × H) |
| 15. C2×#731 | C2×C4 | semidirect |
| 21. C22×3−1+2 | Dic3 | semidirect |
| 25. C22×C3×C9 | Dic3 | semidirect |
| 30. #731 | C22×C4 | semidirect |
| 39. C2×3−1+2 | C2×Dic3 | semidirect |
| 43. C2×C3×C9 | C2×Dic3 | semidirect |
| 60. 3−1+2 | C22×Dic3 | semidirect |
| 62. C3×C9 | C22×Dic3 | semidirect |
| 99. C22 | #66331 | direct (N × H) |
| 105. C2 | C2×#66331 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#731 | 2 |
| C2×#66331 | 2 |
| C22×3−1+2⋊C4 | 3 |
| C22×(C3⋊Dic3)⋊C3 | 3 |
| C22×C3×Dic9 | 3 |