Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
742 conjugacy classes of subgroups; only the 321 normal subgroups are shown.
This group has 742 conjugacy classes of subgroups — too many to draw. Only its 321 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×C9 | C3⋊D3 |
| commutator, hypocenter | C32 | C22×C4×C9 |
| Frattini | C2×C3 | C22×C3×C3⋊D3 |
| Fitting | C23×C32×C9 | C2 |
| socle, prime radical | C23×C33 | C2×C3 |
| cosocle | C2×C33 | C23×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C23×C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C9×C3⋊Dic3 | C2 | direct (N × H) |
| 5. C22×C32×C9 | C4 | semidirect |
| 7. C9×C3⋊Dic3 | C22 | direct (N × H) |
| 13. C2×C32×C9 | C2×C4 | semidirect |
| 15. C22×C3⋊Dic3 | C9 | direct (N × H) |
| 23. C22×C3×C9 | Dic3 | semidirect |
| 26. C32×C9 | C22×C4 | semidirect |
| 30. C2×C3⋊Dic3 | C2×C9 | direct (N × H) |
| 39. C2×C3×C9 | C2×Dic3 | semidirect |
| 45. C22×C32 | C4×C9 | semidirect |
| 49. C3⋊Dic3 | C22×C9 | direct (N × H) |
| 53. C22×C9 | C3⋊Dic3 | direct (N × H) |
| 58. C3×C9 | C22×Dic3 | semidirect |
| 66. C2×C32 | C2×C4×C9 | semidirect |
| 72. C2×C9 | C2×C3⋊Dic3 | direct (N × H) |
| 77. C22×C3 | C9×Dic3 | semidirect |
| 84. C32 | C22×C4×C9 | semidirect |
| 87. C9 | C22×C3⋊Dic3 | direct (N × H) |
| 92. C2×C3 | C2×C9×Dic3 | semidirect |
| 96. C22 | C9×C3⋊Dic3 | direct (N × H) |
| 100. C3 | C22×C9×Dic3 | semidirect |
| 102. C2 | C2×C9×C3⋊Dic3 | 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×C9×C3⋊Dic3 | 2 |
| C22×C3×C3⋊Dic3 | 3 |
| C22×C9×Dic3 | 3 |