Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
288 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C9 | D32 |
| commutator, hypocenter | C32 | C42×C9 |
| Frattini | C22×C3 | C3×D32 |
| Fitting | C22×C32×C9 | C22 |
| socle, prime radical, cosocle | C22×C33 | C22×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C22×C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. C9×C3⋊Dic3 | C4 | semidirect |
| 7. C3×C9×Dic3 | C4 | semidirect |
| 18. Dic32 | C9 | direct (N × H) |
| 31. C9×Dic3 | Dic3 | direct (N × H) |
| 32. C32×C9 | C42 | semidirect |
| 63. C3⋊Dic3 | C4×C9 | semidirect |
| 67. C3×Dic3 | C4×C9 | semidirect |
| 79. C3×C9 | C4×Dic3 | semidirect |
| 118. Dic3 | C9×Dic3 | direct (N × H) |
| 122. C32 | C42×C9 | semidirect |
| 126. C9 | Dic32 | direct (N × H) |
| 147. C3 | C4×C9×Dic3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C9×C3⋊Dic3 | 2 |
| C2×C3×C9×Dic3 | 2 |
| C3×Dic32 | 3 |
| C4×C9×Dic3 | 3 |