Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
650 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 650 conjugacy classes of subgroups — too many to draw. Only its 118 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×C3 | #1113 |
| commutator, hypocenter | C33 | C3×C42 |
| Frattini | C22 | C3×#1113 |
| Fitting, socle, prime radical | C22×C34 | C22 |
| cosocle | C22×C33 | C22×C3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C22×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #88813 | C3 | direct (N × H) |
| 5. C32×C3⋊Dic3 | C4 | semidirect |
| 22. C3×C3⋊Dic3 | Dic3 | semidirect |
| 23. C3×C3⋊Dic3 | C3×C4 | semidirect |
| 26. C34 | C42 | semidirect |
| 62. C3⋊Dic3 | C3×Dic3 | semidirect |
| 74. C33 | C4×Dic3 | semidirect |
| 75. C33 | C3×C42 | semidirect |
| 129. C32 | Dic32 | semidirect |
| 130. C32 | C3×C4×Dic3 | semidirect |
| 150. C3 | C3×Dic32 | semidirect |
| 151. C3 | #88813 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C32×C3⋊Dic3 | 2 |
| #88813 | 3 |
| C3×Dic32 | 3 |