Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
213 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 213 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | D3×A4 |
| commutator, hypocenter | C3×Q8 | C2×C3×C9 |
| Frattini | C2×C3 | C3×D3×A4 |
| Fitting | C33×Q8 | C2×C3 |
| socle | C2×C33 | C2×A4 |
| prime radical | C32×#239 | C3 |
| cosocle | C32×Q8 | C2×C32 |
| LCS | G ▷ C3×Q8 | |
| UCS | C1 ◁ C2×C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×Q8⋊C9 | C2 | semidirect |
| 4. C3⋊(#41⋊C9) | C3 | direct (N × H) |
| 11. C3×Q8⋊C9 | D3 | semidirect |
| 12. C3×Q8⋊C9 | C2×C3 | semidirect |
| 15. C3×#239 | C9 | semidirect |
| 36. Q8⋊C9 | C3×D3 | semidirect |
| 45. #239 | C3×C9 | semidirect |
| 98. C32 | #41⋊C9 | semidirect |
| 119. C3 | C3⋊(#41⋊C9) | direct (N × H) |
| 120. C3 | C3×#41⋊C9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×Q8⋊C9 | 2 |
| C32×#239 | 3 |
| C3×#41⋊C9 | 3 |
| C3⋊(#41⋊C9) | 3 |
| C3×C9×Dic3 | 4 |