Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
456 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 174 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C32 | C22×D9 |
| commutator, cosocle | C2×C9 | C23×C32 |
| Frattini | C2×C3 | C22×C32×D3 |
| Fitting | C32×Q8×C9 | C2 |
| socle | C2×C33 | C22×D3 |
| hypercenter | C32×Q8 | D9 |
| hypocenter | C9 | C32×#41 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2×C32 ◁ C32×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×Q8×C9 | C2 | semidirect |
| 2. C32×D36 | C2 | semidirect |
| 3. C32×C4×D9 | C2 | semidirect |
| 5. C3×#3701 | C3 | direct (N × H) |
| 13. C3×Q8×C9 | C2×C3 | semidirect |
| 14. C3×D36 | C2×C3 | semidirect |
| 15. C3×C4×D9 | C2×C3 | semidirect |
| 20. #3701 | C32 | direct (N × H) |
| 28. C32×C9 | #41 | semidirect |
| 30. C32×Q8 | D9 | semidirect |
| 37. Q8×C9 | C2×C32 | semidirect |
| 38. D36 | C2×C32 | semidirect |
| 39. C4×D9 | C2×C32 | semidirect |
| 60. C3×C9 | C3×#41 | semidirect |
| 62. C3×Q8 | C3×D9 | semidirect |
| 85. C32 | #3701 | direct (N × H) |
| 88. C9 | C32×#41 | semidirect |
| 89. Q8 | C32×D9 | semidirect |
| 101. C3 | C3×#3701 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×Q8×C9 | 2 |
| C32×D36 | 2 |
| C32×C4×D9 | 2 |
| C32×#239 | 3 |
| C3×#3701 | 3 |