Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
262 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32×C8 | D9 |
| commutator, hypocenter | C9 | C2×C32×C8 |
| Frattini | C3×C4 | C2×C32×D3 |
| Fitting | C32×C8×C9 | C2 |
| socle | C2×C33 | C4×D3 |
| prime radical | C2×C32×D9 | C4 |
| cosocle | C4×C9 | C22×C32 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C32×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C32×C8×C9 | C2 | semidirect |
| 3. C32×C9⋊C8 | C2 | semidirect |
| 5. C3×C8×D9 | C3 | direct (N × H) |
| 14. C3×C8×C9 | C2×C3 | semidirect |
| 15. C3×C9⋊C8 | C2×C3 | semidirect |
| 17. C32×D9 | C8 | direct (N × H) |
| 20. C8×D9 | C32 | direct (N × H) |
| 28. C32×C9 | C2×C8 | semidirect |
| 33. C32×C8 | D9 | direct (N × H) |
| 38. C8×C9 | C2×C32 | semidirect |
| 39. C9⋊C8 | C2×C32 | semidirect |
| 44. C3×D9 | C3×C8 | direct (N × H) |
| 60. C3×C9 | C2×C3×C8 | semidirect |
| 65. C3×C8 | C3×D9 | direct (N × H) |
| 75. D9 | C32×C8 | direct (N × H) |
| 85. C32 | C8×D9 | direct (N × H) |
| 88. C9 | C2×C32×C8 | semidirect |
| 91. C8 | C32×D9 | direct (N × H) |
| 101. C3 | C3×C8×D9 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×C4×D9 | 2 |
| C32×C8×C9 | 2 |
| C32×C9⋊C8 | 2 |
| C32×D3×C8 | 3 |
| C3×C8×D9 | 3 |