Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
390 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 90 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×C3×C9 | S4 |
| commutator, hypocenter | A4 | C22×C3×C9 |
| Frattini | C3 | C2×C32×S4 |
| Fitting | C23×C3×C9 | D3 |
| socle | C23×C32 | C3×D3 |
| prime radical | C2×C32×S4 | C3 |
| cosocle | C3×A4 | C22×C32 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C2×C3×C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×C9×A4 | C2 | semidirect |
| 2. C3×C9×S4 | C2 | direct (N × H) |
| 4. C2×C9×S4 | C3 | direct (N × H) |
| 7. C3×C9×A4 | C22 | semidirect |
| 11. C23×C3×C9 | D3 | semidirect |
| 14. C2×C9×A4 | C2×C3 | semidirect |
| 15. C9×S4 | C2×C3 | direct (N × H) |
| 21. C2×C3×S4 | C9 | direct (N × H) |
| 32. C22×C3×C9 | C2×D3 | semidirect |
| 36. C9×A4 | C22×C3 | semidirect |
| 43. C2×C3×A4 | C2×C9 | semidirect |
| 45. C3×S4 | C2×C9 | direct (N × H) |
| 51. C23×C9 | C3×D3 | semidirect |
| 64. C2×C3×C9 | S4 | direct (N × H) |
| 66. C2×S4 | C3×C9 | direct (N × H) |
| 79. C3×A4 | C22×C9 | semidirect |
| 87. C22×C9 | C2×C3×D3 | semidirect |
| 94. C3×C9 | C2×S4 | direct (N × H) |
| 97. C23×C3 | D3×C9 | semidirect |
| 101. C2×A4 | C2×C3×C9 | semidirect |
| 102. S4 | C2×C3×C9 | direct (N × H) |
| 123. C2×C9 | C3×S4 | direct (N × H) |
| 135. C22×C3 | C2×D3×C9 | semidirect |
| 140. A4 | C22×C3×C9 | semidirect |
| 149. C9 | C2×C3×S4 | direct (N × H) |
| 151. C23 | C3×D3×C9 | semidirect |
| 164. C2×C3 | C9×S4 | direct (N × H) |
| 172. C22 | C2×C3×D3×C9 | semidirect |
| 177. C3 | C2×C9×S4 | direct (N × H) |
| 182. C2 | C3×C9×S4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C9×A4 | 2 |
| C3×C9×S4 | 2 |
| C2×C32×S4 | 3 |
| C2×C9×S4 | 3 |
| C2×C3×D4×C9 | 3 |
| C2×C3×D3×C9 | 4 |