Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
250 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 70 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 | C3×C4×C9 |
| Frattini | C2×C3 | C32×S4 |
| Fitting | C23×C3×C9 | D3 |
| socle | C23×C32 | C3×D3 |
| prime radical | C2×C32×A4 | C2×C3 |
| cosocle | C2×C3×A4 | C2×C32 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C2×C3×C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C9×SL(2,ℤ/4) | C3 | direct (N × H) |
| 5. C3×C9×A4 | C4 | semidirect |
| 15. C3×SL(2,ℤ/4) | C9 | direct (N × H) |
| 23. C22×C3×C9 | Dic3 | semidirect |
| 26. C9×A4 | C3×C4 | semidirect |
| 49. SL(2,ℤ/4) | C3×C9 | direct (N × H) |
| 57. C3×A4 | C4×C9 | semidirect |
| 65. C22×C9 | C3×Dic3 | semidirect |
| 73. C3×C9 | SL(2,ℤ/4) | direct (N × H) |
| 99. C22×C3 | C9×Dic3 | semidirect |
| 103. A4 | C3×C4×C9 | semidirect |
| 115. C9 | C3×SL(2,ℤ/4) | direct (N × H) |
| 130. C22 | C3×C9×Dic3 | semidirect |
| 136. C3 | C9×SL(2,ℤ/4) | 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 |
| C32×SL(2,ℤ/4) | 3 |
| C9×SL(2,ℤ/4) | 3 |
| C3×C9×C22⋊C4 | 3 |
| C3×C9×Dic3 | 4 |