Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
130 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 130 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | #4625 |
| commutator, hypocenter | C22⋊A4 | C4×C9 |
| Frattini | C23×C3 | C3×S4 |
| Fitting | C9×#120 | D3 |
| socle | C22×C3 | C3×SL(2,ℤ/4) |
| prime radical, cosocle | C3×#120⋊C3 | C2×C3 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C9×C22⋊A4 | C4 | semidirect |
| 12. #4625 | C9 | direct (N × H) |
| 16. C24×C9 | Dic3 | semidirect |
| 43. C22⋊A4 | C4×C9 | semidirect |
| 80. C24 | C9×Dic3 | semidirect |
| 100. C9 | #4625 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×#120⋊C3 | 2 |
| C3×#4625 | 3 |
| C9×AΣL(1,𝔽16) | 3 |
| C9×SL(2,ℤ/4) | 4 |