Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
115 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 115 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8×C9 | S4 |
| commutator, hypocenter | A4 | C9×C16 |
| Frattini | C3×C8 | C3×S4 |
| Fitting | C22×C8×C9 | D3 |
| socle | C23×C3 | C3×C3⋊C8 |
| prime radical | C2×C3×A4 | C3×C8 |
| cosocle | C3×C8×A4 | C2×C3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C8×C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 13. A4⋊C16 | C9 | direct (N × H) |
| 23. C9×A4 | C16 | semidirect |
| 53. C22×C9 | C3⋊C16 | semidirect |
| 85. A4 | C9×C16 | semidirect |
| 92. C9 | A4⋊C16 | direct (N × H) |
| 105. C22 | C9×C3⋊C16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C8×C9×A4 | 2 |
| C3×A4⋊C16 | 3 |
| C9×C22⋊C16 | 3 |
| C9×C3⋊C16 | 4 |