Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
253 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 253 conjugacy classes of subgroups — too many to draw. Only its 57 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 | C2×C8×C9 |
| Frattini | C3×C4 | C2×C3×S4 |
| Fitting | C22×C8×C9 | D3 |
| socle | C23×C3 | C3×C4×D3 |
| prime radical | C2×C3×S4 | C3×C4 |
| cosocle | C3×C4×A4 | C22×C3 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C8×C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C8×C9×A4 | C2 | semidirect |
| 3. C9×A4⋊C8 | C2 | semidirect |
| 14. C22×C8×C9 | D3 | semidirect |
| 22. C9×S4 | C8 | direct (N × H) |
| 26. C8×S4 | C9 | direct (N × H) |
| 47. C9×A4 | C2×C8 | semidirect |
| 55. C8×A4 | C2×C9 | semidirect |
| 56. A4⋊C8 | C2×C9 | semidirect |
| 82. C8×C9 | S4 | direct (N × H) |
| 113. C22×C9 | D3×C8 | semidirect |
| 122. C22×C8 | D3×C9 | semidirect |
| 133. S4 | C8×C9 | direct (N × H) |
| 179. A4 | C2×C8×C9 | semidirect |
| 190. C9 | C8×S4 | direct (N × H) |
| 205. C8 | C9×S4 | direct (N × H) |
| 218. C22 | D3×C8×C9 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×C9×S4 | 2 |
| C8×C9×A4 | 2 |
| C9×A4⋊C8 | 2 |
| C3×C8×S4 | 3 |
| C8×D4×C9 | 3 |
| D3×C8×C9 | 4 |