Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
197 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 197 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×A42 |
| commutator, socle, cosocle | C24×C3 | C33 |
| Fitting | C24×C9 | C32 |
| prime radical | (C3×A4)⋊A4 | C3 |
| hypercenter | C9 | A42 |
| hypocenter | C24 | #737 |
| LCS | G ▷ C24×C3 ▷ C24 | |
| UCS | C1 ◁ C3 ◁ C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9⋊(C22⋊A4) | C3 | semidirect |
| 2. C9×C22⋊A4 | C3 | semidirect |
| 3. C22×C9⋊A4 | C3 | semidirect |
| 4. #88881 | C3 | semidirect |
| 5. #88881 | C3 | semidirect |
| 6. #88880 | C3 | semidirect |
| 8. (C22⋊A4)⋊C9 | C3 | semidirect |
| 9. A4×C22⋊C9 | C3 | semidirect |
| 31. C9⋊A4 | A4 | semidirect |
| 86. C24 | #737 | semidirect |
| 116. C22 | #66440 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9⋊(C22⋊A4) | 3 |
| C9×C22⋊A4 | 3 |
| C22×C9⋊A4 | 3 |
| #88881 | 3 |
| #88880 | 3 |
| (C3×A4)⋊A4 | 3 |
| (C22⋊A4)⋊C9 | 3 |
| A4×C22⋊C9 | 3 |
| #66440 | 4 |