Finite Group DatabasebetaFinite Group DatabasebetaOrder 1296.
219 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 219 conjugacy classes of subgroups — too many to draw. Only its 52 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×A42 | C3 |
| hypercenter | C9 | A42 |
| hypocenter | C24 | C3×3−1+2 |
| LCS | G ▷ C24×C3 ▷ C24 | |
| UCS | C1 ◁ C3 ◁ C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C9⋊(C22⋊A4) | C3 | semidirect |
| 3. C22×C9⋊A4 | C3 | semidirect |
| 4. C22×C9×A4 | C3 | semidirect |
| 5. #88881 | C3 | semidirect |
| 6. #88880 | C3 | semidirect |
| 7. A4×C22⋊C9 | C3 | semidirect |
| 16. C24×C9 | C32 | semidirect |
| 17. C22⋊(C22⋊C9) | C32 | semidirect |
| 18. C22×C22⋊C9 | C32 | semidirect |
| 19. C22×C22⋊C9 | C32 | semidirect |
| 32. C9⋊A4 | A4 | direct (N × H) |
| 34. C9×A4 | A4 | semidirect |
| 48. C22×A4 | 3−1+2 | semidirect |
| 64. C22×C9 | C3×A4 | semidirect |
| 65. C22×C9 | C3×A4 | semidirect |
| 69. C22⋊C9 | C3×A4 | semidirect |
| 70. C22⋊C9 | C3×A4 | semidirect |
| 90. C24 | C3×3−1+2 | semidirect |
| 103. A4 | C9⋊A4 | direct (N × H) |
| 111. C9 | A42 | semidirect |
| 124. C22 | C3×C9⋊A4 | semidirect |
| 125. C22 | A4×3−1+2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×A42 | 3 |
| C9⋊(C22⋊A4) | 3 |
| C22×C9⋊A4 | 3 |
| C22×C9×A4 | 3 |
| #88881 | 3 |
| #88880 | 3 |
| A4×C22⋊C9 | 3 |
| A4×3−1+2 | 4 |
| C3×C9⋊A4 | 4 |