Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
142 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×A4 |
| commutator, socle, cosocle | C23×C3 | C22×C3 |
| Fitting | C24×C3 | C2×C3 |
| hypercenter | C22 | D3×A4 |
| hypocenter | C22×C3 | C3×D4 |
| LCS | G ▷ C23×C3 ▷ C22×C3 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×A4 | C2 | semidirect |
| 2. C2×D3×A4 | C2 | semidirect |
| 3. Dic3×A4 | C2 | semidirect |
| 4. C22×#67 | C3 | semidirect |
| 10. C24×C3 | C2×C3 | semidirect |
| 11. C23×D3 | C2×C3 | semidirect |
| 14. C22×A4 | D3 | semidirect |
| 19. C22×Dic3 | C2×C3 | semidirect |
| 23. C3×A4 | D4 | semidirect |
| 45. #67 | A4 | direct (N × H) |
| 52. C24 | C3×D3 | semidirect |
| 68. C22×C3 | C3×D4 | semidirect |
| 69. C22×C3 | C2×A4 | semidirect |
| 75. C2×D3 | C2×A4 | semidirect |
| 77. A4 | #67 | direct (N × H) |
| 80. Dic3 | C2×A4 | semidirect |
| 123. C22 | C3×#67 | semidirect |
| 124. C22 | D3×A4 | semidirect |
| 129. C3 | D4×A4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×A4 | 2 |
| C2×D3×A4 | 2 |
| Dic3×A4 | 2 |
| C22×#67 | 3 |
| D4×A4 | 3 |
| C3×#67 | 4 |