Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
163 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 163 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | D3×D4 |
| commutator, cosocle | C3×C4 | C23×C3 |
| Frattini | C4 | C22×C3×D3 |
| Fitting | C32×D8 | C2 |
| socle | C2×C32 | C2×D4 |
| hypercenter | C3×D8 | D3 |
| hypocenter | C3 | C2×C3×D8 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×D3×D4 | C2 | semidirect |
| 2. C32×D8 | C2 | semidirect |
| 3. C3×#213 | C2 | semidirect |
| 4. C3×D24 | C2 | semidirect |
| 5. C3×D3×C8 | C2 | semidirect |
| 7. D3×D8 | C3 | direct (N × H) |
| 9. C32×D4 | C22 | semidirect |
| 11. C3×D12 | C22 | semidirect |
| 13. C32×C8 | C22 | semidirect |
| 14. C3×C3⋊C8 | C22 | semidirect |
| 16. D3×D4 | C2×C3 | semidirect |
| 20. C3×D8 | D3 | direct (N × H) |
| 21. C3×D8 | C2×C3 | semidirect |
| 23. #213 | C2×C3 | semidirect |
| 24. D24 | C2×C3 | semidirect |
| 25. D3×C8 | C2×C3 | semidirect |
| 38. C3×D4 | C2×D3 | semidirect |
| 39. C3×D4 | C22×C3 | semidirect |
| 42. D12 | C22×C3 | semidirect |
| 46. C3×C8 | C2×D3 | semidirect |
| 47. C3×C8 | C22×C3 | semidirect |
| 48. C3⋊C8 | C22×C3 | semidirect |
| 52. C3×D3 | D8 | direct (N × H) |
| 56. D8 | C3×D3 | direct (N × H) |
| 72. C32 | C2×D8 | semidirect |
| 76. D4 | C2×C3×D3 | semidirect |
| 79. C8 | C2×C3×D3 | semidirect |
| 89. D3 | C3×D8 | direct (N × H) |
| 97. C3 | D3×D8 | direct (N × H) |
| 98. C3 | C2×C3×D8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×D3×D4 | 2 |
| C32×D8 | 2 |
| C3×#213 | 2 |
| C3×D24 | 2 |
| C3×D3×C8 | 2 |
| C2×C3×D8 | 3 |
| D3×D8 | 3 |