Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
499 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 499 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D32 |
| commutator, cosocle | C2×C32 | C24 |
| Frattini | C2 | C22×D32 |
| Fitting | C2×C32×C4 | C22 |
| socle | C22×C32 | C23 |
| hypercenter | C2×C4 | D32 |
| hypocenter | C32 | C22×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D32 | C2 | semidirect |
| 2. C2×C4×C3⋊D3 | C2 | semidirect |
| 3. C2×C3×D12 | C2 | semidirect |
| 4. C2×#625 | C2 | semidirect |
| 5. #3802 | C2 | direct (N × H) |
| 8. C22×C3×D3 | C22 | semidirect |
| 10. C2×D32 | C22 | semidirect |
| 11. C2×C32×C4 | C22 | semidirect |
| 12. C2×C3⋊Dic3 | C22 | semidirect |
| 13. C4×C3⋊D3 | C22 | semidirect |
| 14. C3×D12 | C22 | semidirect |
| 15. #625 | C22 | semidirect |
| 20. C2×D12 | D3 | semidirect |
| 24. C2×C3⋊D3 | D4 | semidirect |
| 27. C2×C3×D3 | C23 | semidirect |
| 29. C32×C4 | C23 | semidirect |
| 30. C3⋊Dic3 | C23 | semidirect |
| 37. C22×D3 | C2×D3 | semidirect |
| 40. C2×C3×C4 | C2×D3 | semidirect |
| 44. D12 | C2×D3 | semidirect |
| 47. C2×C32 | C2×D4 | semidirect |
| 49. C3⋊D3 | C2×D4 | semidirect |
| 66. C2×D3 | C22×D3 | semidirect |
| 68. C3×C4 | C22×D3 | semidirect |
| 71. C32 | C22×D4 | semidirect |
| 80. C2×C4 | D32 | semidirect |
| 84. C2×C3 | D3×D4 | semidirect |
| 96. C4 | C2×D32 | semidirect |
| 98. C3 | C2×D3×D4 | semidirect |
| 101. C2 | #3802 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D32 | 2 |
| C2×C4×C3⋊D3 | 2 |
| C2×C3×D12 | 2 |
| C2×#625 | 2 |
| #3802 | 2 |
| C2×D3×D4 | 3 |