Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
359 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 359 conjugacy classes of subgroups — too many to draw. Only its 170 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 | C23×D3 |
| commutator, cosocle | C2×C3 | C24×C3 |
| Frattini | C2 | C23×C3×D3 |
| Fitting | C2×C32×D4 | C2 |
| socle | C2×C32 | C24 |
| hypercenter | C2×C3×D4 | D3 |
| hypocenter | C3 | C3×H(5,𝔽2) |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C32×D4 | C2 | semidirect |
| 2. C2×C3×#67 | C2 | semidirect |
| 3. C3×#237 | C2 | semidirect |
| 4. C3×D3×D4 | C2 | semidirect |
| 5. C3×#235 | C2 | semidirect |
| 7. #1004 | C3 | direct (N × H) |
| 8. C23×C32 | C22 | semidirect |
| 9. C22×C3×D3 | C22 | semidirect |
| 10. C32×D4 | C22 | semidirect |
| 11. C2×C32×C4 | C22 | semidirect |
| 12. C3×#67 | C22 | semidirect |
| 13. C3×#67 | C22 | semidirect |
| 14. C2×C3×Dic3 | C22 | semidirect |
| 15. C3×D12 | C22 | semidirect |
| 16. C3×C4×D3 | C22 | semidirect |
| 17. C3×Dic6 | C22 | semidirect |
| 23. C2×C3×D4 | D3 | semidirect |
| 24. C2×C3×D4 | C2×C3 | semidirect |
| 25. C2×#67 | C2×C3 | semidirect |
| 26. #237 | C2×C3 | semidirect |
| 27. D3×D4 | C2×C3 | semidirect |
| 28. #235 | C2×C3 | semidirect |
| 39. C23×C3 | C2×D3 | semidirect |
| 40. C23×C3 | C22×C3 | semidirect |
| 41. C22×D3 | C22×C3 | semidirect |
| 47. C3×D4 | C2×D3 | semidirect |
| 48. C3×D4 | C22×C3 | semidirect |
| 52. C2×C3×C4 | C2×D3 | semidirect |
| 53. C2×C3×C4 | C22×C3 | semidirect |
| 54. #67 | C22×C3 | semidirect |
| 55. #67 | C22×C3 | semidirect |
| 56. C2×Dic3 | C22×C3 | semidirect |
| 57. D12 | C22×C3 | semidirect |
| 58. C4×D3 | C22×C3 | semidirect |
| 59. Dic6 | C22×C3 | semidirect |
| 68. C2×D4 | C3×D3 | semidirect |
| 87. C32 | H(5,𝔽2) | semidirect |
| 89. C23 | C2×C3×D3 | semidirect |
| 94. D4 | C2×C3×D3 | semidirect |
| 97. C2×C4 | C2×C3×D3 | semidirect |
| 117. C3 | #1004 | direct (N × H) |
| 118. C3 | C3×H(5,𝔽2) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C32×D4 | 2 |
| C2×C3×#67 | 2 |
| C3×#237 | 2 |
| C3×D3×D4 | 2 |
| C3×#235 | 2 |
| C3×H(5,𝔽2) | 3 |
| #1004 | 3 |