Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
348 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 174 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C22×D3 |
| commutator, cosocle | C2×C3 | C24×C3 |
| Frattini | C2 | C23×C3×D3 |
| Fitting | C32×#41 | C2 |
| socle | C2×C32 | C24 |
| hypercenter | C3×#41 | D3 |
| hypocenter | C3 | C2×C3×#41 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C3×C4 ◁ C3×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×#41 | C2 | semidirect |
| 2. C3×#239 | C2 | semidirect |
| 3. C3×D3×Q8 | C2 | semidirect |
| 4. C3×#237 | C2 | semidirect |
| 5. C3×D3×D4 | C2 | semidirect |
| 6. C3×#235 | C2 | semidirect |
| 7. C2×C3×C4×D3 | C2 | semidirect |
| 9. D3×#41 | C3 | direct (N × H) |
| 11. C32×Q8 | C22 | semidirect |
| 12. C32×D4 | C22 | semidirect |
| 13. C2×C32×C4 | C22 | semidirect |
| 14. C3×#67 | C22 | semidirect |
| 15. C2×C3×Dic3 | C22 | semidirect |
| 16. C3×D12 | C22 | semidirect |
| 19. C3×C4×D3 | C22 | semidirect |
| 20. C3×C4×D3 | C22 | semidirect |
| 21. C3×Dic6 | C22 | semidirect |
| 26. C3×#41 | D3 | direct (N × H) |
| 27. C3×#41 | C2×C3 | semidirect |
| 31. #239 | C2×C3 | semidirect |
| 32. D3×Q8 | C2×C3 | semidirect |
| 33. #237 | C2×C3 | semidirect |
| 34. D3×D4 | C2×C3 | semidirect |
| 35. #235 | C2×C3 | semidirect |
| 36. C2×C4×D3 | C2×C3 | semidirect |
| 50. C3×Q8 | C2×D3 | semidirect |
| 51. C3×Q8 | C22×C3 | semidirect |
| 55. C3×D4 | C2×D3 | semidirect |
| 56. C3×D4 | C22×C3 | semidirect |
| 63. C2×C3×C4 | C2×D3 | semidirect |
| 64. C2×C3×C4 | C22×C3 | semidirect |
| 65. #67 | C22×C3 | semidirect |
| 66. C2×Dic3 | C22×C3 | semidirect |
| 67. D12 | C22×C3 | semidirect |
| 70. C4×D3 | C22×C3 | semidirect |
| 71. C4×D3 | C22×C3 | semidirect |
| 72. Dic6 | C22×C3 | semidirect |
| 76. C3×D3 | #41 | direct (N × H) |
| 80. #41 | C3×D3 | direct (N × H) |
| 103. C32 | C2×#41 | semidirect |
| 106. Q8 | C2×C3×D3 | semidirect |
| 109. D4 | C2×C3×D3 | semidirect |
| 115. C2×C4 | C2×C3×D3 | semidirect |
| 125. D3 | C3×#41 | direct (N × H) |
| 135. C3 | D3×#41 | direct (N × H) |
| 136. C3 | C2×C3×#41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#41 | 2 |
| C3×#239 | 2 |
| C3×D3×Q8 | 2 |
| C3×#237 | 2 |
| C3×D3×D4 | 2 |
| C3×#235 | 2 |
| C2×C3×C4×D3 | 2 |
| C2×C3×#41 | 3 |
| D3×#41 | 3 |