Finite Group DatabasebetaFinite Group DatabasebetaOrder 288.
539 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 539 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C3 | C2×D3 |
| commutator, cosocle | C2×C3 | C24×C3 |
| Frattini | C2 | C23×C3×D3 |
| Fitting | C24×C32 | C2 |
| socle | C23×C32 | C22 |
| hypercenter | C24×C3 | D3 |
| hypocenter | C3 | C22×C3×D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C23×C3 ◁ C24×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C32 | C2 | semidirect |
| 2. C23×C3×D3 | C2 | semidirect |
| 3. C2×C3×#67 | C2 | direct (N × H) |
| 4. C22×C3×Dic3 | C2 | semidirect |
| 6. C22×#67 | C3 | direct (N × H) |
| 9. C23×C32 | C22 | semidirect |
| 11. C22×C3×D3 | C22 | semidirect |
| 12. C3×#67 | C22 | direct (N × H) |
| 13. C2×C3×Dic3 | C22 | semidirect |
| 16. C24×C3 | D3 | semidirect |
| 17. C24×C3 | C2×C3 | semidirect |
| 18. C23×D3 | C2×C3 | semidirect |
| 21. C2×#67 | C2×C3 | direct (N × H) |
| 22. C22×Dic3 | C2×C3 | semidirect |
| 24. C22×C32 | D4 | semidirect |
| 26. C22×C32 | C23 | semidirect |
| 28. C2×C3×D3 | C23 | semidirect |
| 29. C3×Dic3 | C23 | semidirect |
| 39. C23×C3 | C2×D3 | semidirect |
| 41. C23×C3 | C22×C3 | semidirect |
| 43. C22×D3 | C22×C3 | semidirect |
| 46. #67 | C22×C3 | direct (N × H) |
| 47. C2×Dic3 | C22×C3 | semidirect |
| 49. C2×C32 | C2×D4 | semidirect |
| 53. C24 | C3×D3 | semidirect |
| 64. C22×C3 | #67 | direct (N × H) |
| 65. C22×C3 | C3×D4 | semidirect |
| 67. C22×C3 | C22×D3 | semidirect |
| 69. C22×C3 | C23×C3 | semidirect |
| 71. C2×D3 | C23×C3 | semidirect |
| 73. Dic3 | C23×C3 | semidirect |
| 74. C32 | C22×D4 | semidirect |
| 79. C23 | C2×C3×D3 | semidirect |
| 88. C2×C3 | C2×#67 | direct (N × H) |
| 89. C2×C3 | C2×C3×D4 | semidirect |
| 96. C22 | C3×#67 | direct (N × H) |
| 98. C22 | C22×C3×D3 | semidirect |
| 101. C3 | C22×#67 | direct (N × H) |
| 102. C3 | C22×C3×D4 | semidirect |
| 105. C2 | C2×C3×#67 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C32 | 2 |
| C23×C3×D3 | 2 |
| C2×C3×#67 | 2 |
| C22×C3×Dic3 | 2 |
| C22×C3×D4 | 3 |
| C22×#67 | 3 |