Finite Group DatabasebetaFinite Group DatabasebetaOrder 432.
452 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C32 | C2×D3 |
| commutator, cosocle | C2×C3 | C23×C32 |
| Frattini | C2 | C22×C32×D3 |
| Fitting | C23×C33 | C2 |
| socle | C22×C33 | C22 |
| hypercenter | C23×C32 | D3 |
| hypocenter | C3 | C2×C32×D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C32 ◁ C23×C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C33 | C2 | semidirect |
| 2. C22×C32×D3 | C2 | semidirect |
| 3. C32×#67 | C2 | direct (N × H) |
| 4. C2×C32×Dic3 | C2 | semidirect |
| 6. C2×C3×#67 | C3 | direct (N × H) |
| 9. C22×C33 | C22 | semidirect |
| 11. C2×C32×D3 | C22 | semidirect |
| 12. C32×Dic3 | C22 | semidirect |
| 15. C23×C32 | D3 | semidirect |
| 16. C23×C32 | C2×C3 | semidirect |
| 17. C22×C3×D3 | C2×C3 | semidirect |
| 20. C3×#67 | C2×C3 | direct (N × H) |
| 21. C2×C3×Dic3 | C2×C3 | semidirect |
| 23. C2×C33 | D4 | semidirect |
| 27. C2×#67 | C32 | direct (N × H) |
| 36. C22×C32 | C2×D3 | semidirect |
| 38. C22×C32 | C22×C3 | semidirect |
| 40. C2×C3×D3 | C22×C3 | semidirect |
| 42. C3×Dic3 | C22×C3 | semidirect |
| 43. C33 | C2×D4 | semidirect |
| 46. C23×C3 | C3×D3 | semidirect |
| 47. C23×C3 | C2×C32 | semidirect |
| 48. C22×D3 | C2×C32 | semidirect |
| 51. #67 | C2×C32 | direct (N × H) |
| 52. C2×Dic3 | C2×C32 | semidirect |
| 59. C2×C32 | #67 | direct (N × H) |
| 60. C2×C32 | C3×D4 | semidirect |
| 73. C22×C3 | C2×C3×D3 | semidirect |
| 75. C22×C3 | C22×C32 | semidirect |
| 77. C2×D3 | C22×C32 | semidirect |
| 79. Dic3 | C22×C32 | semidirect |
| 81. C32 | C2×#67 | direct (N × H) |
| 82. C32 | C2×C3×D4 | semidirect |
| 84. C23 | C32×D3 | semidirect |
| 93. C2×C3 | C3×#67 | direct (N × H) |
| 94. C2×C3 | C32×D4 | semidirect |
| 102. C22 | C2×C32×D3 | semidirect |
| 105. C3 | C2×C3×#67 | direct (N × H) |
| 106. C3 | C2×C32×D4 | semidirect |
| 109. C2 | C32×#67 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C33 | 2 |
| C22×C32×D3 | 2 |
| C32×#67 | 2 |
| C2×C32×Dic3 | 2 |
| C2×C32×D4 | 3 |
| C2×C3×#67 | 3 |