Finite Group DatabasebetaFinite Group DatabasebetaOrder 144.
116 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D32 |
| commutator, socle, cosocle | C2×C32 | C23 |
| Fitting | C22×C32 | C22 |
| hypercenter | C22 | D32 |
| hypocenter | C32 | C2×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×D3 | C2 | semidirect |
| 2. C2×D32 | C2 | semidirect |
| 3. #638 | C2 | semidirect |
| 4. C3×#67 | C2 | semidirect |
| 5. #626 | C2 | semidirect |
| 6. #625 | C2 | semidirect |
| 7. D3×Dic3 | C2 | semidirect |
| 10. C22×C32 | C22 | semidirect |
| 11. C2×C3⋊D3 | C22 | semidirect |
| 14. C2×C3×D3 | C22 | semidirect |
| 15. C2×C3×D3 | C22 | semidirect |
| 17. C3⋊Dic3 | C22 | semidirect |
| 18. C3×Dic3 | C22 | semidirect |
| 22. C22×D3 | D3 | semidirect |
| 30. #67 | D3 | direct (N × H) |
| 39. C3×D3 | D4 | semidirect |
| 46. C22×C3 | C2×D3 | semidirect |
| 47. C22×C3 | C2×D3 | semidirect |
| 55. C2×D3 | C2×D3 | semidirect |
| 56. C2×D3 | C2×D3 | semidirect |
| 61. Dic3 | C2×D3 | semidirect |
| 62. C32 | C2×D4 | semidirect |
| 84. D3 | #67 | direct (N × H) |
| 91. C22 | D32 | semidirect |
| 95. C3 | D3×D4 | semidirect |
| 96. C3 | C2×#67 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×D3 | 2 |
| C2×D32 | 2 |
| #638 | 2 |
| C3×#67 | 2 |
| #626 | 2 |
| #625 | 2 |
| D3×Dic3 | 2 |
| C2×#67 | 3 |
| D3×D4 | 3 |