Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
452 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×#121 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#472 | C2 |
| socle | C2×C3 | C2×#121 |
| hypercenter | #472 | D3 |
| hypocenter | C3 | #3067 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #472 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5773 | C2 | semidirect |
| 2. C2×#912 | C2 | semidirect |
| 3. D3×#131 | C2 | semidirect |
| 4. C3×#472 | C2 | semidirect |
| 5. #5200 | C2 | semidirect |
| 6. #5160 | C2 | semidirect |
| 7. #5143 | C2 | semidirect |
| 8. #4896 | C2 | semidirect |
| 9. D3×#108 | C2 | semidirect |
| 10. #4748 | C2 | semidirect |
| 11. #4745 | C2 | semidirect |
| 16. C2×C3×D8 | C22 | semidirect |
| 20. C2×#214 | C22 | semidirect |
| 36. C2×Dic12 | C22 | semidirect |
| 41. C3×#108 | C22 | semidirect |
| 42. C3×#100 | C22 | semidirect |
| 43. #833 | C22 | semidirect |
| 44. #824 | C22 | semidirect |
| 45. #824 | C22 | semidirect |
| 46. #822 | C22 | semidirect |
| 47. #819 | C22 | semidirect |
| 54. #472 | D3 | semidirect |
| 130. C2×D8 | C2×D3 | semidirect |
| 144. #108 | C2×D3 | semidirect |
| 148. #100 | C2×D3 | semidirect |
| 325. C3 | #3067 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5773 | 2 |
| C2×#912 | 2 |
| D3×#131 | 2 |
| C3×#472 | 2 |
| #5200 | 2 |
| #5160 | 2 |
| #5143 | 2 |
| #4896 | 2 |
| D3×#108 | 2 |
| #4748 | 2 |
| #4745 | 2 |
| #3067 | 3 |