Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
398 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 398 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 | C22 | C22×#67 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#555 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #555 | D3 |
| hypocenter | C3 | #3303 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ #555 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#555 | C2 | semidirect |
| 2. #5551 | C2 | semidirect |
| 3. #5093 | C2 | semidirect |
| 4. #5084 | C2 | semidirect |
| 5. #5041 | C2 | semidirect |
| 6. #5037 | C2 | semidirect |
| 7. #5019 | C2 | semidirect |
| 8. C4×#213 | C2 | semidirect |
| 9. D4×C3⋊C8 | C2 | semidirect |
| 10. #5002 | C2 | semidirect |
| 11. #4996 | C2 | semidirect |
| 42. #555 | D3 | semidirect |
| 254. C3 | #3303 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#555 | 2 |
| #5551 | 2 |
| #5093 | 2 |
| #5084 | 2 |
| #5041 | 2 |
| #5037 | 2 |
| #5019 | 2 |
| C4×#213 | 2 |
| D4×C3⋊C8 | 2 |
| #5002 | 2 |
| #4996 | 2 |
| #3303 | 3 |