Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
406 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 406 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×#556 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #556 | D3 |
| hypocenter | C3 | #3397 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ #556 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#556 | C2 | semidirect |
| 2. #5577 | C2 | semidirect |
| 3. #5085 | C2 | semidirect |
| 4. #5083 | C2 | semidirect |
| 5. #5058 | C2 | semidirect |
| 6. #5057 | C2 | semidirect |
| 7. #5027 | C2 | semidirect |
| 8. #5026 | C2 | semidirect |
| 9. #5024 | C2 | semidirect |
| 10. #5015 | C2 | semidirect |
| 11. C4×#213 | C2 | semidirect |
| 42. #556 | D3 | semidirect |
| 147. C3⋊C8 | #41 | semidirect |
| 258. C3 | #3397 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#556 | 2 |
| #5577 | 2 |
| #5085 | 2 |
| #5083 | 2 |
| #5058 | 2 |
| #5057 | 2 |
| #5027 | 2 |
| #5026 | 2 |
| #5024 | 2 |
| #5015 | 2 |
| C4×#213 | 2 |
| #3397 | 3 |