Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
420 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#466 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #466 | D3 |
| hypocenter | C3 | #3334 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #466 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5611 | C2 | semidirect |
| 2. C3×#466 | C2 | semidirect |
| 3. #5157 | C2 | semidirect |
| 4. Dic3×D8 | C2 | semidirect |
| 5. C8×#67 | C2 | semidirect |
| 6. C2×#818 | C2 | semidirect |
| 7. #5033 | C2 | semidirect |
| 8. D3×#107 | C2 | semidirect |
| 9. #4874 | C2 | semidirect |
| 10. #4774 | C2 | semidirect |
| 11. #4759 | C2 | semidirect |
| 46. #466 | D3 | semidirect |
| 121. #67 | D8 | semidirect |
| 134. C3×C8 | #41 | semidirect |
| 214. C8 | #237 | semidirect |
| 240. C3 | #3334 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5611 | 2 |
| C3×#466 | 2 |
| #5157 | 2 |
| Dic3×D8 | 2 |
| C8×#67 | 2 |
| C2×#818 | 2 |
| #5033 | 2 |
| D3×#107 | 2 |
| #4874 | 2 |
| #4774 | 2 |
| #4759 | 2 |
| #3334 | 3 |