Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
492 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 181 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×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting | C3×#424 | C2 |
| prime radical | #5524 | C2 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #424 | D3 |
| hypocenter | C3 | #2930 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #424 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#907 | C2 | semidirect |
| 2. C22×ΓL(1,𝔽25) | C2 | semidirect |
| 4. C3×#424 | C2 | semidirect |
| 5. #5126 | C2 | semidirect |
| 6. #5110 | C2 | semidirect |
| 7. #4997 | C2 | semidirect |
| 8. #4838 | C2 | semidirect |
| 9. #4836 | C2 | semidirect |
| 22. C2×ΓL(1,𝔽25) | C22 | semidirect |
| 23. C2×D3×C8 | C22 | semidirect |
| 29. C3×C4⋊C8 | C22 | semidirect |
| 31. (C2×D3)⋊C8 | C22 | semidirect |
| 32. #803 | C22 | semidirect |
| 33. C4×C3⋊C8 | C22 | semidirect |
| 39. #424 | D3 | semidirect |
| 68. C2×C3⋊C8 | D4 | semidirect |
| 94. C4⋊C8 | C2×D3 | semidirect |
| 221. C3 | #2930 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#907 | 2 |
| C22×ΓL(1,𝔽25) | 2 |
| #5524 | 2 |
| C3×#424 | 2 |
| #5126 | 2 |
| #5110 | 2 |
| #4997 | 2 |
| #4838 | 2 |
| #4836 | 2 |
| #2930 | 3 |