Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
452 conjugacy classes of subgroups; only the 177 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 177 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×#431 | C2 |
| prime radical | C22×D12 | C4 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #431 | D3 |
| hypocenter | C3 | #2975 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #431 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#431 | C2 | semidirect |
| 3. #4925 | C2 | semidirect |
| 4. #4838 | C2 | semidirect |
| 5. #4726 | C2 | semidirect |
| 6. #4688 | C2 | semidirect |
| 13. C2×ΓL(1,𝔽25) | C22 | semidirect |
| 18. C3×C4⋊C8 | C22 | semidirect |
| 20. C3×C4×C8 | C22 | semidirect |
| 22. (C2×D3)⋊C8 | C22 | semidirect |
| 23. C4⋊(C3⋊C8) | C22 | semidirect |
| 24. C4×C3⋊C8 | C22 | semidirect |
| 29. #431 | D3 | semidirect |
| 64. C4⋊C8 | C2×D3 | semidirect |
| 69. C4×C8 | C2×D3 | semidirect |
| 77. C2×C3×C4 | M16 | semidirect |
| 142. C2×C4 | ΓL(1,𝔽25) | semidirect |
| 171. C3 | #2975 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D12 | 2 |
| C3×#431 | 2 |
| #4925 | 2 |
| #4838 | 2 |
| #4726 | 2 |
| #4688 | 2 |
| #2975 | 3 |