Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
408 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D3 |
| commutator, socle | C23×C3 | C22×C4 |
| Fitting | C3×#336 | C2 |
| prime radical | C22×D12 | C4 |
| cosocle | C22×C3×C4 | C23 |
| hypercenter | #336 | D3 |
| hypocenter | C3 | #1870 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C4 ◁ #336 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×ΓL(1,𝔽25) | C2 | semidirect |
| 3. C2×(C2×D3)⋊C8 | C2 | semidirect |
| 4. C2×(C2×D3)⋊C8 | C2 | semidirect |
| 5. C2×(C2×D3)⋊C8 | C2 | semidirect |
| 6. C3×#336 | C2 | semidirect |
| 7. #4524 | C2 | semidirect |
| 34. #336 | D3 | semidirect |
| 62. C2×C3×C8 | D4 | semidirect |
| 73. C2×C3⋊C8 | D4 | semidirect |
| 108. C22×D3 | M16 | semidirect |
| 114. C2×C3×C4 | M16 | semidirect |
| 174. C2×C8 | D12 | semidirect |
| 244. C2×C4 | ΓL(1,𝔽25) | semidirect |
| 293. C3 | #1870 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×ΓL(1,𝔽25) | 2 |
| C2×C4×D12 | 2 |
| C2×(C2×D3)⋊C8 | 2 |
| C3×#336 | 2 |
| #4524 | 2 |
| #1870 | 3 |