Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
208 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 208 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 | C42 | C22×D3 |
| commutator, socle | C22×C3 | C2×C42 |
| Fitting | C3×#322 | C2 |
| prime radical | C22×D3 | C42 |
| cosocle | C3×C42 | C23 |
| hypercenter | #322 | D3 |
| hypocenter | C3 | #1451 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C42 ◁ #322 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C3×#322 | C2 | semidirect |
| 5. #4455 | C2 | semidirect |
| 6. #4455 | C2 | semidirect |
| 7. #4454 | C2 | semidirect |
| 28. #322 | D3 | semidirect |
| 69. C3×C8 | M16 | semidirect |
| 113. C8 | ΓL(1,𝔽25) | semidirect |
| 128. C3 | #1451 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×ΓL(1,𝔽25) | 2 |
| #4685 | 2 |
| C4×D3×C8 | 2 |
| C3×#322 | 2 |
| #4455 | 2 |
| #4454 | 2 |
| #1451 | 3 |