Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
446 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 446 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×#493 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #493 | D3 |
| hypocenter | C3 | #3184 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #493 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5716 | C2 | semidirect |
| 2. C3×#493 | C2 | semidirect |
| 3. #5159 | C2 | semidirect |
| 4. #5152 | C2 | semidirect |
| 5. #5070 | C2 | semidirect |
| 6. C4×ΓL(1,𝔽25) | C2 | semidirect |
| 7. #4680 | C2 | semidirect |
| 27. #493 | D3 | semidirect |
| 83. C3×C8 | #41 | semidirect |
| 132. C8 | #237 | semidirect |
| 148. C3 | #3184 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5716 | 2 |
| C3×#493 | 2 |
| #5159 | 2 |
| #5152 | 2 |
| #5070 | 2 |
| C4×ΓL(1,𝔽25) | 2 |
| #4680 | 2 |
| #3184 | 3 |