Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
500 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 189 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×#423 | C2 |
| prime radical | #5477 | C2 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #423 | D3 |
| hypocenter | C3 | #2931 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #423 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#907 | C2 | semidirect |
| 3. C3×#423 | C2 | semidirect |
| 4. #5129 | C2 | semidirect |
| 5. C2×C4×C3⋊C8 | C2 | semidirect |
| 6. #4838 | C2 | semidirect |
| 7. #4834 | C2 | semidirect |
| 30. #423 | D3 | semidirect |
| 50. C2×C3⋊C8 | D4 | semidirect |
| 123. C3×C4 | #131 | semidirect |
| 159. C4 | #907 | semidirect |
| 162. C3 | #2931 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#907 | 2 |
| #5477 | 2 |
| C3×#423 | 2 |
| #5129 | 2 |
| C2×C4×C3⋊C8 | 2 |
| #4838 | 2 |
| #4834 | 2 |
| #2931 | 3 |