Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
512 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 512 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C2×C3×C4×D4 | C2 |
| prime radical | C22×C3×D4 | C22 |
| hypercenter | C2×C4×D4 | D3 |
| hypocenter | C3 | #2633 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#870 | C2 | semidirect |
| 4. #5210 | C2 | semidirect |
| 5. #4983 | C2 | semidirect |
| 6. #4953 | C2 | semidirect |
| 7. #4949 | C2 | semidirect |
| 8. #4937 | C2 | semidirect |
| 183. C3 | #2633 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C4×D4 | 2 |
| C2×#870 | 2 |
| #5222 | 2 |
| #5210 | 2 |
| #4983 | 2 |
| #4953 | 2 |
| #4949 | 2 |
| #4937 | 2 |
| #2633 | 3 |