Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
964 conjugacy classes of subgroups; only the 553 normal subgroups are shown.
This group has 964 conjugacy classes of subgroups — too many to draw. Only its 553 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 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C2×C3×C4×Q8 | C2 |
| prime radical | C23×C3 | C24 |
| hypercenter | C2×C4×Q8 | D3 |
| hypocenter | C3 | #3502 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #5570 | C2 | semidirect |
| 4. #5569 | C2 | semidirect |
| 5. Q8×Dic6 | C2 | semidirect |
| 33. C2×C3×Q8 | Q8 | semidirect |
| 83. C2×Q8 | Dic6 | semidirect |
| 136. C3 | #3502 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C4×Q8 | 2 |
| #5812 | 2 |
| #5570 | 2 |
| #5569 | 2 |
| Q8×Dic6 | 2 |
| #5500 | 2 |
| #5470 | 2 |
| #3502 | 3 |