Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1290 conjugacy classes of subgroups; only the 327 normal subgroups are shown.
This group has 1290 conjugacy classes of subgroups — too many to draw. Only its 327 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C24 | C23×D3 |
| commutator | C23×C3 | C23×C4 |
| Fitting | C2×C3×#399 | C2 |
| socle, cosocle | C24×C3 | C24 |
| prime radical | C25×C3 | C23 |
| hypercenter | C2×#399 | D3 |
| hypocenter | C3 | #14618 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C2×#399 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #71564 | C2 | semidirect |
| 5. #69416 | C2 | semidirect |
| 12. C3×#399 | C4 | semidirect |
| 49. #399 | Dic3 | semidirect |
| 252. C3 | #14618 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#399 | 2 |
| #71564 | 2 |
| #71548 | 2 |
| #71545 | 2 |
| #69416 | 2 |
| #14618 | 3 |