Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1422 conjugacy classes of subgroups; only the 451 normal subgroups are shown.
This group has 1422 conjugacy classes of subgroups — too many to draw. Only its 451 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×#383 | C2 |
| socle, prime radical, cosocle | C24×C3 | C24 |
| hypercenter | C2×#383 | D3 |
| hypocenter | C3 | #14710 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C24 ◁ C2×#383 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 9. C3×#383 | C4 | semidirect |
| 36. #383 | Dic3 | semidirect |
| 55. C3×C42 | C4⋊C4 | semidirect |
| 104. C42 | C4⋊Dic3 | semidirect |
| 146. C3 | #14710 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#383 | 2 |
| C2×#4937 | 2 |
| C2×C42×Dic3 | 2 |
| #69407 | 2 |
| #14710 | 3 |