Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1462 conjugacy classes of subgroups; only the 462 normal subgroups are shown.
This group has 1462 conjugacy classes of subgroups — too many to draw. Only its 462 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | C24 |
| commutator | C23 | C23×C3×C4 |
| Frattini, cosocle | C24 | C24×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. #14663 | C3 | direct (N × H) |
| 8. C2×C3×C4⋊Q8 | C4 | semidirect |
| 36. C2×C4⋊Q8 | C3×C4 | semidirect |
| 152. C3 | #14663 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×C4⋊Q8 | 2 |
| C2×C3×#383 | 2 |
| C3×#1446 | 2 |
| C3×#1445 | 2 |
| #14663 | 3 |