Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
240 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 240 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C42 | C22×A4 |
| commutator, hypocenter | Q8 | C3×C4×C8 |
| Fitting | #17316 | C3 |
| socle | C22 | C42×A4 |
| prime radical | C2×#41⋊C3 | C2×C4 |
| cosocle | C4×#41 | C22×C3 |
| hypercenter | #322 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C42 ◁ #322 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #17316 | C3 | semidirect |
| 14. #131⋊C3 | C8 | semidirect |
| 38. #131 | C3×C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C8×#41⋊C3 | 2 |
| C4×#131⋊C3 | 2 |
| #17316 | 3 |
| C3×#322 | 4 |