Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
224 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 224 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, hypercenter | C4×C16 | A4 |
| commutator, hypocenter | Q8 | C3×C4×C8 |
| Frattini | C2×C8 | C22×A4 |
| Fitting | C4×#504 | C3 |
| socle | C22 | C2×C8×A4 |
| prime radical | C2×#41⋊C3 | C2×C4 |
| cosocle | C2×#131 | C22×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C4×C16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C4×#504 | C3 | semidirect |
| 7. #504⋊C3 | C4 | direct (N × H) |
| 24. #504 | C3×C4 | semidirect |
| 130. C4 | #504⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#504⋊C3 | 2 |
| C4×#131⋊C3 | 2 |
| C4×#504 | 3 |
| C3×C4×C16 | 4 |