Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1956 conjugacy classes of subgroups; only the 1114 normal subgroups are shown.
This group has 1956 conjugacy classes of subgroups — too many to draw. Only its 1114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C22×C42 |
| commutator | C4 | C22×C3×C42 |
| Frattini, cosocle | C22×C4 | C24×C3 |
| socle | C2×C3 | C22×#95 |
| prime radical | C22×C3×#41 | C22 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×#338 | C2 | semidirect |
| 3. #20340 | C3 | direct (N × H) |
| 10. C2×#338 | C2×C3 | semidirect |
| 101. C3 | #20340 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#568 | 2 |
| C2×C3×#338 | 2 |
| #20340 | 3 |