Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
268 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 70 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 | #331 |
| commutator | C42 | C2×C3×C8 |
| Frattini, cosocle | C43 | C22×C3 |
| socle | C2×C3 | #1270 |
| prime radical | C3×#532 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×C43 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#1276 | C2 | semidirect |
| 3. #7106 | C3 | direct (N × H) |
| 10. #1276 | C2×C3 | semidirect |
| 13. C3×C4⋊Q8 | C8 | semidirect |
| 14. C3×#127 | C8 | semidirect |
| 56. C4⋊Q8 | C3×C8 | semidirect |
| 57. #127 | C3×C8 | semidirect |
| 164. C3 | #7106 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#1962 | 2 |
| C3×#1276 | 2 |
| #7106 | 3 |