Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
238 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #340 |
| commutator | C2×C8 | C3×C42 |
| Frattini, cosocle | C2×C4×C8 | C22×C3 |
| prime radical | C23×C3 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #7302 | C3 | direct (N × H) |
| 9. C3×C4⋊C16 | C4 | semidirect |
| 14. C3×#334 | C4 | semidirect |
| 37. C4⋊C16 | C3×C4 | semidirect |
| 42. #334 | C3×C4 | semidirect |
| 164. C3 | #7302 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#2163 | 2 |
| C3×C4×#106 | 2 |
| C3×#1566 | 2 |
| #7302 | 3 |