Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
330 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 110 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 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C23×C3 |
| prime radical | C3×#2246 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C82 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#2248 | C2 | semidirect |
| 2. C3×#2247 | C2 | semidirect |
| 4. C3×#1709 | C2 | semidirect |
| 5. C3×C8×C16 | C2 | semidirect |
| 6. #12518 | C3 | direct (N × H) |
| 16. #2248 | C2×C3 | semidirect |
| 17. #2247 | C2×C3 | semidirect |
| 19. #1709 | C2×C3 | semidirect |
| 20. C8×C16 | C2×C3 | semidirect |
| 127. C3 | #12518 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#2248 | 2 |
| C3×#2247 | 2 |
| C3×#2246 | 2 |
| C3×#1709 | 2 |
| C3×C8×C16 | 2 |
| #12518 | 3 |