Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
546 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 546 conjugacy classes of subgroups — too many to draw. Only its 122 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 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| prime radical | C2×C3×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×#494 | C2 | semidirect |
| 3. C2×C3×#362 | C2 | semidirect |
| 5. C3×#1376 | C2 | semidirect |
| 6. #13333 | C3 | direct (N × H) |
| 23. C2×#494 | C2×C3 | semidirect |
| 24. C2×#362 | C2×C3 | semidirect |
| 26. #1376 | C2×C3 | semidirect |
| 275. C3 | #13333 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#510 | 2 |
| C2×C3×#494 | 2 |
| C2×C3×#362 | 2 |
| C3×#1974 | 2 |
| C3×#1376 | 2 |
| #13333 | 3 |