Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
284 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 82 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 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23×C3 |
| prime radical | C23×C3 | D16 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ C22×C3×C16 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×Q64 | C2 | semidirect |
| 4. C3×#1416 | C2 | semidirect |
| 6. #13665 | C3 | direct (N × H) |
| 19. C2×Q64 | C2×C3 | semidirect |
| 21. #1416 | C2×C3 | semidirect |
| 131. C22×C3 | Q64 | semidirect |
| 162. C22 | C3×Q64 | semidirect |
| 170. C3 | #13665 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×Q32 | 2 |
| C2×C3×Q64 | 2 |
| C3×#2216 | 2 |
| C3×#1416 | 2 |
| C3×C22⋊C32 | 2 |
| #13665 | 3 |