Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
288 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 78 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 | C2×C3×#508 | C2 |
| 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 |
| 3. C2×C3×SD64 | C2 | semidirect |
| 4. C3×#2216 | C2 | semidirect |
| 5. C3×#1416 | C2 | semidirect |
| 6. C3×#1415 | C2 | semidirect |
| 7. C3×C22⋊C32 | C2 | semidirect |
| 8. #13666 | C3 | direct (N × H) |
| 25. C2×Q64 | C2×C3 | semidirect |
| 26. C2×SD64 | C2×C3 | semidirect |
| 27. #2216 | C2×C3 | semidirect |
| 28. #1416 | C2×C3 | semidirect |
| 29. #1415 | C2×C3 | semidirect |
| 30. C22⋊C32 | C2×C3 | semidirect |
| 215. C3 | #13666 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#508 | 2 |
| C2×C3×Q64 | 2 |
| C2×C3×SD64 | 2 |
| C3×#2216 | 2 |
| C3×#1416 | 2 |
| C3×#1415 | 2 |
| C3×C22⋊C32 | 2 |
| #13666 | 3 |