Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
698 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 698 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23×C3 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×D16 | C2 | semidirect |
| 2. C2×C3×#493 | C2 | semidirect |
| 3. C2×C3×#357 | C2 | semidirect |
| 4. C3×#1970 | C2 | semidirect |
| 5. C3×#1374 | C2 | semidirect |
| 6. #13325 | C3 | direct (N × H) |
| 21. C22×D16 | C2×C3 | semidirect |
| 22. C2×#493 | C2×C3 | semidirect |
| 23. C2×#357 | C2×C3 | semidirect |
| 24. #1970 | C2×C3 | semidirect |
| 25. #1374 | C2×C3 | semidirect |
| 41. C2×C3×D8 | D4 | semidirect |
| 52. C2×C3×C16 | D4 | semidirect |
| 125. C2×D8 | C3×D4 | semidirect |
| 136. C2×C16 | C3×D4 | semidirect |
| 164. C2×C3×C4 | D16 | semidirect |
| 259. C2×C4 | C3×D16 | semidirect |
| 297. C3 | #13325 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×D16 | 2 |
| C2×C3×#493 | 2 |
| C2×C3×#357 | 2 |
| C3×#1970 | 2 |
| C3×#1374 | 2 |
| #13325 | 3 |