Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
688 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 202 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 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C3 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ C23×C3×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#466 | C2 | semidirect |
| 2. C3×#2233 | C2 | semidirect |
| 3. C3×#2218 | C2 | semidirect |
| 4. C3×#2184 | C2 | semidirect |
| 5. C3×#2112 | C2 | semidirect |
| 6. #33758 | C3 | direct (N × H) |
| 7. C22×C3×D8 | C22 | semidirect |
| 10. C2×C3×D16 | C22 | semidirect |
| 11. C2×C3×M32 | C22 | semidirect |
| 15. C3×#466 | C22 | semidirect |
| 16. C2×C3×#107 | C22 | semidirect |
| 18. C3×#367 | C22 | semidirect |
| 19. C3×#357 | C22 | semidirect |
| 20. C3×C22⋊C16 | C22 | semidirect |
| 21. C2×#466 | C2×C3 | semidirect |
| 22. #2233 | C2×C3 | semidirect |
| 23. #2218 | C2×C3 | semidirect |
| 24. #2184 | C2×C3 | semidirect |
| 25. #2112 | C2×C3 | semidirect |
| 50. C22×D8 | C22×C3 | semidirect |
| 53. C2×D16 | C22×C3 | semidirect |
| 54. C2×M32 | C22×C3 | semidirect |
| 58. #466 | C22×C3 | semidirect |
| 59. C2×#107 | C22×C3 | semidirect |
| 61. #367 | C22×C3 | semidirect |
| 62. #357 | C22×C3 | semidirect |
| 63. C22⋊C16 | C22×C3 | semidirect |
| 239. C3 | #33758 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#466 | 2 |
| C3×#2233 | 2 |
| C3×#2218 | 2 |
| C3×#2184 | 2 |
| C3×#2112 | 2 |
| #33758 | 3 |