Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1708 conjugacy classes of subgroups; only the 315 normal subgroups are shown.
This group has 1708 conjugacy classes of subgroups — too many to draw. Only its 315 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C3×C8 | C2 | semidirect |
| 3. #84031 | C2 | semidirect |
| 4. C2×#5114 | C2 | semidirect |
| 5. #71973 | C2 | semidirect |
| 6. #71972 | C2 | semidirect |
| 7. #71970 | C2 | semidirect |
| 16. C22×#204 | C22 | semidirect |
| 23. C2×#821 | C22 | semidirect |
| 24. C2×#817 | C22 | semidirect |
| 25. C2×#816 | C22 | semidirect |
| 26. #4551 | C22 | semidirect |
| 27. C24×C8 | D3 | semidirect |
| 46. C22×C3×C8 | D4 | semidirect |
| 87. C24×C3 | SD16 | semidirect |
| 138. C22×C8 | #67 | semidirect |
| 190. C3×C8 | #120 | semidirect |
| 194. C24 | #204 | semidirect |
| 236. C22×C3 | #465 | semidirect |
| 281. C8 | #953 | semidirect |
| 295. C22 | #5114 | semidirect |
| 305. C3 | #32929 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C3×C8 | 2 |
| #84032 | 2 |
| #84031 | 2 |
| C2×#5114 | 2 |
| #71973 | 2 |
| #71972 | 2 |
| #71970 | 2 |
| #32929 | 3 |