Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
3350 conjugacy classes of subgroups; only the 827 normal subgroups are shown.
This group has 3350 conjugacy classes of subgroups — too many to draw. Only its 827 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. C22×C3×C4⋊Q8 | C2 | semidirect |
| 2. C23×#216 | C2 | semidirect |
| 5. C2×#5086 | C2 | direct (N × H) |
| 6. C22×#803 | C2 | semidirect |
| 12. C2×C3×C4⋊Q8 | C22 | semidirect |
| 16. C22×#216 | C22 | semidirect |
| 23. #5086 | C22 | direct (N × H) |
| 24. C2×#803 | C22 | semidirect |
| 27. C22×C4⋊Q8 | D3 | semidirect |
| 47. C3×C4⋊Q8 | C23 | semidirect |
| 52. C2×#216 | C23 | semidirect |
| 60. #803 | C23 | semidirect |
| 70. C2×C4⋊Q8 | C2×D3 | semidirect |
| 134. C4⋊Q8 | C22×D3 | semidirect |
| 201. C22×C3 | #497 | semidirect |
| 238. C2×C3 | C2×#497 | semidirect |
| 245. C22 | #5086 | direct (N × H) |
| 254. C3 | C22×#497 | semidirect |
| 256. C2 | C2×#5086 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×C4⋊Q8 | 2 |
| C23×#216 | 2 |
| C23×#215 | 2 |
| C22×#875 | 2 |
| C2×#5086 | 2 |
| C22×#803 | 2 |
| C22×#497 | 3 |