Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
224 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 79 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. C3×C8×Q16 | C2 | semidirect |
| 4. #67765 | C2 | semidirect |
| 6. #67469 | C2 | semidirect |
| 7. #67438 | C2 | semidirect |
| 26. C8×Q16 | D3 | semidirect |
| 71. C3×Q16 | M16 | semidirect |
| 137. Q16 | ΓL(1,𝔽25) | semidirect |
| 206. C3 | #12075 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C8×Q16 | 2 |
| C4×#828 | 2 |
| #68052 | 2 |
| #67765 | 2 |
| #67471 | 2 |
| #67469 | 2 |
| #67438 | 2 |
| #12075 | 3 |