Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
501 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 501 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 |
|---|---|---|
| 2. C2×C8×SL(2,𝔽3) | C2 | semidirect |
| 3. C2×SL(2,𝔽3)⋊C8 | C2 | semidirect |
| 4. #73018 | C2 | direct (N × H) |
| 12. C8×SL(2,𝔽3) | C22 | semidirect |
| 13. SL(2,𝔽3)⋊C8 | C22 | semidirect |
| 14. C2×C8×Q8 | D3 | semidirect |
| 39. C8×Q8 | C2×D3 | semidirect |
| 67. C2×SL(2,𝔽3) | M16 | semidirect |
| 133. SL(2,𝔽3) | C2×M16 | semidirect |
| 139. C2×Q8 | ΓL(1,𝔽25) | semidirect |
| 167. C2×C8 | GL(2,𝔽3) | semidirect |
| 200. Q8 | C2×ΓL(1,𝔽25) | semidirect |
| 225. C8 | C2×GL(2,𝔽3) | semidirect |
| 250. C2 | #73018 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×GL(2,𝔽3) | 2 |
| C2×C8×SL(2,𝔽3) | 2 |
| C2×SL(2,𝔽3)⋊C8 | 2 |
| #73018 | 2 |
| C2×#1583 | 3 |
| C22×ΓL(1,𝔽25) | 4 |