Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
220 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 90 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. ΓL(1,𝔽8)×C4⋊C8 | C2 | semidirect |
| 4. #6389⋊C3 | C2 | semidirect |
| 5. C4⋊(ΓL(1,𝔽8)⋊C8) | C2 | semidirect |
| 6. #89607 | C3 | semidirect |
| 17. C7×C4⋊C8 | C2×C3 | semidirect |
| 19. #6389 | C2×C3 | semidirect |
| 20. C4⋊(C7⋊C8) | C2×C3 | semidirect |
| 89. C4⋊C8 | AGL(1,𝔽7) | semidirect |
| 116. ΓL(1,𝔽8) | #432 | semidirect |
| 154. C7 | C3×#432 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C42×AGL(1,𝔽7) | 2 |
| ΓL(1,𝔽8)×C4⋊C8 | 2 |
| (C2×AGL(1,𝔽7))⋊C8 | 2 |
| #6389⋊C3 | 2 |
| C4⋊(ΓL(1,𝔽8)⋊C8) | 2 |
| #89607 | 3 |
| C3×#432 | 7 |