Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
684 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 684 conjugacy classes of subgroups — too many to draw. Only its 216 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. C23×C4×ΓL(1,𝔽8) | C2 | semidirect |
| 3. C2×(C22⋊Dic7)⋊C3 | C2 | semidirect |
| 4. #6492⋊C3 | C2 | direct (N × H) |
| 5. C2×(C2×AGL(1,𝔽7))⋊C4 | C2 | semidirect |
| 6. C2×(ΓL(1,𝔽8)⋊C4)⋊C4 | C2 | semidirect |
| 7. C2×#6492 | C3 | semidirect |
| 12. C22×C4×ΓL(1,𝔽8) | C22 | semidirect |
| 17. (C22⋊Dic7)⋊C3 | C22 | semidirect |
| 18. (C2×AGL(1,𝔽7))⋊C4 | C22 | semidirect |
| 19. (ΓL(1,𝔽8)⋊C4)⋊C4 | C22 | semidirect |
| 20. C23×C4×C7 | C2×C3 | semidirect |
| 22. C2×C22⋊Dic7 | C2×C3 | semidirect |
| 23. #6492 | C2×C3 | semidirect |
| 24. C2×(C2×D7)⋊C4 | C2×C3 | semidirect |
| 25. C2×Dic7⋊C4 | C2×C3 | semidirect |
| 46. C22×C4×C7 | C22×C3 | semidirect |
| 51. C22⋊Dic7 | C22×C3 | semidirect |
| 52. (C2×D7)⋊C4 | C22×C3 | semidirect |
| 53. Dic7⋊C4 | C22×C3 | semidirect |
| 101. C2×ΓL(1,𝔽8) | #123 | semidirect |
| 106. C23×C4 | AGL(1,𝔽7) | semidirect |
| 138. ΓL(1,𝔽8) | C2×#123 | semidirect |
| 147. C22×C4 | C2×AGL(1,𝔽7) | semidirect |
| 152. C2×C7 | C3×#123 | semidirect |
| 183. C7 | C2×C3×#123 | semidirect |
| 204. C2 | #6492⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C4×ΓL(1,𝔽8) | 2 |
| C22×(C22⋊D7)⋊C3 | 2 |
| C2×(C22⋊Dic7)⋊C3 | 2 |
| #6492⋊C3 | 2 |
| C2×(C2×AGL(1,𝔽7))⋊C4 | 2 |
| C2×(ΓL(1,𝔽8)⋊C4)⋊C4 | 2 |
| C2×#6492 | 3 |
| C2×C3×#123 | 7 |