Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
676 conjugacy classes of subgroups; only the 272 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 272 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)×#117 | C2 | semidirect |
| 3. #6489⋊C3 | C2 | semidirect |
| 4. #6456⋊C3 | C2 | semidirect |
| 5. C22⋊C4×AGL(1,𝔽7) | C2 | semidirect |
| 6. C4×C4⋊AGL(1,𝔽7) | C2 | semidirect |
| 7. #90212 | C3 | semidirect |
| 10. C2×C4⋊AGL(1,𝔽7) | C4 | semidirect |
| 13. C2×C4×AGL(1,𝔽7) | C22 | semidirect |
| 14. C4⋊C4×ΓL(1,𝔽8) | C22 | semidirect |
| 16. C42×ΓL(1,𝔽8) | C22 | semidirect |
| 18. (C2×AGL(1,𝔽7))⋊C4 | C22 | semidirect |
| 19. C4⋊(ΓL(1,𝔽8)⋊C4) | C22 | semidirect |
| 20. C4×ΓL(1,𝔽8)⋊C4 | C22 | semidirect |
| 22. C7×#117 | C2×C3 | semidirect |
| 23. #6489 | C2×C3 | semidirect |
| 24. #6456 | C2×C3 | semidirect |
| 25. D7×C22⋊C4 | C2×C3 | semidirect |
| 26. C4×D28 | C2×C3 | semidirect |
| 30. C22×AGL(1,𝔽7) | C2×C4 | semidirect |
| 32. C2×C4×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 37. C4⋊AGL(1,𝔽7) | C2×C4 | semidirect |
| 38. C4⋊AGL(1,𝔽7) | C2×C4 | semidirect |
| 42. C2×D28 | C3×C4 | semidirect |
| 45. C2×C4×D7 | C22×C3 | semidirect |
| 46. C7×C4⋊C4 | C22×C3 | semidirect |
| 48. C42×C7 | C22×C3 | semidirect |
| 50. (C2×D7)⋊C4 | C22×C3 | semidirect |
| 51. C4⋊Dic7 | C22×C3 | semidirect |
| 52. C4×Dic7 | C22×C3 | semidirect |
| 72. C22×D7 | C2×C3×C4 | semidirect |
| 74. C2×C4×C7 | C2×C3×C4 | semidirect |
| 79. D28 | C2×C3×C4 | semidirect |
| 80. D28 | C2×C3×C4 | semidirect |
| 103. #117 | AGL(1,𝔽7) | semidirect |
| 127. ΓL(1,𝔽8) | #518 | semidirect |
| 135. C4⋊C4 | C2×AGL(1,𝔽7) | semidirect |
| 140. C42 | C2×AGL(1,𝔽7) | semidirect |
| 163. C2×C4 | C4×AGL(1,𝔽7) | semidirect |
| 167. C7 | C3×#518 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4⋊AGL(1,𝔽7) | 2 |
| ΓL(1,𝔽8)×#117 | 2 |
| #6489⋊C3 | 2 |
| #6456⋊C3 | 2 |
| C22⋊C4×AGL(1,𝔽7) | 2 |
| C4×C4⋊AGL(1,𝔽7) | 2 |
| #90212 | 3 |
| C3×#518 | 7 |