Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
1700 conjugacy classes of subgroups; only the 928 normal subgroups are shown.
This group has 1700 conjugacy classes of subgroups — too many to draw. Only its 928 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. C22×Q8×ΓL(1,𝔽8) | C2 | semidirect |
| 2. C2×Q8×AGL(1,𝔽7) | C2 | direct (N × H) |
| 4. C22×ΓL(1,𝔽8)⋊Q8 | C2 | semidirect |
| 5. C22×Q8×D7 | C3 | semidirect |
| 7. C2×Q8×ΓL(1,𝔽8) | C22 | semidirect |
| 10. Q8×AGL(1,𝔽7) | C22 | direct (N × H) |
| 12. C2×ΓL(1,𝔽8)⋊Q8 | C22 | semidirect |
| 13. C22×C7×Q8 | C2×C3 | semidirect |
| 14. C2×Q8×D7 | C2×C3 | semidirect |
| 16. C22×Dic14 | C2×C3 | semidirect |
| 19. C22×AGL(1,𝔽7) | Q8 | direct (N × H) |
| 21. Q8×ΓL(1,𝔽8) | C23 | semidirect |
| 25. ΓL(1,𝔽8)⋊Q8 | C23 | semidirect |
| 27. C2×C7×Q8 | C22×C3 | semidirect |
| 30. Q8×D7 | C22×C3 | semidirect |
| 32. C2×Dic14 | C22×C3 | semidirect |
| 37. C22×ΓL(1,𝔽8) | C2×Q8 | semidirect |
| 39. C2×AGL(1,𝔽7) | C2×Q8 | direct (N × H) |
| 45. C22×D7 | C3×Q8 | semidirect |
| 47. C7×Q8 | C23×C3 | semidirect |
| 51. Dic14 | C23×C3 | semidirect |
| 59. C2×ΓL(1,𝔽8) | C22×Q8 | semidirect |
| 61. AGL(1,𝔽7) | C22×Q8 | direct (N × H) |
| 64. C22×Q8 | AGL(1,𝔽7) | direct (N × H) |
| 66. C22×C7 | C2×C3×Q8 | semidirect |
| 68. C2×D7 | C2×C3×Q8 | semidirect |
| 80. ΓL(1,𝔽8) | C23×Q8 | semidirect |
| 84. C2×Q8 | C2×AGL(1,𝔽7) | direct (N × H) |
| 88. C2×C7 | C22×C3×Q8 | semidirect |
| 90. D7 | C22×C3×Q8 | semidirect |
| 101. Q8 | C22×AGL(1,𝔽7) | direct (N × H) |
| 105. C7 | C23×C3×Q8 | semidirect |
| 111. C22 | Q8×AGL(1,𝔽7) | direct (N × H) |
| 117. C2 | C2×Q8×AGL(1,𝔽7) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8×ΓL(1,𝔽8) | 2 |
| C2×Q8×AGL(1,𝔽7) | 2 |
| C22×C4×AGL(1,𝔽7) | 2 |
| C22×ΓL(1,𝔽8)⋊Q8 | 2 |
| C22×Q8×D7 | 3 |
| C23×C3×Q8 | 7 |