Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
560 conjugacy classes of subgroups; only the 214 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 214 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. C2×Q8×AGL(1,𝔽7) | C2 | semidirect |
| 2. C4×Q8×ΓL(1,𝔽8) | C2 | semidirect |
| 3. #6460⋊C3 | C2 | semidirect |
| 5. #6432⋊C3 | C2 | semidirect |
| 6. Q8×D28 | C3 | semidirect |
| 17. C2×Q8×D7 | C2×C3 | semidirect |
| 18. C4×C7×Q8 | C2×C3 | semidirect |
| 19. #6460 | C2×C3 | semidirect |
| 21. #6432 | C2×C3 | semidirect |
| 24. Q8×ΓL(1,𝔽8) | D4 | semidirect |
| 29. C4⋊AGL(1,𝔽7) | Q8 | direct (N × H) |
| 50. C2×AGL(1,𝔽7) | C2×Q8 | semidirect |
| 53. C4×ΓL(1,𝔽8) | C2×Q8 | semidirect |
| 57. C7×Q8 | C3×D4 | semidirect |
| 62. D28 | C3×Q8 | semidirect |
| 82. C4×Q8 | AGL(1,𝔽7) | semidirect |
| 87. C2×D7 | C2×C3×Q8 | semidirect |
| 90. C4×C7 | C2×C3×Q8 | semidirect |
| 101. ΓL(1,𝔽8) | D4×Q8 | semidirect |
| 126. Q8 | C4⋊AGL(1,𝔽7) | direct (N × H) |
| 133. C7 | C3×D4×Q8 | semidirect |
| 145. C4 | Q8×AGL(1,𝔽7) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×AGL(1,𝔽7) | 2 |
| C4×Q8×ΓL(1,𝔽8) | 2 |
| #6460⋊C3 | 2 |
| C4×C4⋊AGL(1,𝔽7) | 2 |
| #6432⋊C3 | 2 |
| Q8×D28 | 3 |
| C3×D4×Q8 | 7 |