Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
660 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 660 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 |
|---|---|---|
| 2. C22×ΓL(1,𝔽8)⋊Q8 | C2 | semidirect |
| 3. C2×C42×ΓL(1,𝔽8) | C2 | semidirect |
| 4. C2×(C2×AGL(1,𝔽7))⋊C4 | C2 | semidirect |
| 5. #6438⋊C3 | C2 | direct (N × H) |
| 6. C2×#6438 | C3 | semidirect |
| 14. C2×ΓL(1,𝔽8)⋊Q8 | C22 | semidirect |
| 15. C42×ΓL(1,𝔽8) | C22 | semidirect |
| 16. (C2×AGL(1,𝔽7))⋊C4 | C22 | semidirect |
| 18. C22×Dic14 | C2×C3 | semidirect |
| 19. C2×C42×C7 | C2×C3 | semidirect |
| 20. C2×(C2×D7)⋊C4 | C2×C3 | semidirect |
| 21. #6438 | C2×C3 | semidirect |
| 42. C2×Dic14 | C22×C3 | semidirect |
| 43. C42×C7 | C22×C3 | semidirect |
| 44. (C2×D7)⋊C4 | C22×C3 | semidirect |
| 82. C2×ΓL(1,𝔽8) | #124 | semidirect |
| 90. C2×C42 | AGL(1,𝔽7) | semidirect |
| 112. ΓL(1,𝔽8) | C2×#124 | semidirect |
| 122. C42 | C2×AGL(1,𝔽7) | semidirect |
| 123. C2×C7 | C3×#124 | semidirect |
| 148. C7 | C2×C3×#124 | semidirect |
| 164. C2 | #6438⋊C3 | direct (N × H) |
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 |
| C22×ΓL(1,𝔽8)⋊Q8 | 2 |
| C2×C42×ΓL(1,𝔽8) | 2 |
| C2×(C2×AGL(1,𝔽7))⋊C4 | 2 |
| #6438⋊C3 | 2 |
| C2×#6438 | 3 |
| C2×C3×#124 | 7 |