Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
264 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 86 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. C4×D4×ΓL(1,𝔽8) | C2 | semidirect |
| 3. C4×C4⋊AGL(1,𝔽7) | C2 | semidirect |
| 4. #6406⋊C3 | C2 | semidirect |
| 5. #6383⋊C3 | C2 | semidirect |
| 6. #6382⋊C3 | C2 | semidirect |
| 7. #6377⋊C3 | C2 | semidirect |
| 8. #89786 | C3 | semidirect |
| 16. #1141⋊C3 | C4 | semidirect |
| 21. C4×C7×D4 | C2×C3 | semidirect |
| 23. C4×D28 | C2×C3 | semidirect |
| 24. #6406 | C2×C3 | semidirect |
| 25. #6383 | C2×C3 | semidirect |
| 26. #6382 | C2×C3 | semidirect |
| 27. #6377 | C2×C3 | semidirect |
| 32. D4×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 40. C4⋊AGL(1,𝔽7) | C2×C4 | semidirect |
| 43. ΓL(1,𝔽8)⋊C8 | C2×C4 | semidirect |
| 51. #1141 | C3×C4 | semidirect |
| 80. C7×D4 | C2×C3×C4 | semidirect |
| 88. D28 | C2×C3×C4 | semidirect |
| 91. C7⋊C8 | C2×C3×C4 | semidirect |
| 111. C4×D4 | AGL(1,𝔽7) | semidirect |
| 144. ΓL(1,𝔽8) | #442 | semidirect |
| 180. D4 | C4×AGL(1,𝔽7) | semidirect |
| 190. C7 | C3×#442 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×D4×ΓL(1,𝔽8) | 2 |
| C2×#1141⋊C3 | 2 |
| C4×C4⋊AGL(1,𝔽7) | 2 |
| #6406⋊C3 | 2 |
| #6383⋊C3 | 2 |
| #6382⋊C3 | 2 |
| #6377⋊C3 | 2 |
| #89786 | 3 |
| C3×#442 | 7 |