Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
248 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 72 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×Q16×ΓL(1,𝔽8) | C2 | semidirect |
| 3. #6507⋊C3 | C2 | semidirect |
| 4. C2×#1144⋊C3 | C2 | semidirect |
| 7. #6389⋊C3 | C2 | semidirect |
| 8. #89962 | C3 | semidirect |
| 22. C2×C7×Q16 | C2×C3 | semidirect |
| 24. #6507 | C2×C3 | semidirect |
| 25. C2×#1144 | C2×C3 | semidirect |
| 28. #6389 | C2×C3 | semidirect |
| 109. C2×Q16 | AGL(1,𝔽7) | semidirect |
| 140. ΓL(1,𝔽8) | #497 | semidirect |
| 182. C7 | C3×#497 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q16×ΓL(1,𝔽8) | 2 |
| #6510⋊C3 | 2 |
| #6507⋊C3 | 2 |
| C2×#1144⋊C3 | 2 |
| C2×#1143⋊C3 | 2 |
| C2×#1132⋊C3 | 2 |
| #6389⋊C3 | 2 |
| #89962 | 3 |
| C3×#497 | 7 |