Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
472 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 472 conjugacy classes of subgroups — too many to draw. Only its 174 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×Q8×ΓL(1,𝔽8) | C2 | semidirect |
| 2. #6510⋊C3 | C2 | semidirect |
| 3. #6457⋊C3 | C2 | semidirect |
| 4. #6456⋊C3 | C2 | semidirect |
| 5. #6438⋊C3 | C2 | semidirect |
| 7. C4×C4⋊AGL(1,𝔽7) | C2 | semidirect |
| 8. C4×ΓL(1,𝔽8)⋊Q8 | C2 | semidirect |
| 9. #90276 | C3 | semidirect |
| 26. C4×C7×Q8 | C2×C3 | semidirect |
| 27. #6510 | C2×C3 | semidirect |
| 28. #6457 | C2×C3 | semidirect |
| 29. #6456 | C2×C3 | semidirect |
| 30. #6438 | C2×C3 | semidirect |
| 32. C4×D28 | C2×C3 | semidirect |
| 33. C4×Dic14 | C2×C3 | semidirect |
| 124. C4×Q8 | AGL(1,𝔽7) | semidirect |
| 152. ΓL(1,𝔽8) | #559 | semidirect |
| 198. C7 | C3×#559 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×Q8×ΓL(1,𝔽8) | 2 |
| #6510⋊C3 | 2 |
| #6457⋊C3 | 2 |
| #6456⋊C3 | 2 |
| #6438⋊C3 | 2 |
| #10620710 | 2 |
| C4×C4⋊AGL(1,𝔽7) | 2 |
| C4×ΓL(1,𝔽8)⋊Q8 | 2 |
| #90276 | 3 |
| C3×#559 | 7 |