Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
284 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 66 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. ΓL(1,𝔽8)×#137 | C2 | semidirect |
| 3. #6507⋊C3 | C2 | semidirect |
| 4. #6505⋊C3 | C2 | semidirect |
| 5. #6411⋊C3 | C2 | semidirect |
| 6. #6399⋊C3 | C2 | semidirect |
| 7. #6398⋊C3 | C2 | semidirect |
| 8. #89977 | C3 | semidirect |
| 27. C7×#137 | C2×C3 | semidirect |
| 28. #6507 | C2×C3 | semidirect |
| 29. #6505 | C2×C3 | semidirect |
| 30. #6411 | C2×C3 | semidirect |
| 31. #6399 | C2×C3 | semidirect |
| 32. #6398 | C2×C3 | semidirect |
| 136. #137 | AGL(1,𝔽7) | semidirect |
| 173. ΓL(1,𝔽8) | #456 | semidirect |
| 227. C7 | C3×#456 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10621519 | 2 |
| ΓL(1,𝔽8)×#137 | 2 |
| #6507⋊C3 | 2 |
| #6505⋊C3 | 2 |
| #6411⋊C3 | 2 |
| #6399⋊C3 | 2 |
| #6398⋊C3 | 2 |
| #89977 | 3 |
| C3×#456 | 7 |