Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
452 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 200 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. ΓL(1,𝔽8)×#125 | C2 | semidirect |
| 4. #6453⋊C3 | C2 | semidirect |
| 5. #6452⋊C3 | C2 | semidirect |
| 7. #6433⋊C3 | C2 | semidirect |
| 8. D7×#125 | C3 | semidirect |
| 22. C7×#125 | C2×C3 | semidirect |
| 25. #6453 | C2×C3 | semidirect |
| 26. #6452 | C2×C3 | semidirect |
| 28. #6433 | C2×C3 | semidirect |
| 95. AGL(1,𝔽7) | #125 | direct (N × H) |
| 99. #125 | AGL(1,𝔽7) | direct (N × H) |
| 122. ΓL(1,𝔽8) | C2×#125 | semidirect |
| 138. D7 | C3×#125 | semidirect |
| 158. C7 | C2×C3×#125 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#125 | 2 |
| C4⋊C4×AGL(1,𝔽7) | 2 |
| #6453⋊C3 | 2 |
| #6452⋊C3 | 2 |
| C42×AGL(1,𝔽7) | 2 |
| #6433⋊C3 | 2 |
| D7×#125 | 3 |
| C2×C3×#125 | 7 |