Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
288 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 288 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. ΓL(1,𝔽8)×#124 | C2 | semidirect |
| 2. C2×#1143⋊C3 | C2 | semidirect |
| 3. C2×#1141⋊C3 | C2 | semidirect |
| 4. #10620710 | C2 | semidirect |
| 5. #6377⋊C3 | C2 | semidirect |
| 6. #89830 | C3 | semidirect |
| 16. C7×#124 | C2×C3 | semidirect |
| 17. C2×#1143 | C2×C3 | semidirect |
| 18. C2×#1141 | C2×C3 | semidirect |
| 19. #6437 | C2×C3 | semidirect |
| 20. #6377 | C2×C3 | semidirect |
| 32. ΓL(1,𝔽8)⋊C8 | D4 | semidirect |
| 66. C7⋊C8 | C3×D4 | semidirect |
| 83. #124 | AGL(1,𝔽7) | semidirect |
| 104. ΓL(1,𝔽8) | #496 | semidirect |
| 137. C7 | C3×#496 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#124 | 2 |
| C2×#1143⋊C3 | 2 |
| C2×#1141⋊C3 | 2 |
| #10620710 | 2 |
| #6377⋊C3 | 2 |
| #89830 | 3 |
| C3×#496 | 7 |