Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
472 conjugacy classes of subgroups; only the 225 normal subgroups are shown.
This group has 472 conjugacy classes of subgroups — too many to draw. Only its 225 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)×#124 | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#123 | C2 | semidirect |
| 4. C4×Q8×ΓL(1,𝔽8) | C2 | semidirect |
| 5. C4×D4×ΓL(1,𝔽8) | C2 | semidirect |
| 6. C7×#559 | C3 | semidirect |
| 18. C7×#124 | C2×C3 | semidirect |
| 19. C7×#123 | C2×C3 | semidirect |
| 20. C4×C7×Q8 | C2×C3 | semidirect |
| 21. C4×C7×D4 | C2×C3 | semidirect |
| 54. #559 | ΓL(1,𝔽8) | direct (N × H) |
| 78. #124 | C2×ΓL(1,𝔽8) | semidirect |
| 79. #123 | C2×ΓL(1,𝔽8) | semidirect |
| 80. C4×Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 81. C4×D4 | C2×ΓL(1,𝔽8) | semidirect |
| 98. ΓL(1,𝔽8) | #559 | direct (N × H) |
| 128. C7 | C3×#559 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#127 | 2 |
| ΓL(1,𝔽8)×#124 | 2 |
| ΓL(1,𝔽8)×#123 | 2 |
| C4×Q8×ΓL(1,𝔽8) | 2 |
| C4×D4×ΓL(1,𝔽8) | 2 |
| C7×#559 | 3 |
| C3×#559 | 7 |