Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
496 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 496 conjugacy classes of subgroups — too many to draw. Only its 237 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)×2−1+4 | C2 | semidirect |
| 2. ΓL(1,𝔽8)×#137 | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#135 | C2 | semidirect |
| 4. C2×Q16×ΓL(1,𝔽8) | C2 | semidirect |
| 5. ΓL(1,𝔽8)×#131 | C2 | semidirect |
| 6. C7×#578 | C3 | semidirect |
| 18. C7×2−1+4 | C2×C3 | semidirect |
| 19. C7×#137 | C2×C3 | semidirect |
| 20. C7×#135 | C2×C3 | semidirect |
| 21. C2×C7×Q16 | C2×C3 | semidirect |
| 22. C7×#131 | C2×C3 | semidirect |
| 56. #578 | ΓL(1,𝔽8) | direct (N × H) |
| 82. 2−1+4 | C2×ΓL(1,𝔽8) | semidirect |
| 83. #137 | C2×ΓL(1,𝔽8) | semidirect |
| 84. #135 | C2×ΓL(1,𝔽8) | semidirect |
| 85. C2×Q16 | C2×ΓL(1,𝔽8) | semidirect |
| 86. #131 | C2×ΓL(1,𝔽8) | semidirect |
| 103. ΓL(1,𝔽8) | #578 | direct (N × H) |
| 134. C7 | C3×#578 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×2−1+4 | 2 |
| ΓL(1,𝔽8)×#137 | 2 |
| ΓL(1,𝔽8)×#135 | 2 |
| C2×Q16×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#131 | 2 |
| C7×#578 | 3 |
| C3×#578 | 7 |