Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
532 conjugacy classes of subgroups; only the 363 normal subgroups are shown.
This group has 532 conjugacy classes of subgroups — too many to draw. Only its 363 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)×#131 | C2 | direct (N × H) |
| 3. C2×M16×ΓL(1,𝔽8) | C2 | semidirect |
| 4. C22×C8×ΓL(1,𝔽8) | C2 | semidirect |
| 5. C2×C7×#131 | C3 | semidirect |
| 11. M16×ΓL(1,𝔽8) | C22 | semidirect |
| 14. C2×C8×ΓL(1,𝔽8) | C22 | semidirect |
| 16. C7×#131 | C2×C3 | semidirect |
| 17. C2×C7×M16 | C2×C3 | semidirect |
| 18. C22×C7×C8 | C2×C3 | semidirect |
| 34. C7×M16 | C22×C3 | semidirect |
| 37. C2×C7×C8 | C22×C3 | semidirect |
| 48. C2×#131 | ΓL(1,𝔽8) | direct (N × H) |
| 68. C2×ΓL(1,𝔽8) | #131 | direct (N × H) |
| 71. #131 | C2×ΓL(1,𝔽8) | direct (N × H) |
| 72. C2×M16 | C2×ΓL(1,𝔽8) | semidirect |
| 73. C22×C8 | C2×ΓL(1,𝔽8) | semidirect |
| 89. ΓL(1,𝔽8) | C2×#131 | direct (N × H) |
| 95. M16 | C22×ΓL(1,𝔽8) | semidirect |
| 98. C2×C8 | C22×ΓL(1,𝔽8) | semidirect |
| 100. C2×C7 | C3×#131 | semidirect |
| 117. C7 | C2×C3×#131 | semidirect |
| 129. C2 | ΓL(1,𝔽8)×#131 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#41×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#131 | 2 |
| C2×M16×ΓL(1,𝔽8) | 2 |
| C22×C8×ΓL(1,𝔽8) | 2 |
| C2×C7×#131 | 3 |
| C2×C3×#131 | 7 |