Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
596 conjugacy classes of subgroups; only the 387 normal subgroups are shown.
This group has 596 conjugacy classes of subgroups — too many to draw. Only its 387 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. C2×M16×ΓL(1,𝔽8) | C2 | direct (N × H) |
| 3. C22×C8×ΓL(1,𝔽8) | C2 | semidirect |
| 4. C22×C7×M16 | C3 | semidirect |
| 10. M16×ΓL(1,𝔽8) | C22 | direct (N × H) |
| 11. C2×C8×ΓL(1,𝔽8) | C22 | semidirect |
| 13. C2×C7×M16 | C2×C3 | semidirect |
| 14. C22×C7×C8 | C2×C3 | semidirect |
| 24. C8×ΓL(1,𝔽8) | C23 | semidirect |
| 30. C7×M16 | C22×C3 | semidirect |
| 31. C2×C7×C8 | C22×C3 | semidirect |
| 36. C22×ΓL(1,𝔽8) | M16 | direct (N × H) |
| 42. C22×M16 | ΓL(1,𝔽8) | direct (N × H) |
| 51. C7×C8 | C23×C3 | semidirect |
| 60. C2×ΓL(1,𝔽8) | C2×M16 | direct (N × H) |
| 63. C2×M16 | C2×ΓL(1,𝔽8) | direct (N × H) |
| 64. C22×C8 | C2×ΓL(1,𝔽8) | semidirect |
| 66. C22×C7 | C3×M16 | semidirect |
| 81. ΓL(1,𝔽8) | C22×M16 | direct (N × H) |
| 87. M16 | C22×ΓL(1,𝔽8) | direct (N × H) |
| 88. C2×C8 | C22×ΓL(1,𝔽8) | semidirect |
| 90. C2×C7 | C2×C3×M16 | semidirect |
| 107. C8 | C23×ΓL(1,𝔽8) | semidirect |
| 108. C7 | C22×C3×M16 | semidirect |
| 113. C22 | M16×ΓL(1,𝔽8) | direct (N × H) |
| 121. C2 | C2×M16×ΓL(1,𝔽8) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C4×ΓL(1,𝔽8) | 2 |
| C2×M16×ΓL(1,𝔽8) | 2 |
| C22×C8×ΓL(1,𝔽8) | 2 |
| C22×C7×M16 | 3 |
| C22×C3×M16 | 7 |