Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
159 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 159 conjugacy classes of subgroups — too many to draw. Only its 24 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. C22×AΓL(1,𝔽8) | C2 | semidirect |
| 2. C4×AΓL(1,𝔽8) | C2 | semidirect |
| 3. D4×AGL(1,𝔽8) | C3 | semidirect |
| 6. C22×AGL(1,𝔽8) | C2×C3 | semidirect |
| 7. C4×AGL(1,𝔽8) | C2×C3 | semidirect |
| 9. AΓL(1,𝔽8) | D4 | direct (N × H) |
| 20. C23×D4 | ΓL(1,𝔽8) | semidirect |
| 21. AGL(1,𝔽8) | C3×D4 | semidirect |
| 33. C25 | C2×ΓL(1,𝔽8) | semidirect |
| 37. C23×C4 | C2×ΓL(1,𝔽8) | semidirect |
| 80. C23 | D4×ΓL(1,𝔽8) | semidirect |
| 84. D4 | AΓL(1,𝔽8) | direct (N × H) |
| 101. C22 | C2×AΓL(1,𝔽8) | semidirect |
| 103. C4 | C2×AΓL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×AΓL(1,𝔽8) | 2 |
| C4×AΓL(1,𝔽8) | 2 |
| D4×AGL(1,𝔽8) | 3 |
| C2×D4×A4 | 7 |
| D4×ΓL(1,𝔽8) | 8 |