Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
208 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 75 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)×#137 | C2 | semidirect |
| 3. C2×SD16×ΓL(1,𝔽8) | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#131 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×#108 | C2 | semidirect |
| 6. ΓL(1,𝔽8)×#101 | C2 | semidirect |
| 7. ΓL(1,𝔽8)×#100 | C2 | semidirect |
| 8. C7×#471 | C3 | semidirect |
| 24. C7×#137 | C2×C3 | semidirect |
| 26. C2×C7×SD16 | C2×C3 | semidirect |
| 27. C7×#131 | C2×C3 | semidirect |
| 28. C7×#108 | C2×C3 | semidirect |
| 29. C7×#101 | C2×C3 | semidirect |
| 30. C7×#100 | C2×C3 | semidirect |
| 78. #471 | ΓL(1,𝔽8) | direct (N × H) |
| 114. #137 | C2×ΓL(1,𝔽8) | semidirect |
| 116. C2×SD16 | C2×ΓL(1,𝔽8) | semidirect |
| 117. #131 | C2×ΓL(1,𝔽8) | semidirect |
| 118. #108 | C2×ΓL(1,𝔽8) | semidirect |
| 119. #101 | C2×ΓL(1,𝔽8) | semidirect |
| 120. #100 | C2×ΓL(1,𝔽8) | semidirect |
| 145. ΓL(1,𝔽8) | #471 | direct (N × H) |
| 189. C7 | C3×#471 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#137 | 2 |
| ΓL(1,𝔽8)×AGL(1,ℤ/8) | 2 |
| C2×SD16×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#131 | 2 |
| ΓL(1,𝔽8)×#108 | 2 |
| ΓL(1,𝔽8)×#101 | 2 |
| ΓL(1,𝔽8)×#100 | 2 |
| C7×#471 | 3 |
| C3×#471 | 7 |