Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
308 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 117 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 | semidirect |
| 3. ΓL(1,𝔽8)×#117 | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#104 | C2 | semidirect |
| 5. C7×#421 | C3 | semidirect |
| 8. #41×ΓL(1,𝔽8) | C4 | semidirect |
| 10. C2×Q8×ΓL(1,𝔽8) | C4 | semidirect |
| 12. C2×D4×ΓL(1,𝔽8) | C4 | semidirect |
| 16. M16×ΓL(1,𝔽8) | C22 | semidirect |
| 20. C42×ΓL(1,𝔽8) | C22 | semidirect |
| 22. C2×C7×M16 | C2×C3 | semidirect |
| 23. C7×#117 | C2×C3 | semidirect |
| 24. C7×#104 | C2×C3 | semidirect |
| 47. C7×#41 | C3×C4 | semidirect |
| 49. C2×C7×Q8 | C3×C4 | semidirect |
| 51. C2×C7×D4 | C3×C4 | semidirect |
| 55. C7×M16 | C22×C3 | semidirect |
| 59. C42×C7 | C22×C3 | semidirect |
| 76. #421 | ΓL(1,𝔽8) | direct (N × H) |
| 117. C2×M16 | C2×ΓL(1,𝔽8) | semidirect |
| 118. #117 | C2×ΓL(1,𝔽8) | semidirect |
| 119. #104 | C2×ΓL(1,𝔽8) | semidirect |
| 151. ΓL(1,𝔽8) | #421 | direct (N × H) |
| 154. #41 | C4×ΓL(1,𝔽8) | semidirect |
| 156. C2×Q8 | C4×ΓL(1,𝔽8) | semidirect |
| 158. C2×D4 | C4×ΓL(1,𝔽8) | semidirect |
| 162. M16 | C22×ΓL(1,𝔽8) | semidirect |
| 166. C42 | C22×ΓL(1,𝔽8) | semidirect |
| 203. C7 | C3×#421 | semidirect |
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 |
| C2×M16×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#117 | 2 |
| ΓL(1,𝔽8)×#104 | 2 |
| C7×#421 | 3 |
| C3×#421 | 7 |