Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
172 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 39 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)×#99 | C2 | semidirect |
| 3. C7×#353 | C3 | semidirect |
| 5. C2×D4×ΓL(1,𝔽8) | C4 | semidirect |
| 8. C42×ΓL(1,𝔽8) | C4 | semidirect |
| 10. C7×#99 | C2×C3 | semidirect |
| 21. C2×C7×D4 | C3×C4 | semidirect |
| 24. C42×C7 | C3×C4 | semidirect |
| 35. #353 | ΓL(1,𝔽8) | direct (N × H) |
| 54. #99 | C2×ΓL(1,𝔽8) | semidirect |
| 71. ΓL(1,𝔽8) | #353 | direct (N × H) |
| 73. C2×D4 | C4×ΓL(1,𝔽8) | semidirect |
| 76. C42 | C4×ΓL(1,𝔽8) | semidirect |
| 97. C7 | C3×#353 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#127 | 2 |
| ΓL(1,𝔽8)×#99 | 2 |
| C7×#353 | 3 |
| C3×#353 | 7 |