Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
192 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 192 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 |
|---|---|---|
| 3. ΓL(1,𝔽8)×#107 | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#106 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×#103 | C2 | semidirect |
| 6. ΓL(1,𝔽8)×#103 | C2 | semidirect |
| 8. C7×#485 | C3 | semidirect |
| 22. C7×#107 | C2×C3 | semidirect |
| 23. C7×#106 | C2×C3 | semidirect |
| 24. C7×#103 | C2×C3 | semidirect |
| 25. C7×#103 | C2×C3 | semidirect |
| 70. #485 | ΓL(1,𝔽8) | direct (N × H) |
| 101. #107 | C2×ΓL(1,𝔽8) | semidirect |
| 102. #106 | C2×ΓL(1,𝔽8) | semidirect |
| 103. #103 | C2×ΓL(1,𝔽8) | semidirect |
| 104. #103 | C2×ΓL(1,𝔽8) | semidirect |
| 130. ΓL(1,𝔽8) | #485 | direct (N × H) |
| 170. C7 | C3×#485 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×C22⋊Q8 | 2 |
| ΓL(1,𝔽8)×#117 | 2 |
| ΓL(1,𝔽8)×#107 | 2 |
| ΓL(1,𝔽8)×#106 | 2 |
| ΓL(1,𝔽8)×#103 | 2 |
| ΓL(1,𝔽8)×C22⋊C8 | 2 |
| C7×#485 | 3 |
| C3×#485 | 7 |