Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
392 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 213 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)×C4⋊Q8 | C2 | semidirect |
| 2. ΓL(1,𝔽8)×#126 | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#125 | C2 | semidirect |
| 6. ΓL(1,𝔽8)×C22⋊Q8 | C2 | semidirect |
| 7. C7×#563 | C3 | semidirect |
| 22. C7×C4⋊Q8 | C2×C3 | semidirect |
| 23. C7×#126 | C2×C3 | semidirect |
| 24. C7×#125 | C2×C3 | semidirect |
| 27. C7×C22⋊Q8 | C2×C3 | semidirect |
| 67. #563 | ΓL(1,𝔽8) | direct (N × H) |
| 95. C4⋊Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 96. #126 | C2×ΓL(1,𝔽8) | semidirect |
| 97. #125 | C2×ΓL(1,𝔽8) | semidirect |
| 100. C22⋊Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 119. ΓL(1,𝔽8) | #563 | direct (N × H) |
| 155. C7 | C3×#563 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×C4⋊Q8 | 2 |
| ΓL(1,𝔽8)×#126 | 2 |
| ΓL(1,𝔽8)×#125 | 2 |
| ΓL(1,𝔽8)×#124 | 2 |
| ΓL(1,𝔽8)×#123 | 2 |
| ΓL(1,𝔽8)×C22⋊Q8 | 2 |
| C7×#563 | 3 |
| C3×#563 | 7 |