Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
384 conjugacy classes of subgroups; only the 219 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 219 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 |
| 3. ΓL(1,𝔽8)×#125 | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#125 | C2 | semidirect |
| 6. C4×Q8×ΓL(1,𝔽8) | C2 | semidirect |
| 8. C7×#541 | C3 | semidirect |
| 22. C7×C4⋊Q8 | C2×C3 | semidirect |
| 24. C7×#125 | C2×C3 | semidirect |
| 25. C7×#125 | C2×C3 | semidirect |
| 27. C4×C7×Q8 | C2×C3 | semidirect |
| 68. #541 | ΓL(1,𝔽8) | direct (N × H) |
| 94. C4⋊Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 96. #125 | C2×ΓL(1,𝔽8) | semidirect |
| 97. #125 | C2×ΓL(1,𝔽8) | semidirect |
| 99. C4×Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 119. ΓL(1,𝔽8) | #541 | direct (N × H) |
| 154. C7 | C3×#541 | 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)×C22⋊Q8 | 2 |
| C4×Q8×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#117 | 2 |
| C7×#541 | 3 |
| C3×#541 | 7 |