Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
580 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 580 conjugacy classes of subgroups — too many to draw. Only its 248 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 |
|---|---|---|
| 6. #6451⋊C3 | C2 | direct (N × H) |
| 7. C2×#6451 | C3 | semidirect |
| 24. #6451 | C2×C3 | semidirect |
| 28. C2×C4×ΓL(1,𝔽8) | Q8 | semidirect |
| 33. C2×ΓL(1,𝔽8)⋊C4 | Q8 | semidirect |
| 60. C4×ΓL(1,𝔽8) | C2×Q8 | semidirect |
| 63. ΓL(1,𝔽8)⋊C4 | C2×Q8 | semidirect |
| 67. C2×C4×C7 | C3×Q8 | semidirect |
| 72. C2×Dic7 | C3×Q8 | semidirect |
| 88. C2×ΓL(1,𝔽8) | C4⋊Q8 | semidirect |
| 103. C4×C7 | C2×C3×Q8 | semidirect |
| 106. Dic7 | C2×C3×Q8 | semidirect |
| 119. ΓL(1,𝔽8) | C2×C4⋊Q8 | semidirect |
| 131. C2×C7 | C3×C4⋊Q8 | semidirect |
| 153. C2×C4 | ΓL(1,𝔽8)⋊Q8 | semidirect |
| 156. C7 | C2×C3×C4⋊Q8 | semidirect |
| 169. C4 | C2×ΓL(1,𝔽8)⋊Q8 | semidirect |
| 171. C2 | #6451⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×ΓL(1,𝔽8)⋊Q8 | 2 |
| C2×C4⋊C4×ΓL(1,𝔽8) | 2 |
| C2×C4⋊(ΓL(1,𝔽8)⋊C4) | 2 |
| C2×(ΓL(1,𝔽8)⋊C4)⋊C4 | 2 |
| C2×C4×ΓL(1,𝔽8)⋊C4 | 2 |
| #6451⋊C3 | 2 |
| C2×#6451 | 3 |
| C2×C3×C4⋊Q8 | 7 |