Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
324 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 136 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 |
|---|---|---|
| 4. #6409⋊C3 | C2 | direct (N × H) |
| 5. C2×#6409 | C3 | semidirect |
| 8. C2×Q8×ΓL(1,𝔽8) | C4 | semidirect |
| 20. #6409 | C2×C3 | semidirect |
| 25. Q8×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 37. C2×C7×Q8 | C3×C4 | semidirect |
| 63. C7×Q8 | C2×C3×C4 | semidirect |
| 84. C2×ΓL(1,𝔽8) | #103 | semidirect |
| 114. ΓL(1,𝔽8) | C2×#103 | semidirect |
| 117. C2×Q8 | ΓL(1,𝔽8)⋊C4 | semidirect |
| 126. C2×C7 | C3×#103 | semidirect |
| 142. Q8 | C2×ΓL(1,𝔽8)⋊C4 | semidirect |
| 152. C7 | C2×C3×#103 | semidirect |
| 167. C2 | #6409⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8×ΓL(1,𝔽8) | 2 |
| C2×C4⋊(ΓL(1,𝔽8)⋊C4) | 2 |
| C22×ΓL(1,𝔽8)⋊C8 | 2 |
| #6409⋊C3 | 2 |
| C2×#6409 | 3 |
| C2×C3×#103 | 7 |