Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
180 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 64 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. C2×Q16×ΓL(1,𝔽8) | C2 | semidirect |
| 3. #6402⋊C3 | C2 | direct (N × H) |
| 4. C2×ΓL(1,𝔽8)⋊C16 | C2 | semidirect |
| 5. C2×#6402 | C3 | semidirect |
| 9. Q16×ΓL(1,𝔽8) | C22 | semidirect |
| 13. ΓL(1,𝔽8)⋊C16 | C22 | semidirect |
| 14. C2×C7×Q16 | C2×C3 | semidirect |
| 16. #6402 | C2×C3 | semidirect |
| 17. C2×C7⋊C16 | C2×C3 | semidirect |
| 31. C7×Q16 | C22×C3 | semidirect |
| 35. C7⋊C16 | C22×C3 | semidirect |
| 64. C2×ΓL(1,𝔽8) | SD32 | semidirect |
| 67. C2×Q16 | AGL(1,𝔽7) | semidirect |
| 86. ΓL(1,𝔽8) | C2×SD32 | semidirect |
| 90. Q16 | C2×AGL(1,𝔽7) | semidirect |
| 95. C2×C7 | C3×SD32 | semidirect |
| 113. C7 | C2×C3×SD32 | semidirect |
| 125. C2 | #6402⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q16×ΓL(1,𝔽8) | 2 |
| C2×D56⋊C3 | 2 |
| #6402⋊C3 | 2 |
| C2×ΓL(1,𝔽8)⋊C16 | 2 |
| C2×#6402 | 3 |
| C2×C3×SD32 | 7 |