Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
404 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 404 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 |
|---|---|---|
| 2. C2×C4⋊(ΓL(1,𝔽8)⋊C4) | C2 | semidirect |
| 3. C22×ΓL(1,𝔽8)⋊C8 | C2 | semidirect |
| 4. #6406⋊C3 | C2 | direct (N × H) |
| 5. C2×#6406 | C3 | semidirect |
| 9. C2×D4×ΓL(1,𝔽8) | C4 | semidirect |
| 14. C4⋊(ΓL(1,𝔽8)⋊C4) | C22 | semidirect |
| 16. C2×ΓL(1,𝔽8)⋊C8 | C22 | semidirect |
| 18. C2×C4⋊Dic7 | C2×C3 | semidirect |
| 19. C22×C7⋊C8 | C2×C3 | semidirect |
| 20. #6406 | C2×C3 | semidirect |
| 29. D4×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 40. C2×C7×D4 | C3×C4 | semidirect |
| 45. C4⋊Dic7 | C22×C3 | semidirect |
| 47. C2×C7⋊C8 | C22×C3 | semidirect |
| 72. C7×D4 | C2×C3×C4 | semidirect |
| 92. C2×ΓL(1,𝔽8) | #102 | semidirect |
| 127. ΓL(1,𝔽8) | C2×#102 | semidirect |
| 131. C2×D4 | ΓL(1,𝔽8)⋊C4 | semidirect |
| 140. C2×C7 | C3×#102 | semidirect |
| 163. D4 | C2×ΓL(1,𝔽8)⋊C4 | semidirect |
| 171. C7 | C2×C3×#102 | semidirect |
| 191. C2 | #6406⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D4×ΓL(1,𝔽8) | 2 |
| C2×C4⋊(ΓL(1,𝔽8)⋊C4) | 2 |
| C22×ΓL(1,𝔽8)⋊C8 | 2 |
| #6406⋊C3 | 2 |
| C2×#6406 | 3 |
| C2×C3×#102 | 7 |