Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
644 conjugacy classes of subgroups; only the 240 normal subgroups are shown.
This group has 644 conjugacy classes of subgroups — too many to draw. Only its 240 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. C22×ΓL(1,𝔽8)⋊Q8 | C2 | semidirect |
| 4. C2×C4⋊(ΓL(1,𝔽8)⋊C4) | C2 | semidirect |
| 5. #6487⋊C3 | C2 | direct (N × H) |
| 6. C2×(ΓL(1,𝔽8)⋊C4)⋊C4 | C2 | semidirect |
| 7. C2×#6487 | C3 | semidirect |
| 16. C2×ΓL(1,𝔽8)⋊Q8 | C22 | semidirect |
| 18. C4⋊(ΓL(1,𝔽8)⋊C4) | C22 | semidirect |
| 19. (ΓL(1,𝔽8)⋊C4)⋊C4 | C22 | semidirect |
| 21. C22×Dic14 | C2×C3 | semidirect |
| 23. C2×C4⋊Dic7 | C2×C3 | semidirect |
| 24. #6487 | C2×C3 | semidirect |
| 25. C2×Dic7⋊C4 | C2×C3 | semidirect |
| 29. C23×ΓL(1,𝔽8) | Q8 | semidirect |
| 50. C2×Dic14 | C22×C3 | semidirect |
| 52. C4⋊Dic7 | C22×C3 | semidirect |
| 53. Dic7⋊C4 | C22×C3 | semidirect |
| 65. C22×ΓL(1,𝔽8) | C2×Q8 | semidirect |
| 75. C23×C7 | C3×Q8 | semidirect |
| 101. C2×ΓL(1,𝔽8) | C22⋊Q8 | semidirect |
| 116. C22×C7 | C2×C3×Q8 | semidirect |
| 138. ΓL(1,𝔽8) | C2×C22⋊Q8 | semidirect |
| 152. C2×C7 | C3×C22⋊Q8 | semidirect |
| 170. C23 | ΓL(1,𝔽8)⋊Q8 | semidirect |
| 183. C7 | C2×C3×C22⋊Q8 | semidirect |
| 193. C22 | C2×ΓL(1,𝔽8)⋊Q8 | semidirect |
| 203. C2 | #6487⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C4×ΓL(1,𝔽8) | 2 |
| C22×ΓL(1,𝔽8)⋊Q8 | 2 |
| C2×(C22⋊Dic7)⋊C3 | 2 |
| C2×C4⋊(ΓL(1,𝔽8)⋊C4) | 2 |
| #6487⋊C3 | 2 |
| C2×(ΓL(1,𝔽8)⋊C4)⋊C4 | 2 |
| C2×#6487 | 3 |
| C2×C3×C22⋊Q8 | 7 |