Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
216 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 87 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. ΓL(1,𝔽8)×C4⋊Q8 | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#106 | C2 | semidirect |
| 4. ΓL(1,𝔽8)×C4⋊C8 | C2 | semidirect |
| 6. C7×#476 | C3 | semidirect |
| 16. C7×C4⋊Q8 | C2×C3 | semidirect |
| 18. C7×#106 | C2×C3 | semidirect |
| 19. C7×C4⋊C8 | C2×C3 | semidirect |
| 25. D4×ΓL(1,𝔽8) | Q8 | semidirect |
| 57. #476 | ΓL(1,𝔽8) | direct (N × H) |
| 61. C7×D4 | C3×Q8 | semidirect |
| 82. C4⋊Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 84. #106 | C2×ΓL(1,𝔽8) | semidirect |
| 85. C4⋊C8 | C2×ΓL(1,𝔽8) | semidirect |
| 108. ΓL(1,𝔽8) | #476 | direct (N × H) |
| 135. D4 | Q8×ΓL(1,𝔽8) | semidirect |
| 143. C7 | C3×#476 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×C4⋊Q8 | 2 |
| C4×D4×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#106 | 2 |
| ΓL(1,𝔽8)×C4⋊C8 | 2 |
| ΓL(1,𝔽8)×#102 | 2 |
| C7×#476 | 3 |
| C3×#476 | 7 |