Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
256 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 256 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. C2×SD16×ΓL(1,𝔽8) | C2 | semidirect |
| 3. C4×Q8×ΓL(1,𝔽8) | C2 | semidirect |
| 4. ΓL(1,𝔽8)×C4⋊C8 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×#102 | C2 | semidirect |
| 6. C7×#460 | C3 | semidirect |
| 17. C2×C7×SD16 | C2×C3 | semidirect |
| 19. C4×C7×Q8 | C2×C3 | semidirect |
| 20. C7×C4⋊C8 | C2×C3 | semidirect |
| 21. C7×#102 | C2×C3 | semidirect |
| 25. Q8×ΓL(1,𝔽8) | D4 | semidirect |
| 58. C4×ΓL(1,𝔽8) | SD16 | semidirect |
| 61. #460 | ΓL(1,𝔽8) | direct (N × H) |
| 64. C7×Q8 | C3×D4 | semidirect |
| 89. C2×SD16 | C2×ΓL(1,𝔽8) | semidirect |
| 91. C4×Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 92. C4⋊C8 | C2×ΓL(1,𝔽8) | semidirect |
| 93. #102 | C2×ΓL(1,𝔽8) | semidirect |
| 101. C4×C7 | C3×SD16 | semidirect |
| 117. ΓL(1,𝔽8) | #460 | direct (N × H) |
| 144. Q8 | D4×ΓL(1,𝔽8) | semidirect |
| 155. C7 | C3×#460 | semidirect |
| 167. C4 | SD16×ΓL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×SD16×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×#127 | 2 |
| C4×Q8×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×C4⋊C8 | 2 |
| ΓL(1,𝔽8)×#102 | 2 |
| C7×#460 | 3 |
| C3×#460 | 7 |