Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
376 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 376 conjugacy classes of subgroups — too many to draw. Only its 93 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×SD16×ΓL(1,𝔽8) | C2 | semidirect |
| 3. ΓL(1,𝔽8)×C22⋊Q8 | C2 | semidirect |
| 4. ΓL(1,𝔽8)×#102 | C2 | semidirect |
| 5. ΓL(1,𝔽8)×C22⋊C8 | C2 | semidirect |
| 6. C7×#450 | C3 | semidirect |
| 8. C2×Q8×ΓL(1,𝔽8) | C22 | semidirect |
| 14. C2×C8×ΓL(1,𝔽8) | C22 | semidirect |
| 16. C4⋊C4×ΓL(1,𝔽8) | C22 | semidirect |
| 19. C2×C7×SD16 | C2×C3 | semidirect |
| 20. C7×C22⋊Q8 | C2×C3 | semidirect |
| 21. C7×#102 | C2×C3 | semidirect |
| 22. C7×C22⋊C8 | C2×C3 | semidirect |
| 40. C2×C7×Q8 | C22×C3 | semidirect |
| 46. C2×C7×C8 | C22×C3 | semidirect |
| 48. C7×C4⋊C4 | C22×C3 | semidirect |
| 61. C22×ΓL(1,𝔽8) | SD16 | semidirect |
| 67. #450 | ΓL(1,𝔽8) | direct (N × H) |
| 100. C2×SD16 | C2×ΓL(1,𝔽8) | semidirect |
| 101. C22⋊Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 102. #102 | C2×ΓL(1,𝔽8) | semidirect |
| 103. C22⋊C8 | C2×ΓL(1,𝔽8) | semidirect |
| 110. C22×C7 | C3×SD16 | semidirect |
| 131. ΓL(1,𝔽8) | #450 | direct (N × H) |
| 133. C2×Q8 | C22×ΓL(1,𝔽8) | semidirect |
| 139. C2×C8 | C22×ΓL(1,𝔽8) | semidirect |
| 141. C4⋊C4 | C22×ΓL(1,𝔽8) | semidirect |
| 175. C7 | C3×#450 | semidirect |
| 187. C22 | SD16×ΓL(1,𝔽8) | semidirect |
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×SD16×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×C22⋊Q8 | 2 |
| ΓL(1,𝔽8)×#102 | 2 |
| ΓL(1,𝔽8)×C22⋊C8 | 2 |
| C7×#450 | 3 |
| C3×#450 | 7 |