Finite Group DatabasebetaFinite Group DatabasebetaOrder 2688.
1600 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 1600 conjugacy classes of subgroups — too many to draw. Only its 239 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. H(5,𝔽2)×AGL(1,𝔽7) | C2 | semidirect |
| 2. #90344⋊C3 | C2 | semidirect |
| 3. ΓL(1,𝔽8)×#458 | C2 | semidirect |
| 4. #13052144 | C2 | semidirect |
| 5. #89825⋊C3 | C2 | semidirect |
| 6. #99×AGL(1,𝔽7) | C2 | semidirect |
| 7. #89513⋊C3 | C2 | semidirect |
| 8. #11737756 | C3 | semidirect |
| 16. C22×C4⋊AGL(1,𝔽7) | C22 | semidirect |
| 17. ΓL(1,𝔽8)×#123 | C22 | semidirect |
| 18. #6500⋊C3 | C22 | semidirect |
| 19. #6492⋊C3 | C22 | semidirect |
| 24. #6445⋊C3 | C22 | semidirect |
| 25. C22⋊C4×AGL(1,𝔽7) | C22 | semidirect |
| 26. ΓL(1,𝔽8)×#99 | C22 | semidirect |
| 27. #6408⋊C3 | C22 | semidirect |
| 28. #6380⋊C3 | C22 | semidirect |
| 29. D7×H(5,𝔽2) | C2×C3 | semidirect |
| 30. #90344 | C2×C3 | semidirect |
| 31. C7×#458 | C2×C3 | semidirect |
| 32. #90016 | C2×C3 | semidirect |
| 33. #89825 | C2×C3 | semidirect |
| 34. D7×#99 | C2×C3 | semidirect |
| 35. #89513 | C2×C3 | semidirect |
| 38. C23×AGL(1,𝔽7) | D4 | semidirect |
| 40. C23×AGL(1,𝔽7) | C23 | semidirect |
| 44. C22×C4×ΓL(1,𝔽8) | C23 | semidirect |
| 48. C2×(C22⋊D7)⋊C3 | D4 | semidirect |
| 65. C2×C4⋊AGL(1,𝔽7) | D4 | semidirect |
| 68. C2×C4×AGL(1,𝔽7) | D4 | semidirect |
| 71. C22⋊C4×ΓL(1,𝔽8) | C23 | semidirect |
| 72. (C22⋊Dic7)⋊C3 | C23 | semidirect |
| 85. C22×D28 | C22×C3 | semidirect |
| 86. C7×#123 | C22×C3 | semidirect |
| 87. #6500 | C22×C3 | semidirect |
| 88. #6492 | C22×C3 | semidirect |
| 93. #6445 | C22×C3 | semidirect |
| 94. D7×C22⋊C4 | C22×C3 | semidirect |
| 95. C7×#99 | C22×C3 | semidirect |
| 96. #6408 | C22×C3 | semidirect |
| 97. #6380 | C22×C3 | semidirect |
| 145. C23×D7 | C3×D4 | semidirect |
| 147. C23×D7 | C23×C3 | semidirect |
| 151. C22×C4×C7 | C23×C3 | semidirect |
| 155. C2×C22⋊D7 | C3×D4 | semidirect |
| 172. C2×D28 | C3×D4 | semidirect |
| 175. C2×C4×D7 | C3×D4 | semidirect |
| 178. C7×C22⋊C4 | C23×C3 | semidirect |
| 179. C22⋊Dic7 | C23×C3 | semidirect |
| 230. #458 | AGL(1,𝔽7) | semidirect |
| 330. #123 | C2×AGL(1,𝔽7) | semidirect |
| 339. #99 | C2×AGL(1,𝔽7) | semidirect |
| 400. ΓL(1,𝔽8) | #3026 | semidirect |
| 432. C22×C4 | C22×AGL(1,𝔽7) | semidirect |
| 441. C22⋊C4 | C22×AGL(1,𝔽7) | semidirect |
| 509. C7 | C3×#3026 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(5,𝔽2)×AGL(1,𝔽7) | 2 |
| #90344⋊C3 | 2 |
| ΓL(1,𝔽8)×#458 | 2 |
| #13052144 | 2 |
| #89825⋊C3 | 2 |
| #99×AGL(1,𝔽7) | 2 |
| #89513⋊C3 | 2 |
| #11737756 | 3 |
| C3×#3026 | 7 |