Finite Group DatabasebetaFinite Group DatabasebetaOrder 2688.
8844 conjugacy classes of subgroups; only the 249 normal subgroups are shown.
This group has 8844 conjugacy classes of subgroups — too many to draw. Only its 249 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D3×AGL(1,𝔽8) |
| commutator, cosocle, hypocenter | C23×C3 | C24×C7 |
| Frattini | C1 | G |
| Fitting, socle | C26×C3 | C2×C7 |
| LCS | G ▷ C23×C3 | |
| UCS | C1 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C3×AGL(1,𝔽8) | C2 | semidirect |
| 2. C22×D3×AGL(1,𝔽8) | C2 | direct (N × H) |
| 4. C22×C3×AGL(1,𝔽8) | C22 | semidirect |
| 5. C2×D3×AGL(1,𝔽8) | C22 | direct (N × H) |
| 7. C23×AGL(1,𝔽8) | D3 | direct (N × H) |
| 8. C26×D3 | C7 | semidirect |
| 10. C2×C3×AGL(1,𝔽8) | C23 | semidirect |
| 11. D3×AGL(1,𝔽8) | C23 | direct (N × H) |
| 13. C22×AGL(1,𝔽8) | C2×D3 | direct (N × H) |
| 14. C26×C3 | C2×C7 | semidirect |
| 18. C25×D3 | C2×C7 | semidirect |
| 21. C3×AGL(1,𝔽8) | C24 | semidirect |
| 25. C2×AGL(1,𝔽8) | C22×D3 | direct (N × H) |
| 28. C25×C3 | C22×C7 | semidirect |
| 37. C24×D3 | C22×C7 | semidirect |
| 44. C26 | D3×C7 | semidirect |
| 47. AGL(1,𝔽8) | C23×D3 | direct (N × H) |
| 53. C24×C3 | C23×C7 | semidirect |
| 68. C23×D3 | AGL(1,𝔽8) | direct (N × H) |
| 69. C23×D3 | C23×C7 | semidirect |
| 83. C25 | C2×D3×C7 | semidirect |
| 94. C23×C3 | C2×AGL(1,𝔽8) | semidirect |
| 95. C23×C3 | C24×C7 | semidirect |
| 107. C22×D3 | C2×AGL(1,𝔽8) | direct (N × H) |
| 130. C24 | C22×D3×C7 | semidirect |
| 138. C22×C3 | C22×AGL(1,𝔽8) | semidirect |
| 144. C2×D3 | C22×AGL(1,𝔽8) | direct (N × H) |
| 165. C23 | D3×AGL(1,𝔽8) | direct (N × H) |
| 166. C23 | C23×D3×C7 | semidirect |
| 170. C2×C3 | C23×AGL(1,𝔽8) | semidirect |
| 172. D3 | C23×AGL(1,𝔽8) | direct (N × H) |
| 184. C22 | C2×D3×AGL(1,𝔽8) | direct (N × H) |
| 185. C3 | C24×AGL(1,𝔽8) | semidirect |
| 190. C2 | C22×D3×AGL(1,𝔽8) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×AGL(1,𝔽8) | 2 |
| C22×D3×AGL(1,𝔽8) | 2 |
| C24×AGL(1,𝔽8) | 3 |
| C26×D3 | 7 |
| C23×D3×C7 | 8 |