Finite Group DatabasebetaFinite Group DatabasebetaOrder 3024.
836 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 836 conjugacy classes of subgroups — too many to draw. Only its 208 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #67356⋊C2 |
| commutator, cosocle, hypocenter | C7×C9 | C24×C3 |
| Frattini | C3 | C22×D3×AGL(1,𝔽7) |
| Fitting | C22×C7×C9 | C22×C3 |
| socle | C22×C3×C7 | C2×C3×D3 |
| LCS | G ▷ C7×C9 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#67366 | C2 | semidirect |
| 2. C22×#67361 | C2 | semidirect |
| 3. C22×#67356 | C2 | semidirect |
| 4. C2×#67356⋊C2 | C2 | direct (N × H) |
| 6. C22×D7×D9 | C3 | semidirect |
| 7. C22×#4427 | C22 | semidirect |
| 8. C2×#67366 | C22 | semidirect |
| 9. C2×#67361 | C22 | semidirect |
| 10. C2×#67356 | C22 | semidirect |
| 11. #67356⋊C2 | C22 | direct (N × H) |
| 16. C22×D63 | C2×C3 | semidirect |
| 17. C22×C7×D9 | C2×C3 | semidirect |
| 18. C22×C9×D7 | C2×C3 | semidirect |
| 19. C2×D7×D9 | C2×C3 | semidirect |
| 22. C2×#4427 | C23 | semidirect |
| 23. #67366 | C23 | semidirect |
| 24. #67361 | C23 | semidirect |
| 25. #67356 | C23 | semidirect |
| 33. C22×C7×C9 | C22×C3 | semidirect |
| 34. C2×D63 | C22×C3 | semidirect |
| 35. C2×C7×D9 | C22×C3 | semidirect |
| 36. C2×C9×D7 | C22×C3 | semidirect |
| 38. D7×D9 | C22×C3 | semidirect |
| 44. #4427 | C24 | semidirect |
| 59. C2×C7×C9 | C23×C3 | semidirect |
| 60. D63 | C23×C3 | semidirect |
| 61. C9×D7 | C23×C3 | semidirect |
| 62. C7×D9 | C23×C3 | semidirect |
| 89. C22×D9 | AGL(1,𝔽7) | semidirect |
| 91. C7×C9 | C24×C3 | semidirect |
| 96. C22×D7 | AGL(1,ℤ/9) | semidirect |
| 119. C22×C9 | C2×AGL(1,𝔽7) | semidirect |
| 122. C2×D9 | C2×AGL(1,𝔽7) | semidirect |
| 124. C22×C7 | C2×AGL(1,ℤ/9) | semidirect |
| 127. C2×D7 | C2×AGL(1,ℤ/9) | semidirect |
| 146. C2×C9 | C22×AGL(1,𝔽7) | semidirect |
| 148. D9 | C22×AGL(1,𝔽7) | semidirect |
| 151. C2×C7 | C22×AGL(1,ℤ/9) | semidirect |
| 153. D7 | C22×AGL(1,ℤ/9) | semidirect |
| 166. C9 | C23×AGL(1,𝔽7) | semidirect |
| 171. C7 | C23×AGL(1,ℤ/9) | semidirect |
| 184. C22 | #67356⋊C2 | direct (N × H) |
| 190. C2 | C2×#67356⋊C2 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#67366 | 2 |
| C22×#67361 | 2 |
| C22×#67356 | 2 |
| C2×#67356⋊C2 | 2 |
| C22×D3×AGL(1,𝔽7) | 3 |
| C22×D7×D9 | 3 |
| C23×AGL(1,ℤ/9) | 7 |