Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
880 conjugacy classes of subgroups; only the 213 normal subgroups are shown.
This group has 880 conjugacy classes of subgroups — too many to draw. Only its 213 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. C23×(C3⋊D3)⋊C3 | C2 | semidirect |
| 2. C2×D4×H(3,𝔽3) | C2 | semidirect |
| 3. C2×#638⋊C3 | C2 | semidirect |
| 4. D4×(C3⋊D3)⋊C3 | C2 | direct (N × H) |
| 5. C2×C4⋊((C3⋊D3)⋊C3) | C2 | semidirect |
| 6. C2×C4×(C3⋊D3)⋊C3 | C2 | semidirect |
| 7. C2×D4×C3⋊D3 | C3 | semidirect |
| 9. C23×H(3,𝔽3) | C22 | semidirect |
| 12. C22×(C3⋊D3)⋊C3 | C22 | semidirect |
| 13. C22×(C3⋊D3)⋊C3 | C22 | semidirect |
| 14. D4×H(3,𝔽3) | C22 | semidirect |
| 15. C2×C4×H(3,𝔽3) | C22 | semidirect |
| 16. #638⋊C3 | C22 | semidirect |
| 17. C2×(C3⋊Dic3)⋊C3 | C22 | semidirect |
| 18. C4⋊((C3⋊D3)⋊C3) | C22 | semidirect |
| 19. C4×(C3⋊D3)⋊C3 | C22 | semidirect |
| 20. C23×C3⋊D3 | C2×C3 | semidirect |
| 23. C2×C32×D4 | D3 | semidirect |
| 24. C2×C32×D4 | C2×C3 | semidirect |
| 25. C2×#638 | C2×C3 | semidirect |
| 26. D4×C3⋊D3 | C2×C3 | semidirect |
| 27. C2×C4⋊(C3⋊D3) | C2×C3 | semidirect |
| 28. C2×C4×C3⋊D3 | C2×C3 | semidirect |
| 35. C22×H(3,𝔽3) | C23 | semidirect |
| 38. C2×(C3⋊D3)⋊C3 | D4 | direct (N × H) |
| 40. C2×(C3⋊D3)⋊C3 | C23 | semidirect |
| 41. C4×H(3,𝔽3) | C23 | semidirect |
| 42. (C3⋊Dic3)⋊C3 | C23 | semidirect |
| 47. C23×C32 | C2×D3 | semidirect |
| 48. C23×C32 | C22×C3 | semidirect |
| 51. C22×C3⋊D3 | C22×C3 | semidirect |
| 52. C22×C3⋊D3 | C22×C3 | semidirect |
| 58. C32×D4 | C2×D3 | semidirect |
| 59. C32×D4 | C22×C3 | semidirect |
| 61. C2×C32×C4 | C2×D3 | semidirect |
| 62. C2×C32×C4 | C22×C3 | semidirect |
| 63. #638 | C22×C3 | semidirect |
| 64. C2×C3⋊Dic3 | C22×C3 | semidirect |
| 65. C4⋊(C3⋊D3) | C22×C3 | semidirect |
| 66. C4×C3⋊D3 | C22×C3 | semidirect |
| 72. C2×H(3,𝔽3) | C2×D4 | semidirect |
| 75. (C3⋊D3)⋊C3 | C2×D4 | direct (N × H) |
| 101. C22×C32 | C22×D3 | semidirect |
| 103. C22×C32 | C23×C3 | semidirect |
| 106. C2×C3⋊D3 | C3×D4 | semidirect |
| 108. C2×C3⋊D3 | C23×C3 | semidirect |
| 115. C32×C4 | C22×D3 | semidirect |
| 116. C32×C4 | C23×C3 | semidirect |
| 117. C3⋊Dic3 | C23×C3 | semidirect |
| 120. H(3,𝔽3) | C22×D4 | semidirect |
| 162. C2×C32 | D3×D4 | semidirect |
| 163. C2×C32 | C2×C3×D4 | semidirect |
| 167. C3⋊D3 | C2×C3×D4 | semidirect |
| 174. C2×D4 | (C3⋊D3)⋊C3 | direct (N × H) |
| 211. C32 | C2×D3×D4 | semidirect |
| 212. C32 | C22×C3×D4 | semidirect |
| 217. C23 | C2×(C3⋊D3)⋊C3 | semidirect |
| 220. D4 | C2×(C3⋊D3)⋊C3 | direct (N × H) |
| 223. C2×C4 | C2×(C3⋊D3)⋊C3 | semidirect |
| 249. C22 | C22×(C3⋊D3)⋊C3 | semidirect |
| 251. C4 | C22×(C3⋊D3)⋊C3 | semidirect |
| 259. C2 | D4×(C3⋊D3)⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×(C3⋊D3)⋊C3 | 2 |
| C2×D4×H(3,𝔽3) | 2 |
| C2×#638⋊C3 | 2 |
| D4×(C3⋊D3)⋊C3 | 2 |
| C2×C4⋊((C3⋊D3)⋊C3) | 2 |
| C2×C4×(C3⋊D3)⋊C3 | 2 |
| C2×D4×C3⋊D3 | 3 |
| C2×C3×D3×D4 | 3 |