Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
621 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 621 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C23×(C3⋊D3)⋊C3 |
| commutator, cosocle | C2×C32 | C24×C3 |
| Frattini, socle | C2×C3 | C23×C3×D3 |
| Fitting | #41×H(3,𝔽3) | C2 |
| hypercenter | #41 | (C3⋊D3)⋊C3 |
| hypocenter | C32 | C3×H(5,𝔽2) |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ #41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #41×H(3,𝔽3) | C2 | semidirect |
| 2. #88722 | C2 | semidirect |
| 3. D4×(C3⋊D3)⋊C3 | C2 | semidirect |
| 4. #3828⋊C3 | C2 | semidirect |
| 5. C2×C4⋊((C3⋊D3)⋊C3) | C2 | semidirect |
| 6. #64366 | C3 | semidirect |
| 8. C22×(C3⋊D3)⋊C3 | C22 | semidirect |
| 9. Q8×H(3,𝔽3) | C22 | semidirect |
| 10. D4×H(3,𝔽3) | C22 | semidirect |
| 11. C2×C4×H(3,𝔽3) | C22 | semidirect |
| 12. #638⋊C3 | C22 | semidirect |
| 13. C4⋊((C3⋊D3)⋊C3) | C22 | semidirect |
| 14. C4⋊((C3⋊D3)⋊C3) | C22 | semidirect |
| 15. C4×(C3⋊D3)⋊C3 | C22 | semidirect |
| 16. #634⋊C3 | C22 | semidirect |
| 18. C32×#41 | D3 | semidirect |
| 19. C32×#41 | C2×C3 | semidirect |
| 20. #3832 | C2×C3 | semidirect |
| 21. D4×C3⋊D3 | C2×C3 | semidirect |
| 22. #3828 | C2×C3 | semidirect |
| 23. C2×C4⋊(C3⋊D3) | C2×C3 | semidirect |
| 37. C22×C3⋊D3 | C22×C3 | semidirect |
| 40. C32×Q8 | C2×D3 | semidirect |
| 41. C32×Q8 | C22×C3 | semidirect |
| 43. C32×D4 | C2×D3 | semidirect |
| 44. C32×D4 | C22×C3 | semidirect |
| 46. C2×C32×C4 | C2×D3 | semidirect |
| 47. C2×C32×C4 | C22×C3 | semidirect |
| 48. #638 | C22×C3 | semidirect |
| 49. C4⋊(C3⋊D3) | C22×C3 | semidirect |
| 50. C4⋊(C3⋊D3) | C22×C3 | semidirect |
| 51. C4×C3⋊D3 | C22×C3 | semidirect |
| 52. #634 | C22×C3 | semidirect |
| 93. H(3,𝔽3) | H(5,𝔽2) | semidirect |
| 134. #41 | (C3⋊D3)⋊C3 | semidirect |
| 159. C32 | #1009 | semidirect |
| 160. C32 | C3×H(5,𝔽2) | semidirect |
| 163. Q8 | C2×(C3⋊D3)⋊C3 | semidirect |
| 167. D4 | C2×(C3⋊D3)⋊C3 | semidirect |
| 169. C2×C4 | C2×(C3⋊D3)⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #41×H(3,𝔽3) | 2 |
| #88722 | 2 |
| D4×(C3⋊D3)⋊C3 | 2 |
| #3828⋊C3 | 2 |
| C2×C4⋊((C3⋊D3)⋊C3) | 2 |
| #64366 | 3 |
| C3×#1009 | 3 |