Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
614 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 614 conjugacy classes of subgroups — too many to draw. Only its 191 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×(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 | C2×C3×#41 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ #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. Q8×(C3⋊D3)⋊C3 | C2 | semidirect |
| 4. #3830⋊C3 | C2 | semidirect |
| 5. D4×(C3⋊D3)⋊C3 | C2 | semidirect |
| 6. #3828⋊C3 | C2 | semidirect |
| 7. C2×C4×(C3⋊D3)⋊C3 | C2 | semidirect |
| 8. #41×C3⋊D3 | C3 | semidirect |
| 11. Q8×H(3,𝔽3) | C22 | semidirect |
| 12. D4×H(3,𝔽3) | C22 | semidirect |
| 13. C2×C4×H(3,𝔽3) | C22 | semidirect |
| 14. #638⋊C3 | C22 | semidirect |
| 15. C2×(C3⋊Dic3)⋊C3 | C22 | semidirect |
| 16. C4⋊((C3⋊D3)⋊C3) | C22 | semidirect |
| 19. C4×(C3⋊D3)⋊C3 | C22 | semidirect |
| 20. C4×(C3⋊D3)⋊C3 | C22 | semidirect |
| 21. #634⋊C3 | C22 | semidirect |
| 23. C32×#41 | D3 | semidirect |
| 24. C32×#41 | C2×C3 | semidirect |
| 25. #3832 | C2×C3 | semidirect |
| 26. Q8×C3⋊D3 | C2×C3 | semidirect |
| 27. #3830 | C2×C3 | semidirect |
| 28. D4×C3⋊D3 | C2×C3 | semidirect |
| 29. #3828 | C2×C3 | semidirect |
| 30. C2×C4×C3⋊D3 | C2×C3 | semidirect |
| 51. C32×Q8 | C2×D3 | semidirect |
| 52. C32×Q8 | C22×C3 | semidirect |
| 54. C32×D4 | C2×D3 | semidirect |
| 55. C32×D4 | C22×C3 | semidirect |
| 57. C2×C32×C4 | C2×D3 | semidirect |
| 58. C2×C32×C4 | C22×C3 | semidirect |
| 59. #638 | C22×C3 | semidirect |
| 60. C2×C3⋊Dic3 | C22×C3 | semidirect |
| 61. C4⋊(C3⋊D3) | C22×C3 | semidirect |
| 64. C4×C3⋊D3 | C22×C3 | semidirect |
| 65. C4×C3⋊D3 | C22×C3 | semidirect |
| 66. #634 | C22×C3 | semidirect |
| 78. (C3⋊D3)⋊C3 | #41 | direct (N × H) |
| 121. H(3,𝔽3) | C2×#41 | semidirect |
| 168. C3⋊D3 | C3×#41 | semidirect |
| 174. #41 | (C3⋊D3)⋊C3 | direct (N × H) |
| 206. C32 | D3×#41 | semidirect |
| 207. C32 | C2×C3×#41 | semidirect |
| 210. Q8 | C2×(C3⋊D3)⋊C3 | semidirect |
| 213. D4 | C2×(C3⋊D3)⋊C3 | semidirect |
| 219. 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 |
| Q8×(C3⋊D3)⋊C3 | 2 |
| #3830⋊C3 | 2 |
| D4×(C3⋊D3)⋊C3 | 2 |
| #3828⋊C3 | 2 |
| C2×C4×(C3⋊D3)⋊C3 | 2 |
| #41×C3⋊D3 | 3 |
| C3×D3×#41 | 3 |