Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
2296 conjugacy classes of subgroups; only the 529 normal subgroups are shown.
This group has 2296 conjugacy classes of subgroups — too many to draw. Only its 529 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⋊(C3⋊D3) |
| commutator, socle, cosocle | C2×C33 | C24 |
| Frattini | C2 | C23×C3⋊(C3⋊D3) |
| Fitting | C33×#41 | C2 |
| hypercenter | #41 | C3⋊(C3⋊D3) |
| hypocenter | C33 | C2×#41 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C4 ◁ #41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C33×#41 | C2 | semidirect |
| 2. #89080 | C2 | semidirect |
| 3. Q8×C3⋊(C3⋊D3) | C2 | semidirect |
| 4. #89078 | C2 | semidirect |
| 5. D4×C3⋊(C3⋊D3) | C2 | semidirect |
| 6. #89076 | C2 | semidirect |
| 7. C2×C4×C3⋊(C3⋊D3) | C2 | semidirect |
| 10. C33×Q8 | C22 | semidirect |
| 11. C33×D4 | C22 | semidirect |
| 12. C2×C33×C4 | C22 | semidirect |
| 13. #6312 | C22 | semidirect |
| 14. C2×C3⋊(C3⋊Dic3) | C22 | semidirect |
| 15. C4⋊(C3⋊(C3⋊D3)) | C22 | semidirect |
| 18. C4×C3⋊(C3⋊D3) | C22 | semidirect |
| 19. C4×C3⋊(C3⋊D3) | C22 | semidirect |
| 20. #6308 | C22 | semidirect |
| 21. C32×#41 | D3 | semidirect |
| 38. C32×Q8 | C2×D3 | semidirect |
| 39. C32×D4 | C2×D3 | semidirect |
| 40. C2×C32×C4 | C2×D3 | semidirect |
| 52. C3⋊(C3⋊D3) | #41 | direct (N × H) |
| 53. C3×#41 | C3⋊D3 | semidirect |
| 69. C33 | C2×#41 | semidirect |
| 71. C3×Q8 | C2×C3⋊D3 | semidirect |
| 72. C3×D4 | C2×C3⋊D3 | semidirect |
| 73. C2×C3×C4 | C2×C3⋊D3 | semidirect |
| 89. #41 | C3⋊(C3⋊D3) | direct (N × H) |
| 101. C32 | D3×#41 | semidirect |
| 104. Q8 | C2×C3⋊(C3⋊D3) | semidirect |
| 107. D4 | C2×C3⋊(C3⋊D3) | semidirect |
| 113. C2×C4 | C2×C3⋊(C3⋊D3) | semidirect |
| 126. C3 | #41×C3⋊D3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×#41 | 2 |
| #89080 | 2 |
| Q8×C3⋊(C3⋊D3) | 2 |
| #89078 | 2 |
| D4×C3⋊(C3⋊D3) | 2 |
| #89076 | 2 |
| C2×C4×C3⋊(C3⋊D3) | 2 |
| #41×C3⋊D3 | 3 |