Finite Group DatabasebetaFinite Group DatabasebetaOrder 864.
2324 conjugacy classes of subgroups; only the 527 normal subgroups are shown.
This group has 2324 conjugacy classes of subgroups — too many to draw. Only its 527 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C2×C33 | C24 |
| Fitting | C33×#41 | C2 |
| hypercenter | #41 | C3⋊(C3⋊D3) |
| hypocenter | C33 | H(5,𝔽2) |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ #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. D4×C3⋊(C3⋊D3) | C2 | semidirect |
| 4. #89076 | C2 | semidirect |
| 5. C2×C4⋊(C3⋊(C3⋊D3)) | C2 | semidirect |
| 7. C22×C3⋊(C3⋊D3) | C22 | semidirect |
| 8. C33×Q8 | C22 | semidirect |
| 9. C33×D4 | C22 | semidirect |
| 10. C2×C33×C4 | C22 | semidirect |
| 11. #6312 | C22 | semidirect |
| 12. C4⋊(C3⋊(C3⋊D3)) | C22 | semidirect |
| 13. C4⋊(C3⋊(C3⋊D3)) | C22 | semidirect |
| 14. C4×C3⋊(C3⋊D3) | C22 | semidirect |
| 15. #6308 | C22 | semidirect |
| 16. C32×#41 | D3 | semidirect |
| 29. C32×Q8 | C2×D3 | semidirect |
| 30. C32×D4 | C2×D3 | semidirect |
| 31. C2×C32×C4 | C2×D3 | semidirect |
| 40. C3×#41 | C3⋊D3 | semidirect |
| 52. C33 | H(5,𝔽2) | semidirect |
| 54. C3×Q8 | C2×C3⋊D3 | semidirect |
| 55. C3×D4 | C2×C3⋊D3 | semidirect |
| 56. C2×C3×C4 | C2×C3⋊D3 | semidirect |
| 67. #41 | C3⋊(C3⋊D3) | semidirect |
| 76. C32 | #1009 | semidirect |
| 79. Q8 | C2×C3⋊(C3⋊D3) | semidirect |
| 83. D4 | C2×C3⋊(C3⋊D3) | semidirect |
| 85. C2×C4 | C2×C3⋊(C3⋊D3) | semidirect |
| 95. C3 | #64366 | 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 |
| D4×C3⋊(C3⋊D3) | 2 |
| #89076 | 2 |
| C2×C4⋊(C3⋊(C3⋊D3)) | 2 |
| #64366 | 3 |