Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
11340 conjugacy classes of subgroups; only the 2211 normal subgroups are shown.
This group has 11340 conjugacy classes of subgroups — too many to draw. Only its 2211 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 | C24×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C2×C33 | C25 |
| Fitting | C33×2−1+4 | C2 |
| hypercenter | 2−1+4 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #585 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ 2−1+4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C33×2−1+4 | C2 | semidirect |
| 2. #11724684 | C2 | semidirect |
| 3. #41×C3⋊(C3⋊D3) | C2 | semidirect |
| 4. #11724682 | C2 | semidirect |
| 5. C2×#89080 | C2 | semidirect |
| 7. C33×#41 | C22 | semidirect |
| 8. C2×C33×Q8 | C22 | semidirect |
| 9. #89080 | C22 | semidirect |
| 10. #89080 | C22 | semidirect |
| 11. Q8×C3⋊(C3⋊D3) | C22 | semidirect |
| 12. #89078 | C22 | semidirect |
| 13. D4×C3⋊(C3⋊D3) | C22 | semidirect |
| 14. #89076 | C22 | semidirect |
| 15. C2×C4⋊(C3⋊(C3⋊D3)) | C22 | semidirect |
| 16. C2×C4×C3⋊(C3⋊D3) | C22 | semidirect |
| 17. C32×2−1+4 | D3 | semidirect |
| 33. C32×#41 | C2×D3 | semidirect |
| 34. C2×C32×Q8 | C2×D3 | semidirect |
| 49. C3×2−1+4 | C3⋊D3 | semidirect |
| 68. C3×#41 | C2×C3⋊D3 | semidirect |
| 69. C2×C3×Q8 | C2×C3⋊D3 | semidirect |
| 85. 2−1+4 | C3⋊(C3⋊D3) | semidirect |
| 89. C33 | #585 | semidirect |
| 107. #41 | C2×C3⋊(C3⋊D3) | semidirect |
| 109. C2×Q8 | C2×C3⋊(C3⋊D3) | semidirect |
| 119. C32 | #5968 | semidirect |
| 139. C3 | #10596760 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×2−1+4 | 2 |
| #11724684 | 2 |
| #41×C3⋊(C3⋊D3) | 2 |
| #11724682 | 2 |
| C2×#89080 | 2 |
| #10596760 | 3 |