Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
3640 conjugacy classes of subgroups; only the 611 normal subgroups are shown.
This group has 3640 conjugacy classes of subgroups — too many to draw. Only its 611 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C22×C33 | C24 |
| Fitting | C33×#120 | C2 |
| hypercenter | #120 | C3⋊(C3⋊D3) |
| hypocenter | C33 | H(3,𝔽4) |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ #120 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C33×#120 | C2 | semidirect |
| 2. #11723994 | C2 | semidirect |
| 3. #11723978 | C2 | semidirect |
| 4. #11723977 | C2 | semidirect |
| 5. #11723926 | C2 | semidirect |
| 7. C24×C33 | C22 | semidirect |
| 8. C23×C3⋊(C3⋊D3) | C22 | semidirect |
| 11. C2×#6308 | C22 | semidirect |
| 12. C33×C22⋊C4 | C22 | semidirect |
| 13. #88864 | C22 | semidirect |
| 14. #88864 | C22 | semidirect |
| 15. #88859 | C22 | semidirect |
| 16. C4×C3⋊(C3⋊Dic3) | C22 | semidirect |
| 17. C32×#120 | D3 | semidirect |
| 33. C24×C32 | C2×D3 | semidirect |
| 38. C32×C22⋊C4 | C2×D3 | semidirect |
| 52. C3×#120 | C3⋊D3 | semidirect |
| 72. C24×C3 | C2×C3⋊D3 | semidirect |
| 77. C3×C22⋊C4 | C2×C3⋊D3 | semidirect |
| 95. #120 | C3⋊(C3⋊D3) | semidirect |
| 96. C33 | H(3,𝔽4) | semidirect |
| 112. C24 | C2×C3⋊(C3⋊D3) | semidirect |
| 119. C22⋊C4 | C2×C3⋊(C3⋊D3) | semidirect |
| 130. C32 | #5594 | semidirect |
| 154. C3 | #10596065 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×#120 | 2 |
| #11723994 | 2 |
| #11723978 | 2 |
| #11723977 | 2 |
| #11723926 | 2 |
| #10596065 | 3 |