Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
3500 conjugacy classes of subgroups; only the 667 normal subgroups are shown.
This group has 3500 conjugacy classes of subgroups — too many to draw. Only its 667 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 | C2×C33×C22⋊C4 | C2 |
| hypercenter | C2×C22⋊C4 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #538 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C33×C22⋊C4 | C2 | semidirect |
| 2. #11723994 | C2 | semidirect |
| 3. #11723971 | C2 | semidirect |
| 4. C4×#6312 | C2 | semidirect |
| 5. #11723965 | C2 | semidirect |
| 6. #11723926 | C2 | semidirect |
| 7. #11723925 | C2 | semidirect |
| 8. #11723924 | C2 | semidirect |
| 9. #11723920 | C2 | semidirect |
| 17. C2×C4⋊(C3⋊(C3⋊D3)) | C22 | semidirect |
| 18. C2×C4×C3⋊(C3⋊D3) | C22 | semidirect |
| 19. C2×#6308 | C22 | semidirect |
| 25. #88859 | C22 | semidirect |
| 26. #88859 | C22 | semidirect |
| 27. C4⋊(C3⋊(C3⋊Dic3)) | C22 | semidirect |
| 28. C4⋊(C3⋊(C3⋊Dic3)) | C22 | semidirect |
| 29. #88857 | C22 | semidirect |
| 30. #88857 | C22 | semidirect |
| 31. C4×C3⋊(C3⋊Dic3) | C22 | semidirect |
| 32. C2×C32×C22⋊C4 | D3 | semidirect |
| 100. C2×C3×C22⋊C4 | C3⋊D3 | semidirect |
| 182. C2×C22⋊C4 | C3⋊(C3⋊D3) | semidirect |
| 183. C33 | #538 | semidirect |
| 248. C32 | #5494 | semidirect |
| 292. C3 | #10595965 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C33×C22⋊C4 | 2 |
| #11723994 | 2 |
| #11723971 | 2 |
| C4×#6312 | 2 |
| #11723965 | 2 |
| #11723926 | 2 |
| #11723925 | 2 |
| #11723924 | 2 |
| #11723920 | 2 |
| #10595965 | 3 |