Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1859 conjugacy classes of subgroups; only the 301 normal subgroups are shown.
This group has 1859 conjugacy classes of subgroups — too many to draw. Only its 301 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. H(3,𝔽3)×#117 | C2 | semidirect |
| 3. #11721826 | C2 | semidirect |
| 4. #11721761 | C2 | semidirect |
| 5. C22⋊C4×#272 | C2 | semidirect |
| 6. C4×C4⋊#272 | C2 | semidirect |
| 10. C2×C4⋊#272 | C4 | semidirect |
| 13. C2×C4×#272 | C22 | semidirect |
| 14. C4⋊C4×H(3,𝔽3) | C22 | semidirect |
| 16. C42×H(3,𝔽3) | C22 | semidirect |
| 18. #88542 | C22 | semidirect |
| 19. C4⋊#1084 | C22 | semidirect |
| 20. C4×#1084 | C22 | semidirect |
| 22. C32×#117 | D3 | semidirect |
| 29. C22×#272 | C2×C4 | semidirect |
| 31. C2×C4×H(3,𝔽3) | C2×C4 | semidirect |
| 36. C4⋊#272 | C2×C4 | semidirect |
| 37. C4⋊#272 | C2×C4 | semidirect |
| 47. C32×C4⋊C4 | C2×D3 | semidirect |
| 49. C32×C42 | C2×D3 | semidirect |
| 80. C2×C32×C4 | C4×D3 | semidirect |
| 133. #117 | #272 | semidirect |
| 135. H(3,𝔽3) | #518 | semidirect |
| 171. C4⋊C4 | C2×#272 | semidirect |
| 176. C42 | C2×#272 | semidirect |
| 197. C32 | #5521 | semidirect |
| 206. C2×C4 | C4×#272 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4⋊#272 | 2 |
| H(3,𝔽3)×#117 | 2 |
| #11721826 | 2 |
| #11721761 | 2 |
| C22⋊C4×#272 | 2 |
| C4×C4⋊#272 | 2 |
| C3×#5521 | 3 |