Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
3211 conjugacy classes of subgroups; only the 635 normal subgroups are shown.
This group has 3211 conjugacy classes of subgroups — too many to draw. Only its 635 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. C22×#634⋊C3 | C2 | semidirect |
| 4. C22×#6196 | C2 | semidirect |
| 5. C2×#88651 | C2 | direct (N × H) |
| 12. C2×#634⋊C3 | C22 | semidirect |
| 14. C2×#6196 | C22 | semidirect |
| 15. #88651 | C22 | direct (N × H) |
| 16. C22×#634 | D3 | semidirect |
| 24. C22×#272 | Q8 | semidirect |
| 30. #634⋊C3 | C23 | semidirect |
| 32. #6196 | C23 | semidirect |
| 40. C2×#634 | C2×D3 | semidirect |
| 50. C22×H(3,𝔽3) | C2×Q8 | semidirect |
| 52. C2×#272 | C2×Q8 | semidirect |
| 78. #634 | C22×D3 | semidirect |
| 88. C2×H(3,𝔽3) | C22×Q8 | semidirect |
| 90. #272 | C22×Q8 | semidirect |
| 117. C22×C32 | D3×Q8 | semidirect |
| 134. H(3,𝔽3) | C23×Q8 | semidirect |
| 165. C2×C32 | C2×D3×Q8 | semidirect |
| 200. C32 | C22×D3×Q8 | semidirect |
| 222. C22 | #88651 | direct (N × H) |
| 231. C2 | C2×#88651 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×#272 | 2 |
| C22×#634⋊C3 | 2 |
| C22×#6199 | 2 |
| C22×#6196 | 2 |
| C2×#88651 | 2 |
| C22×D3×Dic6 | 3 |