Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1059 conjugacy classes of subgroups; only the 292 normal subgroups are shown.
This group has 1059 conjugacy classes of subgroups — too many to draw. Only its 292 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 |
|---|---|---|
| 3. C2×#3756⋊C3 | C2 | direct (N × H) |
| 5. C22×#3756 | C3 | semidirect |
| 14. #3756⋊C3 | C22 | direct (N × H) |
| 21. C2×#3756 | C2×C3 | semidirect |
| 56. #3756 | C22×C3 | semidirect |
| 60. C22×H(3,𝔽3) | Q16 | semidirect |
| 106. C2×H(3,𝔽3) | C2×Q16 | semidirect |
| 141. C22×C32 | #216 | semidirect |
| 142. C22×C32 | C3×Q16 | semidirect |
| 160. H(3,𝔽3) | C22×Q16 | semidirect |
| 199. C2×C32 | C2×#216 | semidirect |
| 200. C2×C32 | C2×C3×Q16 | semidirect |
| 236. C32 | C22×#216 | semidirect |
| 237. C32 | C22×C3×Q16 | semidirect |
| 256. C22 | #3756⋊C3 | direct (N × H) |
| 266. C2 | C2×#3756⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8×H(3,𝔽3) | 2 |
| C22×#634⋊C3 | 2 |
| C2×#3756⋊C3 | 2 |
| C22×(C3⋊(C3⋊C8))⋊C3 | 2 |
| C22×#3756 | 3 |
| C22×C3×#216 | 3 |