Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
919 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 919 conjugacy classes of subgroups — too many to draw. Only its 183 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×#6188 | C2 | semidirect |
| 3. C2×#88627 | C2 | direct (N × H) |
| 9. C2×#6188 | C22 | semidirect |
| 10. #88627 | C22 | direct (N × H) |
| 20. #6188 | C23 | semidirect |
| 27. C22×H(3,𝔽3) | M16 | semidirect |
| 47. C2×H(3,𝔽3) | C2×M16 | semidirect |
| 72. H(3,𝔽3) | C22×M16 | semidirect |
| 132. C22 | #88627 | direct (N × H) |
| 140. C2 | C2×#88627 | 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×#6188 | 2 |
| C2×#88627 | 2 |
| C22×C3×M16 | 9 |