Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
4675 conjugacy classes of subgroups; only the 1133 normal subgroups are shown.
This group has 4675 conjugacy classes of subgroups — too many to draw. Only its 1133 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. C22×#6229 | C2 | direct (N × H) |
| 8. C2×#6229 | C22 | direct (N × H) |
| 12. C23×H(3,𝔽3) | Q8 | semidirect |
| 16. #6229 | C23 | direct (N × H) |
| 23. C22×H(3,𝔽3) | C2×Q8 | semidirect |
| 33. C23×C32 | Dic6 | semidirect |
| 39. C2×H(3,𝔽3) | C22×Q8 | semidirect |
| 49. C22×C32 | C2×Dic6 | semidirect |
| 56. H(3,𝔽3) | C23×Q8 | semidirect |
| 67. C2×C32 | C22×Dic6 | semidirect |
| 81. C32 | C23×Dic6 | semidirect |
| 82. C23 | #6229 | direct (N × H) |
| 91. C22 | C2×#6229 | direct (N × H) |
| 97. C2 | C22×#6229 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C4×H(3,𝔽3) | 2 |
| C23×#1084 | 2 |
| C22×#6229 | 2 |
| C23×C3×Dic6 | 3 |