Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
715 conjugacy classes of subgroups; only the 229 normal subgroups are shown.
This group has 715 conjugacy classes of subgroups — too many to draw. Only its 229 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 |
|---|---|---|
| 4. #11720529 | C2 | direct (N × H) |
| 13. C2×#6180 | C4 | semidirect |
| 28. #6180 | C2×C4 | semidirect |
| 68. C2×H(3,𝔽3) | #107 | semidirect |
| 103. H(3,𝔽3) | C2×#107 | semidirect |
| 124. C2×C32 | #807 | semidirect |
| 155. C32 | C2×#807 | semidirect |
| 185. C2 | #11720529 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4⋊C4×H(3,𝔽3) | 2 |
| C2×C4⋊#1084 | 2 |
| C22×#6180 | 2 |
| #11720529 | 2 |
| C2×C3×#807 | 3 |