Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
781 conjugacy classes of subgroups; only the 265 normal subgroups are shown.
This group has 781 conjugacy classes of subgroups — too many to draw. Only its 265 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. #11720520 | C2 | semidirect |
| 4. C4×#6180 | C2 | semidirect |
| 11. #88538 | C4 | direct (N × H) |
| 24. #6180 | C2×C4 | semidirect |
| 39. C4×H(3,𝔽3) | M16 | semidirect |
| 79. C32×C4 | #208 | semidirect |
| 86. H(3,𝔽3) | C4×M16 | semidirect |
| 127. C32 | C4×#208 | semidirect |
| 146. C4 | #88538 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C42×H(3,𝔽3) | 2 |
| C2×#88538 | 2 |
| #11720520 | 2 |
| C4×#6180 | 2 |
| C3×C4×#208 | 3 |