Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
671 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 671 conjugacy classes of subgroups — too many to draw. Only its 181 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. C2×#88538 | C2 | semidirect |
| 3. #11720573 | C2 | semidirect |
| 4. C4⋊#6180 | C2 | semidirect |
| 5. C4×#6180 | C2 | semidirect |
| 31. #6180 | D4 | semidirect |
| 49. C4×H(3,𝔽3) | M16 | semidirect |
| 102. C32×C4 | #208 | semidirect |
| 110. H(3,𝔽3) | #436 | semidirect |
| 164. C32 | #5012 | semidirect |
| 190. C4 | #88538 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×D4×H(3,𝔽3) | 2 |
| C2×#88538 | 2 |
| #11720573 | 2 |
| C4⋊#6180 | 2 |
| C4×#6180 | 2 |
| C3×#5012 | 3 |