Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1089 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 1089 conjugacy classes of subgroups — too many to draw. Only its 151 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 |
|---|---|---|
| 1. C22×D4×H(3,𝔽3) | C2 | semidirect |
| 2. C2×#88545 | C2 | semidirect |
| 3. #11721830 | C2 | semidirect |
| 4. #11720578 | C2 | semidirect |
| 5. #11720573 | C2 | semidirect |
| 13. C2×C4⋊#272 | C22 | semidirect |
| 16. C4⋊#1084 | C22 | semidirect |
| 17. C2×#6180 | C22 | semidirect |
| 18. C22×C32×D4 | D3 | semidirect |
| 31. D4×H(3,𝔽3) | D4 | semidirect |
| 58. C22×H(3,𝔽3) | D8 | semidirect |
| 84. C32×D4 | #67 | semidirect |
| 127. C22×C32 | #213 | semidirect |
| 135. C22×D4 | #272 | semidirect |
| 140. H(3,𝔽3) | #447 | semidirect |
| 218. C32 | #5217 | semidirect |
| 229. D4 | #6235 | semidirect |
| 256. C22 | #88545 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D4×H(3,𝔽3) | 2 |
| C2×#88545 | 2 |
| #11721830 | 2 |
| #11720578 | 2 |
| #11720573 | 2 |
| C3×#5217 | 3 |