Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
675 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 675 conjugacy classes of subgroups — too many to draw. Only its 120 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. H(3,𝔽3)×#127 | C2 | semidirect |
| 2. C2×#3753⋊C3 | C2 | semidirect |
| 3. #11721385 | C2 | semidirect |
| 4. C4×(C3⋊(C3⋊C8))⋊C3 | C2 | semidirect |
| 5. #10592772 | C3 | semidirect |
| 15. C32×#127 | D3 | semidirect |
| 16. C32×#127 | C2×C3 | semidirect |
| 17. C2×#3753 | C2×C3 | semidirect |
| 18. #64083 | C2×C3 | semidirect |
| 21. C4×C3⋊(C3⋊C8) | C2×C3 | semidirect |
| 31. (C3⋊(C3⋊C8))⋊C3 | D4 | semidirect |
| 57. C4×H(3,𝔽3) | D8 | semidirect |
| 93. C3⋊(C3⋊C8) | C3×D4 | semidirect |
| 139. C32×C4 | #213 | semidirect |
| 140. C32×C4 | C3×D8 | semidirect |
| 145. #127 | (C3⋊D3)⋊C3 | semidirect |
| 147. H(3,𝔽3) | #493 | semidirect |
| 225. C32 | #5073 | semidirect |
| 226. C32 | C3×#493 | semidirect |
| 256. C4 | #3753⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(3,𝔽3)×#127 | 2 |
| C2×#3753⋊C3 | 2 |
| #11721385 | 2 |
| C4×(C3⋊(C3⋊C8))⋊C3 | 2 |
| #10592772 | 3 |
| C3×#5073 | 3 |