Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
715 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 715 conjugacy classes of subgroups — too many to draw. Only its 165 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. C4×C8×H(3,𝔽3) | C2 | semidirect |
| 5. #11720548 | C2 | semidirect |
| 6. C8×#1084 | C2 | semidirect |
| 7. #11720520 | C2 | semidirect |
| 18. #88527 | C4 | semidirect |
| 19. C8×#272 | C4 | semidirect |
| 23. C32×C4×C8 | D3 | semidirect |
| 124. C4×C8 | #272 | semidirect |
| 125. H(3,𝔽3) | #405 | semidirect |
| 178. C32 | #4689 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#88527 | 2 |
| C2×C8×#272 | 2 |
| #11721718 | 2 |
| C4×C8×H(3,𝔽3) | 2 |
| #11720548 | 2 |
| C8×#1084 | 2 |
| #11720520 | 2 |
| C3×#4689 | 3 |