Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
539 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 539 conjugacy classes of subgroups — too many to draw. Only its 73 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. D8×#272 | C2 | semidirect |
| 2. H(3,𝔽3)×D16 | C2 | semidirect |
| 3. #11720565 | C2 | semidirect |
| 4. #11720512 | C2 | semidirect |
| 5. C16×#272 | C2 | semidirect |
| 8. D8×H(3,𝔽3) | C22 | semidirect |
| 10. #88529 | C22 | semidirect |
| 12. C16×H(3,𝔽3) | C22 | semidirect |
| 13. #88386 | C22 | semidirect |
| 15. C32×D16 | D3 | semidirect |
| 29. C32×D8 | C2×D3 | semidirect |
| 33. C32×C16 | C2×D3 | semidirect |
| 62. #272 | D16 | direct (N × H) |
| 86. D16 | #272 | direct (N × H) |
| 88. H(3,𝔽3) | C2×D16 | semidirect |
| 110. D8 | C2×#272 | semidirect |
| 113. C16 | C2×#272 | semidirect |
| 126. C32 | D3×D16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D8×#272 | 2 |
| H(3,𝔽3)×D16 | 2 |
| #11720565 | 2 |
| #11720512 | 2 |
| C16×#272 | 2 |
| C3×D3×D16 | 3 |