Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1023 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 1023 conjugacy classes of subgroups — too many to draw. Only its 113 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. M16×#272 | C2 | semidirect |
| 3. H(3,𝔽3)×#100 | C2 | semidirect |
| 4. #11720579 | C2 | semidirect |
| 5. #11720560 | C2 | semidirect |
| 14. M16×H(3,𝔽3) | C22 | semidirect |
| 15. #88538 | C22 | semidirect |
| 20. C32×#100 | D3 | semidirect |
| 47. C32×M16 | C2×D3 | semidirect |
| 94. #272 | #100 | direct (N × H) |
| 133. #100 | #272 | direct (N × H) |
| 134. H(3,𝔽3) | C2×#100 | semidirect |
| 180. M16 | C2×#272 | semidirect |
| 206. C32 | D3×#100 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D4×#272 | 2 |
| M16×#272 | 2 |
| H(3,𝔽3)×#100 | 2 |
| #11720579 | 2 |
| #11720560 | 2 |
| C3×D3×#100 | 3 |