Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1213 conjugacy classes of subgroups; only the 169 normal subgroups are shown.
This group has 1213 conjugacy classes of subgroups — too many to draw. Only its 169 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. #11722577 | C2 | direct (N × H) |
| 2. C2×#41⋊H(3,𝔽3) | C2 | semidirect |
| 3. C2×C32×H(5,𝔽2) | C3 | semidirect |
| 4. C2×C3×#994 | C3 | semidirect |
| 6. #41⋊H(3,𝔽3) | C22 | semidirect |
| 7. C2×Q8⋊H(3,𝔽3) | C22 | semidirect |
| 9. C32×H(5,𝔽2) | C2×C3 | semidirect |
| 13. C3×#994 | C2×C3 | semidirect |
| 14. C2×C3×#41⋊C3 | C2×C3 | semidirect |
| 17. Q8⋊H(3,𝔽3) | C23 | semidirect |
| 34. C3×#41⋊C3 | C22×C3 | semidirect |
| 35. C2×C3×SL(2,𝔽3) | C22×C3 | semidirect |
| 65. C3×SL(2,𝔽3) | C23×C3 | semidirect |
| 66. C2×H(5,𝔽2) | H(3,𝔽3) | semidirect |
| 106. H(5,𝔽2) | C2×H(3,𝔽3) | semidirect |
| 143. C2×C32 | #994 | semidirect |
| 174. C32 | C2×#994 | semidirect |
| 207. C2 | #11722577 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11722577 | 2 |
| C2×#41⋊H(3,𝔽3) | 2 |
| C2×C32×H(5,𝔽2) | 3 |
| C2×C3×#994 | 3 |
| C2×Q8×H(3,𝔽3) | 4 |