Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
891 conjugacy classes of subgroups; only the 405 normal subgroups are shown.
This group has 891 conjugacy classes of subgroups — too many to draw. Only its 405 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 |
|---|---|---|
| 3. C4×#6180 | C2 | direct (N × H) |
| 9. C2×#6180 | C4 | direct (N × H) |
| 15. C2×C4×H(3,𝔽3) | C8 | semidirect |
| 20. #6180 | C2×C4 | direct (N × H) |
| 33. C4×H(3,𝔽3) | C2×C8 | semidirect |
| 42. C2×C32×C4 | C3⋊C8 | semidirect |
| 49. C2×H(3,𝔽3) | C4×C8 | semidirect |
| 67. C32×C4 | C2×C3⋊C8 | semidirect |
| 73. H(3,𝔽3) | C2×C4×C8 | semidirect |
| 88. C2×C32 | C4×C3⋊C8 | semidirect |
| 109. C32 | C2×C4×C3⋊C8 | semidirect |
| 111. C2×C4 | #6180 | direct (N × H) |
| 126. C4 | C2×#6180 | direct (N × H) |
| 131. C2 | C4×#6180 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C42×H(3,𝔽3) | 2 |
| C22×#6180 | 2 |
| C4×#6180 | 2 |
| C2×C3×C4×C3⋊C8 | 3 |