Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
880 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 880 conjugacy classes of subgroups — too many to draw. Only its 189 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C32×#120 |
| commutator, Frattini, cosocle | C23×C3 | C23×C32 |
| prime radical | H(3,𝔽3)×H(5,𝔽2) | C2 |
| LCS | G ▷ C23×C3 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C23×H(3,𝔽3) ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. H(3,𝔽3)×#123 | C2 | semidirect |
| 3. H(3,𝔽3)×#99 | C2 | semidirect |
| 4. C32×#458 | C3 | semidirect |
| 8. C22×C4×H(3,𝔽3) | C22 | semidirect |
| 11. C22⋊C4×H(3,𝔽3) | C22 | semidirect |
| 13. C32×#123 | C2×C3 | semidirect |
| 14. C32×#99 | C2×C3 | semidirect |
| 30. C22×C32×C4 | C22×C3 | semidirect |
| 33. C32×C22⋊C4 | C22×C3 | semidirect |
| 58. #458 | H(3,𝔽3) | direct (N × H) |
| 85. #123 | C2×H(3,𝔽3) | semidirect |
| 86. #99 | C2×H(3,𝔽3) | semidirect |
| 87. H(3,𝔽3) | #458 | direct (N × H) |
| 114. C22×C4 | C22×H(3,𝔽3) | semidirect |
| 117. C22⋊C4 | C22×H(3,𝔽3) | semidirect |
| 134. C32 | C3×#458 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(3,𝔽3)×H(5,𝔽2) | 2 |
| H(3,𝔽3)×#123 | 2 |
| H(3,𝔽3)×#99 | 2 |
| C32×#458 | 3 |