Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
2783 conjugacy classes of subgroups; only the 455 normal subgroups are shown.
This group has 2783 conjugacy classes of subgroups — too many to draw. Only its 455 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23×C3 | C23×C32 |
| commutator | C22×C3 | C22×C32×C4 |
| prime radical | C25×H(3,𝔽3) | C2 |
| LCS | G ▷ C22×C3 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C22⋊C4×H(3,𝔽3) | C2 | semidirect |
| 3. C32×#379 | C3 | semidirect |
| 5. C24×H(3,𝔽3) | C4 | semidirect |
| 7. C22×C4×H(3,𝔽3) | C22 | semidirect |
| 10. C2×C32×C22⋊C4 | C2×C3 | semidirect |
| 21. C24×C32 | C3×C4 | semidirect |
| 23. C22×C32×C4 | C22×C3 | semidirect |
| 28. C22×H(3,𝔽3) | C22⋊C4 | semidirect |
| 43. #379 | H(3,𝔽3) | direct (N × H) |
| 60. C22×C32 | C3×C22⋊C4 | semidirect |
| 65. C2×C22⋊C4 | C2×H(3,𝔽3) | semidirect |
| 66. H(3,𝔽3) | #379 | direct (N × H) |
| 85. C24 | C4×H(3,𝔽3) | semidirect |
| 87. C22×C4 | C22×H(3,𝔽3) | semidirect |
| 103. C32 | C3×#379 | semidirect |
| 120. C22 | C22⋊C4×H(3,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×H(3,𝔽3) | 2 |
| C2×C22⋊C4×H(3,𝔽3) | 2 |
| C32×#379 | 3 |