Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1045 conjugacy classes of subgroups; only the 315 normal subgroups are shown.
This group has 1045 conjugacy classes of subgroups — too many to draw. Only its 315 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C3×C4 | C23×C32 |
| commutator, socle | C22×C3 | C22×C32×C4 |
| prime radical | C24×H(3,𝔽3) | C4 |
| LCS | G ▷ C22×C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×M16×H(3,𝔽3) | C2 | semidirect |
| 3. H(3,𝔽3)×C22⋊C8 | C2 | semidirect |
| 4. C32×#407 | C3 | semidirect |
| 12. C2×C8×H(3,𝔽3) | C22 | semidirect |
| 14. C2×C32×M16 | C2×C3 | semidirect |
| 15. C32×C22⋊C8 | C2×C3 | semidirect |
| 37. C2×C32×C8 | C22×C3 | semidirect |
| 43. C22×H(3,𝔽3) | M16 | semidirect |
| 66. #407 | H(3,𝔽3) | direct (N × H) |
| 92. C22×C32 | C3×M16 | semidirect |
| 98. C2×M16 | C2×H(3,𝔽3) | semidirect |
| 99. C22⋊C8 | C2×H(3,𝔽3) | semidirect |
| 100. H(3,𝔽3) | #407 | direct (N × H) |
| 136. C2×C8 | C22×H(3,𝔽3) | semidirect |
| 157. C32 | C3×#407 | semidirect |
| 183. C22 | M16×H(3,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C4×H(3,𝔽3) | 2 |
| C2×M16×H(3,𝔽3) | 2 |
| H(3,𝔽3)×C22⋊C8 | 2 |
| C32×#407 | 3 |