Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
459 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 459 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C42 | #157 |
| commutator, hypocenter | H(3,𝔽3) | C2×C4×C8 |
| Frattini, socle | C22×C3 | C22×#157 |
| Fitting | C42×H(3,𝔽3) | C4 |
| prime radical | C22×#272 | C2×C4 |
| cosocle | C2×#1084 | C23 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C4×#6188 | C2 | semidirect |
| 7. #88626 | C4 | direct (N × H) |
| 17. C4×#272 | C8 | semidirect |
| 19. #6188 | C2×C4 | semidirect |
| 30. C4×H(3,𝔽3) | C2×C8 | semidirect |
| 31. #1084 | C2×C8 | semidirect |
| 47. #272 | C4×C8 | semidirect |
| 70. H(3,𝔽3) | C2×C4×C8 | semidirect |
| 130. C4 | #88626 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#88626 | 2 |
| C42×#272 | 2 |
| C4×#6188 | 2 |
| C2×C3×C4×C8 | 9 |