Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
1086 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 1086 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23×C32 | S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C23×C32 |
| Frattini | C2 | C22×C32×S4 |
| Fitting | C22×C32×Q8 | D3 |
| prime radical | C22×C32×SL(2,𝔽3) | C2 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C23×C32 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C32×#226 | C2 | direct (N × H) |
| 3. C22×C3×#226 | C3 | direct (N × H) |
| 7. C32×#226 | C22 | direct (N × H) |
| 12. C2×C3×#226 | C2×C3 | direct (N × H) |
| 19. C22×#226 | C32 | direct (N × H) |
| 30. C3×#226 | C22×C3 | direct (N × H) |
| 41. C2×#226 | C2×C32 | direct (N × H) |
| 72. #226 | C22×C32 | direct (N × H) |
| 80. C22×C32 | #226 | direct (N × H) |
| 111. C2×C32 | C2×#226 | direct (N × H) |
| 126. C22×C3 | C3×#226 | direct (N × H) |
| 134. C32 | C22×#226 | direct (N × H) |
| 149. C2×C3 | C2×C3×#226 | direct (N × H) |
| 151. C22 | C32×#226 | direct (N × H) |
| 158. C3 | C22×C3×#226 | direct (N × H) |
| 159. C2 | C2×C32×#226 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C32×SL(2,𝔽3) | 2 |
| C2×C32×#226 | 2 |
| C22×C3×#226 | 3 |
| C22×C32×Q16 | 3 |
| C22×C32×Dic3 | 4 |