Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
232 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4×C9 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C2×C4×C9 |
| Frattini, socle | C22×C3 | C2×C3×S4 |
| Fitting | C4×Q8×C9 | D3 |
| prime radical, cosocle | C2×C3×SL(2,𝔽3) | C22×C3 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C4×C9 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. C9×#226 | C4 | direct (N × H) |
| 24. C4×#226 | C9 | direct (N × H) |
| 81. #226 | C4×C9 | direct (N × H) |
| 104. C4×C9 | #226 | direct (N × H) |
| 165. C9 | C4×#226 | direct (N × H) |
| 192. C4 | C9×#226 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C9×#226 | 2 |
| C4×C9×SL(2,𝔽3) | 2 |
| C9×#859 | 2 |
| C3×C4×#226 | 3 |
| C4×C9×Q16 | 3 |
| C4×C9×Dic3 | 4 |