Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
543 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 543 conjugacy classes of subgroups — too many to draw. Only its 42 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 | D9×C22⋊A4 |
| commutator, Fitting, cosocle, hypocenter | C22×Q8×C9 | C2×C3 |
| Frattini | C2×C3 | D3×C22⋊A4 |
| socle | C23×C3 | D3×A4 |
| LCS | G ▷ C22×Q8×C9 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×C22⋊SL(2,𝔽3) | C2 | semidirect |
| 3. C22×Q8×D9 | C3 | semidirect |
| 7. C22×Q8×C9 | C2×C3 | semidirect |
| 27. Q8×D9 | A4 | semidirect |
| 39. C22⋊SL(2,𝔽3) | D9 | direct (N × H) |
| 47. C22×D9 | SL(2,𝔽3) | semidirect |
| 50. Q8×C9 | C2×A4 | semidirect |
| 78. C22×C9 | C2×SL(2,𝔽3) | semidirect |
| 91. C22×Q8 | C3×D9 | semidirect |
| 119. D9 | C22⋊SL(2,𝔽3) | direct (N × H) |
| 147. C9 | C2×C22⋊SL(2,𝔽3) | semidirect |
| 156. Q8 | A4×D9 | semidirect |
| 177. C22 | D9×SL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×C22⋊SL(2,𝔽3) | 2 |
| D3×C22⋊SL(2,𝔽3) | 3 |
| C22×Q8×D9 | 3 |
| C2×A4×D9 | 4 |
| D9×SL(2,𝔽3) | 4 |