Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
282 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×AGL(1,𝔽9) |
| commutator, hypocenter | C33 | C22×C16 |
| Frattini | C2 | C2×D3×AGL(1,𝔽9) |
| Fitting, socle | C22×C33 | C2×C8 |
| prime radical | C22×C3⋊(C3⋊D3) | C8 |
| cosocle | C3×#622 | C23 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3⋊#3771 | C2 | semidirect |
| 3. C2×C3×#3771 | C2 | semidirect |
| 4. #11724092 | C2 | direct (N × H) |
| 11. C3⋊#3771 | C22 | semidirect |
| 12. C3×#3771 | C22 | semidirect |
| 15. C2×#3771 | D3 | semidirect |
| 33. #3771 | C2×D3 | semidirect |
| 35. C2×C3⋊(C3⋊D3) | C16 | semidirect |
| 59. C2×C33 | C2×C16 | semidirect |
| 61. C3⋊(C3⋊D3) | C2×C16 | semidirect |
| 89. C33 | C22×C16 | semidirect |
| 108. C2×C32 | D3×C16 | semidirect |
| 137. C32 | C2×D3×C16 | semidirect |
| 150. C2×C3 | #64213 | semidirect |
| 163. C3 | C2×#64213 | semidirect |
| 165. C2 | #11724092 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#88924 | 2 |
| C2×C3⋊#3771 | 2 |
| C2×C3×#3771 | 2 |
| #11724092 | 2 |
| C2×#64213 | 3 |
| C2×D3×C16 | 9 |