Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
266 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 84 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×C32×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. D3×#3771 | C2 | direct (N × H) |
| 11. C3⋊#3771 | C22 | semidirect |
| 12. C3×#3771 | C22 | semidirect |
| 16. C2×#3771 | D3 | direct (N × H) |
| 33. #3771 | C2×D3 | direct (N × H) |
| 35. C2×C32×D3 | C16 | semidirect |
| 59. C2×C33 | C2×C16 | semidirect |
| 61. C32×D3 | C2×C16 | semidirect |
| 88. C33 | C22×C16 | semidirect |
| 108. C2×C32 | D3×C16 | semidirect |
| 123. C2×D3 | #3771 | direct (N × H) |
| 134. C32 | C2×D3×C16 | semidirect |
| 148. C2×C3 | C2×#3771 | semidirect |
| 150. D3 | C2×#3771 | direct (N × H) |
| 159. C3 | C22×#3771 | semidirect |
| 161. C2 | D3×#3771 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D3×#622 | 2 |
| C2×C3⋊#3771 | 2 |
| C2×C3×#3771 | 2 |
| D3×#3771 | 2 |
| C22×#3771 | 3 |
| C2×D3×C16 | 9 |