Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
266 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D3×AGL(1,𝔽9) |
| commutator, hypocenter | C33 | C22×C16 |
| Frattini | C2 | C2×D3×AGL(1,𝔽9) |
| Fitting | C33×C4 | C2×C8 |
| socle | C2×C33 | C22×C8 |
| prime radical | C2×D3×C3⋊D3 | C8 |
| cosocle | C3×#622 | C23 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11724237 | C2 | semidirect |
| 3. C3×#64213 | C2 | semidirect |
| 4. #11724092 | C2 | semidirect |
| 5. D3×#3771 | C2 | semidirect |
| 14. C3⋊#3771 | C22 | semidirect |
| 15. C3×#3771 | C22 | semidirect |
| 20. #64213 | D3 | direct (N × H) |
| 42. #3771 | C2×D3 | semidirect |
| 46. D3×C3⋊D3 | C16 | semidirect |
| 77. C3⋊(C3⋊D3) | C2×C16 | semidirect |
| 78. C3×C3⋊D3 | C2×C16 | semidirect |
| 79. C32×D3 | C2×C16 | semidirect |
| 114. C33 | C22×C16 | semidirect |
| 141. C3⋊D3 | D3×C16 | semidirect |
| 174. C32 | C2×D3×C16 | semidirect |
| 195. D3 | #64213 | direct (N × H) |
| 206. C3 | C2×#64213 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D3×#3787 | 2 |
| #11724237 | 2 |
| C3×#64213 | 2 |
| #11724092 | 2 |
| D3×#3771 | 2 |
| C2×#64213 | 3 |
| C2×D3×C16 | 9 |