Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
326 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×AΓL(1,𝔽9) |
| commutator, cosocle | C2×C3×#157 | C23 |
| Fitting, socle | C2×C33 | C2×SD16 |
| prime radical | #11724124 | C2 |
| hypocenter | C33 | #460 |
| LCS | G ▷ C2×C3×#157 ▷ C3×C3⋊D3 ▷ C33 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#89092 | C2 | semidirect |
| 2. C2×C3×AΓL(1,𝔽9) | C2 | semidirect |
| 3. Dic3×PSU(3,𝔽2) | C2 | semidirect |
| 5. #11724104 | C2 | semidirect |
| 6. GO+(4,𝔽2)⋊Dic3 | C2 | semidirect |
| 7. Dic3⋊AGL(1,𝔽9) | C2 | semidirect |
| 23. C2×AΓL(1,𝔽9) | D3 | semidirect |
| 36. C3×PSU(3,𝔽2) | D4 | semidirect |
| 74. C3⋊(C3⋊Dic3) | SD16 | semidirect |
| 75. C32×Dic3 | SD16 | semidirect |
| 92. PSU(3,𝔽2) | #67 | semidirect |
| 164. C33 | #460 | semidirect |
| 246. Dic3 | AΓL(1,𝔽9) | semidirect |
| 249. C32 | #5163 | semidirect |
| 299. C3 | #10596410 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#89092 | 2 |
| C2×C3×AΓL(1,𝔽9) | 2 |
| Dic3×PSU(3,𝔽2) | 2 |
| #11724124 | 2 |
| #11724104 | 2 |
| GO+(4,𝔽2)⋊Dic3 | 2 |
| Dic3⋊AGL(1,𝔽9) | 2 |
| #10596410 | 3 |
| #5163 | 9 |