Finite Group DatabasebetaFinite Group DatabasebetaOrder 1728.
318 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 46 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 | #11724116 | C2 |
| hypocenter | C33 | #465 |
| 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×D3×AGL(1,𝔽9) | C2 | semidirect |
| 3. #11724149 | C2 | semidirect |
| 5. #11724104 | C2 | semidirect |
| 6. #11724100 | C2 | semidirect |
| 7. C3×PSU(3,𝔽2)⋊C4 | C2 | semidirect |
| 27. PSU(3,𝔽2)⋊C4 | D3 | semidirect |
| 41. C3⋊AGL(1,𝔽9) | D4 | semidirect |
| 61. C2×C3⋊(C3⋊D3) | SD16 | semidirect |
| 64. C2×C32×D3 | SD16 | semidirect |
| 163. C33 | #465 | semidirect |
| 236. C2×D3 | AΓL(1,𝔽9) | semidirect |
| 248. C32 | #4807 | semidirect |
| 298. C3 | #10596416 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#89092 | 2 |
| C2×D3×AGL(1,𝔽9) | 2 |
| #11724149 | 2 |
| #11724116 | 2 |
| #11724104 | 2 |
| #11724100 | 2 |
| C3×PSU(3,𝔽2)⋊C4 | 2 |
| #10596416 | 3 |
| #4807 | 9 |