Finite Group DatabasebetaFinite Group DatabasebetaOrder 2160.
116 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #67195 |
| commutator, hypocenter | C32×C5 | C2×C3×C8 |
| Frattini | C1 | G |
| Fitting, socle | C2×C33×C5 | C8 |
| prime radical | C2×C3×C5×C3⋊D3 | C4 |
| cosocle | #4379 | C22×C3 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C2×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#67195 | C2 | direct (N × H) |
| 3. C2×#67195 | C3 | direct (N × H) |
| 9. #67195 | C2×C3 | direct (N × H) |
| 10. C2×C33×C5 | C8 | semidirect |
| 11. C3×C5×C3⋊D3 | C8 | semidirect |
| 21. C33×C5 | C2×C8 | semidirect |
| 29. C2×C32×C5 | C3×C8 | semidirect |
| 30. C5×C3⋊D3 | C3×C8 | semidirect |
| 41. C2×C33 | #170 | semidirect |
| 42. C3×C3⋊D3 | #170 | semidirect |
| 47. C32×C5 | C2×C3×C8 | semidirect |
| 58. C2×C3×C5 | AGL(1,𝔽9) | semidirect |
| 61. C33 | C2×#170 | semidirect |
| 69. C2×C32 | C3×#170 | semidirect |
| 70. C3⋊D3 | C3×#170 | semidirect |
| 77. C3×C5 | C2×AGL(1,𝔽9) | semidirect |
| 84. C2×C5 | C3×AGL(1,𝔽9) | semidirect |
| 87. C32 | C2×C3×#170 | semidirect |
| 94. C2×C3 | #67195 | direct (N × H) |
| 97. C5 | C2×C3×AGL(1,𝔽9) | semidirect |
| 103. C3 | C2×#67195 | direct (N × H) |
| 106. C2 | C3×#67195 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#4379 | 2 |
| C3×#67195 | 2 |
| C2×#67195 | 3 |
| C2×C3×AGL(1,𝔽9) | 5 |
| C2×C3×#170 | 9 |