Finite Group DatabasebetaFinite Group DatabasebetaOrder 1512.
180 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 180 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×C3⋊AGL(1,𝔽7) |
| commutator, hypocenter | C32×C7 | C2×C3×C4 |
| Fitting, socle, cosocle | C2×C32×C7 | C22×C3 |
| prime radical | C2×C3×C3⋊AGL(1,𝔽7) | C2 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3⋊(C3⋊(ΓL(1,𝔽8)⋊C4)) | C2 | semidirect |
| 3. C3×Dic3×ΓL(1,𝔽8) | C2 | semidirect |
| 4. Dic3×D21 | C3 | semidirect |
| 8. C3×C3⋊AGL(1,𝔽7) | C4 | semidirect |
| 13. C3⋊Dic21 | C2×C3 | semidirect |
| 14. C3×C7×Dic3 | C2×C3 | semidirect |
| 20. Dic3×ΓL(1,𝔽8) | D3 | semidirect |
| 22. C32×ΓL(1,𝔽8) | C2×C4 | semidirect |
| 27. C3×D21 | C3×C4 | semidirect |
| 35. C3⋊AGL(1,𝔽7) | Dic3 | direct (N × H) |
| 48. C7×Dic3 | C3×D3 | semidirect |
| 55. C32×C7 | C2×C3×C4 | semidirect |
| 60. C3×ΓL(1,𝔽8) | C4×D3 | semidirect |
| 61. C3×ΓL(1,𝔽8) | C2×Dic3 | semidirect |
| 68. D21 | C3×Dic3 | semidirect |
| 87. C3×Dic3 | AGL(1,𝔽7) | semidirect |
| 96. C3×C7 | C3×C4×D3 | semidirect |
| 97. C3×C7 | C2×C3×Dic3 | semidirect |
| 101. ΓL(1,𝔽8) | D3×Dic3 | semidirect |
| 126. Dic3 | C3⋊AGL(1,𝔽7) | direct (N × H) |
| 133. C32 | C4×AGL(1,𝔽7) | semidirect |
| 135. C7 | C3×D3×Dic3 | semidirect |
| 155. C3 | Dic3×AGL(1,𝔽7) | semidirect |
| 156. C3 | C4×C3⋊AGL(1,𝔽7) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C3⋊AGL(1,𝔽7) | 2 |
| C3⋊(C3⋊(ΓL(1,𝔽8)⋊C4)) | 2 |
| C3×Dic3×ΓL(1,𝔽8) | 2 |
| Dic3×D21 | 3 |
| C4×C3⋊AGL(1,𝔽7) | 3 |
| Dic3×AGL(1,𝔽7) | 3 |
| C3×D3×Dic3 | 7 |