Finite Group DatabasebetaFinite Group DatabasebetaOrder 1512.
188 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 42 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 | D32×ΓL(1,𝔽8) |
| commutator, Fitting, socle, cosocle | C2×C32×C7 | C22×C3 |
| hypocenter | C32×C7 | C3×D4 |
| LCS | G ▷ C2×C32×C7 ▷ C32×C7 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×D3×ΓL(1,𝔽8) | C2 | semidirect |
| 2. ΓL(1,𝔽8)×C3⋊Dic3 | C2 | semidirect |
| 3. C7×#625 | C3 | semidirect |
| 7. C2×C3×D3×C7 | C2×C3 | semidirect |
| 10. C2×D3×ΓL(1,𝔽8) | D3 | semidirect |
| 11. C7×C3⋊Dic3 | C2×C3 | semidirect |
| 15. C32×ΓL(1,𝔽8) | D4 | semidirect |
| 30. C2×D3×C7 | C3×D3 | semidirect |
| 37. #625 | ΓL(1,𝔽8) | direct (N × H) |
| 38. C32×C7 | C3×D4 | semidirect |
| 42. C3×ΓL(1,𝔽8) | #67 | semidirect |
| 58. C2×C3×D3 | C2×ΓL(1,𝔽8) | semidirect |
| 59. C3⋊Dic3 | C2×ΓL(1,𝔽8) | semidirect |
| 68. C3×C7 | C3×#67 | semidirect |
| 71. ΓL(1,𝔽8) | #625 | direct (N × H) |
| 86. C2×D3 | D3×ΓL(1,𝔽8) | semidirect |
| 94. C32 | D4×ΓL(1,𝔽8) | semidirect |
| 96. C7 | C3×#625 | semidirect |
| 112. C3 | ΓL(1,𝔽8)×#67 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×D3×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×C3⋊Dic3 | 2 |
| C7×#625 | 3 |
| ΓL(1,𝔽8)×#67 | 3 |
| C3×#625 | 7 |