Finite Group DatabasebetaFinite Group Databasebeta← ΓL(1,𝔽8)×B(4,𝔽2)⋊C3 #72276694
Order 4032.
292 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 28 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 | ΓL(1,𝔽8)×#120⋊C3 |
| commutator, cosocle, hypocenter | C7×H(5,𝔽2) | C2×C32 |
| Frattini | C23 | C2×A4×ΓL(1,𝔽8) |
| Fitting | C7×B(4,𝔽2) | C32 |
| socle | C2×C7 | C3×#120⋊C3 |
| LCS | G ▷ C7×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. ΓL(1,𝔽8)×#997 | C2 | semidirect |
| 2. B(4,𝔽2)⋊ΓL(1,𝔽8) | C3 | semidirect |
| 3. C7×B(4,𝔽2)⋊C3 | C3 | semidirect |
| 4. ΓL(1,𝔽8)×B(4,𝔽2) | C3 | semidirect |
| 6. C7⋊#997 | C2×C3 | semidirect |
| 7. C7×#997 | C2×C3 | semidirect |
| 8. ΓL(1,𝔽8)×H(5,𝔽2) | C2×C3 | semidirect |
| 16. C7×B(4,𝔽2) | C32 | semidirect |
| 19. C24×ΓL(1,𝔽8) | A4 | semidirect |
| 30. C7×H(5,𝔽2) | C2×C32 | semidirect |
| 35. B(4,𝔽2)⋊C3 | ΓL(1,𝔽8) | direct (N × H) |
| 59. C24×C7 | C3×A4 | semidirect |
| 68. #997 | C2×ΓL(1,𝔽8) | semidirect |
| 91. B(4,𝔽2) | C3×ΓL(1,𝔽8) | semidirect |
| 125. H(5,𝔽2) | C2×C3×ΓL(1,𝔽8) | semidirect |
| 160. ΓL(1,𝔽8) | B(4,𝔽2)⋊C3 | direct (N × H) |
| 163. C24 | A4×ΓL(1,𝔽8) | semidirect |
| 203. C7 | C3×B(4,𝔽2)⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| ΓL(1,𝔽8)×#997 | 2 |
| B(4,𝔽2)⋊ΓL(1,𝔽8) | 3 |
| C7×B(4,𝔽2)⋊C3 | 3 |
| ΓL(1,𝔽8)×B(4,𝔽2) | 3 |
| C22×A4×ΓL(1,𝔽8) | 4 |
| C3×B(4,𝔽2)⋊C3 | 7 |