Finite Group DatabasebetaFinite Group DatabasebetaOrder 4032.
794 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 794 conjugacy classes of subgroups — too many to draw. Only its 53 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×C22⋊(C22⋊ΓL(1,𝔽8)) |
| commutator, Fitting, cosocle, hypocenter | C3×C7×H(5,𝔽2) | C2×C3 |
| socle | C2×C3×C7 | C2×C22⋊A4 |
| LCS | G ▷ C3×C7×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×C7⋊#997 | C2 | semidirect |
| 2. D3×C7×H(5,𝔽2) | C3 | semidirect |
| 6. C3×C7×H(5,𝔽2) | C2×C3 | semidirect |
| 12. C7⋊#997 | D3 | direct (N × H) |
| 19. C23×D3×C7 | A4 | semidirect |
| 26. D3×C7×Q8 | A4 | semidirect |
| 39. C7×H(5,𝔽2) | C3×D3 | semidirect |
| 42. D3×H(5,𝔽2) | ΓL(1,𝔽8) | semidirect |
| 45. C23×C3×C7 | C2×A4 | semidirect |
| 54. C3×C7×Q8 | C2×A4 | semidirect |
| 83. C3×H(5,𝔽2) | C2×ΓL(1,𝔽8) | semidirect |
| 114. C23×C7 | D3×A4 | semidirect |
| 116. C7×Q8 | D3×A4 | semidirect |
| 125. C23×D3 | C22⋊ΓL(1,𝔽8) | semidirect |
| 132. D3×Q8 | C22⋊ΓL(1,𝔽8) | semidirect |
| 143. D3×C7 | #997 | semidirect |
| 151. H(5,𝔽2) | D3×ΓL(1,𝔽8) | semidirect |
| 165. C23×C3 | C2×C22⋊ΓL(1,𝔽8) | semidirect |
| 174. C3×Q8 | C2×C22⋊ΓL(1,𝔽8) | semidirect |
| 185. C3×C7 | C2×#997 | semidirect |
| 228. C23 | D3×C22⋊ΓL(1,𝔽8) | semidirect |
| 230. Q8 | D3×C22⋊ΓL(1,𝔽8) | semidirect |
| 238. C7 | D3×#997 | semidirect |
| 245. D3 | C7⋊#997 | direct (N × H) |
| 258. C3 | C2×C7⋊#997 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×C7⋊#997 | 2 |
| D3×C7×H(5,𝔽2) | 3 |
| C2×C7⋊#997 | 3 |
| C2×D3×C22⋊ΓL(1,𝔽8) | 4 |
| D3×C7⋊SL(2,𝔽3) | 4 |
| D3×#997 | 7 |