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Here is an interactive visualization of the 27 lines on a Clebsch diagonal cubic surface rendered:

Click button to open the graphic in a separate full window. The figure can be rotated by clicking and dragging with a mouse or swiping on a touch device, as well as zoomed in or out with a mouse wheel or pinching on a touch device.

As stated in the post, the isosurface is defined by

81(x3 +y3 +z3) -189(x2y +x2z +xy2 +xz2 +y2z +yz2) +54xyz +126(xy+xz +yz) -9(x2 +y2 +z2) -9(x+y +z) +1 =0