In most cases, we also obtain simple surface singularities or minimal singularities, though often with more complicated branching than occurs in the classical types. There are, however, six singularities that do not occur in the classical types. Three of these are unibranch non-normal singularities: an 816311811&_mathId=si1096.gif&_user=111111111&_pii=S0001870816311811&_rdoc=1&_issn=00018708&md5=a422179ec9cd78512ff8bd83eb976bb5" title="Click to view the MathML source">SL2(C)-variety whose normalization is 816311811&_mathId=si3.gif&_user=111111111&_pii=S0001870816311811&_rdoc=1&_issn=00018708&md5=8f0b4286c80131b1d4952f5e789ccf64" title="Click to view the MathML source">A2, an 816311811&_mathId=si151.gif&_user=111111111&_pii=S0001870816311811&_rdoc=1&_issn=00018708&md5=c10425f22db25af9fd876c94610b094e" title="Click to view the MathML source">Sp4(C)-variety whose normalization is 816311811&_mathId=si5.gif&_user=111111111&_pii=S0001870816311811&_rdoc=1&_issn=00018708&md5=bfa576f2fa56742f4c3f670f27846a37" title="Click to view the MathML source">A4, and a two-dimensional variety whose normalization is the simple surface singularity 816311811&_mathId=si6.gif&_user=111111111&_pii=S0001870816311811&_rdoc=1&_issn=00018708&md5=76245941e5801d8cf653ccda9a151af8" title="Click to view the MathML source">A3. In addition, there are three 4-dimensional isolated singularities each appearing once. We also study an intrinsic symmetry action on the singularities, extending Slodowy's work for the singularity of the nilpotent cone at a point in the subregular orbit.