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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311811&_mathId=si1096.gif&_user=111111111&_pii=S0001870816311811&_rdoc=1&_issn=00018708&md5=a422179ec9cd78512ff8bd83eb976bb5" title="Click to view the MathML source">SL2(C)class="mathContainer hidden">class="mathCode">-variety whose normalization is class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311811&_mathId=si3.gif&_user=111111111&_pii=S0001870816311811&_rdoc=1&_issn=00018708&md5=8f0b4286c80131b1d4952f5e789ccf64" title="Click to view the MathML source">A2class="mathContainer hidden">class="mathCode">, an class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311811&_mathId=si151.gif&_user=111111111&_pii=S0001870816311811&_rdoc=1&_issn=00018708&md5=c10425f22db25af9fd876c94610b094e" title="Click to view the MathML source">Sp4(C)class="mathContainer hidden">class="mathCode">-variety whose normalization is class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311811&_mathId=si5.gif&_user=111111111&_pii=S0001870816311811&_rdoc=1&_issn=00018708&md5=bfa576f2fa56742f4c3f670f27846a37" title="Click to view the MathML source">A4class="mathContainer hidden">class="mathCode">, and a two-dimensional variety whose normalization is the simple surface singularity class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816311811&_mathId=si6.gif&_user=111111111&_pii=S0001870816311811&_rdoc=1&_issn=00018708&md5=76245941e5801d8cf653ccda9a151af8" title="Click to view the MathML source">A3class="mathContainer hidden">class="mathCode">. 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.