for G in AllSmallGroups(60) do
  Conj:=ConjugacyClassesSubgroups(LatticeSubgroups(G));
  SubgroupSizes:=Set(Conj,C->Size(C[1]));
  Print(StructureDescription(G)," ",IsAbelian(G)," ",SubgroupSizes,"n");
od;

C5 x (C3 : C4) false [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ]
C3 x (C5 : C4) false [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ]
C15 : C4 false [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ]
C60 true [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ]
A5 false [ 1, 2, 3, 4, 5, 6, 10, 12, 60 ]
C3 x (C5 : C4) false [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ]
C15 : C4 false [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ]
S3 x D10 false [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ]
C5 x A4 false [ 1, 2, 3, 4, 5, 10, 12, 15, 20, 60 ]
C6 x D10 false [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ]
C10 x S3 false [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ]
D60 false [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ]
C30 x C2 true [ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 ]