The book visual group theory by Nathan Carter does a great job explaining the Direct product with the help of multiplication table.
But it does not discuss the quotient process in terms of the multiplication table.
I think if we allign the group in order of a subgroup we are dealing with, then the multiplication table of the group will be partitioned into several parts, one will correspond to the table of the subgroup and the others, its cosets.
Now we take each of them and collapse them into a single point in the table; thus, we obtain the table of the quotient group.
Is my visualization correct?
See the above diagram, does it make the partitions prominent?
If each partition is now collapsed to a single point it would look like a table for $\mathbb Z_2$.