I solved this problem by writing out the different numbers I then wrote out the factors for each number and used the numbers gained from that in the next stage. I then went through and picked out the most common numbers, I counted if there were four occurrences of the number and wrote that number down as a 'multiples of x'. I then went back and took the less common numbers and counted each individually until I had gone through each number and written each valid one down. I then thought what other groups I knew and I thought about prime numbers and the Fibonacci sequence. I wrote out the Fibonacci sequence up to 144 and then looked back to see if there were four numbers that matched my list, there were so I wrote down Fibonacci numbers. I then had a look at my list and The first few prime numbers were in the list, so I wrote down prime numbers as well. I then realised that I had missed the obvious even and odds so I quickly checked that there were four of each and added that to my list. I ended up with eleven number families:
Multiples of 2
Multiples of 3
Multiples of 4
Multiples of 7
Multiples of 8
Multiples of 13
Multiples of 17
Odd numbers
Even numbers
Prime numbers
Fibonacci numbers