This is how I thought about this problem not entirely sure it is correct
if$ Z_2\oplus Z_2 \oplus Z_2 = G $ then $ |G|=8 $
by def a subgroup must contain I, I = identity
since $ |G-I|=7 $ there are 7 subgroups $ (I,a) $ where a is each of the 7 remaining elements of G
