Yes, that first group has twelve elements of order 4.
The second group only has eight elements of order four, namely, 7, 43, 57, 93, 107, 143, 157, 193. This shows that these two groups are not the same.
From the classification of finite abelian groups, the only possibility for this second group is Z_4 x Z_2 x Z_2. (Because there are elements of order 4, no elements of order 8, the group has order 16, and it's not Z_4 x Z_4)
Last edited by vonLipwig
on Tue Sep 11, 2012 8:37 pm, edited 1 time in total.