The set of all binary vectors with 12 components forms a field with 2^12 elements containing 000000000000 and another 65*63 elements. Is it possible to partition these elements into 63 subgroups of 65 vectors so that each of them is closed under binary addition (XOR)?

For example, if we had 4 components instead of 12, the field contains 0000 and another 3*5 elements, we can partition them as follows:

0000

0001
0010
0011

0100
1000
1100

0101
1010
1111

0110
1011
1101

0111
1001
1110

so that, whenever we add two elements from the same group, we get another from the same group. We have five subgroups closed under XOR that only share the element 0000. Could we do the same with vectors with 12 components? Is there some theorem that governs this property for more components?

Thank you!