The following groups are not direct products of smaller groups:

group 12 of order 64
group 16 of order 64
group 29 of order 64
group 31 of order 64
group 38 of order 64
group 40 of order 64
group 44 of order 64
group 45 of order 64
group 47 of order 64
group 49 of order 64
group 51 of order 64
group 52 of order 64 dihedral(64)
group 54 of order 64 quaternion(64)
group 61 of order 64
group 70 of order 64
group 112 of order 64
group 124 of order 64
group 128 of order 64
group 140 of order 64
group 147 of order 64
group 174 of order 64
group 185 of order 64
group 189 of order 64
group 257 of order 64

Here are some product groups:

group 2 of order 64, gp8_1 x gp8_1
group 26 of order 64, gp4_1 x gp16_1
group 50 of order 64, ab 2 x gp32_1
group 55 of order 64, gp4_1 x gp16_2
group 58 of order 64, gp4_1 x gp16_3
group 59 of order 64, gp4_1 x gp16_2
group 83 of order 64, gp4_1 x gp16_2
group 85 of order 64, gp4_1 x gp16_2
group 87 of order 64, gp4_1 x gp16_2
group 95 of order 64, ab 2 x gp32_9
group 103 of order 64, ab 2 x gp32_12
group 107 of order 64, ab 2 x gp32_14
group 110 of order 64, ab 2 x gp32_15
group 115 of order 64, gp8_1 x gp8_3
group 118 of order 64, gp4_1 x gp16_7
group 120 of order 64, gp4_1 x gp16_9
group 126 of order 64, gp8_1 x gp8_4
group 183 of order 64, ab 2 x gp32_16
group 184 of order 64, ab 2 x gp32_17
group 186 of order 64, ab 2 x gp32_18
group 188 of order 64, ab 2 x gp32_20
group 192 of order 64, ab 2 x gp32_21
group 194 of order 64, ab 2 x gp32_23
group 196 of order 64, ab 2 x gp32_25
group 197 of order 64, ab 2 x gp32_26
group 198 of order 64, gp4_1 x gp16_13
group 203 of order 64, ab 2 x gp32_28
group 226 of order 64, gp8_3 x gp8_3
group 246 of order 64, ab 2 x gp32_36
group 247 of order 64, ab 2 x gp32_37
group 248 of order 64, ab 2 x gp32_38
group 250 of order 64, ab 2 x gp32_39
group 252 of order 64, ab 2 x gp32_41
group 253 of order 64, ab 2 x gp32_42
group 261 of order 64, ab 2 x gp32_46
group 262 of order 64, ab 2 x gp32_47
group 263 of order 64, ab 2 x gp32_48

(This list is made of nearly all the products of two groups G and H of orders multiplying to 64, such that we have carried the computations through for both G and H (there are no redundancies). However a few such products are missing: strangely enough, the complexity has exploded in these cases for GxH. It is of course possible to use the Kunneth formula to recover most of the information (but not the computation of Milnor constants, for example).)