The integers between 1 and 100 that contain 4 as a digit are: 4, 14, 24, 34, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 54, 64, 74, 84, and 94. There are 19 such integers in total.

Out of these, the integers divisible by 4 are: 4, 24, 40, 44, 48, 64, and 84. This gives a total of 7 integers divisible by 4.

Therefore, the number of integers that are not divisible by 4 is: $19 - 7 = 12$.