To find the number of three-digit odd numbers divisible by 3 using the digits 3, 4, 5, and 6 with repetition allowed, we need to consider the possible combinations of these digits.

Step 1: Determine the possible choices for the hundreds place
Since we need odd numbers, the hundreds place cannot be 4 or 6. So, the possible choices for the hundreds place are 3 and 5.

Step 2: Determine the possible choices for the tens place
Since repetition is allowed, any of the digits can be used in the tens place. So, the possible choices for the tens place are 3, 4, 5, and 6.

Step 3: Determine the possible choices for the units place
To be divisible by 3, the sum of the digits of the number must be divisible by 3. Since we have the digits 3, 4, 5, and 6, the possible combinations that give a sum divisible by 3 are:

- 3, 3, 3 (sum = 9)
- 4, 4, 4 (sum = 12)
- 5, 5, 5 (sum = 15)
- 6, 6, 6 (sum = 18)
- 3, 4, 6 (sum = 13)
- 3, 6, 6 (sum = 15)
- 4, 4, 6 (sum = 14)
- 4, 5, 5 (sum = 14)
- 4, 6, 6 (sum = 16)
- 5, 5, 6 (sum = 16)

Step 4: Calculate the total number of possible combinations
To calculate the total number of possible combinations, we multiply the number of choices for each place value.

- Choices for the hundreds place: 2 (3 or 5)
- Choices for the tens place: 4 (3, 4, 5, or 6)
- Choices for the units place: 10 (as calculated in Step 3)

Total number of possible combinations = 2 * 4 * 10 = 80

Step 5: Determine the number of three-digit odd numbers
Out of the 80 possible combinations, we need to determine the number of three-digit odd numbers. Since the units place can only be an odd digit (3 or 5), we need to exclude the combinations where the units place is an even digit (4 or 6).

- Choices for the units place (odd): 2 (3 or 5)
- Choices for the units place (even): 2 (4 or 6)

Number of three-digit odd numbers = 2 * 4 * 2 = 16

Hence, the correct answer is option 'C' (11).