Are there any normal and Toeplitz matrices which are not circulant?

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The way I understand it currently, an Euler diagram can be made of normal, circulant, and Toeplitz matrices which looks like this:

enter image description here My question: is $\left( \text{Normal} \, \cap \, \text{Toeplitz} \right) - \text{Circulant} = Ø$?