Regex to validate password strength

You can do these checks using positive look ahead assertions:

^(?=.*[A-Z].*[A-Z])(?=.*[!@#$&*])(?=.*[0-9].*[0-9])(?=.*[a-z].*[a-z].*[a-z]).{8}$

Rubular link

Explanation:

^                         Start anchor
(?=.*[A-Z].*[A-Z])        Ensure string has two uppercase letters.
(?=.*[!@#$&*])            Ensure string has one special case letter.
(?=.*[0-9].*[0-9])        Ensure string has two digits.
(?=.*[a-z].*[a-z].*[a-z]) Ensure string has three lowercase letters.
.{8}                      Ensure string is of length 8.
$                         End anchor.

You should also consider changing some of your rules to:

1. Add more special characters i.e. %, ^, (, ), -, _, +, and period. I'm adding all the special characters that you missed above the number signs in US keyboards. Escape the ones regex uses.
2. Make the password 8 or more characters. Not just a static number 8.

With the above improvements, and for more flexibility and readability, I would modify the regex to.

^(?=(.*[a-z]){3,})(?=(.*[A-Z]){2,})(?=(.*[0-9]){2,})(?=(.*[!@#$%^&*()\-__+.]){1,}).{8,}$

Basic Explanation

(?=(.*RULE){MIN_OCCURANCES,})     

Each rule block is shown by (?=(){}). The rule and number of occurrences can then be easily specified and tested separately, before getting combined

Detailed Explanation

^                               start anchor
(?=(.*[a-z]){3,})               lowercase letters. {3,} indicates that you want 3 of this group
(?=(.*[A-Z]){2,})               uppercase letters. {2,} indicates that you want 2 of this group
(?=(.*[0-9]){2,})               numbers. {2,} indicates that you want 2 of this group
(?=(.*[!@#$%^&*()\-__+.]){1,})  all the special characters in the [] fields. The ones used by regex are escaped by using the \ or the character itself. {1,} is redundant, but good practice, in case you change that to more than 1 in the future. Also keeps all the groups consistent
{8,}                            indicates that you want 8 or more
$                               end anchor

And lastly, for testing purposes here is a robulink with the above regex

You can use zero-length positive look-aheads to specify each of your constraints separately:

(?=.{8,})(?=.*\p{Lu}.*\p{Lu})(?=.*[!@#$&*])(?=.*[0-9])(?=.*\p{Ll}.*\p{Ll})

If your regex engine doesn't support the \p notation and pure ASCII is enough, then you can replace \p{Lu} with [A-Z] and \p{Ll} with [a-z].