If in class diagram I have a parent Account and three childs Child1, Child2, and Admin. The Admin has control over Child1 and Child2 accounts. Will the relation be like this?
Class Diagram - Inheritance Relationship
class diagram
The diagram is is close to what you said.  But, just going by the names, Child1 and Child2 should not be classes. Those names are instances of Child.  Also, according to the zero one infinity rule, in a design, there is no such thing as 2.  Everything is either zero, one, or some arbitrary user configurable number.  That's what the * you used is. A design with a 2 in it should raise your eyebrows.