A subgroup of a group is a subset of the group  a) closed relative to the group operation,  and  
                                                b) closed relative taking the opposite (reciprocal) elements.

So, if  G  is a group and  H  is a subgroup in  G,  then  H is a subset of  G  and for any two elements  "a"  and  "b"  of  H  
their product  a*b  belongs to  H,  and for any element  "a"  of  H  its opposite  -a  (or its reciprocal  )  belongs to H.

Examples.

All even integer numbers form the subgroup in the group of all integers for addition.

All integers multiple 3 form the subgroup in the group of all integers for addition.

All integer numbers form the subgroup in the group of all real numbers for addition.

All complex numbers with the modulus 1,  {z of C| |z| = 1},  form the subgroup in the multiplicative group of all complex non-zero numbers 
    {z of C| z =/= 0}  for multiplication.

In accordance with the definition, you should check that
   
   a) it is a subset in the given group;
   b) for any two elements "a" and "b" of the subset their product (sum, composition) belongs to the subset;
   c) for any element "a" of the subset its opposite "-a" (or ) belongs to the subset.

The simplest way to generate a subgroup is to take ANY element g of the group and collect all elements {ng}  (or {g^n}), 
       n = 0, +/-1, +/-2, . . . for all integer n. 

Notice that the element  (or ) is the zero (neutral, or unit) element of the group. It must be present in any subgroup.

The group described in this section, is minimal subgroup generated by the element "g" of G.

You can create larger subgroups by generating them using any two, three . . . elements of the group and taking all their linear combinations/compositions.