theory – classification minimal 2-group with special property

I would greatly appreciate it if you kindly give me some advice to tackle the below situation.

Let G be non-abelian 2-group

G=<a, b, c | a^{2^n}=b^{2^m}=c^2=1, (a,b)=c, (a,c)=1, (b,c)=1> where n+mgep3

Moreover, every proper subgroup of G is abelian and cd(G)={1,2}. Also |G’|=1 and Phi(G)=Z(G) is of index 4. Are there any normal subgroup N of G such that G/N is a non-abelian group isomorphic to one of the following groups

Dihedral group, Semi Dihedral group, Q_8