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As we know, Alternating Groups $A_n$ of degree $n\leq 5$, Dihedral Groups $D_{2n}$ of order $2n$ (for odd natural number $n$) and Cyclic Groups $C_n$ of order $n$ (for positive integer $n$) have subgroups of same order conjugate. Are such (finite) groups classified? Is there some more groups of this type?

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Why single out the cyclic groups of prime order? It holds for any finite cyclic group. – Tobias Kildetoft Sep 4 '13 at 7:32
Oops. I just missed it. It is true for all positive integer $n$. Thanks @TobiasKildetoft. – Shodharthi Sep 4 '13 at 7:36
up vote 5 down vote accepted

I haven't read it, but a classification is claimed in this article:

Robert W. van der Waall (September 2012), "The classification of the finite groups whose subgroups of equal order are conjugate", Indagationes Mathematicae 23 (3) 448–478

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