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In this Ph.D dissertation,we investigated some problems related to finite group G under the condition that some given subgroups of G are embedded subgroups and supplemented subgroups.This dissertation consists of six chapters.Throughout this dissertation,all groups under consideration are finite.The first chapter comprises research background,motivation and the main results in this dissertation.The second chapter presents some basic terminologies,basic definitions and some known conclusions which will be used in this dissertation.In Chapter 3,we study the relationship between nearly σ-embedded subgroups and finite groups.By using the nearly σ-embedded subgroups,we obtain the conditions under which a group is soluble and σj-nilpotent（that is,p-nilpotent for any prime p ∈σj）.In particular,a new characterization of a group to be supersoluble is obtained by studying the nearly a-embedded property of subgroups.Some known results are generalized.Chapter 4 aims to study the influence of weakly σ-permutably embedded subgroups on the structure of finite groups.We first introduce the concept of weakly σ-permutably embedded subgroups and discuss the basic properties of them.We study the theory of finite groups under the condition that some given subgroups are weakly a-permutably embedded and obtain some new characterization of a group to be supersoluble.In particular,we show that under which conditions a normal subgroup of G is hypercyclically embedded.These results unify and generalize some known achievements.In Chapter 5,we introduce the new concept of weakly σ-quasinormal subgroups,and investigate their influence on finite groups.We establish some new criteria for a group to be supersoluble or more general belongs to a saturated formation that contains all supersoluble groups.Chapter 6 is devoted to analysis the effect of nearly SΦ-normal subgroups on the theory of finite groups.The concept is to obtain some conditions under which a finite group G is p-nilpotent and belongs to solubly saturated formation that contains all supersoluble groups.Moreover,we find a new condition under which a normal subgroup is hypercyclically embedded in G.