Template:Subgroup property see examples embed

Below are some examples of a proper nontrivial subgroup that satisfy the property [[{{{1}}}]].

	Group part	Subgroup part	Quotient part
2-Sylow subgroup of general linear group:GL(2,3)	General linear group:GL(2,3)	Semidihedral group:SD16
A3 in A4	Alternating group:A4	Cyclic group:Z3
A3 in A5	Alternating group:A5	Cyclic group:Z3
A3 in S3	Symmetric group:S3	Cyclic group:Z3	Cyclic group:Z2
A3 in S4	Symmetric group:S4	Cyclic group:Z3
A4 in A5	Alternating group:A5	Alternating group:A4
A4 in S4	Symmetric group:S4	Alternating group:A4	Cyclic group:Z2
Center of M16	M16	Cyclic group:Z4	Klein four-group
Center of central product of D8 and Z4	Central product of D8 and Z4	Cyclic group:Z4	Klein four-group
Center of dihedral group:D16	Dihedral group:D16	Cyclic group:Z2	Dihedral group:D8
Center of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group
Center of direct product of D8 and Z2	Direct product of D8 and Z2	Klein four-group	Klein four-group
Center of nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Klein four-group	Klein four-group
Center of quaternion group	Quaternion group	Cyclic group:Z2	Klein four-group
Center of semidihedral group:SD16	Semidihedral group:SD16	Cyclic group:Z2	Dihedral group:D8
Center of special linear group:SL(2,3)	Special linear group:SL(2,3)	Cyclic group:Z2	Alternating group:A4
Center of special linear group:SL(2,5)	Special linear group:SL(2,5)	Cyclic group:Z2	Alternating group:A5
Center of unitriangular matrix group:UT(3,p)	Unitriangular matrix group:UT(3,p)	Group of prime order	Elementary abelian group of prime-square order
Central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Cyclic group:Z2	Quaternion group
Cyclic four-subgroups of symmetric group:S4	Symmetric group:S4	Cyclic group:Z4
Cyclic maximal subgroup of dihedral group:D16	Dihedral group:D16	Cyclic group:Z8	Cyclic group:Z2
Cyclic maximal subgroup of dihedral group:D8	Dihedral group:D8	Cyclic group:Z4	Cyclic group:Z2
Cyclic maximal subgroup of semidihedral group:SD16	Semidihedral group:SD16	Cyclic group:Z8	Cyclic group:Z2
Cyclic maximal subgroups of quaternion group	Quaternion group	Cyclic group:Z4	Cyclic group:Z2
D8 in A6	Alternating group:A6	Dihedral group:D8
D8 in D16	Dihedral group:D16	Dihedral group:D8	Cyclic group:Z2
D8 in S4	Symmetric group:S4	Dihedral group:D8
D8 in SD16	Semidihedral group:SD16	Dihedral group:D8	Cyclic group:Z2
Derived subgroup of M16	M16	Cyclic group:Z2	Direct product of Z4 and Z2
Derived subgroup of dihedral group:D16	Dihedral group:D16	Cyclic group:Z4	Klein four-group
Derived subgroup of nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Cyclic group:Z2	Direct product of Z4 and Z2
Diagonally embedded Z4 in direct product of Z8 and Z2	Direct product of Z8 and Z2	Cyclic group:Z4	Cyclic group:Z4
Direct product of Z4 and Z2 in M16	M16	Direct product of Z4 and Z2	Cyclic group:Z2
First agemo subgroup of direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z2	Klein four-group
First omega subgroup of direct product of Z4 and Z2	Direct product of Z4 and Z2	Klein four-group	Cyclic group:Z2
Group of integers in group of rational numbers	Group of rational numbers	Group of integers	Group of rational numbers modulo integers
Klein four-subgroup of M16	M16	Klein four-group	Cyclic group:Z4
Klein four-subgroup of alternating group:A4	Alternating group:A4	Klein four-group	Cyclic group:Z3
Klein four-subgroup of alternating group:A5	Alternating group:A5	Klein four-group
Klein four-subgroups of dihedral group:D8	Dihedral group:D8	Klein four-group	Cyclic group:Z2
Non-central Z4 in M16	M16	Cyclic group:Z4	Cyclic group:Z4
Non-characteristic order two subgroups of direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z2	Cyclic group:Z4
Non-normal Klein four-subgroups of symmetric group:S4	Symmetric group:S4	Klein four-group
Non-normal subgroups of M16	M16	Cyclic group:Z2
Non-normal subgroups of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2
Normal Klein four-subgroup of symmetric group:S4	Symmetric group:S4	Klein four-group	Symmetric group:S3
Q8 in SD16	Semidihedral group:SD16	Quaternion group	Cyclic group:Z2
Q8 in central product of D8 and Z4	Central product of D8 and Z4	Quaternion group	Cyclic group:Z2
S2 in S3	Symmetric group:S3	Cyclic group:Z2
S2 in S4	Symmetric group:S4	Cyclic group:Z2
subgroups satisfying this property

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Below are some examples of a proper nontrivial subgroup that does not satisfy the property [[{{{1}}}]].

	Group part	Subgroup part	Quotient part
2-Sylow subgroup of general linear group:GL(2,3)	General linear group:GL(2,3)	Semidihedral group:SD16
A3 in A4	Alternating group:A4	Cyclic group:Z3
A3 in A5	Alternating group:A5	Cyclic group:Z3
A3 in S3	Symmetric group:S3	Cyclic group:Z3	Cyclic group:Z2
A3 in S4	Symmetric group:S4	Cyclic group:Z3
A4 in A5	Alternating group:A5	Alternating group:A4
A4 in S4	Symmetric group:S4	Alternating group:A4	Cyclic group:Z2
Center of M16	M16	Cyclic group:Z4	Klein four-group
Center of central product of D8 and Z4	Central product of D8 and Z4	Cyclic group:Z4	Klein four-group
Center of dihedral group:D16	Dihedral group:D16	Cyclic group:Z2	Dihedral group:D8
Center of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group
Center of direct product of D8 and Z2	Direct product of D8 and Z2	Klein four-group	Klein four-group
Center of semidihedral group:SD16	Semidihedral group:SD16	Cyclic group:Z2	Dihedral group:D8
Center of unitriangular matrix group:UT(3,p)	Unitriangular matrix group:UT(3,p)	Group of prime order	Elementary abelian group of prime-square order
Central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Cyclic group:Z2	Quaternion group
Cyclic four-subgroups of symmetric group:S4	Symmetric group:S4	Cyclic group:Z4
Cyclic maximal subgroup of dihedral group:D16	Dihedral group:D16	Cyclic group:Z8	Cyclic group:Z2
Cyclic maximal subgroup of dihedral group:D8	Dihedral group:D8	Cyclic group:Z4	Cyclic group:Z2
Cyclic maximal subgroup of semidihedral group:SD16	Semidihedral group:SD16	Cyclic group:Z8	Cyclic group:Z2
Cyclic maximal subgroups of quaternion group	Quaternion group	Cyclic group:Z4	Cyclic group:Z2
D8 in A6	Alternating group:A6	Dihedral group:D8
D8 in D16	Dihedral group:D16	Dihedral group:D8	Cyclic group:Z2
D8 in S4	Symmetric group:S4	Dihedral group:D8
D8 in SD16	Semidihedral group:SD16	Dihedral group:D8	Cyclic group:Z2
Derived subgroup of M16	M16	Cyclic group:Z2	Direct product of Z4 and Z2
Derived subgroup of dihedral group:D16	Dihedral group:D16	Cyclic group:Z4	Klein four-group
Derived subgroup of nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Cyclic group:Z2	Direct product of Z4 and Z2
Diagonally embedded Z4 in direct product of Z8 and Z2	Direct product of Z8 and Z2	Cyclic group:Z4	Cyclic group:Z4
Direct product of Z4 and Z2 in M16	M16	Direct product of Z4 and Z2	Cyclic group:Z2
First agemo subgroup of direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z2	Klein four-group
First omega subgroup of direct product of Z4 and Z2	Direct product of Z4 and Z2	Klein four-group	Cyclic group:Z2
Group of integers in group of rational numbers	Group of rational numbers	Group of integers	Group of rational numbers modulo integers
Klein four-subgroup of M16	M16	Klein four-group	Cyclic group:Z4
Klein four-subgroup of alternating group:A4	Alternating group:A4	Klein four-group	Cyclic group:Z3
Klein four-subgroup of alternating group:A5	Alternating group:A5	Klein four-group
Klein four-subgroups of dihedral group:D8	Dihedral group:D8	Klein four-group	Cyclic group:Z2
Non-central Z4 in M16	M16	Cyclic group:Z4	Cyclic group:Z4
Non-characteristic order two subgroups of direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z2	Cyclic group:Z4
Non-normal Klein four-subgroups of symmetric group:S4	Symmetric group:S4	Klein four-group
Non-normal subgroups of M16	M16	Cyclic group:Z2
Non-normal subgroups of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2
Normal Klein four-subgroup of symmetric group:S4	Symmetric group:S4	Klein four-group	Symmetric group:S3
Q8 in SD16	Semidihedral group:SD16	Quaternion group	Cyclic group:Z2
Q8 in central product of D8 and Z4	Central product of D8 and Z4	Quaternion group	Cyclic group:Z2
S2 in S3	Symmetric group:S3	Cyclic group:Z2
S2 in S4	Symmetric group:S4	Cyclic group:Z2
S3 in S4	Symmetric group:S4	Symmetric group:S3
SL(2,3) in GL(2,3)	General linear group:GL(2,3)	Special linear group:SL(2,3)	Cyclic group:Z2
Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Cyclic group:Z2	Dihedral group:D8
Subgroup generated by double transposition in symmetric group:S4	Symmetric group:S4	Cyclic group:Z2
subgroups dissatisfying property {{{1}}}

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