This article gives specific information, namely, subgroup structure, about a family of groups, namely: groups of order 8.
View subgroup structure of group families  View subgroup structure of groups of a particular order View other specific information about groups of order 8
The list
Subgroup/quotient relationships
Subgroup relationships
Quotient relationships
Numerical information on counts of subgroups by isomorphism type
FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (group of prime power order)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so all subgroups have prime power orders)order of quotient group divides order of group (and equals index of corresponding normal subgroup, so all quotients have prime power orders)
prime power order implies not centerless  prime power order implies nilpotent  prime power order implies center is normalitylarge
size of conjugacy class of subgroups divides index of center
congruence condition on number of subgroups of given prime power order: The total number of subgroups of any fixed prime power order is congruent to 1 mod the prime.
Number of subgroups per isomorphism type
The number in each column is the number of subgroups in the given group of that isomorphism type:
Number of conjugacy classes of subgroups per isomorphism type
The number in each column is the number of conjugacy classes of subgroups in the given group of that isomorphism type:
Number of normal subgroups per isomorphism type
Number of automorphism classes of subgroups per isomorphism type
The number in each column is the number of automorphism classes of subgroups in the given group of that isomorphism type:
Number of characteristic subgroups per isomorphism type
Numerical information on counts of subgroups by order
Number of subgroups per order
Note that by the congruence condition on number of subgroups of given prime power order, all the counts of total number of subgroups as well as number of normal subgroups are congruent to 1 modulo the prime , and hence are odd numbers.
Number of abelian subgroups per order
This is identical to the above table, because all groups of order 2 or 4 are abelian.
Subgroups of order 2
The table below provides information on the counts of subgroups of order 2. Note the following:
General assertion 
Implication for the counts in this case

congruence condition on number of subgroups of given prime power order 
The number of subgroups of order 2 is odd. The number of normal subgroups of order 2 is odd. The number of pcoreautomorphisminvariant subgroups (which in this case means the number of subgroups invariant under automorphisms in the 2core of the automorphism group) of order 2 is odd.

In a group of prime power order , the normal subgroups of prime order are precisely the subgroups of prime order inside the socle, which is the first omega subgroup of the center, and is elementary abelian of order where is the rank of the center. (See minimal normal implies central in nilpotent). The number of normal subgroups of prime order is thus , where . For a nonabelian group, . 
In our case, the number of normal subgroups of order 2 is , which must be one of the numbers 1,3,7. For a nonabelian group, the socle cannot have order 16 or 32, so the number of normal subgroups of order 2 is exactly one.

Group 
Second part of GAP ID 
HallSenior number 
HallSenior symbol 
Nilpotency class 
Number of subgroups of order 2 
Number of normal subgroups of order 2 
Number of 2coreautomorphisminvariant subgroups of order 2 (must be odd) 
Number of 2automorphisminvariant subgroups of order 2 
Number of characteristic subgroups of order 2

cyclic group:Z8 
1 
3 

1 
1 
1 
1 
1 
1

direct product of Z4 and Z2 
2 
2 

1 
3 
3 
1 
1 
1

dihedral group:D8 
3 
4 

2 
5 
1 
1 
1 
1

quaternion group 
4 
5 

2 
1 
1 
1 
1 
1

elementary abelian group:E8 
5 
1 

1 
7 
7 
7 
0 
0

Subgroups of order 4
The table below provides information on the counts of subgroups of order 2. Note the following:
Group 
Second part of GAP ID 
HallSenior number 
HallSenior symbol 
Nilpotency class 
Number of subgroups of order 4 
Number of normal subgroups of order 4 
Number of 2coreautomorphisminvariant subgroups of order 4 (must be odd) 
Number of 2automorphisminvariant subgroups of order 4 
Number of characteristic subgroups of order 4

cyclic group:Z8 
1 
3 

1 
1 
1 
1 
1 
1

direct product of Z4 and Z2 
2 
2 

1 
3 
3 
1 
1 
1

dihedral group:D8 
3 
4 

2 
3 
3 
1 
1 
1

quaternion group 
4 
5 

2 
3 
3 
3 
0 
0

elementary abelian group:E8 
5 
1 

1 
7 
7 
7 
0 
0

Possibilities for maximal subgroups
Subgroupdefining functions