See symmetric group:S3. We take the symmetric group on the set of size three.
Elements
See element structure of symmetric group:S3 for full details.
Review the multiplication table in cycle decomposition notation:
[SHOW MORE]
Review the multiplication table in oneline notation:
[SHOW MORE]
Element 
123 
213 
132 
321 
231 
312

123 
123 
213 
132 
321 
231 
312

213 
213 
123 
231 
312 
132 
321

132 
132 
312 
123 
231 
321 
213

321 
321 
231 
312 
123 
213 
132

231 
231 
321 
213 
132 
312 
123

312 
312 
132 
321 
213 
123 
231

Review the conjugacy class structure:
[SHOW MORE]
Partition 
Partition in grouped form 
Verbal description of cycle type 
Elements with the cycle type in cycle decomposition notation 
Elements with the cycle type in oneline notation 
Size of conjugacy class 
Formula for size 
Even or odd? If even, splits? If splits, real in alternating group? 
Element order 
Formula calculating element order

1 + 1 + 1 
1 (3 times) 
three fixed points 
 the identity element 
123 
1 

even; no 
1 

2 + 1 
2 (1 time), 1 (1 time) 
transposition in symmetric group:S3: one 2cycle, one fixed point 
, , 
213, 321, 132 
3 

odd 
2 

3 
3 (1 time) 
3cycle in symmetric group:S3: one 3cycle 
, 
231, 312 
2 

even; yes; no 
3 

Total (3 rows  3 being the number of unordered integer partitions of 3) 
 
 
 
 
6 (equals 3!, the size of the symmetric group) 
 
odd: 3 even;no: 1 even; yes; no: 2 
order 1: 1, order 2: 3, order 3: 2 


Subgroups
Summary table on the structure of subgroups:
[SHOW MORE]
Quick summary
Item 
Value

Number of subgroups 
6 Compared with : 1,2,6,30,156,1455,11300, 151221

Number of conjugacy classes of subgroups 
4 Compared with : 1,2,4,11,19,56,96,296,554,1593

Number of automorphism classes of subgroups 
4 Compared with : 1,2,4,11,19,37,96,296,554,1593

Isomorphism classes of Sylow subgroups and the corresponding Sylow numbers and fusion systems 
2Sylow: cyclic group:Z2, Sylow number is 3, fusion system is the trivial one 3Sylow: cyclic group:Z3, Sylow number is 1, fusion system is noninner fusion system for cyclic group:Z3

Hall subgroups 
Given that the order has only two distinct prime factors, the Hall subgroups are the whole group, trivial subgroup, and Sylow subgroups. Interestingly, all subgroups are Hall subgroups, because the order is a squarefree number

maximal subgroups 
maximal subgroups have order 2 (S2 in S3) and 3 (A3 in S3).

normal subgroups 
There are three normal subgroups: the trivial subgroup, the whole group, and A3 in S3.

Table classifying subgroups up to automorphisms
For more information on each automorphism type, follow the link.