# Subgroup structure of symmetric group:S8

## Contents

View subgroup structure of particular groups | View other specific information about symmetric group:S8

This article discusses the subgroup structure of symmetric group:S8, which is the symmetric group on the set $\{ 1, 2,3,4,5,6,7,8\}$. The group has order 40320.

## Tables for quick information

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite group)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so both divide it), |order of quotient group divides order of group (and equals index of corresponding normal subgroup)
Sylow subgroups exist, Sylow implies order-dominating, congruence condition on Sylow numbers|congruence condition on number of subgroups of given prime power order
normal Hall implies permutably complemented, Hall retract implies order-conjugate

### Quick summary

Item Value
Number of subgroups 151221
Compared with $S_n, n=1,2,\dots$: 1,2,6,30,156,1455,11300,151221
Number of conjugacy classes of subgroups 296
Compared with $S_n, n=1,2,\dots$: 1,2,4,11,19,56,96,296,554,1593,...
Number of automorphism classes of subgroups 96
Compared with $S_n, n=1,2,\dots$: 1,2,4,11,19,37,96,296,554,1593,...