Subgroup structure of symmetric group:S8

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This article gives specific information, namely, subgroup structure, about a particular group, namely: symmetric group:S8.
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

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,...