Subgroup structure of semidihedral group:SD16

This article gives specific information, namely, subgroup structure, about a particular group, namely: semidihedral group:SD16.
View subgroup structure of particular groups | View other specific information about semidihedral group:SD16

We are interested in the group $G=SD_{16}$, the semidihedral group:SD16, given by the presentation:

$\langle a,x\mid a^{8}=x^{2}=e,xax^{-1}=a^{3}\rangle$

where $e$ denotes the identity element.

The group has 16 elements:

$\{e,a,a^{2},a^{3},a^{4},a^{5},a^{6},a^{7},x,ax,a^{2}x,a^{3}x,a^{4}x,a^{5}x,a^{6}x,a^{7}x\}$

Tables for quick information

Table classifying subgroups up to automorphisms

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 normality-large
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.

Automorphism class of subgroups	List of subgroups	Isomorphism class	Order of subgroups	Index of subgroups	Number of conjugacy classes (=1 iff automorph-conjugate subgroup)	Size of each conjugacy class (=1 iff normal subgroup)	Total number of subgroups (=1 iff characteristic subgroup)	Isomorphism class of quotient (if subgroup is normal)	Subnormal depth (if proper and normal, this equals 1)	Nilpotency class
trivial subgroup	$\{e\}$	trivial group	1	16	1	1	1	semidihedral group:SD16	1	0
center of semidihedral group:SD16	$\{e,a^{4}\}$	cyclic group:Z2	2	8	1	1	1	dihedral group:D8	1	1
non-normal order two subgroups of semidihedral group:SD16	$\{e,x\},\{e,a^{2}x\},\{e,a^{4}x\},\{e,a^{6}x\}$	cyclic group:Z2	2	8	1	4	4	--	3	1
derived subgroup of semidihedral group:SD16	$\{e,a^{2},a^{4},a^{6}\}$	cyclic group:Z4	4	4	1	1	1	Klein four-group	1	1
non-normal order four cyclic subgroups of semidihedral group:SD16	$\{e,ax,a^{4},a^{5}x\}$, $\{e,a^{3}x,a^{4},a^{7}x\}$	cyclic group:Z4	4	4	1	2	2	--	2	1
Klein four-subgroups of semidihedral group:SD16	$\{e,x,a^{4},a^{4}x\}$, $\{e,a^{2}x,a^{4},a^{6}x\}$	Klein four-group	4	4	1	2	2	--	2	1
Z8 in SD16	$\langle a\rangle$	cyclic group:Z8	8	2	1	1	1	cyclic group:Z2	1	1
D8 in SD16	$\langle a^{2},x\rangle$	dihedral group:D8	8	2	1	1	1	cyclic group:Z2	1	2
Q8 in SD16	$\langle a^{2},ax\rangle$	quaternion group	8	2	1	1	1	cyclic group:Z2	1	2
whole group	all elements	semidihedral group:SD16	16	1	1	1	1	trivial group	0	3
Total (10 rows)	--	--	--	--	10	--	15	--	--	--