Group #492778 — subgroups

Subgroups — Q₁₆×Q₃₂ #492778

Order 512.

392 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Q₃₂	C₂	1	2	yes	maximal
2	Q₁₆²	C₂	1	2	yes	maximal
3	#12521	C₂	1	1	yes	maximal, characteristic
4	#12458	C₂	1	2	yes	maximal
5	#12448	C₂	1	2	yes	maximal
6	#12410	C₂	1	4	yes	maximal
7	C₁₆×Q₁₆	C₂	1	1	yes	maximal, characteristic
8	C₈×Q₃₂	C₂	1	1	yes	maximal, characteristic
9	Q₈×Q₁₆	—	2	2	no	—
10	Q₈×Q₁₆	C₂²	1	4	yes	—
11	#2252	—	2	2	no	—
12	#2252	C₂²	1	2	yes	—
13	#2252	C₂²	1	2	yes	—
14	#2227	—	2	4	no	—
15	#2227	C₂²	1	4	yes	—
16	Q₈×C₁₆	C₂²	1	2	yes	—
17	C₄×Q₃₂	—	2	2	no	—
18	C₄×Q₃₂	C₂²	1	1	yes	characteristic
19	#1713	C₂²	1	2	yes	—
20	#1676	—	2	2	no	—
21	#1676	C₂²	1	4	yes	—
22	#1633	—	2	4	no	—
23	C₈×Q₁₆	C₂²	1	2	yes	—
24	C₈×Q₁₆	C₂²	1	1	yes	characteristic
25	#1371	C₂²	1	1	yes	characteristic
26	#1370	C₂²	1	1	yes	characteristic
27	#1365	C₂²	1	4	yes	—
28	Q₈⋊C₁₆	C₂²	1	2	yes	—
29	#1332	C₂²	1	2	yes	—
30	C₈×C₁₆	C₂²	1	1	yes	characteristic
31	Q₈²	—	2	4	no	—
32	C₂×Q₃₂	D₄	1	1	yes	characteristic
33	#500	—	2	4	no	—
34	#500	—	2	2	no	—
35	#500	—	4	2	no	—
36	#500	C₂³	1	4	yes	—
37	#494	—	2	2	no	—
38	#477	—	2	4	no	—
39	#477	—	2	4	no	—
40	#477	—	4	4	no	—
41	#477	—	4	4	no	—
42	C₈×Q₈	—	2	2	no	—
43	C₈×Q₈	C₂³	1	2	yes	—
44	C₄×Q₁₆	—	2	4	no	—
45	C₄×Q₁₆	—	4	2	no	—
46	C₄×Q₁₆	D₄	1	1	yes	characteristic
47	C₄×Q₁₆	C₂³	1	2	yes	—
48	#366	—	2	2	no	—
49	#366	—	2	2	no	—
50	#366	—	2	4	no	—
51	#366	—	4	2	no	—
52	#366	D₄	1	1	yes	characteristic
53	C₄⋊C₁₆	—	2	2	no	—
54	C₄⋊C₁₆	C₂³	1	2	yes	—
55	#358	—	2	4	no	—
56	#358	D₄	1	2	yes	—
57	C₄×C₁₆	—	2	2	no	—
58	C₄×C₁₆	C₂³	1	1	yes	characteristic
59	C₄×C₁₆	C₂³	1	1	yes	characteristic
60	#335	D₄	1	1	yes	characteristic
61	#335	C₂³	1	2	yes	—
62	#333	—	2	4	no	—
63	Q₈⋊C₈	—	2	2	no	—
64	Q₈⋊C₈	D₄	1	2	yes	—
65	C₈²	C₂³	1	1	yes	characteristic
66	C₂×Q₁₆	—	8	2	no	—
67	C₂×Q₁₆	D₈	1	1	yes	characteristic
68	C₂×Q₁₆	C₂×D₄	1	2	yes	—
69	C₄⋊Q₈	—	2	4	no	—
70	C₄⋊Q₈	—	4	4	no	—
71	C₄⋊Q₈	—	4	4	no	—
72	C₄⋊Q₈	—	8	4	no	—
73	C₄×Q₈	—	2	2	no	—
74	C₄×Q₈	—	4	4	no	—
75	C₄×Q₈	—	4	4	no	—
76	C₄×Q₈	C₂×D₄	1	2	yes	—
77	Q₃₂	Q₁₆	1	4	yes	direct product factor
78	C₂×C₁₆	—	2	1	no	—
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
81	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
82	#107	—	2	4	no	—
83	#107	—	2	4	no	—
84	#107	—	4	4	no	—
85	#107	—	4	2	no	—
86	#107	—	4	2	no	—
87	#107	—	4	4	no	—
88	#107	D₈	1	1	yes	characteristic
89	#107	C₂×D₄	1	2	yes	—
90	C₄⋊C₈	—	2	2	no	—
91	C₄⋊C₈	—	2	2	no	—
92	C₄⋊C₈	—	4	2	no	—
93	C₄⋊C₈	C₂×D₄	1	2	yes	—
94	#103	—	2	2	no	—
95	#103	—	4	4	no	—
96	#103	—	4	4	no	—
97	#103	D₈	1	2	yes	—
98	C₄×C₈	—	2	2	no	—
99	C₄×C₈	—	4	2	no	—
100	C₄×C₈	C₂×D₄	1	1	yes	characteristic
101	C₄×C₈	C₂×D₄	1	1	yes	characteristic
102	C₄×C₈	C₂⁴	1	1	yes	characteristic
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	4	4	no	—
105	C₂×Q₈	—	8	4	no	—
106	C₂×Q₈	C₂×D₈	1	2	yes	—
107	Q₁₆	—	2	2	no	—
108	Q₁₆	—	16	4	no	—
109	Q₁₆	Q₃₂	1	4	yes	direct product factor
110	Q₁₆	C₂×Q₁₆	1	4	yes	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	4	2	no	—
114	C₂×C₈	C₂×D₈	1	1	yes	characteristic
115	C₂×C₈	C₂×D₈	1	1	yes	characteristic
116	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
118	C₄⋊C₄	—	2	2	no	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	4	no	—
122	C₄⋊C₄	—	4	4	no	—
123	C₄⋊C₄	—	8	4	no	—
124	C₄⋊C₄	—	8	4	no	—
125	C₄⋊C₄	C₂×D₈	1	2	yes	—
126	C₄²	—	2	2	no	—
127	C₄²	—	4	2	no	—
128	C₄²	—	8	4	no	—
129	C₄²	C₂²×D₄	1	1	yes	characteristic
130	C₁₆	—	2	2	no	—
131	C₁₆	—	2	1	no	—
132	C₁₆	—	4	2	no	—
133	C₁₆	C₂×Q₁₆	1	2	yes	—
134	Q₈	—	2	2	no	—
135	Q₈	—	2	4	no	—
136	Q₈	—	4	2	no	—
137	Q₈	—	16	4	no	—
138	Q₈	—	32	4	no	—
139	Q₈	C₂×Q₃₂	1	4	yes	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	2	no	—
144	C₂×C₄	—	8	4	no	—
145	C₂×C₄	D₄²	1	1	yes	characteristic
146	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
147	C₂×C₄	#576	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	4	2	no	—
152	C₈	—	8	2	no	—
153	C₈	C₂×Q₃₂	1	2	yes	—
154	C₈	C₂²×Q₁₆	1	1	yes	characteristic
155	C₈	#578	1	1	yes	characteristic
156	C₂²	D₄×D₈	1	1	yes	characteristic
157	C₄	—	2	4	no	—
158	C₄	—	2	1	no	—
159	C₄	—	4	4	no	—
160	C₄	—	4	2	no	—
161	C₄	—	8	2	no	—
162	C₄	—	16	4	no	—
163	C₄	D₄×Q₁₆	1	1	yes	characteristic
164	C₄	#3293	1	1	yes	characteristic
165	C₄	C₂²×Q₃₂	1	1	yes	characteristic
166	C₄	#3417	1	1	yes	characteristic
167	C₂	D₈×Q₁₆	1	1	yes	characteristic
168	C₂	D₄×Q₃₂	1	1	yes	characteristic
169	C₂	#33858	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Q₃₂

C₂

yes

maximal

Q₁₆²

C₂

yes

maximal

#12521

C₂

yes

maximal, characteristic

#12458

C₂

yes

maximal

#12448

C₂

yes

maximal

#12410

C₂

yes

maximal

C₁₆×Q₁₆

C₂

yes

maximal, characteristic

C₈×Q₃₂

C₂

yes

maximal, characteristic

Q₈×Q₁₆

—

Q₈×Q₁₆

C₂²

yes

—

#2252

—

#2252

C₂²

yes

—

#2252

C₂²

yes

—

#2227

—

#2227

C₂²

yes

—

Q₈×C₁₆

C₂²

yes

—

C₄×Q₃₂

—

C₄×Q₃₂

C₂²

yes

characteristic

#1713

C₂²

yes

—

#1676

—

#1676

C₂²

yes

—

#1633

—

C₈×Q₁₆

C₂²

yes

—

C₈×Q₁₆

C₂²

yes

characteristic

#1371

C₂²

yes

characteristic

#1370

C₂²

yes

characteristic

#1365

C₂²

yes

—

Q₈⋊C₁₆

C₂²

yes

—

#1332

C₂²

yes

—

C₈×C₁₆

C₂²

yes

characteristic

Q₈²

—

C₂×Q₃₂

D₄

yes

—

—

—

C₂³

yes

—

—

—

—

—

—

C₈×Q₈

—

C₈×Q₈

C₂³

yes

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₄×Q₁₆

D₄

yes

characteristic

C₄×Q₁₆

C₂³

yes

—

—

—

—

—

D₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂³

yes

—

#358

—

#358

D₄

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂³

yes

characteristic

C₄×C₁₆

C₂³

yes

characteristic

#335

D₄

yes

characteristic

#335

C₂³

yes

—

#333

—

Q₈⋊C₈

—

Q₈⋊C₈

D₄

yes

—

C₈²

C₂³

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

D₈

yes

characteristic

C₂×Q₁₆

C₂×D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₄

yes

—

Q₃₂

Q₁₆

yes

direct product factor

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

—

—

—

—

—

—

D₈

yes

characteristic

#107

C₂×D₄

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄

yes

—

—

—

—

D₈

yes

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂×D₄

yes

characteristic

101

C₄×C₈

C₂×D₄

yes

characteristic

102

C₄×C₈

C₂⁴

yes

characteristic

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

C₂×D₈

yes

—

107

Q₁₆

—

108

Q₁₆

—

109

Q₁₆

Q₃₂

yes

direct product factor

110

Q₁₆

C₂×Q₁₆

yes

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₂×D₈

yes

characteristic

115

C₂×C₈

C₂×D₈

yes

characteristic

116

C₂×C₈

C₂²×D₄

yes

characteristic

117

C₂×C₈

H(5,𝔽₂)

yes

characteristic

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

C₂×D₈

yes

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₄²

C₂²×D₄

yes

characteristic

130

C₁₆

—

131

C₁₆

—

132

C₁₆

—

133

C₁₆

C₂×Q₁₆

yes

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

137

Q₈

—

138

Q₈

—

139

Q₈

C₂×Q₃₂

yes

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

D₄²

yes

characteristic

146

C₂×C₄

C₂²×D₈

yes

characteristic

147

C₂×C₄

#576

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

C₂×Q₃₂

yes

—

154

C₈

C₂²×Q₁₆

yes

characteristic

155

C₈

#578

yes

characteristic

156

C₂²

D₄×D₈

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

D₄×Q₁₆

yes

characteristic

164

C₄

#3293

yes

characteristic

165

C₄

C₂²×Q₃₂

yes

characteristic

166

C₄

#3417

yes

characteristic

167

C₂

D₈×Q₁₆

yes

characteristic

168

C₂

D₄×Q₃₂

yes

characteristic

169

C₂

#33858

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂²

D₄×D₈

commutator, Frattini, cosocle

C₄×C₈

C₂⁴

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆ ◁ G

LECS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
77. Q₃₂	Q₁₆	direct (N × H)
109. Q₁₆	Q₃₂	direct (N × H)

H ≅ G/N

type

77. Q₃₂

Q₁₆

direct (N × H)

109. Q₁₆

Q₃₂

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
Q₈×Q₃₂	2
Q₁₆²	2
#12521	2
#12458	2
#12448	2
#12410	2
C₁₆×Q₁₆	2
C₈×Q₃₂	2

subgroup

index

Q₈×Q₃₂

Q₁₆²

C₁₆×Q₁₆

C₈×Q₃₂

Subgroups — Q₁₆×Q₃₂ #492778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q16×Q32 #492778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×Q₃₂ #492778