Group #134859 — subgroups

Subgroups — #134859

Order 512.

223 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 223 conjugacy classes of subgroups — too many to draw. Only its 86 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	#33474	C₂	1	1	yes	maximal, characteristic
2	#12024	C₂	1	1	yes	maximal, characteristic
3	#12024	C₂	1	1	yes	maximal, characteristic
4	#8633	C₂	1	2	yes	maximal
5	#7246	C₂	1	2	yes	maximal
6	#2960	C₂²	1	1	yes	characteristic
7	#2183	—	2	1	no	—
8	#2183	—	2	1	no	—
9	Q₈×C₁₆	—	2	1	no	—
10	Q₈×C₁₆	—	2	1	no	—
11	#2151	C₄	1	2	yes	—
12	#2150	C₂²	1	1	yes	characteristic
13	#2108	C₄	1	2	yes	—
14	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
15	#1568	C₂²	1	2	yes	—
16	#1568	C₂²	1	2	yes	—
17	#1537	—	2	2	no	—
18	#1372	—	2	1	no	—
19	#1372	—	2	1	no	—
20	#1371	—	2	1	no	—
21	#1371	—	2	1	no	—
22	#1307	—	2	2	no	—
23	#533	D₄	1	1	yes	characteristic
24	C₂×M₃₂	C₂×C₄	1	2	yes	—
25	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
27	#446	—	2	1	no	—
28	#446	—	2	1	no	—
29	C₈×Q₈	—	2	1	no	—
30	C₈×Q₈	—	2	1	no	—
31	#432	—	2	2	no	—
32	#432	—	2	2	no	—
33	#424	Q₈	1	2	yes	—
34	#423	D₄	1	1	yes	characteristic
35	#405	C₂×C₄	1	2	yes	—
36	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
37	C₄⋊C₁₆	—	2	2	no	—
38	C₄⋊C₁₆	—	2	2	no	—
39	C₄⋊C₁₆	—	2	1	no	—
40	C₄⋊C₁₆	—	2	1	no	—
41	#346	—	2	2	no	—
42	C₄×C₁₆	—	2	2	no	—
43	C₄×C₁₆	D₄	1	1	yes	characteristic
44	C₄×C₁₆	D₄	1	1	yes	characteristic
45	C₄×C₁₆	D₄	1	1	yes	characteristic
46	C₄×C₁₆	D₄	1	1	yes	characteristic
47	#335	—	2	2	no	—
48	#335	—	2	2	no	—
49	#334	—	2	2	no	—
50	#334	—	2	2	no	—
51	#330	—	4	4	no	—
52	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
53	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
55	C₄⋊Q₈	Q₁₆	1	1	yes	characteristic
56	C₄⋊Q₈	Q₁₆	1	1	yes	characteristic
57	#125	D₈	1	2	yes	—
58	C₂²⋊Q₈	—	2	1	no	—
59	C₂²⋊Q₈	—	2	1	no	—
60	C₄×Q₈	—	2	1	no	—
61	C₄×Q₈	—	2	1	no	—
62	#117	C₄⋊C₄	1	2	yes	—
63	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
64	M₃₂	—	4	2	no	—
65	C₂×C₁₆	—	2	2	no	—
66	C₂×C₁₆	—	2	2	no	—
67	C₂×C₁₆	—	4	1	no	—
68	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
72	C₄⋊C₈	—	2	2	no	—
73	C₄⋊C₈	—	2	2	no	—
74	C₄⋊C₈	—	2	1	no	—
75	C₄⋊C₈	—	2	1	no	—
76	C₄⋊C₈	—	4	2	no	—
77	C₄⋊C₈	—	4	2	no	—
78	C₂²⋊C₈	—	4	2	no	—
79	C₂²⋊C₈	—	4	2	no	—
80	#97	—	2	2	no	—
81	#97	—	4	2	no	—
82	#97	—	4	2	no	—
83	C₄×C₈	—	2	2	no	—
84	C₄×C₈	C₂×D₄	1	1	yes	characteristic
85	C₄×C₈	C₂×D₄	1	1	yes	characteristic
86	C₄×C₈	#41	1	1	yes	characteristic
87	C₄×C₈	#41	1	1	yes	characteristic
88	C₂×Q₈	—	2	1	no	—
89	C₂×Q₈	—	2	1	no	—
90	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₄	#135	1	1	yes	characteristic
92	C₂²×C₄	#135	1	1	yes	characteristic
93	M₁₆	—	4	2	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	4	1	no	—
98	C₂×C₈	—	8	2	no	—
99	C₂×C₈	—	8	2	no	—
100	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₈	#117	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	#107	1	2	yes	—
109	C₄⋊C₄	#107	1	2	yes	—
110	C₂²⋊C₄	—	2	2	no	—
111	C₄²	—	2	2	no	—
112	C₄²	#121	1	1	yes	characteristic
113	C₄²	#121	1	1	yes	characteristic
114	C₄²	C₂×D₈	1	1	yes	characteristic
115	C₄²	C₂×Q₁₆	1	1	yes	characteristic
116	C₁₆	—	2	1	no	—
117	C₁₆	—	2	1	no	—
118	C₁₆	—	2	1	no	—
119	C₁₆	—	4	2	no	—
120	C₁₆	#101	1	2	yes	—
121	C₂³	#427	1	1	yes	characteristic
122	Q₈	—	4	2	no	—
123	Q₈	—	4	2	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	#380	1	1	yes	characteristic
131	C₂×C₄	C₂×#107	1	1	yes	characteristic
132	C₂×C₄	#466	1	1	yes	characteristic
133	C₂×C₄	#466	1	1	yes	characteristic
134	C₂×C₄	#467	1	1	yes	characteristic
135	C₂×C₄	#467	1	1	yes	characteristic
136	C₈	—	2	1	no	—
137	C₈	—	2	1	no	—
138	C₈	—	2	1	no	—
139	C₈	—	4	2	no	—
140	C₈	—	16	2	no	—
141	C₈	—	16	2	no	—
142	C₈	C₂×#101	1	1	yes	characteristic
143	C₈	#413	1	1	yes	characteristic
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#1808	1	1	yes	characteristic
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	4	2	no	—
154	C₄	—	4	2	no	—
155	C₄	#1801	1	1	yes	characteristic
156	C₄	#1813	1	1	yes	characteristic
157	C₂	—	4	1	no	—
158	C₂	#8617	1	1	yes	characteristic
159	C₂	#12841	1	1	yes	characteristic
160	C₂	#12841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33474

C₂

yes

maximal, characteristic

#12024

C₂

yes

maximal, characteristic

#12024

C₂

yes

maximal, characteristic

#8633

C₂

yes

maximal

#7246

C₂

yes

maximal

#2960

C₂²

yes

characteristic

#2183

—

#2183

—

Q₈×C₁₆

—

Q₈×C₁₆

—

#2151

C₄

yes

—

#2150

C₂²

yes

characteristic

#2108

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1568

C₂²

yes

—

#1568

C₂²

yes

—

—

—

—

—

—

—

D₄

yes

characteristic

C₂×M₃₂

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

#446

—

#446

—

C₈×Q₈

—

C₈×Q₈

—

#432

—

#432

—

#424

Q₈

yes

—

#423

D₄

yes

characteristic

#405

C₂×C₄

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

—

—

—

—

—

C₂×M₁₆

C₄⋊C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₄⋊Q₈

Q₁₆

yes

characteristic

C₄⋊Q₈

Q₁₆

yes

characteristic

#125

D₈

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

C₄⋊C₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

C₄×C₈

yes

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×C₄⋊C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

101

C₂×C₈

#117

yes

characteristic

102

C₂×C₈

C₂²⋊Q₈

yes

characteristic

103

C₂×C₈

C₂²⋊Q₈

yes

characteristic

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

#107

yes

—

109

C₄⋊C₄

#107

yes

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₄²

#121

yes

characteristic

113

C₄²

#121

yes

characteristic

114

C₄²

C₂×D₈

yes

characteristic

115

C₄²

C₂×Q₁₆

yes

characteristic

116

C₁₆

—

117

C₁₆

—

118

C₁₆

—

119

C₁₆

—

120

C₁₆

#101

yes

—

121

C₂³

#427

yes

characteristic

122

Q₈

—

123

Q₈

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#380

yes

characteristic

131

C₂×C₄

C₂×#107

yes

characteristic

132

C₂×C₄

#466

yes

characteristic

133

C₂×C₄

#466

yes

characteristic

134

C₂×C₄

#467

yes

characteristic

135

C₂×C₄

#467

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

C₂×#101

yes

characteristic

143

C₈

#413

yes

characteristic

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#1808

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#1801

yes

characteristic

156

C₄

#1813

yes

characteristic

157

C₂

—

158

C₂

#8617

yes

characteristic

159

C₂

#12841

yes

characteristic

160

C₂

#12841

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1808
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#427
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1808

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#427

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#33474	2
#12024	2
#8633	2
#7246	2

subgroup

index

Subgroups — #134859

Conjugacy classes of subgroups

Maximal subgroups