Group #134856 — subgroups

Subgroups — #134856

Order 512.

234 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 88 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	C₂×#2183	C₂	1	1	yes	maximal, characteristic
2	#12024	C₂	1	1	yes	maximal, characteristic
3	#12024	C₂	1	1	yes	maximal, characteristic
4	#8623	C₂	1	2	yes	maximal
5	#7242	C₂	1	2	yes	maximal
6	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
7	#2183	—	2	1	no	—
8	#2183	—	2	1	no	—
9	#2183	—	2	1	no	—
10	#2183	—	2	1	no	—
11	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
12	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×#346	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	#1467	—	2	2	no	—
18	#1372	—	2	1	no	—
19	#1372	—	2	1	no	—
20	#1371	—	2	1	no	—
21	#1371	—	2	1	no	—
22	#1290	—	2	2	no	—
23	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
24	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₁₆	C₂×C₄	1	2	yes	—
26	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₈×Q₈	—	2	1	no	—
28	C₈×Q₈	—	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	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
34	C₂×C₄⋊C₈	Q₈	1	2	yes	—
35	C₂×C₄×C₈	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	#346	—	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	#324	—	4	4	no	—
52	C₂²×Q₈	Q₁₆	1	1	yes	characteristic
53	C₂²×C₈	C₄⋊C₄	1	2	yes	—
54	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
56	C₄×Q₈	—	2	1	no	—
57	C₄×Q₈	—	2	1	no	—
58	C₄×Q₈	—	2	1	no	—
59	C₄×Q₈	—	2	1	no	—
60	C₂×C₄⋊C₄	D₈	1	2	yes	—
61	C₂×C₄⋊C₄	Q₁₆	1	1	yes	characteristic
62	C₂×C₄²	C₄⋊C₄	1	2	yes	—
63	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	—	2	2	no	—
66	C₂×C₁₆	—	4	1	no	—
67	C₂×C₁₆	—	4	2	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	—	4	2	no	—
81	#97	—	4	2	no	—
82	C₄×C₈	—	2	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	C₂×Q₈	—	4	2	no	—
91	C₂²×C₄	#107	1	2	yes	—
92	C₂²×C₄	#107	1	2	yes	—
93	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
95	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	4	1	no	—
100	C₂×C₈	—	4	2	no	—
101	C₂×C₈	—	8	2	no	—
102	C₂×C₈	—	8	2	no	—
103	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₈	#117	1	1	yes	characteristic
105	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	2	2	no	—
113	C₄²	#121	1	1	yes	characteristic
114	C₄²	#121	1	1	yes	characteristic
115	C₄²	#135	1	1	yes	characteristic
116	C₄²	#135	1	1	yes	characteristic
117	C₁₆	—	2	1	no	—
118	C₁₆	—	2	1	no	—
119	C₁₆	—	2	1	no	—
120	C₁₆	—	4	2	no	—
121	C₁₆	#101	1	2	yes	—
122	C₂³	C₂×#107	1	1	yes	characteristic
123	Q₈	—	4	2	no	—
124	Q₈	—	4	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	#380	1	1	yes	characteristic
134	C₂×C₄	#427	1	1	yes	characteristic
135	C₂×C₄	#466	1	1	yes	characteristic
136	C₂×C₄	#466	1	1	yes	characteristic
137	C₂×C₄	#467	1	1	yes	characteristic
138	C₂×C₄	#467	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	4	2	no	—
143	C₈	—	16	2	no	—
144	C₈	—	16	2	no	—
145	C₈	C₂×#101	1	1	yes	characteristic
146	C₈	#413	1	1	yes	characteristic
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	#1808	1	1	yes	characteristic
150	C₂²	#2161	1	2	yes	—
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	4	2	no	—
157	C₄	—	4	2	no	—
158	C₄	#1801	1	1	yes	characteristic
159	C₄	#1813	1	1	yes	characteristic
160	C₂	—	2	2	no	—
161	C₂	#8617	1	1	yes	characteristic
162	C₂	#12838	1	1	yes	characteristic
163	C₂	#12839	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2183

C₂

yes

maximal, characteristic

#12024

C₂

yes

maximal, characteristic

#12024

C₂

yes

maximal, characteristic

#8623

C₂

yes

maximal

#7242

C₂

yes

maximal

C₂×C₈×Q₈

C₂²

yes

—

—

—

—

C₂×C₄⋊C₁₆

C₄

yes

—

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×#346

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1568

C₂²

yes

—

#1568

C₂²

yes

—

—

—

—

—

—

—

C₂×C₄×Q₈

D₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

#432

—

#432

—

C₂×C₄⋊C₈

D₄

yes

characteristic

C₂×C₄⋊C₈

Q₈

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

#346

—

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

—

—

—

—

—

C₂²×Q₈

Q₁₆

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

D₈

yes

—

C₂×C₄⋊C₄

Q₁₆

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

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

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

C₄×C₈

yes

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

#107

yes

—

C₂²×C₄

#107

yes

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

104

C₂×C₈

#117

yes

characteristic

105

C₂×C₈

C₂²⋊Q₈

yes

characteristic

106

C₂×C₈

C₂²⋊Q₈

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄²

—

112

C₄²

—

113

C₄²

#121

yes

characteristic

114

C₄²

#121

yes

characteristic

115

C₄²

#135

yes

characteristic

116

C₄²

#135

yes

characteristic

117

C₁₆

—

118

C₁₆

—

119

C₁₆

—

120

C₁₆

—

121

C₁₆

#101

yes

—

122

C₂³

C₂×#107

yes

characteristic

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

#380

yes

characteristic

134

C₂×C₄

#427

yes

characteristic

135

C₂×C₄

#466

yes

characteristic

136

C₂×C₄

#466

yes

characteristic

137

C₂×C₄

#467

yes

characteristic

138

C₂×C₄

#467

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

C₂×#101

yes

characteristic

146

C₈

#413

yes

characteristic

147

C₂²

—

148

C₂²

—

149

C₂²

#1808

yes

characteristic

150

C₂²

#2161

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

#1801

yes

characteristic

159

C₄

#1813

yes

characteristic

160

C₂

—

161

C₂

#8617

yes

characteristic

162

C₂

#12838

yes

characteristic

163

C₂

#12839

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1808
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#107
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ 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

C₂×#107

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ 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
C₂×#2183	2
#12024	2
#8623	2
#7242	2

subgroup

index

Subgroups — #134856

Conjugacy classes of subgroups

Maximal subgroups