Group #134853 — subgroups

Subgroups — #134853

Order 512.

252 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 96 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	#12023	C₂	1	1	yes	maximal, characteristic
3	#12022	C₂	1	1	yes	maximal, characteristic
4	#8617	C₂	1	2	yes	maximal
5	#7241	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	#1801	—	2	2	no	—
16	#1567	C₂²	1	2	yes	—
17	#1567	C₂²	1	2	yes	—
18	#1372	—	2	1	no	—
19	#1372	—	2	1	no	—
20	#1371	—	2	1	no	—
21	#1371	—	2	1	no	—
22	#1292	—	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	#423	—	2	2	no	—
32	#423	—	2	2	no	—
33	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
34	C₂×C₄⋊C₈	Q₈	1	2	yes	—
35	C₂×#101	—	4	2	no	—
36	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
37	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
38	C₄⋊C₁₆	—	2	2	no	—
39	C₄⋊C₁₆	—	2	2	no	—
40	C₄⋊C₁₆	—	2	1	no	—
41	C₄⋊C₁₆	—	2	1	no	—
42	#346	—	2	2	no	—
43	#346	—	2	2	no	—
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	C₄×C₁₆	D₄	1	1	yes	characteristic
48	#343	—	4	2	no	—
49	#335	—	2	2	no	—
50	#335	—	2	2	no	—
51	#335	—	2	2	no	—
52	#335	—	2	2	no	—
53	C₂²×Q₈	D₈	1	1	yes	characteristic
54	C₂×M₁₆	—	4	2	no	—
55	C₂×M₁₆	—	4	2	no	—
56	C₂²×C₈	C₄⋊C₄	1	2	yes	—
57	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
59	C₄×Q₈	—	2	1	no	—
60	C₄×Q₈	—	2	1	no	—
61	C₄×Q₈	—	2	1	no	—
62	C₄×Q₈	—	2	1	no	—
63	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
64	C₂×C₄⋊C₄	Q₁₆	1	2	yes	—
65	C₂×C₄²	C₄⋊C₄	1	2	yes	—
66	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
67	C₂×C₁₆	—	2	2	no	—
68	C₂×C₁₆	—	2	2	no	—
69	C₂×C₁₆	—	4	1	no	—
70	C₂×C₁₆	—	4	2	no	—
71	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
75	C₄⋊C₈	—	2	2	no	—
76	C₄⋊C₈	—	2	2	no	—
77	C₄⋊C₈	—	2	1	no	—
78	C₄⋊C₈	—	2	1	no	—
79	C₄⋊C₈	—	4	2	no	—
80	C₄⋊C₈	—	4	2	no	—
81	C₄⋊C₈	—	4	2	no	—
82	C₄⋊C₈	—	4	2	no	—
83	#101	—	8	4	no	—
84	C₄×C₈	—	2	2	no	—
85	C₄×C₈	—	2	2	no	—
86	C₄×C₈	C₂×D₄	1	1	yes	characteristic
87	C₄×C₈	C₂×D₄	1	1	yes	characteristic
88	C₄×C₈	#41	1	1	yes	characteristic
89	C₄×C₈	#41	1	1	yes	characteristic
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂²×C₄	#107	1	2	yes	—
95	C₂²×C₄	#107	1	2	yes	—
96	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
98	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
99	M₁₆	—	8	2	no	—
100	M₁₆	—	8	2	no	—
101	M₁₆	—	8	2	no	—
102	M₁₆	—	8	2	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	4	1	no	—
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	—	8	2	no	—
109	C₂×C₈	—	8	2	no	—
110	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₈	#117	1	1	yes	characteristic
112	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄²	—	2	2	no	—
119	C₄²	—	2	2	no	—
120	C₄²	#121	1	1	yes	characteristic
121	C₄²	#121	1	1	yes	characteristic
122	C₄²	#135	1	1	yes	characteristic
123	C₄²	#135	1	1	yes	characteristic
124	C₁₆	—	2	1	no	—
125	C₁₆	—	2	1	no	—
126	C₁₆	—	2	1	no	—
127	C₁₆	—	4	2	no	—
128	C₁₆	#101	1	2	yes	complemented
129	C₂³	C₂×#107	1	1	yes	characteristic
130	Q₈	—	4	1	no	—
131	Q₈	—	4	1	no	—
132	Q₈	—	4	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	#380	1	1	yes	characteristic
141	C₂×C₄	#427	1	1	yes	characteristic
142	C₂×C₄	#466	1	1	yes	characteristic
143	C₂×C₄	#466	1	1	yes	characteristic
144	C₂×C₄	#467	1	1	yes	characteristic
145	C₂×C₄	#467	1	1	yes	characteristic
146	C₂×C₄	#509	1	2	yes	—
147	C₂×C₄	#510	1	2	yes	—
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	4	2	no	—
152	C₈	—	16	2	no	—
153	C₈	—	16	2	no	—
154	C₈	C₂×#101	1	1	yes	characteristic
155	C₈	#413	1	1	yes	characteristic
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	#1808	1	1	yes	characteristic
159	C₂²	#2159	1	2	yes	—
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	4	2	no	—
164	C₄	—	4	2	no	—
165	C₄	#1801	1	1	yes	characteristic
166	C₄	#1813	1	1	yes	characteristic
167	C₄	#2218	1	1	yes	characteristic
168	C₄	#2219	1	1	yes	characteristic
169	C₄	#2220	1	2	yes	—
170	C₂	—	2	2	no	—
171	C₂	#8617	1	1	yes	characteristic
172	C₂	#12836	1	1	yes	characteristic
173	C₂	#12840	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2183

C₂

yes

maximal, characteristic

#12023

C₂

yes

maximal, characteristic

#12022

C₂

yes

maximal, characteristic

#8617

C₂

yes

maximal

#7241

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

#1801

—

#1567

C₂²

yes

—

#1567

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

—

#423

—

#423

—

C₂×C₄⋊C₈

D₄

yes

characteristic

C₂×C₄⋊C₈

Q₈

yes

—

C₂×#101

—

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

D₈

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

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

characteristic

C₂×C₄⋊C₄

Q₁₆

yes

—

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

—

C₄⋊C₈

—

C₄⋊C₈

—

#101

—

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

—

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

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

111

C₂×C₈

#117

yes

characteristic

112

C₂×C₈

C₂²⋊Q₈

yes

characteristic

113

C₂×C₈

C₂²⋊Q₈

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

#121

yes

characteristic

121

C₄²

#121

yes

characteristic

122

C₄²

#135

yes

characteristic

123

C₄²

#135

yes

characteristic

124

C₁₆

—

125

C₁₆

—

126

C₁₆

—

127

C₁₆

—

128

C₁₆

#101

yes

complemented

129

C₂³

C₂×#107

yes

characteristic

130

Q₈

—

131

Q₈

—

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

#380

yes

characteristic

141

C₂×C₄

#427

yes

characteristic

142

C₂×C₄

#466

yes

characteristic

143

C₂×C₄

#466

yes

characteristic

144

C₂×C₄

#467

yes

characteristic

145

C₂×C₄

#467

yes

characteristic

146

C₂×C₄

#509

yes

—

147

C₂×C₄

#510

yes

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

C₂×#101

yes

characteristic

155

C₈

#413

yes

characteristic

156

C₂²

—

157

C₂²

—

158

C₂²

#1808

yes

characteristic

159

C₂²

#2159

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

#1801

yes

characteristic

166

C₄

#1813

yes

characteristic

167

C₄

#2218

yes

characteristic

168

C₄

#2219

yes

characteristic

169

C₄

#2220

yes

—

170

C₂

—

171

C₂

#8617

yes

characteristic

172

C₂

#12836

yes

characteristic

173

C₂

#12840

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

N	H ≅ G/N	type
128. C₁₆	#101	semidirect

H ≅ G/N

type

128. C₁₆

#101

semidirect

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

subgroup	index
C₂×#2183	2
#12023	2
#12022	2
#8617	2
#7241	2

subgroup

index

Subgroups — #134853

Conjugacy classes of subgroups

Complements

Maximal subgroups