Group #146126 — subgroups

Subgroups — #146126

Order 512.

174 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 56 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	#12032	C₂	1	1	yes	maximal, characteristic
2	#11640	C₂	1	1	yes	maximal, characteristic
3	#8689	C₂	1	1	yes	maximal, characteristic
4	#8641	C₂	1	1	yes	maximal, characteristic
5	#7292	C₂	1	1	yes	maximal, characteristic
6	#7223	C₂	1	1	yes	maximal, characteristic
7	#7219	C₂	1	1	yes	maximal, characteristic
8	#2960	C₂²	1	1	yes	characteristic
9	#2163	C₂²	1	1	yes	characteristic
10	#2162	C₂²	1	1	yes	characteristic
11	#1837	C₂²	1	1	yes	characteristic
12	#1567	C₂²	1	1	yes	characteristic
13	#1566	C₂²	1	1	yes	characteristic
14	C₂×#335	C₂²	1	1	yes	characteristic
15	#1561	—	2	1	no	—
16	#1550	—	2	1	no	—
17	#1371	—	2	2	no	—
18	Q₈⋊C₁₆	—	2	2	no	—
19	#1309	—	2	1	no	—
20	#1292	—	2	1	no	—
21	#533	D₄	1	1	yes	characteristic
22	#446	—	2	1	no	—
23	#446	—	2	1	no	—
24	C₈×Q₈	—	2	1	no	—
25	C₈×Q₈	—	2	1	no	—
26	#424	—	2	1	no	—
27	#423	—	2	1	no	—
28	#423	—	2	1	no	—
29	#423	—	2	1	no	—
30	#423	D₄	1	1	yes	characteristic
31	C₂×C₄⋊C₈	—	2	1	no	—
32	#405	—	2	1	no	—
33	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
34	C₄⋊C₁₆	—	2	1	no	—
35	C₄⋊C₁₆	—	2	1	no	—
36	C₂²⋊C₁₆	—	2	1	no	—
37	C₂²⋊C₁₆	—	2	1	no	—
38	#346	—	2	1	no	—
39	C₄×C₁₆	D₄	1	2	yes	—
40	#341	—	2	1	no	—
41	#341	—	2	1	no	—
42	#335	—	2	1	no	—
43	#335	—	2	1	no	—
44	#335	—	2	1	no	—
45	#335	—	2	1	no	—
46	#334	—	2	1	no	—
47	#334	—	2	1	no	—
48	#333	—	4	1	no	—
49	#332	—	4	1	no	—
50	#332	—	4	1	no	—
51	#332	—	4	1	no	—
52	C₂×M₁₆	—	2	1	no	—
53	C₂×M₁₆	—	4	1	no	—
54	C₂×M₁₆	—	4	1	no	—
55	C₂×M₁₆	—	4	1	no	—
56	C₂²×C₈	—	4	1	no	—
57	C₂²×C₈	#41	1	1	yes	characteristic
58	C₂²×C₈	#41	1	1	yes	characteristic
59	C₄⋊Q₈	D₈	1	1	yes	characteristic
60	C₄⋊Q₈	D₈	1	1	yes	characteristic
61	#125	—	2	1	no	—
62	C₂²⋊Q₈	—	2	1	no	—
63	C₂²⋊Q₈	—	2	1	no	—
64	C₄×Q₈	—	2	1	no	—
65	C₄×Q₈	—	2	1	no	—
66	#117	—	2	1	no	—
67	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
68	C₂×C₁₆	—	2	2	no	—
69	C₂×C₁₆	—	4	1	no	—
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	2	1	no	—
72	C₄⋊C₈	—	4	1	no	—
73	C₄⋊C₈	—	4	1	no	—
74	C₄⋊C₈	—	4	1	no	—
75	C₄⋊C₈	—	4	1	no	—
76	C₄⋊C₈	—	4	1	no	—
77	C₄⋊C₈	—	4	1	no	—
78	C₄⋊C₈	—	4	1	no	—
79	C₄⋊C₈	—	4	1	no	—
80	C₄⋊C₈	—	4	1	no	—
81	C₄⋊C₈	—	4	1	no	—
82	#97	—	4	1	no	—
83	C₄×C₈	—	4	1	no	—
84	C₄×C₈	C₂×D₄	1	1	yes	characteristic
85	C₄×C₈	#41	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	#123	1	1	yes	characteristic
91	C₂²×C₄	#135	1	1	yes	characteristic
92	C₂²×C₄	#137	1	1	yes	characteristic
93	M₁₆	—	8	1	no	—
94	M₁₆	—	8	1	no	—
95	M₁₆	—	8	1	no	—
96	M₁₆	—	16	1	no	—
97	M₁₆	—	16	1	no	—
98	C₂×C₈	—	4	1	no	—
99	C₂×C₈	—	4	1	no	—
100	C₂×C₈	—	4	1	no	—
101	C₂×C₈	—	8	1	no	—
102	C₂×C₈	—	8	1	no	—
103	C₂×C₈	—	8	1	no	—
104	C₂×C₈	—	8	1	no	—
105	C₂×C₈	—	16	1	no	—
106	C₂×C₈	#123	1	1	yes	characteristic
107	C₂×C₈	#124	1	1	yes	characteristic
108	C₂×C₈	#126	1	1	yes	characteristic
109	C₂×C₈	#126	1	1	yes	characteristic
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	4	1	no	—
116	C₂²⋊C₄	—	4	1	no	—
117	C₄²	—	4	1	no	—
118	C₄²	#121	1	1	yes	characteristic
119	C₄²	#123	1	1	yes	characteristic
120	C₄²	C₂×D₈	1	1	yes	characteristic
121	C₄²	#137	1	1	yes	characteristic
122	C₁₆	—	4	2	no	—
123	C₁₆	—	8	1	no	—
124	C₂³	#485	1	1	yes	characteristic
125	Q₈	—	4	1	no	—
126	Q₈	—	4	1	no	—
127	Q₈	—	8	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	#397	1	1	yes	characteristic
136	C₂×C₄	#466	1	1	yes	characteristic
137	C₂×C₄	#470	1	1	yes	characteristic
138	C₂×C₄	#480	1	1	yes	characteristic
139	C₂×C₄	#480	1	1	yes	characteristic
140	C₂×C₄	#485	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	8	1	no	—
146	C₈	—	16	1	no	—
147	C₈	—	16	1	no	—
148	C₈	—	16	1	no	—
149	C₈	—	16	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	#2075	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	8	1	no	—
159	C₄	—	8	1	no	—
160	C₄	#2080	1	1	yes	characteristic
161	C₄	#2082	1	1	yes	characteristic
162	C₄	#2221	1	1	yes	characteristic
163	C₄	#2222	1	1	yes	characteristic
164	C₂	—	4	1	no	—
165	C₂	#10639	1	1	yes	characteristic
166	C₂	#13495	1	1	yes	characteristic
167	C₂	#13497	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12032

C₂

yes

maximal, characteristic

#11640

C₂

yes

maximal, characteristic

#8689

C₂

yes

maximal, characteristic

#8641

C₂

yes

maximal, characteristic

#7292

C₂

yes

maximal, characteristic

#7223

C₂

yes

maximal, characteristic

#7219

C₂

yes

maximal, characteristic

#2960

C₂²

yes

characteristic

#2163

C₂²

yes

characteristic

#2162

C₂²

yes

characteristic

#1837

C₂²

yes

characteristic

#1567

C₂²

yes

characteristic

#1566

C₂²

yes

characteristic

C₂×#335

C₂²

yes

—

—

—

Q₈⋊C₁₆

—

#1309

—

#1292

—

#533

D₄

yes

characteristic

#446

—

#446

—

C₈×Q₈

—

C₈×Q₈

—

—

—

—

—

D₄

yes

characteristic

C₂×C₄⋊C₈

—

#405

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₄×C₁₆

D₄

yes

—

—

—

—

—

—

—

—

—

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

yes

C₂²×C₈

yes

C₄⋊Q₈

D₈

yes

characteristic

C₄⋊Q₈

D₈

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

#123

yes

characteristic

107

C₂×C₈

#124

yes

characteristic

108

C₂×C₈

#126

yes

characteristic

109

C₂×C₈

#126

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

#121

yes

characteristic

119

C₄²

#123

yes

characteristic

120

C₄²

C₂×D₈

yes

characteristic

121

C₄²

#137

yes

characteristic

122

C₁₆

—

123

C₁₆

—

124

C₂³

#485

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

#397

yes

characteristic

136

C₂×C₄

#466

yes

characteristic

137

C₂×C₄

#470

yes

characteristic

138

C₂×C₄

#480

yes

characteristic

139

C₂×C₄

#480

yes

characteristic

140

C₂×C₄

#485

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

#2075

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#2080

yes

characteristic

161

C₄

#2082

yes

characteristic

162

C₄

#2221

yes

characteristic

163

C₄

#2222

yes

characteristic

164

C₂

—

165

C₂

#10639

yes

characteristic

166

C₂

#13495

yes

characteristic

167

C₂

#13497

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2075
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#485
LCS	G ▷ C₂×C₄×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

#2075

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#485

LCS

G ▷ C₂×C₄×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
#12032	2
#11640	2
#8689	2
#8641	2
#7292	2
#7223	2
#7219	2

subgroup

index

Subgroups — #146126

Conjugacy classes of subgroups

Maximal subgroups