Group #146178 — subgroups

Subgroups — #146178

Order 512.

194 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 74 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	#13160	C₂	1	1	yes	maximal, characteristic
2	#12892	C₂	1	1	yes	maximal, characteristic
3	#9628	C₂	1	1	yes	maximal, characteristic
4	#9627	C₂	1	1	yes	maximal, characteristic
5	#7276	C₂	1	1	yes	maximal, characteristic
6	#7276	C₂	1	1	yes	maximal, characteristic
7	#7157	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₁₆	—	2	1	no	—
9	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×#346	C₂²	1	1	yes	characteristic
11	#1848	C₂²	1	1	yes	characteristic
12	#1848	C₂²	1	1	yes	characteristic
13	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	#1571	—	2	1	no	—
15	#1570	—	2	1	no	—
16	#1564	C₂²	1	1	yes	characteristic
17	#1564	C₂²	1	1	yes	characteristic
18	#1381	—	2	1	no	—
19	#1381	—	2	1	no	—
20	#1374	—	2	1	no	—
21	#1303	—	2	1	no	—
22	#1303	—	2	1	no	—
23	C₂²×C₁₆	—	2	1	no	—
24	C₂²×C₁₆	D₄	1	1	yes	characteristic
25	C₂²×C₁₆	D₄	1	1	yes	characteristic
26	#431	—	2	1	no	—
27	#431	—	2	1	no	—
28	C₂×#108	—	2	1	no	—
29	C₂×#108	—	2	1	no	—
30	C₂×#108	—	2	1	no	—
31	C₂×#108	—	2	1	no	—
32	#423	—	2	1	no	—
33	#423	—	2	1	no	—
34	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
35	C₂×C₄×C₈	—	2	1	no	—
36	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
37	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
38	C₄⋊C₁₆	—	2	1	no	—
39	C₄⋊C₁₆	—	2	1	no	—
40	C₄⋊C₁₆	—	4	2	no	—
41	#346	—	2	1	no	—
42	#346	—	2	1	no	—
43	#336	—	2	1	no	—
44	#335	—	2	1	no	—
45	#335	—	2	1	no	—
46	#335	—	4	2	no	—
47	#334	—	4	2	no	—
48	#334	Q₈	1	2	yes	—
49	#334	Q₈	1	2	yes	—
50	C₂×M₁₆	—	4	1	no	—
51	C₂×M₁₆	—	4	1	no	—
52	C₂²×C₈	—	2	1	no	—
53	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₈	#41	1	1	yes	characteristic
55	C₂×C₄⋊C₄	—	2	1	no	—
56	C₂×C₄²	—	2	1	no	—
57	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
58	C₂×C₁₆	—	2	1	no	—
59	C₂×C₁₆	—	2	1	no	—
60	C₂×C₁₆	—	2	1	no	—
61	C₂×C₁₆	—	2	1	no	—
62	C₂×C₁₆	—	2	1	no	—
63	C₂×C₁₆	—	2	1	no	—
64	C₂×C₁₆	—	4	1	no	—
65	C₂×C₁₆	—	4	1	no	—
66	C₂×C₁₆	—	4	1	no	—
67	#108	—	4	1	no	—
68	#108	—	4	1	no	—
69	#108	—	4	1	no	—
70	#108	—	4	1	no	—
71	#108	—	8	1	no	—
72	#108	—	8	1	no	—
73	C₄⋊C₈	—	2	2	no	—
74	C₄⋊C₈	—	2	2	no	—
75	C₄⋊C₈	—	2	1	no	—
76	C₄⋊C₈	—	2	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	C₂²⋊C₈	—	4	1	no	—
82	C₂²⋊C₈	—	4	1	no	—
83	C₄×C₈	—	4	1	no	—
84	C₄×C₈	—	4	1	no	—
85	C₄×C₈	—	4	1	no	—
86	C₄×C₈	—	4	1	no	—
87	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
88	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
89	C₄×C₈	#41	1	1	yes	characteristic
90	C₄×C₈	#41	1	1	yes	characteristic
91	#95	Q₁₆	1	1	yes	characteristic
92	#95	Q₁₆	1	1	yes	characteristic
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	2	1	no	—
95	C₂²×C₄	#121	1	1	yes	characteristic
96	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
97	C₂²×C₄	#135	1	1	yes	characteristic
98	M₁₆	—	8	1	no	—
99	M₁₆	—	8	1	no	—
100	M₁₆	—	8	1	no	—
101	M₁₆	—	8	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	4	1	no	—
105	C₂×C₈	—	4	1	no	—
106	C₂×C₈	—	4	1	no	—
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	—	8	1	no	—
110	C₂×C₈	—	8	1	no	—
111	C₂×C₈	#123	1	1	yes	characteristic
112	C₂×C₈	#125	1	1	yes	characteristic
113	C₂×C₈	#125	1	1	yes	characteristic
114	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₈	#135	1	1	yes	characteristic
116	C₂×C₈	#135	1	1	yes	characteristic
117	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₈	#137	1	1	yes	characteristic
119	C₄⋊C₄	—	4	2	no	—
120	C₄²	—	4	1	no	—
121	C₄²	—	4	1	no	—
122	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
123	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
124	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₁₆	—	4	1	no	—
127	C₁₆	—	4	1	no	—
128	C₁₆	—	8	1	no	—
129	C₂³	#467	1	1	yes	characteristic
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	#398	1	1	yes	characteristic
139	C₂×C₄	#463	1	1	yes	characteristic
140	C₂×C₄	#464	1	1	yes	characteristic
141	C₂×C₄	#469	1	1	yes	characteristic
142	C₂×C₄	#471	1	1	yes	characteristic
143	C₂×C₄	#471	1	1	yes	characteristic
144	C₂×C₄	#477	1	1	yes	characteristic
145	C₂×C₄	#477	1	1	yes	characteristic
146	C₂×C₄	#478	1	1	yes	characteristic
147	C₂×C₄	#478	1	1	yes	characteristic
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	8	2	no	—
151	C₈	—	8	2	no	—
152	C₈	—	8	1	no	—
153	C₈	—	16	1	no	—
154	C₈	—	16	1	no	—
155	C₈	#482	1	1	yes	characteristic
156	C₈	#485	1	1	yes	characteristic
157	C₈	#499	1	1	yes	characteristic
158	C₈	#501	1	1	yes	characteristic
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	#1683	1	1	yes	characteristic
162	C₂²	#1684	1	1	yes	characteristic
163	C₂²	#2087	1	1	yes	characteristic
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	4	2	no	—
169	C₄	—	8	1	no	—
170	C₄	#1701	1	1	yes	characteristic
171	C₄	#1702	1	1	yes	characteristic
172	C₄	#2085	1	1	yes	characteristic
173	C₄	#2090	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	#10656	1	1	yes	characteristic
177	C₂	#13519	1	1	yes	characteristic
178	C₂	#13520	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13160

C₂

yes

maximal, characteristic

#12892

C₂

yes

maximal, characteristic

#9628

C₂

yes

maximal, characteristic

#9627

C₂

yes

maximal, characteristic

#7276

C₂

yes

maximal, characteristic

#7276

C₂

yes

maximal, characteristic

#7157

C₂

yes

maximal, characteristic

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×#346

C₂²

yes

characteristic

#1848

C₂²

yes

characteristic

#1848

C₂²

yes

characteristic

C₈×C₄⋊C₄

C₂²

yes

—

—

C₂²

yes

characteristic

#1564

C₂²

yes

—

—

—

—

—

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

—

—

—

—

—

—

—

—

C₂×C₄⋊C₈

D₄

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄×C₄⋊C₄

D₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

—

—

—

—

Q₈

yes

—

#334

Q₈

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

Q₁₆

yes

characteristic

#95

Q₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

C₂²×C₄

#135

yes

characteristic

M₁₆

—

M₁₆

—

100

M₁₆

—

101

M₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

#123

yes

characteristic

112

C₂×C₈

#125

yes

characteristic

113

C₂×C₈

#125

yes

characteristic

114

C₂×C₈

C₄⋊Q₈

yes

characteristic

115

C₂×C₈

#135

yes

characteristic

116

C₂×C₈

#135

yes

characteristic

117

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

118

C₂×C₈

#137

yes

characteristic

119

C₄⋊C₄

—

120

C₄²

—

121

C₄²

—

122

C₄²

C₂²⋊Q₈

yes

characteristic

123

C₄²

C₂²⋊Q₈

yes

characteristic

124

C₄²

AGL(1,ℤ/8)

yes

characteristic

125

C₄²

AGL(1,ℤ/8)

yes

characteristic

126

C₁₆

—

127

C₁₆

—

128

C₁₆

—

129

C₂³

#467

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#398

yes

characteristic

139

C₂×C₄

#463

yes

characteristic

140

C₂×C₄

#464

yes

characteristic

141

C₂×C₄

#469

yes

characteristic

142

C₂×C₄

#471

yes

characteristic

143

C₂×C₄

#471

yes

characteristic

144

C₂×C₄

#477

yes

characteristic

145

C₂×C₄

#477

yes

characteristic

146

C₂×C₄

#478

yes

characteristic

147

C₂×C₄

#478

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

#482

yes

characteristic

156

C₈

#485

yes

characteristic

157

C₈

#499

yes

characteristic

158

C₈

#501

yes

characteristic

159

C₂²

—

160

C₂²

—

161

C₂²

#1683

yes

characteristic

162

C₂²

#1684

yes

characteristic

163

C₂²

#2087

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

#1701

yes

characteristic

171

C₄

#1702

yes

characteristic

172

C₄

#2085

yes

characteristic

173

C₄

#2090

yes

characteristic

174

C₂

—

175

C₂

—

176

C₂

#10656

yes

characteristic

177

C₂

#13519

yes

characteristic

178

C₂

#13520

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#469
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2087
prime radical	C₂³	#467
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#469

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ 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
#13160	2
#12892	2
#9628	2
#9627	2
#7276	2
#7157	2

subgroup

index

Subgroups — #146178

Conjugacy classes of subgroups

Maximal subgroups