Group #135114 — subgroups

Subgroups — #135114

Order 512.

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

normal subgroups only

This group has 252 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	#17958	C₂	1	1	yes	maximal, characteristic
2	#12013	C₂	1	1	yes	maximal, characteristic
3	#11865	C₂	1	2	yes	maximal
4	#8824	C₂	1	1	yes	maximal, characteristic
5	#7223	C₂	1	2	yes	maximal
6	#3170	C₂²	1	1	yes	characteristic
7	#2960	—	2	1	no	—
8	#2163	C₂²	1	2	yes	—
9	#1837	C₂²	1	2	yes	—
10	#1591	—	2	1	no	—
11	#1591	C₄	1	2	yes	—
12	C₈×Q₁₆	—	2	1	no	—
13	C₈×Q₁₆	C₄	1	2	yes	—
14	#1567	—	2	1	no	—
15	#1567	C₂²	1	1	yes	characteristic
16	#1566	—	2	1	no	—
17	#1493	—	2	1	no	—
18	C₂×C₈²	C₂²	1	1	yes	characteristic
19	#1370	—	2	1	no	—
20	#1367	—	2	1	no	—
21	#1350	—	2	4	no	—
22	#533	—	2	1	no	—
23	#500	Q₈	1	1	yes	characteristic
24	#494	Q₈	1	1	yes	characteristic
25	#487	—	2	1	no	—
26	#467	—	2	1	no	—
27	#467	D₄	1	2	yes	—
28	#446	—	2	2	no	—
29	#446	—	4	1	no	—
30	C₈×Q₈	—	2	2	no	—
31	C₈×Q₈	—	4	1	no	—
32	C₄×Q₁₆	—	2	1	no	—
33	C₄×Q₁₆	C₂×C₄	1	2	yes	—
34	#424	—	4	1	no	—
35	#423	—	2	1	no	—
36	#423	—	2	2	no	—
37	#405	—	4	1	no	—
38	C₂×C₄×C₈	D₄	1	1	yes	characteristic
39	C₂×C₄×C₈	D₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
41	C₄⋊C₁₆	—	2	2	no	—
42	C₂²⋊C₁₆	—	2	2	no	—
43	#346	—	2	2	no	—
44	#341	—	2	2	no	—
45	#341	—	2	2	no	—
46	#335	—	2	1	no	—
47	#335	—	4	1	no	—
48	#335	—	4	1	no	—
49	#335	C₂×C₄	1	2	yes	—
50	#334	—	4	2	no	—
51	Q₈⋊C₈	—	2	2	no	—
52	Q₈⋊C₈	—	2	2	no	—
53	C₈²	C₂×C₄	1	1	yes	characteristic
54	C₈²	C₂×C₄	1	1	yes	characteristic
55	C₈²	D₄	1	1	yes	characteristic
56	C₈²	D₄	1	1	yes	characteristic
57	C₂×Q₁₆	—	2	1	no	—
58	C₂×Q₁₆	C₄⋊C₄	1	2	yes	—
59	C₂×M₁₆	—	4	1	no	—
60	C₂×M₁₆	—	4	2	no	—
61	C₂²×C₈	—	2	1	no	—
62	C₂²×C₈	SD₁₆	1	1	yes	characteristic
63	C₂²×C₈	SD₁₆	1	1	yes	characteristic
64	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₈	#41	1	1	yes	characteristic
66	C₄⋊Q₈	—	2	1	no	—
67	C₄⋊Q₈	—	2	1	no	—
68	#125	—	4	1	no	—
69	C₂²⋊Q₈	—	2	2	no	—
70	C₂²⋊Q₈	—	4	1	no	—
71	C₄×Q₈	—	2	2	no	—
72	C₄×Q₈	—	4	1	no	—
73	#117	—	4	1	no	—
74	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
75	C₂×C₁₆	—	4	2	no	—
76	#107	—	2	1	no	—
77	#107	C₄⋊C₄	1	2	yes	—
78	C₄⋊C₈	—	2	2	no	—
79	C₄⋊C₈	—	4	2	no	—
80	C₄⋊C₈	—	4	2	no	—
81	C₄⋊C₈	—	4	1	no	—
82	C₄⋊C₈	—	4	2	no	—
83	C₄⋊C₈	—	4	2	no	—
84	#103	—	2	2	no	—
85	#103	—	2	2	no	—
86	#97	—	4	2	no	—
87	C₄×C₈	—	2	1	no	—
88	C₄×C₈	—	4	2	no	—
89	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×C₈	D₈	1	1	yes	characteristic
94	C₄×C₈	D₈	1	1	yes	characteristic
95	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₄×C₈	C₂×D₄	1	1	yes	characteristic
97	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
98	C₄×C₈	#41	1	1	yes	characteristic
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	4	1	no	—
101	C₂²×C₄	#121	1	1	yes	characteristic
102	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
103	C₂²×C₄	#137	1	1	yes	characteristic
104	Q₁₆	—	4	2	no	—
105	Q₁₆	—	4	2	no	—
106	M₁₆	—	8	2	no	—
107	M₁₆	—	8	2	no	—
108	M₁₆	—	8	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	—	4	1	no	—
114	C₂×C₈	—	4	1	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	8	2	no	—
118	C₂×C₈	#102	1	1	yes	characteristic
119	C₂×C₈	#102	1	1	yes	characteristic
120	C₂×C₈	#102	1	1	yes	characteristic
121	C₂×C₈	#102	1	1	yes	characteristic
122	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₈	#117	1	1	yes	characteristic
124	C₂×C₈	#121	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
126	C₄⋊C₄	—	2	2	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	4	1	no	—
130	C₄⋊C₄	—	4	1	no	—
131	C₄⋊C₄	—	4	1	no	—
132	C₂²⋊C₄	—	4	2	no	—
133	C₄²	—	4	2	no	—
134	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
136	C₄²	C₂×D₈	1	1	yes	characteristic
137	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₁₆	—	8	2	no	—
139	C₂³	#461	1	1	yes	characteristic
140	Q₈	—	4	2	no	—
141	Q₈	—	4	2	no	—
142	Q₈	—	8	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	#380	1	1	yes	characteristic
151	C₂×C₄	C₂×#102	1	1	yes	characteristic
152	C₂×C₄	#417	1	1	yes	characteristic
153	C₂×C₄	#459	1	1	yes	characteristic
154	C₂×C₄	#474	1	1	yes	characteristic
155	C₂×C₄	#476	1	1	yes	characteristic
156	C₈	—	2	2	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	2	1	no	—
164	C₈	—	4	1	no	—
165	C₈	—	8	2	no	—
166	C₈	—	16	2	no	—
167	C₈	#362	1	2	yes	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	1	no	—
171	C₂²	#1802	1	1	yes	characteristic
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	8	2	no	—
177	C₄	—	8	2	no	—
178	C₄	#1812	1	1	yes	characteristic
179	C₄	#1814	1	1	yes	characteristic
180	C₄	C₂×#362	1	1	yes	characteristic
181	C₄	#2148	1	1	yes	characteristic
182	C₂	—	4	1	no	—
183	C₂	#8676	1	1	yes	characteristic
184	C₂	#12819	1	1	yes	characteristic
185	C₂	#12822	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17958

C₂

yes

maximal, characteristic

#12013

C₂

yes

maximal, characteristic

#11865

C₂

yes

maximal

#8824

C₂

yes

maximal, characteristic

#7223

C₂

yes

maximal

#3170

C₂²

yes

characteristic

#2960

—

#2163

C₂²

yes

—

#1837

C₂²

yes

—

#1591

—

#1591

C₄

yes

—

C₈×Q₁₆

—

C₈×Q₁₆

C₄

yes

—

#1567

—

#1567

C₂²

yes

characteristic

#1566

—

#1493

—

C₂×C₈²

C₂²

yes

—

—

—

—

Q₈

yes

characteristic

#494

Q₈

yes

—

—

D₄

yes

—

#446

—

#446

—

C₈×Q₈

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

C₂×C₄

yes

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

—

—

—

—

—

—

C₂×C₄

yes

—

#334

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

C₄⋊C₄

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

SD₁₆

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

#107

—

#107

C₄⋊C₄

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

#121

yes

characteristic

102

C₂²×C₄

C₂×SD₁₆

yes

characteristic

103

C₂²×C₄

#137

yes

characteristic

104

Q₁₆

—

105

Q₁₆

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

#102

yes

characteristic

119

C₂×C₈

#102

yes

characteristic

120

C₂×C₈

#102

yes

characteristic

121

C₂×C₈

#102

yes

characteristic

122

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

123

C₂×C₈

#117

yes

characteristic

124

C₂×C₈

#121

yes

characteristic

125

C₂×C₈

C₂²⋊Q₈

yes

characteristic

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

—

134

C₄²

C₂×C₂²⋊C₄

yes

characteristic

135

C₄²

C₂²⋊Q₈

yes

characteristic

136

C₄²

C₂×D₈

yes

characteristic

137

C₄²

AGL(1,ℤ/8)

yes

characteristic

138

C₁₆

—

139

C₂³

#461

yes

characteristic

140

Q₈

—

141

Q₈

—

142

Q₈

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

#380

yes

characteristic

151

C₂×C₄

C₂×#102

yes

characteristic

152

C₂×C₄

#417

yes

characteristic

153

C₂×C₄

#459

yes

characteristic

154

C₂×C₄

#474

yes

characteristic

155

C₂×C₄

#476

yes

characteristic

156

C₈

—

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

166

C₈

—

167

C₈

#362

yes

—

168

C₂²

—

169

C₂²

—

170

C₂²

—

171

C₂²

#1802

yes

characteristic

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

#1812

yes

characteristic

179

C₄

#1814

yes

characteristic

180

C₄

C₂×#362

yes

characteristic

181

C₄

#2148

yes

characteristic

182

C₂

—

183

C₂

#8676

yes

characteristic

184

C₂

#12819

yes

characteristic

185

C₂

#12822

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1802
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#461
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1802

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#461

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ 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
#17958	2
#12013	2
#11865	2
#8824	2
#7223	2

subgroup

index

Subgroups — #135114

Conjugacy classes of subgroups

Maximal subgroups