Group #146232 — subgroups

Subgroups — #146232

Order 512.

287 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 92 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	#13212	C₂	1	1	yes	maximal, characteristic
2	#13212	C₂	1	1	yes	maximal, characteristic
3	#12903	C₂	1	1	yes	maximal, characteristic
4	#12886	C₂	1	1	yes	maximal, characteristic
5	#9839	C₂	1	1	yes	maximal, characteristic
6	#8826	C₂	1	1	yes	maximal, characteristic
7	#7139	C₂	1	1	yes	maximal, characteristic
8	C₂×#494	C₂²	1	1	yes	characteristic
9	C₂×#365	—	2	2	no	—
10	C₂×#362	—	2	2	no	—
11	C₂×#362	—	2	2	no	—
12	C₄×M₃₂	C₂²	1	1	yes	characteristic
13	C₄×M₃₂	C₂²	1	1	yes	characteristic
14	#1990	—	2	2	no	—
15	#1848	C₂²	1	1	yes	characteristic
16	#1848	C₂²	1	1	yes	characteristic
17	#1844	C₂²	1	1	yes	characteristic
18	#1567	—	2	2	no	—
19	C₂×#322	C₂²	1	1	yes	characteristic
20	#1383	—	2	2	no	—
21	C₂²×Q₁₆	—	2	1	no	—
22	C₂²×Q₁₆	—	2	1	no	—
23	C₂×C₄⋊Q₈	—	2	1	no	—
24	C₂×M₃₂	D₄	1	2	yes	—
25	C₂×M₃₂	D₄	1	2	yes	—
26	#494	—	2	1	no	—
27	#494	—	2	1	no	—
28	#494	—	2	1	no	—
29	#494	Q₈	1	2	yes	—
30	C₂×#108	—	2	1	no	—
31	C₂×#108	—	2	1	no	—
32	C₂×#108	—	2	1	no	—
33	C₂×#108	—	2	1	no	—
34	C₂×#106	—	2	1	no	—
35	C₂×#106	—	2	1	no	—
36	#423	—	2	1	no	—
37	#423	—	2	1	no	—
38	C₂×C₄×C₈	D₄	1	2	yes	—
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	#384	—	2	1	no	—
41	#365	—	4	4	no	—
42	#362	—	4	4	no	—
43	#362	—	4	4	no	—
44	#346	—	2	1	no	—
45	#346	—	2	1	no	—
46	C₄×C₁₆	—	2	1	no	—
47	C₄×C₁₆	—	2	1	no	—
48	#340	—	4	4	no	—
49	#335	—	4	4	no	—
50	#322	—	2	1	no	—
51	#322	—	2	1	no	—
52	C₂²×Q₈	—	4	1	no	—
53	C₂×Q₁₆	—	2	2	no	—
54	C₂×Q₁₆	—	2	2	no	—
55	C₂×Q₁₆	—	4	1	no	—
56	C₂×Q₁₆	—	4	1	no	—
57	C₂×Q₁₆	—	4	2	no	—
58	C₂×Q₁₆	—	4	2	no	—
59	C₂×Q₁₆	—	4	2	no	—
60	C₂×Q₁₆	—	4	2	no	—
61	C₂×M₁₆	—	4	1	no	—
62	C₂×M₁₆	—	4	1	no	—
63	C₂²×C₈	SD₁₆	1	4	yes	—
64	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	—	4	1	no	—
68	C₄⋊Q₈	—	4	2	no	—
69	C₂×C₄⋊C₄	—	4	2	no	—
70	C₂×C₄⋊C₄	—	4	1	no	—
71	C₂×C₄⋊C₄	—	4	2	no	—
72	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
73	M₃₂	—	2	2	no	—
74	M₃₂	—	2	2	no	—
75	M₃₂	—	2	2	no	—
76	M₃₂	D₈	1	4	yes	—
77	C₂×C₁₆	—	2	2	no	—
78	C₂×C₁₆	—	2	2	no	—
79	#108	—	4	2	no	—
80	#108	—	4	2	no	—
81	#108	—	4	2	no	—
82	#108	—	4	2	no	—
83	#106	—	4	2	no	—
84	#106	—	4	2	no	—
85	C₄⋊C₈	—	4	2	no	—
86	C₄⋊C₈	—	4	2	no	—
87	C₄×C₈	—	2	2	no	—
88	C₄×C₈	—	2	2	no	—
89	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
90	C₄×C₈	#41	1	1	yes	characteristic
91	C₄×C₈	#41	1	1	yes	characteristic
92	C₄×C₈	#41	1	1	yes	characteristic
93	C₂×Q₈	—	4	2	no	—
94	C₂×Q₈	—	8	1	no	—
95	C₂×Q₈	—	8	1	no	—
96	C₂×Q₈	—	8	1	no	—
97	C₂×Q₈	—	8	1	no	—
98	C₂×Q₈	—	8	1	no	—
99	C₂²×C₄	—	8	1	no	—
100	C₂²×C₄	—	8	1	no	—
101	C₂²×C₄	#127	1	1	yes	characteristic
102	C₂²×C₄	C₂×SD₁₆	1	2	yes	—
103	Q₁₆	—	4	4	no	—
104	Q₁₆	—	4	4	no	—
105	Q₁₆	—	8	2	no	—
106	Q₁₆	—	8	2	no	—
107	M₁₆	—	8	1	no	—
108	M₁₆	—	8	1	no	—
109	M₁₆	—	8	1	no	—
110	M₁₆	—	8	1	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	—	4	4	no	—
113	C₂×C₈	—	8	1	no	—
114	C₂×C₈	—	8	1	no	—
115	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₈	#123	1	1	yes	characteristic
118	C₂×C₈	#123	1	1	yes	characteristic
119	C₂×C₈	C₂×D₈	1	2	yes	—
120	C₂×C₈	#135	1	2	yes	—
121	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
122	C₂×C₈	#137	1	2	yes	—
123	C₄⋊C₄	—	4	4	no	—
124	C₄⋊C₄	—	8	4	no	—
125	C₄⋊C₄	—	8	2	no	—
126	C₄⋊C₄	—	8	1	no	—
127	C₄⋊C₄	—	8	1	no	—
128	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
129	C₄²	#123	1	1	yes	characteristic
130	C₄²	AGL(1,ℤ/8)	1	2	yes	—
131	C₁₆	—	2	4	no	—
132	C₁₆	—	4	2	no	—
133	C₂³	#492	1	1	yes	characteristic
134	Q₈	—	8	2	no	—
135	Q₈	—	8	2	no	—
136	Q₈	—	16	1	no	—
137	Q₈	—	16	1	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	—	16	1	no	—
141	C₂×C₄	—	16	1	no	—
142	C₂×C₄	—	16	2	no	—
143	C₂×C₄	—	16	1	no	—
144	C₂×C₄	#399	1	1	yes	characteristic
145	C₂×C₄	#476	1	2	yes	—
146	C₂×C₄	#481	1	2	yes	—
147	C₂×C₄	#493	1	1	yes	characteristic
148	C₂×C₄	#495	1	1	yes	characteristic
149	C₂×C₄	#496	1	1	yes	characteristic
150	C₂×C₄	#496	1	1	yes	characteristic
151	C₂×C₄	#497	1	1	yes	characteristic
152	C₈	—	4	4	no	—
153	C₈	—	16	1	no	—
154	C₈	—	16	1	no	—
155	C₈	#474	1	2	yes	—
156	C₈	#478	1	2	yes	—
157	C₈	#480	1	2	yes	—
158	C₈	#482	1	2	yes	—
159	C₂²	—	2	2	no	—
160	C₂²	#1706	1	1	yes	characteristic
161	C₂²	#1714	1	1	yes	characteristic
162	C₂²	#2094	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	16	2	no	—
165	C₄	—	32	1	no	—
166	C₄	#1720	1	1	yes	characteristic
167	C₄	#1721	1	1	yes	characteristic
168	C₄	#2092	1	1	yes	characteristic
169	C₄	#2093	1	1	yes	characteristic
170	C₄	#2239	1	4	yes	—
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	#10672	1	1	yes	characteristic
174	C₂	#13536	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13212

C₂

yes

maximal, characteristic

#13212

C₂

yes

maximal, characteristic

#12903

C₂

yes

maximal, characteristic

#12886

C₂

yes

maximal, characteristic

#9839

C₂

yes

maximal, characteristic

#8826

C₂

yes

maximal, characteristic

#7139

C₂

yes

maximal, characteristic

C₂×#494

C₂²

yes

—

—

—

C₄×M₃₂

C₂²

yes

characteristic

C₄×M₃₂

C₂²

yes

characteristic

#1990

—

#1848

C₂²

yes

characteristic

#1848

C₂²

yes

characteristic

#1844

C₂²

yes

characteristic

#1567

—

C₂×#322

C₂²

yes

characteristic

#1383

—

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

D₄

yes

—

C₂×M₃₂

D₄

yes

—

—

—

—

Q₈

yes

—

—

—

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

—

—

—

—

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

SD₁₆

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

D₈

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

#127

yes

characteristic

102

C₂²×C₄

C₂×SD₁₆

yes

—

103

Q₁₆

—

104

Q₁₆

—

105

Q₁₆

—

106

Q₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₂²⋊Q₈

yes

characteristic

116

C₂×C₈

C₂²⋊Q₈

yes

characteristic

117

C₂×C₈

#123

yes

characteristic

118

C₂×C₈

#123

yes

characteristic

119

C₂×C₈

C₂×D₈

yes

—

120

C₂×C₈

#135

yes

—

121

C₂×C₈

AGL(1,ℤ/8)

yes

—

122

C₂×C₈

#137

yes

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄²

C₂²⋊Q₈

yes

characteristic

129

C₄²

#123

yes

characteristic

130

C₄²

AGL(1,ℤ/8)

yes

—

131

C₁₆

—

132

C₁₆

—

133

C₂³

#492

yes

characteristic

134

Q₈

—

135

Q₈

—

136

Q₈

—

137

Q₈

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

#399

yes

characteristic

145

C₂×C₄

#476

yes

—

146

C₂×C₄

#481

yes

—

147

C₂×C₄

#493

yes

characteristic

148

C₂×C₄

#495

yes

characteristic

149

C₂×C₄

#496

yes

characteristic

150

C₂×C₄

#496

yes

characteristic

151

C₂×C₄

#497

yes

characteristic

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

#474

yes

—

156

C₈

#478

yes

—

157

C₈

#480

yes

—

158

C₈

#482

yes

—

159

C₂²

—

160

C₂²

#1706

yes

characteristic

161

C₂²

#1714

yes

characteristic

162

C₂²

#2094

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

#1720

yes

characteristic

167

C₄

#1721

yes

characteristic

168

C₄

#2092

yes

characteristic

169

C₄

#2093

yes

characteristic

170

C₄

#2239

yes

—

171

C₂

—

172

C₂

—

173

C₂

#10672

yes

characteristic

174

C₂

#13536

yes

—

	subgroup	G/H
center, socle	C₂²	#2094
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#492
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2094

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#492

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ 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
#13212	2
#12903	2
#12886	2
#9839	2
#8826	2
#7139	2

subgroup

index

Subgroups — #146232

Conjugacy classes of subgroups

Maximal subgroups