Group #145532 — subgroups

Subgroups — #145532

Order 512.

328 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 78 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×#2247	C₂	1	1	yes	maximal, characteristic
2	#13262	C₂	1	2	yes	maximal
3	#13161	C₂	1	1	yes	maximal, characteristic
4	C₂×#1358	C₂	1	2	yes	maximal
5	#7160	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₃₂	—	2	1	no	—
7	C₂²×Q₃₂	—	2	1	no	—
8	C₂×#494	C₂²	1	2	yes	—
9	#2247	—	2	4	no	—
10	C₂×#364	C₂²	1	1	yes	characteristic
11	C₂×#364	C₂²	1	2	yes	—
12	#2151	—	2	1	no	—
13	C₂×#362	—	2	2	no	—
14	#2141	—	2	2	no	—
15	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
16	#1772	C₂²	1	1	yes	characteristic
17	#1379	—	2	1	no	—
18	#1358	—	2	4	no	—
19	C₂²×Q₁₆	—	2	2	no	—
20	C₂²×Q₁₆	—	2	2	no	—
21	C₂×C₄⋊Q₈	—	2	2	no	—
22	C₂×Q₃₂	—	2	4	no	—
23	C₂×Q₃₂	—	2	4	no	—
24	C₂×Q₃₂	—	4	2	no	—
25	C₂×Q₃₂	—	4	2	no	—
26	C₂×Q₃₂	—	4	2	no	—
27	C₂×Q₃₂	—	4	2	no	—
28	C₂×M₃₂	—	2	2	no	—
29	C₂×M₃₂	—	2	1	no	—
30	C₂²×C₁₆	D₄	1	1	yes	characteristic
31	C₂²×C₁₆	D₄	1	1	yes	characteristic
32	#494	—	2	2	no	—
33	#494	—	2	2	no	—
34	#494	—	2	2	no	—
35	#494	D₄	1	4	yes	—
36	C₂×#108	D₄	1	1	yes	characteristic
37	#427	D₄	1	1	yes	characteristic
38	#405	—	2	1	no	—
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	#364	—	2	2	no	—
41	#364	—	2	2	no	—
42	#364	—	2	2	no	—
43	C₄⋊C₁₆	—	4	2	no	—
44	#362	—	4	4	no	—
45	#358	—	4	4	no	—
46	C₄×C₁₆	—	2	2	no	—
47	#339	—	2	1	no	—
48	C₂²×Q₈	—	4	2	no	—
49	C₂×Q₁₆	—	2	4	no	—
50	C₂×Q₁₆	—	2	4	no	—
51	C₂×Q₁₆	—	4	2	no	—
52	C₂×Q₁₆	—	4	2	no	—
53	C₂×Q₁₆	—	4	2	no	—
54	C₂×Q₁₆	—	4	2	no	—
55	C₂×Q₁₆	—	4	2	no	—
56	C₂×Q₁₆	—	4	2	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	—	2	1	no	—
62	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
64	C₄⋊Q₈	—	2	4	no	—
65	C₄⋊Q₈	—	4	2	no	—
66	C₄⋊Q₈	—	8	2	no	—
67	#117	—	2	1	no	—
68	C₂×C₄⋊C₄	—	8	2	no	—
69	C₂×C₄²	#41	1	1	yes	characteristic
70	Q₃₂	—	4	8	no	—
71	Q₃₂	—	4	8	no	—
72	M₃₂	—	4	1	no	—
73	M₃₂	—	4	2	no	—
74	M₃₂	—	4	2	no	—
75	M₃₂	—	4	1	no	—
76	C₂×C₁₆	—	2	1	no	—
77	C₂×C₁₆	—	2	1	no	—
78	C₂×C₁₆	—	2	1	no	—
79	C₂×C₁₆	—	2	1	no	—
80	C₂×C₁₆	—	4	2	no	—
81	C₂×C₁₆	—	4	1	no	—
82	C₂×C₁₆	D₈	1	2	yes	—
83	C₂×C₁₆	SD₁₆	1	2	yes	—
84	#108	—	2	2	no	—
85	#107	D₈	1	2	yes	—
86	#106	—	2	1	no	—
87	#97	—	4	1	no	—
88	C₄×C₈	—	4	1	no	—
89	C₄×C₈	C₂×D₄	1	2	yes	—
90	C₄×C₈	#41	1	2	yes	—
91	C₂×Q₈	—	4	4	no	—
92	C₂×Q₈	—	8	2	no	—
93	C₂×Q₈	—	8	2	no	—
94	C₂×Q₈	—	8	2	no	—
95	C₂×Q₈	—	8	2	no	—
96	C₂×Q₈	—	8	2	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	—	8	2	no	—
99	C₂²×C₄	#121	1	1	yes	characteristic
100	Q₁₆	—	4	4	no	—
101	Q₁₆	—	4	4	no	—
102	Q₁₆	—	4	4	no	—
103	Q₁₆	—	4	4	no	—
104	Q₁₆	—	8	2	no	—
105	Q₁₆	—	8	2	no	—
106	Q₁₆	—	8	2	no	—
107	Q₁₆	—	8	2	no	—
108	M₁₆	—	4	1	no	—
109	M₁₆	—	4	1	no	—
110	C₂×C₈	—	4	1	no	—
111	C₂×C₈	#120	1	1	yes	characteristic
112	C₂×C₈	#121	1	2	yes	—
113	C₂×C₈	#124	1	1	yes	characteristic
114	C₂×C₈	C₂×D₈	1	1	yes	characteristic
115	C₂×C₈	C₂×D₈	1	1	yes	characteristic
116	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₈	#135	1	1	yes	characteristic
118	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	8	4	no	—
123	C₄⋊C₄	—	8	4	no	—
124	C₂²⋊C₄	—	4	1	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	4	1	no	—
127	C₄²	#123	1	2	yes	—
128	C₁₆	—	2	2	no	—
129	C₁₆	—	2	2	no	—
130	C₁₆	—	8	1	no	—
131	C₁₆	—	8	2	no	—
132	C₂³	#472	1	1	yes	characteristic
133	Q₈	—	8	4	no	—
134	Q₈	—	8	4	no	—
135	Q₈	—	16	2	no	—
136	Q₈	—	16	2	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	8	4	no	—
141	C₂×C₄	—	16	2	no	—
142	C₂×C₄	—	16	2	no	—
143	C₂×C₄	#394	1	1	yes	characteristic
144	C₂×C₄	#459	1	1	yes	characteristic
145	C₂×C₄	#460	1	1	yes	characteristic
146	C₂×C₄	#466	1	1	yes	characteristic
147	C₂×C₄	#469	1	1	yes	characteristic
148	C₂×C₄	#472	1	1	yes	characteristic
149	C₈	—	8	1	no	—
150	C₈	#447	1	1	yes	characteristic
151	C₈	#450	1	1	yes	characteristic
152	C₈	#459	1	2	yes	—
153	C₈	#463	1	2	yes	—
154	C₈	#486	1	1	yes	characteristic
155	C₈	#490	1	1	yes	characteristic
156	C₂²	—	2	2	no	—
157	C₂²	#1667	1	1	yes	characteristic
158	C₂²	#1688	1	1	yes	characteristic
159	C₂²	#2045	1	1	yes	characteristic
160	C₄	—	2	4	no	—
161	C₄	—	8	2	no	—
162	C₄	—	16	4	no	—
163	C₄	#1668	1	1	yes	characteristic
164	C₄	#1687	1	1	yes	characteristic
165	C₄	#2041	1	1	yes	characteristic
166	C₄	#2054	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#10536	1	1	yes	characteristic
170	C₂	#12437	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2247

C₂

yes

maximal, characteristic

#13262

C₂

yes

maximal

#13161

C₂

yes

maximal, characteristic

C₂×#1358

C₂

yes

maximal

#7160

C₂

yes

maximal, characteristic

C₂²×Q₃₂

—

C₂²×Q₃₂

—

C₂×#494

C₂²

yes

—

#2247

—

C₂×#364

C₂²

yes

characteristic

C₂×#364

C₂²

yes

—

#2151

—

C₂×#362

—

#2141

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1772

C₂²

yes

characteristic

#1379

—

#1358

—

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

—

—

—

D₄

yes

—

C₂×#108

D₄

yes

characteristic

#427

D₄

yes

characteristic

#405

—

C₂×C₄×C₈

C₂³

yes

—

—

—

C₄⋊C₁₆

—

#362

—

#358

—

C₄×C₁₆

—

#339

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

#41

yes

characteristic

Q₃₂

—

Q₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₈

yes

—

C₂×C₁₆

SD₁₆

yes

—

#108

—

#107

D₈

yes

—

#106

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

100

Q₁₆

—

101

Q₁₆

—

102

Q₁₆

—

103

Q₁₆

—

104

Q₁₆

—

105

Q₁₆

—

106

Q₁₆

—

107

Q₁₆

—

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₂×C₈

#120

yes

characteristic

112

C₂×C₈

#121

yes

—

113

C₂×C₈

#124

yes

characteristic

114

C₂×C₈

C₂×D₈

yes

characteristic

115

C₂×C₈

C₂×D₈

yes

characteristic

116

C₂×C₈

C₂×SD₁₆

yes

characteristic

117

C₂×C₈

#135

yes

characteristic

118

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

119

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

120

C₂×C₈

AGL(1,ℤ/8)

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

—

127

C₄²

#123

yes

—

128

C₁₆

—

129

C₁₆

—

130

C₁₆

—

131

C₁₆

—

132

C₂³

#472

yes

characteristic

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

#394

yes

characteristic

144

C₂×C₄

#459

yes

characteristic

145

C₂×C₄

#460

yes

characteristic

146

C₂×C₄

#466

yes

characteristic

147

C₂×C₄

#469

yes

characteristic

148

C₂×C₄

#472

yes

characteristic

149

C₈

—

150

C₈

#447

yes

characteristic

151

C₈

#450

yes

characteristic

152

C₈

#459

yes

—

153

C₈

#463

yes

—

154

C₈

#486

yes

characteristic

155

C₈

#490

yes

characteristic

156

C₂²

—

157

C₂²

#1667

yes

characteristic

158

C₂²

#1688

yes

characteristic

159

C₂²

#2045

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#1668

yes

characteristic

164

C₄

#1687

yes

characteristic

165

C₄

#2041

yes

characteristic

166

C₄

#2054

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#10536

yes

characteristic

170

C₂

#12437

yes

—

	subgroup	G/H
center, socle	C₂²	#1667
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1667

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#472

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ 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
C₂×#2247	2
#13262	2
#13161	2
C₂×#1358	2
#7160	2

subgroup

index

Subgroups — #145532

Conjugacy classes of subgroups

Maximal subgroups