Group #153355 — subgroups

Subgroups — #153355

Order 512.

223 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 223 conjugacy classes of subgroups — too many to draw. Only its 80 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×#1416	C₂	1	2	yes	maximal
2	#13236	C₂	1	1	yes	maximal, characteristic
3	#12905	C₂	1	1	yes	maximal, characteristic
4	#7471	C₂	1	2	yes	maximal
5	#7465	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₃₂	C₂²	1	1	yes	characteristic
7	C₂²×C₃₂	C₂²	1	2	yes	—
8	C₂×#366	—	2	1	no	—
9	C₂×#366	C₂²	1	2	yes	—
10	C₂×#366	C₂²	1	1	yes	characteristic
11	C₂×#358	—	2	1	no	—
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#1969	—	2	1	no	—
14	#1845	—	2	1	no	—
15	#1416	—	2	4	no	—
16	C₂²×Q₁₆	—	2	1	no	—
17	C₂×Q₃₂	—	2	2	no	—
18	C₂×Q₃₂	—	2	2	no	—
19	C₂×Q₃₂	—	2	1	no	—
20	C₂×Q₃₂	D₄	1	2	yes	—
21	C₂²×C₁₆	D₄	1	1	yes	characteristic
22	C₂²×C₁₆	C₂³	1	1	yes	characteristic
23	C₂×#107	—	2	2	no	—
24	C₂×#107	—	2	1	no	—
25	C₂×#107	—	4	1	no	—
26	C₂×#103	—	4	1	no	—
27	C₂×C₄×C₈	D₄	1	1	yes	characteristic
28	#391	—	4	1	no	—
29	#384	—	4	1	no	—
30	C₂×C₃₂	—	2	2	no	—
31	C₂×C₃₂	—	2	2	no	—
32	C₂×C₃₂	—	2	2	no	—
33	#366	—	2	2	no	—
34	#366	—	2	2	no	—
35	#366	—	2	2	no	—
36	#366	—	4	2	no	—
37	#358	—	4	2	no	—
38	C₄×C₁₆	—	2	2	no	—
39	C₂²×Q₈	—	4	1	no	—
40	C₂×Q₁₆	—	2	2	no	—
41	C₂×Q₁₆	—	4	1	no	—
42	C₂×Q₁₆	—	4	1	no	—
43	C₂×Q₁₆	—	4	1	no	—
44	C₂×Q₁₆	—	4	1	no	—
45	C₂×Q₁₆	—	4	1	no	—
46	C₂²×C₈	D₈	1	1	yes	characteristic
47	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄	—	4	2	no	—
49	C₂×C₄⋊C₄	—	4	1	no	—
50	C₂×C₄⋊C₄	—	8	1	no	—
51	C₂×C₄⋊C₄	—	8	1	no	—
52	C₂×C₄²	D₈	1	1	yes	characteristic
53	Q₃₂	—	2	4	no	—
54	Q₃₂	—	4	2	no	—
55	C₂×C₁₆	—	2	1	no	—
56	C₂×C₁₆	—	2	1	no	—
57	C₂×C₁₆	SD₁₆	1	2	yes	—
58	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
59	C₂×C₁₆	#41	1	2	yes	—
60	C₂×C₁₆	#41	1	2	yes	—
61	C₂×C₁₆	#41	1	1	yes	characteristic
62	#107	—	4	2	no	—
63	#107	—	4	1	no	—
64	#107	—	4	1	no	—
65	#107	—	4	2	no	—
66	#107	—	8	2	no	—
67	#103	—	8	2	no	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	—	2	1	no	—
70	#95	—	8	1	no	—
71	#95	—	8	1	no	—
72	C₃₂	—	4	2	no	—
73	C₂×Q₈	—	4	2	no	—
74	C₂×Q₈	—	8	1	no	—
75	C₂×Q₈	—	8	1	no	—
76	C₂×Q₈	—	8	1	no	—
77	C₂×Q₈	—	8	1	no	—
78	C₂×Q₈	—	8	1	no	—
79	C₂²×C₄	—	8	1	no	—
80	C₂²×C₄	—	8	1	no	—
81	C₂²×C₄	—	8	2	no	—
82	C₂²×C₄	D₁₆	1	1	yes	characteristic
83	C₂²×C₄	D₁₆	1	1	yes	characteristic
84	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
85	Q₁₆	—	4	2	no	—
86	Q₁₆	—	4	2	no	—
87	Q₁₆	—	8	1	no	—
88	Q₁₆	—	8	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	SD₃₂	1	2	yes	—
92	C₂×C₈	#121	1	1	yes	characteristic
93	C₂×C₈	#123	1	1	yes	characteristic
94	C₂×C₈	#124	1	1	yes	characteristic
95	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₈	#137	1	1	yes	characteristic
98	C₄⋊C₄	—	8	2	no	—
99	C₄⋊C₄	—	8	1	no	—
100	C₄⋊C₄	—	8	1	no	—
101	C₄⋊C₄	—	8	2	no	—
102	C₄⋊C₄	—	16	2	no	—
103	C₄⋊C₄	—	16	2	no	—
104	C₄²	—	2	1	no	—
105	C₄²	—	2	1	no	—
106	C₁₆	—	2	2	no	—
107	C₁₆	#123	1	2	yes	—
108	C₁₆	#126	1	2	yes	—
109	C₂³	C₂×D₁₆	1	1	yes	characteristic
110	Q₈	—	8	2	no	—
111	Q₈	—	8	2	no	—
112	Q₈	—	16	1	no	—
113	Q₈	—	16	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	8	2	no	—
119	C₂×C₄	—	16	1	no	—
120	C₂×C₄	—	16	1	no	—
121	C₂×C₄	—	16	2	no	—
122	C₂×C₄	—	16	1	no	—
123	C₂×C₄	—	16	2	no	—
124	C₂×C₄	—	16	2	no	—
125	C₂×C₄	—	16	1	no	—
126	C₂×C₄	—	16	1	no	—
127	C₂×C₄	SD₆₄	1	2	yes	—
128	C₂×C₄	Q₆₄	1	2	yes	—
129	C₂×C₄	#459	1	1	yes	characteristic
130	C₂×C₄	#480	1	1	yes	characteristic
131	C₂×C₄	#486	1	1	yes	characteristic
132	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
133	C₂×C₄	#509	1	1	yes	characteristic
134	C₂×C₄	#510	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	#397	1	1	yes	characteristic
137	C₈	#461	1	1	yes	characteristic
138	C₈	#481	1	1	yes	characteristic
139	C₈	#489	1	1	yes	characteristic
140	C₂²	#2206	1	1	yes	characteristic
141	C₂²	#2232	1	1	yes	characteristic
142	C₂²	#2240	1	1	yes	characteristic
143	C₂²	C₂×SD₆₄	1	1	yes	characteristic
144	C₂²	C₂×Q₆₄	1	1	yes	characteristic
145	C₂²	#2263	1	1	yes	characteristic
146	C₂²	#2264	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	16	2	no	—
150	C₄	—	32	2	no	—
151	C₄	—	32	1	no	—
152	C₄	#2071	1	1	yes	characteristic
153	C₄	#2208	1	1	yes	characteristic
154	C₄	#2233	1	1	yes	characteristic
155	C₄	#2243	1	1	yes	characteristic
156	C₂	#13480	1	1	yes	characteristic
157	C₂	#13686	1	1	yes	characteristic
158	C₂	#13688	1	1	yes	characteristic
159	C₂	#13714	1	1	yes	characteristic
160	C₂	#13715	1	1	yes	characteristic
161	C₂	#13720	1	1	yes	characteristic
162	C₂	#13722	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1416

C₂

yes

maximal

#13236

C₂

yes

maximal, characteristic

#12905

C₂

yes

maximal, characteristic

#7471

C₂

yes

maximal

#7465

C₂

yes

maximal, characteristic

C₂²×Q₃₂

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

—

C₂×#366

—

C₂×#366

C₂²

yes

—

C₂×#366

C₂²

yes

characteristic

C₂×#358

—

C₂×C₄×C₁₆

C₂²

yes

—

—

—

C₂²×Q₁₆

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

D₄

yes

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

#391

—

#384

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

—

—

—

—

—

C₄×C₁₆

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₈

yes

characteristic

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

SD₁₆

yes

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

#41

yes

—

C₂×C₁₆

#41

yes

—

C₂×C₁₆

yes

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

#95

—

#95

—

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₆

yes

characteristic

C₂²×C₄

D₁₆

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

SD₃₂

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂×SD₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄²

—

105

C₄²

—

106

C₁₆

—

107

C₁₆

#123

yes

—

108

C₁₆

#126

yes

—

109

C₂³

C₂×D₁₆

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

SD₆₄

yes

—

128

C₂×C₄

Q₆₄

yes

—

129

C₂×C₄

#459

yes

characteristic

130

C₂×C₄

#480

yes

characteristic

131

C₂×C₄

#486

yes

characteristic

132

C₂×C₄

C₂×SD₃₂

yes

characteristic

133

C₂×C₄

#509

yes

characteristic

134

C₂×C₄

#510

yes

characteristic

135

C₈

—

136

C₈

#397

yes

characteristic

137

C₈

#461

yes

characteristic

138

C₈

#481

yes

characteristic

139

C₈

#489

yes

characteristic

140

C₂²

#2206

yes

characteristic

141

C₂²

#2232

yes

characteristic

142

C₂²

#2240

yes

characteristic

143

C₂²

C₂×SD₆₄

yes

characteristic

144

C₂²

C₂×Q₆₄

yes

characteristic

145

C₂²

#2263

yes

characteristic

146

C₂²

#2264

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#2071

yes

characteristic

153

C₄

#2208

yes

characteristic

154

C₄

#2233

yes

characteristic

155

C₄

#2243

yes

characteristic

156

C₂

#13480

yes

characteristic

157

C₂

#13686

yes

characteristic

158

C₂

#13688

yes

characteristic

159

C₂

#13714

yes

characteristic

160

C₂

#13715

yes

characteristic

161

C₂

#13720

yes

characteristic

162

C₂

#13722

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×D₁₆

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

LCS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#1416	2
#13236	2
#12905	2
#7471	2
#7465	2

subgroup

index

Subgroups — #153355

Conjugacy classes of subgroups

Maximal subgroups