Group #483595 — subgroups

Subgroups — #483595

Order 512.

474 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 218 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×#2176	C₂	1	1	yes	maximal, characteristic
2	#17856	C₂	1	2	yes	maximal
3	#12066	C₂	1	8	yes	maximal, complemented
4	#11999	C₂	1	1	yes	maximal, characteristic
5	#11600	C₂	1	2	yes	maximal
6	#11405	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₃₂	C₄	1	1	yes	characteristic
8	#2982	—	2	2	no	—
9	C₂×C₄×Q₁₆	C₂²	1	2	yes	—
10	#2176	C₄	1	4	yes	—
11	#2176	C₂²	1	4	yes	—
12	C₂×#367	C₄	1	1	yes	characteristic
13	C₂×#358	C₄	1	2	yes	—
14	C₄×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×#346	C₂²	1	1	yes	characteristic
16	#1579	C₂²	1	8	yes	—
17	C₈×Q₁₆	—	2	4	no	—
18	#1564	C₂²	1	2	yes	—
19	#1479	—	2	2	no	—
20	#1450	C₂²	1	1	yes	characteristic
21	#1368	C₂²	1	4	yes	—
22	#1334	C₂²	1	8	yes	—
23	#1313	C₂²	1	4	yes	—
24	C₂²×Q₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₄×Q₈	—	2	2	no	—
26	C₂×Q₃₂	C₈	1	8	yes	complemented
27	C₂×Q₃₂	C₂×C₄	1	4	yes	—
28	C₂×M₃₂	—	2	1	no	—
29	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
31	C₈×Q₈	—	2	8	no	—
32	C₈×Q₈	—	4	4	no	—
33	C₄×Q₁₆	—	2	2	no	—
34	C₄×Q₁₆	—	2	2	no	—
35	C₄×Q₁₆	C₂×C₄	1	4	yes	—
36	C₄×Q₁₆	C₂³	1	4	yes	—
37	#431	—	2	2	no	—
38	#431	—	4	4	no	—
39	#431	D₄	1	2	yes	—
40	C₂×#107	C₂×C₄	1	2	yes	—
41	C₂×#103	—	2	2	no	—
42	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
43	#367	C₂×C₄	1	4	yes	—
44	#358	C₂×C₄	1	8	yes	—
45	#346	C₂×C₄	1	2	yes	—
46	#346	C₂³	1	2	yes	—
47	#346	C₂³	1	2	yes	—
48	C₄×C₁₆	—	2	1	no	—
49	#335	—	2	2	no	—
50	#334	C₂³	1	4	yes	—
51	Q₈⋊C₈	—	2	8	no	—
52	#322	C₂³	1	2	yes	—
53	C₈²	D₄	1	2	yes	—
54	C₂²×Q₈	—	2	2	no	—
55	C₂×Q₁₆	—	2	2	no	—
56	C₂×Q₁₆	—	2	4	no	—
57	C₂×Q₁₆	C₂×C₈	1	8	yes	—
58	C₂×Q₁₆	C₂²×C₄	1	4	yes	—
59	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₈	#41	1	1	yes	characteristic
61	C₄×Q₈	—	2	4	no	—
62	C₄×Q₈	—	2	4	no	—
63	C₄×Q₈	—	4	2	no	—
64	C₄×Q₈	—	4	2	no	—
65	C₂×C₄⋊C₄	—	2	2	no	—
66	C₂×C₄⋊C₄	—	4	2	no	—
67	C₂×C₄²	—	4	2	no	—
68	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
69	Q₃₂	C₂×C₈	1	16	yes	—
70	M₃₂	—	4	2	no	—
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	C₂×C₈	1	4	yes	—
74	C₂×C₁₆	C₂²×C₄	1	2	yes	—
75	C₂×C₁₆	C₂²×C₄	1	2	yes	—
76	#107	—	2	2	no	—
77	#107	C₂²×C₄	1	4	yes	—
78	C₄⋊C₈	—	2	2	no	—
79	C₄⋊C₈	—	2	4	no	—
80	C₄⋊C₈	—	4	8	no	—
81	#103	—	2	4	no	—
82	#103	—	2	4	no	—
83	C₂²⋊C₈	—	2	2	no	—
84	C₂²⋊C₈	—	4	2	no	—
85	C₄×C₈	—	4	8	no	—
86	C₄×C₈	—	4	2	no	—
87	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₄×C₈	C₂×D₄	1	4	yes	—
89	C₄×C₈	C₂×D₄	1	1	yes	characteristic
90	C₄×C₈	#41	1	1	yes	characteristic
91	C₄×C₈	C₂⁴	1	1	yes	characteristic
92	C₂×Q₈	—	2	4	no	—
93	C₂×Q₈	—	2	8	no	—
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	4	4	no	—
96	C₂²×C₄	—	4	2	no	—
97	C₂²×C₄	—	4	2	no	—
98	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
99	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
100	Q₁₆	—	2	4	no	—
101	Q₁₆	—	2	8	no	—
102	Q₁₆	C₂²×C₈	1	8	yes	—
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	4	no	—
105	C₂×C₈	—	4	4	no	—
106	C₂×C₈	C₄×D₄	1	1	yes	characteristic
107	C₂×C₈	C₄×D₄	1	1	yes	characteristic
108	C₂×C₈	C₂²×C₈	1	2	yes	—
109	C₂×C₈	#131	1	2	yes	—
110	C₂×C₈	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₈	C₂×#41	1	1	yes	characteristic
112	C₄⋊C₄	—	2	4	no	—
113	C₄⋊C₄	—	4	4	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	4	4	no	—
116	C₄²	—	4	4	no	—
117	C₄²	—	4	4	no	—
118	C₄²	C₄×D₄	1	1	yes	characteristic
119	C₄²	AGL(1,ℤ/8)	1	2	yes	—
120	C₄²	C₂²×D₄	1	1	yes	characteristic
121	C₁₆	—	2	2	no	—
122	C₁₆	—	4	2	no	—
123	C₁₆	C₂²×C₈	1	4	yes	—
124	C₂³	#442	1	1	yes	characteristic
125	Q₈	—	2	8	no	—
126	Q₈	—	4	8	no	—
127	Q₈	—	4	4	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	8	no	—
130	C₂×C₄	—	4	4	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	16	2	no	—
133	C₂×C₄	C₈×D₄	1	2	yes	—
134	C₂×C₄	#442	1	2	yes	—
135	C₂×C₄	#442	1	1	yes	characteristic
136	C₂×C₄	#444	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
138	C₂×C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	4	4	no	—
142	C₈	—	8	4	no	—
143	C₈	C₈×D₄	1	2	yes	—
144	C₈	C₂³×C₈	1	1	yes	characteristic
145	C₈	C₂×#131	1	1	yes	characteristic
146	C₂²	—	2	2	no	—
147	C₂²	#1580	1	2	yes	—
148	C₂²	C₂×#442	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	8	no	—
152	C₄	—	8	4	no	—
153	C₄	#1580	1	2	yes	—
154	C₄	C₂×C₈×D₄	1	1	yes	characteristic
155	C₄	C₂×#444	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	C₂×#1580	1	1	yes	characteristic
158	C₂	#33470	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2176

C₂

yes

maximal, characteristic

#17856

C₂

yes

maximal

#12066

C₂

yes

maximal, complemented

#11999

C₂

yes

maximal, characteristic

#11600

C₂

yes

maximal

#11405

C₂

yes

maximal, characteristic

C₂²×Q₃₂

C₄

yes

characteristic

#2982

—

C₂×C₄×Q₁₆

C₂²

yes

—

#2176

C₄

yes

—

#2176

C₂²

yes

—

C₂×#367

C₄

yes

characteristic

C₂×#358

C₄

yes

—

C₄×M₃₂

C₂²

yes

characteristic

C₂×#346

C₂²

yes

characteristic

#1579

C₂²

yes

—

C₈×Q₁₆

—

#1564

C₂²

yes

—

#1479

—

#1450

C₂²

yes

characteristic

#1368

C₂²

yes

—

#1334

C₂²

yes

—

#1313

C₂²

yes

—

C₂²×Q₁₆

C₂×C₄

yes

—

C₂×C₄×Q₈

—

C₂×Q₃₂

C₈

yes

complemented

C₂×Q₃₂

C₂×C₄

yes

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₄×Q₁₆

C₂×C₄

yes

—

C₄×Q₁₆

C₂³

yes

—

#431

—

#431

—

#431

D₄

yes

—

C₂×#107

C₂×C₄

yes

—

C₂×#103

—

C₂×C₄×C₈

C₂³

yes

characteristic

#367

C₂×C₄

yes

—

#358

C₂×C₄

yes

—

#346

C₂×C₄

yes

—

#346

C₂³

yes

—

#346

C₂³

yes

—

C₄×C₁₆

—

#335

—

#334

C₂³

yes

—

Q₈⋊C₈

—

#322

C₂³

yes

—

C₈²

D₄

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×C₈

yes

—

C₂×Q₁₆

C₂²×C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

Q₃₂

C₂×C₈

yes

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₈

yes

—

C₂×C₁₆

C₂²×C₄

yes

—

C₂×C₁₆

C₂²×C₄

yes

—

#107

—

#107

C₂²×C₄

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#103

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₄×C₈

C₂⁴

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

—

100

Q₁₆

—

101

Q₁₆

—

102

Q₁₆

C₂²×C₈

yes

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₄×D₄

yes

characteristic

107

C₂×C₈

C₄×D₄

yes

characteristic

108

C₂×C₈

C₂²×C₈

yes

—

109

C₂×C₈

#131

yes

—

110

C₂×C₈

C₂³×C₄

yes

characteristic

111

C₂×C₈

C₂×#41

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄²

—

117

C₄²

—

118

C₄²

C₄×D₄

yes

characteristic

119

C₄²

AGL(1,ℤ/8)

yes

—

120

C₄²

C₂²×D₄

yes

characteristic

121

C₁₆

—

122

C₁₆

—

123

C₁₆

C₂²×C₈

yes

—

124

C₂³

#442

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₈×D₄

yes

—

134

C₂×C₄

#442

yes

—

135

C₂×C₄

#442

yes

characteristic

136

C₂×C₄

#444

yes

—

137

C₂×C₄

C₂×C₄×D₄

yes

characteristic

138

C₂×C₄

C₂×AGL(1,ℤ/8)

yes

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

C₈×D₄

yes

—

144

C₈

C₂³×C₈

yes

characteristic

145

C₈

C₂×#131

yes

characteristic

146

C₂²

—

147

C₂²

#1580

yes

—

148

C₂²

C₂×#442

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#1580

yes

—

154

C₄

C₂×C₈×D₄

yes

characteristic

155

C₄

C₂×#444

yes

characteristic

156

C₂

—

157

C₂

C₂×#1580

yes

characteristic

158

C₂

#33470

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₈	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#442
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#442

commutator

C₈

C₂³×C₈

Frattini, cosocle

C₄×C₈

C₂⁴

prime radical

C₂³

#442

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G

LECS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
3. #12066	C₂	semidirect
26. C₂×Q₃₂	C₈	semidirect

H ≅ G/N

type

3. #12066

C₂

semidirect

26. C₂×Q₃₂

C₈

semidirect

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

subgroup	index
C₂×#2176	2
#17856	2
#12066	2
#11999	2
#11600	2
#11405	2

subgroup

index

Subgroups — #483595

Conjugacy classes of subgroups

Complements

Maximal subgroups