Group #496144 — subgroups

Subgroups — #496144

Order 512.

480 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 266 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	#33474	C₂	1	2	yes	maximal
2	#21735	C₂	1	1	yes	maximal, characteristic
3	#13267	C₂	1	2	yes	maximal
4	#13162	C₂	1	4	yes	maximal
5	#12901	C₂	1	1	yes	maximal, characteristic
6	#12664	C₂	1	4	yes	maximal
7	#12539	C₂	1	1	yes	maximal, characteristic
8	#3162	C₄	1	1	yes	characteristic
9	#2960	C₂²	1	2	yes	—
10	#2183	—	2	4	no	—
11	Q₈×C₁₆	—	2	4	no	—
12	C₂×#364	C₂²	1	4	yes	—
13	#2151	C₂²	1	4	yes	—
14	#2150	C₂²	1	2	yes	—
15	#2108	C₂²	1	4	yes	—
16	C₂×#346	C₂²	1	2	yes	—
17	C₂×#346	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₁₆	C₂²	1	2	yes	—
19	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
20	#1995	C₄	1	2	yes	—
21	#1946	C₄	1	4	yes	—
22	#1849	C₄	1	1	yes	characteristic
23	#1772	C₂²	1	4	yes	—
24	#1726	C₂²	1	1	yes	characteristic
25	#1379	C₂²	1	8	yes	—
26	#533	C₂×C₄	1	2	yes	—
27	C₂×M₃₂	C₂³	1	4	yes	—
28	C₂²×C₁₆	C₂³	1	1	yes	characteristic
29	C₂²×C₁₆	C₂³	1	2	yes	—
30	C₂²×C₁₆	C₂³	1	1	yes	characteristic
31	#501	C₈	1	8	yes	—
32	#446	—	2	4	no	—
33	C₈×Q₈	—	2	4	no	—
34	C₂×#108	C₂×C₄	1	4	yes	—
35	#427	C₂×C₄	1	4	yes	—
36	#424	C₂×C₄	1	4	yes	—
37	#423	C₂×C₄	1	2	yes	—
38	#405	C₂³	1	4	yes	—
39	C₂×#97	C₂×C₄	1	2	yes	—
40	C₂×#97	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂³	1	2	yes	—
43	#376	C₂×C₄	1	1	yes	characteristic
44	#364	—	2	8	no	—
45	C₄⋊C₁₆	—	2	8	no	—
46	C₄⋊C₁₆	—	2	4	no	—
47	#346	—	2	4	no	—
48	#346	—	2	2	no	—
49	#346	—	2	4	no	—
50	C₄×C₁₆	—	2	4	no	—
51	C₄×C₁₆	D₄	1	2	yes	—
52	C₄×C₁₆	D₄	1	2	yes	—
53	C₄×C₁₆	Q₈	1	8	yes	complemented
54	#339	C₂×C₄	1	8	yes	—
55	C₂×M₁₆	C₂²×C₄	1	4	yes	—
56	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₈	C₂²×C₄	1	2	yes	—
58	C₂²×C₈	C₂⁴	1	1	yes	characteristic
59	C₄⋊Q₈	C₂×C₈	1	4	yes	—
60	#125	C₂×C₈	1	4	yes	—
61	C₂²⋊Q₈	—	2	4	no	—
62	C₄×Q₈	—	2	4	no	—
63	#117	C₂²×C₄	1	4	yes	—
64	C₂×C₄²	C₂²×C₄	1	2	yes	—
65	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
66	M₃₂	—	2	8	no	—
67	C₂×C₁₆	—	2	2	no	—
68	C₂×C₁₆	—	2	2	no	—
69	C₂×C₁₆	—	2	2	no	—
70	C₂×C₁₆	—	2	4	no	—
71	C₂×C₁₆	C₂×D₄	1	2	yes	—
72	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
73	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
74	C₂×C₁₆	C₂×Q₈	1	4	yes	—
75	C₂×C₁₆	C₂×Q₈	1	4	yes	—
76	#108	—	2	8	no	—
77	#107	C₂×C₈	1	8	yes	—
78	#106	C₂×C₈	1	8	yes	—
79	C₄⋊C₈	—	2	8	no	—
80	C₄⋊C₈	—	2	4	no	—
81	#97	—	2	4	no	—
82	#97	—	2	4	no	—
83	#97	C₂×C₈	1	4	yes	—
84	C₄×C₈	—	2	4	no	—
85	C₄×C₈	C₂×D₄	1	2	yes	—
86	C₄×C₈	C₂×Q₈	1	4	yes	—
87	C₄×C₈	#41	1	2	yes	—
88	C₄×C₈	#41	1	4	yes	—
89	#95	—	2	2	no	—
90	C₂×Q₈	—	2	4	no	—
91	C₂²×C₄	—	2	2	no	—
92	C₂²×C₄	#131	1	2	yes	—
93	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
94	M₁₆	—	2	8	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	C₄×D₄	1	2	yes	—
98	C₂×C₈	C₄×Q₈	1	4	yes	—
99	C₂×C₈	C₂²×C₈	1	4	yes	—
100	C₂×C₈	#131	1	4	yes	—
101	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₈	C₂²×Q₈	1	2	yes	—
103	C₂×C₈	C₂×#41	1	2	yes	—
104	C₂×C₈	C₂×#41	1	1	yes	characteristic
105	C₄⋊C₄	—	2	8	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	C₂²×C₈	1	8	yes	—
108	C₂²⋊C₄	—	2	4	no	—
109	C₄²	—	2	4	no	—
110	C₄²	—	2	2	no	—
111	C₄²	C₄×D₄	1	2	yes	—
112	C₄²	C₄×Q₈	1	4	yes	—
113	C₄²	C₂²×C₈	1	2	yes	—
114	C₁₆	—	2	2	no	—
115	C₁₆	—	2	4	no	—
116	C₁₆	—	4	4	no	—
117	C₁₆	C₄⋊Q₈	1	2	yes	—
118	C₁₆	C₄⋊Q₈	1	2	yes	—
119	C₂³	C₂×#131	1	1	yes	characteristic
120	Q₈	—	4	8	no	—
121	C₂×C₄	—	2	8	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	4	no	—
125	C₂×C₄	C₈×D₄	1	4	yes	—
126	C₂×C₄	C₂×C₄×D₄	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄×Q₈	1	2	yes	—
128	C₂×C₄	C₂³×C₈	1	1	yes	characteristic
129	C₂×C₄	C₂×#131	1	2	yes	—
130	C₈	—	2	2	no	—
131	C₈	—	4	4	no	—
132	C₈	C₈×Q₈	1	8	yes	—
133	C₈	C₂×C₄⋊Q₈	1	1	yes	characteristic
134	C₈	#532	1	1	yes	characteristic
135	C₈	#533	1	2	yes	—
136	C₂²	—	2	2	no	—
137	C₂²	C₂×C₈×D₄	1	1	yes	characteristic
138	C₂²	#2960	1	2	yes	—
139	C₄	—	2	4	no	—
140	C₄	—	4	4	no	—
141	C₄	—	4	8	no	—
142	C₄	C₄×C₄⋊Q₈	1	1	yes	characteristic
143	C₄	#2931	1	1	yes	characteristic
144	C₄	C₂×C₈×Q₈	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	C₈×C₄⋊Q₈	1	1	yes	characteristic
147	C₂	#33441	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33474

C₂

yes

maximal

#21735

C₂

yes

maximal, characteristic

#13267

C₂

yes

maximal

#13162

C₂

yes

maximal

#12901

C₂

yes

maximal, characteristic

#12664

C₂

yes

maximal

#12539

C₂

yes

maximal, characteristic

#3162

C₄

yes

characteristic

#2960

C₂²

yes

—

#2183

—

Q₈×C₁₆

—

C₂×#364

C₂²

yes

—

#2151

C₂²

yes

—

#2150

C₂²

yes

—

#2108

C₂²

yes

—

C₂×#346

C₂²

yes

—

C₂×#346

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1995

C₄

yes

—

#1946

C₄

yes

—

#1849

C₄

yes

characteristic

#1772

C₂²

yes

—

#1726

C₂²

yes

characteristic

#1379

C₂²

yes

—

#533

C₂×C₄

yes

—

C₂×M₃₂

C₂³

yes

—

C₂²×C₁₆

C₂³

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

C₂²×C₁₆

C₂³

yes

characteristic

#501

C₈

yes

—

#446

—

C₈×Q₈

—

C₂×#108

C₂×C₄

yes

—

#427

C₂×C₄

yes

—

#424

C₂×C₄

yes

—

#423

C₂×C₄

yes

—

#405

C₂³

yes

—

C₂×#97

C₂×C₄

yes

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

#376

C₂×C₄

yes

characteristic

#364

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

—

C₄×C₁₆

D₄

yes

—

C₄×C₁₆

Q₈

yes

complemented

#339

C₂×C₄

yes

—

C₂×M₁₆

C₂²×C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

—

C₂²×C₈

C₂⁴

yes

characteristic

C₄⋊Q₈

C₂×C₈

yes

—

#125

C₂×C₈

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂²×C₄

yes

—

C₂×C₄²

C₂²×C₄

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

C₂×Q₈

yes

—

#108

—

#107

C₂×C₈

yes

—

#106

C₂×C₈

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

—

#97

C₂×C₈

yes

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

C₄×C₈

#41

yes

—

#95

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#131

yes

—

C₂²×C₄

C₂³×C₄

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₄

yes

—

C₂×C₈

C₄×Q₈

yes

—

C₂×C₈

C₂²×C₈

yes

—

100

C₂×C₈

#131

yes

—

101

C₂×C₈

C₂²×D₄

yes

characteristic

102

C₂×C₈

C₂²×Q₈

yes

—

103

C₂×C₈

C₂×#41

yes

—

104

C₂×C₈

C₂×#41

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂²×C₈

yes

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₄²

C₄×D₄

yes

—

112

C₄²

C₄×Q₈

yes

—

113

C₄²

C₂²×C₈

yes

—

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

117

C₁₆

C₄⋊Q₈

yes

—

118

C₁₆

C₄⋊Q₈

yes

—

119

C₂³

C₂×#131

yes

characteristic

120

Q₈

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₈×D₄

yes

—

126

C₂×C₄

C₂×C₄×D₄

yes

characteristic

127

C₂×C₄

C₂×C₄×Q₈

yes

—

128

C₂×C₄

C₂³×C₈

yes

characteristic

129

C₂×C₄

C₂×#131

yes

—

130

C₈

—

131

C₈

—

132

C₈

C₈×Q₈

yes

—

133

C₈

C₂×C₄⋊Q₈

yes

characteristic

134

C₈

#532

yes

characteristic

135

C₈

#533

yes

—

136

C₂²

—

137

C₂²

C₂×C₈×D₄

yes

characteristic

138

C₂²

#2960

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

C₄×C₄⋊Q₈

yes

characteristic

143

C₄

#2931

yes

characteristic

144

C₄

C₂×C₈×Q₈

yes

—

145

C₂

—

146

C₂

C₈×C₄⋊Q₈

yes

characteristic

147

C₂

#33441

yes

—

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂³	C₂×#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₁₆

C₂×D₄

commutator

C₂×C₄

C₂³×C₈

Frattini, cosocle

C₂²×C₈

C₂⁴

socle

C₂²

C₂×C₈×D₄

C₂³

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
53. C₄×C₁₆	Q₈	semidirect

H ≅ G/N

type

53. C₄×C₁₆

Q₈

semidirect

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

subgroup	index
#33474	2
#21735	2
#13267	2
#13162	2
#12901	2
#12664	2
#12539	2

subgroup

index

Subgroups — #496144

Conjugacy classes of subgroups

Complements

Maximal subgroups