Group #140409 — subgroups

Subgroups — #140409

Order 512.

213 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 213 conjugacy classes of subgroups — too many to draw. Only its 114 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	#12015	C₂	1	2	yes	maximal
2	C₂×#1367	C₂	1	1	yes	maximal, characteristic
3	#11410	C₂	1	1	yes	maximal, characteristic
4	C₈×#108	C₂	1	1	yes	maximal, characteristic
5	#7209	C₂	1	2	yes	maximal
6	#2162	C₂²	1	1	yes	characteristic
7	#2162	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
9	C₄×#108	C₄	1	1	yes	characteristic
10	#1564	C₄	1	1	yes	characteristic
11	C₈×M₁₆	C₂²	1	2	yes	—
12	C₂×C₈²	C₂²	1	1	yes	characteristic
13	#1370	—	2	2	no	—
14	#1367	—	2	2	no	—
15	#1367	—	2	1	no	—
16	#1367	—	2	1	no	—
17	#1312	—	2	1	no	—
18	C₈×C₁₆	—	2	1	no	—
19	#1278	C₄	1	2	yes	—
20	C₂²×C₁₆	—	2	2	no	—
21	#431	C₂×C₄	1	2	yes	—
22	C₂×#108	C₈	1	1	yes	characteristic
23	C₂×#108	C₈	1	1	yes	characteristic
24	C₄×M₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	C₄⋊C₁₆	—	2	2	no	—
29	C₄⋊C₁₆	—	2	2	no	—
30	C₄⋊C₁₆	—	2	1	no	—
31	C₄⋊C₁₆	D₄	1	2	yes	—
32	C₂²⋊C₁₆	—	2	1	no	—
33	C₂²⋊C₁₆	Q₈	1	2	yes	—
34	C₄×C₁₆	—	2	1	no	—
35	C₄×C₁₆	—	2	1	no	—
36	#341	C₈	1	2	yes	—
37	#335	—	2	1	no	—
38	#334	—	2	1	no	—
39	#322	—	2	2	no	—
40	C₈²	—	2	2	no	—
41	C₈²	D₄	1	1	yes	characteristic
42	C₈²	D₄	1	1	yes	characteristic
43	C₈²	Q₈	1	1	yes	characteristic
44	C₈²	Q₈	1	1	yes	characteristic
45	C₂×M₁₆	C₂×C₈	1	2	yes	—
46	C₂×M₁₆	C₂×C₈	1	2	yes	—
47	C₂²×C₈	—	2	1	no	—
48	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₈	#41	1	1	yes	characteristic
52	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₁₆	—	2	2	no	—
54	C₂×C₁₆	—	2	2	no	—
55	C₂×C₁₆	—	4	2	no	—
56	C₂×C₁₆	—	8	2	no	—
57	#108	—	2	2	no	—
58	#108	C₁₆	1	4	yes	complemented
59	C₄⋊C₈	—	2	2	no	—
60	C₂²⋊C₈	—	2	2	no	—
61	#97	—	2	2	no	—
62	C₄×C₈	—	2	2	no	—
63	C₄×C₈	—	2	2	no	—
64	C₄×C₈	—	2	1	no	—
65	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
66	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
67	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
68	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
69	C₄×C₈	M₁₆	1	1	yes	characteristic
70	C₄×C₈	M₁₆	1	1	yes	characteristic
71	C₄×C₈	C₂×D₄	1	1	yes	characteristic
72	C₄×C₈	C₂×D₄	1	1	yes	characteristic
73	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
74	C₄×C₈	#41	1	1	yes	characteristic
75	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
76	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
77	C₂²×C₄	#131	1	1	yes	characteristic
78	M₁₆	—	2	2	no	—
79	M₁₆	—	2	2	no	—
80	M₁₆	—	2	2	no	—
81	M₁₆	C₂×C₁₆	1	4	yes	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	4	1	no	—
90	C₂×C₈	—	4	2	no	—
91	C₂×C₈	—	4	2	no	—
92	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
93	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
94	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
95	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
96	C₂×C₈	C₂×C₁₆	1	2	yes	—
97	C₂×C₈	M₃₂	1	2	yes	—
98	C₂×C₈	#121	1	1	yes	characteristic
99	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₈	#125	1	1	yes	characteristic
101	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
102	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₄²	C₄×D₄	1	1	yes	characteristic
104	C₄²	C₂×M₁₆	1	1	yes	characteristic
105	C₄²	#131	1	1	yes	characteristic
106	C₁₆	—	4	2	no	—
107	C₁₆	—	4	2	no	—
108	C₁₆	—	8	2	no	—
109	C₂³	C₈×Q₈	1	1	yes	characteristic
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	#389	1	1	yes	characteristic
114	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₂×C₄	#424	1	1	yes	characteristic
116	C₂×C₄	C₈×D₄	1	1	yes	characteristic
117	C₂×C₄	#436	1	1	yes	characteristic
118	C₂×C₄	#446	1	1	yes	characteristic
119	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
120	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
121	C₂×C₄	#504	1	1	yes	characteristic
122	C₂×C₄	#504	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	—	2	1	no	—
125	C₈	—	2	1	no	—
126	C₈	—	2	4	no	—
127	C₈	—	2	1	no	—
128	C₈	—	2	1	no	—
129	C₈	—	2	1	no	—
130	C₈	—	4	1	no	—
131	C₈	—	4	2	no	—
132	C₈	—	8	2	no	—
133	C₈	C₄⋊C₁₆	1	2	yes	—
134	C₈	C₄⋊C₁₆	1	2	yes	—
135	C₈	#471	1	2	yes	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	#1939	1	1	yes	characteristic
139	C₂²	Q₈×C₁₆	1	1	yes	characteristic
140	C₂²	#2183	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	#1951	1	1	yes	characteristic
145	C₄	#1953	1	1	yes	characteristic
146	C₄	C₂×C₄⋊C₁₆	1	1	yes	characteristic
147	C₄	#2151	1	1	yes	characteristic
148	C₄	D₄×C₁₆	1	1	yes	characteristic
149	C₄	#2169	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	2	1	no	—
152	C₂	#9607	1	1	yes	characteristic
153	C₂	#13159	1	1	yes	characteristic
154	C₂	#13164	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12015

C₂

yes

maximal

C₂×#1367

C₂

yes

maximal, characteristic

#11410

C₂

yes

maximal, characteristic

C₈×#108

C₂

yes

maximal, characteristic

#7209

C₂

yes

maximal

#2162

C₂²

yes

characteristic

#2162

C₂²

yes

characteristic

C₂×C₄⋊C₁₆

C₂²

yes

—

C₄×#108

C₄

yes

characteristic

#1564

C₄

yes

characteristic

C₈×M₁₆

C₂²

yes

—

C₂×C₈²

C₂²

yes

—

—

—

—

—

C₈×C₁₆

—

#1278

C₄

yes

—

C₂²×C₁₆

—

#431

C₂×C₄

yes

—

C₂×#108

C₈

yes

characteristic

C₂×#108

C₈

yes

characteristic

C₄×M₁₆

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

D₄

yes

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

Q₈

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

#341

C₈

yes

—

#335

—

#334

—

#322

—

C₈²

—

C₈²

D₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂×M₁₆

C₂×C₈

yes

—

C₂×M₁₆

C₂×C₈

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

#108

C₁₆

yes

complemented

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂²×C₄

C₄×Q₈

yes

characteristic

C₂²×C₄

C₂²×C₈

yes

characteristic

C₂²×C₄

#131

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₂×C₁₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₈

yes

characteristic

C₂×C₈

C₄⋊C₈

yes

characteristic

C₂×C₈

C₄⋊C₈

yes

characteristic

C₂×C₈

C₄⋊C₈

yes

characteristic

C₂×C₈

C₂×C₁₆

yes

—

C₂×C₈

M₃₂

yes

—

C₂×C₈

#121

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

characteristic

100

C₂×C₈

#125

yes

characteristic

101

C₂×C₈

C₄⋊Q₈

yes

characteristic

102

C₄²

C₂×C₄⋊C₄

yes

characteristic

103

C₄²

C₄×D₄

yes

characteristic

104

C₄²

C₂×M₁₆

yes

characteristic

105

C₄²

#131

yes

characteristic

106

C₁₆

—

107

C₁₆

—

108

C₁₆

—

109

C₂³

C₈×Q₈

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

#389

yes

characteristic

114

C₂×C₄

C₂×C₄⋊C₈

yes

characteristic

115

C₂×C₄

#424

yes

characteristic

116

C₂×C₄

C₈×D₄

yes

characteristic

117

C₂×C₄

#436

yes

characteristic

118

C₂×C₄

#446

yes

characteristic

119

C₂×C₄

C₂²×C₁₆

yes

characteristic

120

C₂×C₄

C₂×M₃₂

yes

characteristic

121

C₂×C₄

#504

yes

characteristic

122

C₂×C₄

#504

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

C₄⋊C₁₆

yes

—

134

C₈

C₄⋊C₁₆

yes

—

135

C₈

#471

yes

—

136

C₂²

—

137

C₂²

—

138

C₂²

#1939

yes

characteristic

139

C₂²

Q₈×C₁₆

yes

characteristic

140

C₂²

#2183

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#1951

yes

characteristic

145

C₄

#1953

yes

characteristic

146

C₄

C₂×C₄⋊C₁₆

yes

characteristic

147

C₄

#2151

yes

characteristic

148

C₄

D₄×C₁₆

yes

characteristic

149

C₄

#2169

yes

characteristic

150

C₂

—

151

C₂

—

152

C₂

#9607

yes

characteristic

153

C₂

#13159

yes

characteristic

154

C₂

#13164

yes

characteristic

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1939
prime radical	C₂³	C₈×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

#121

commutator

C₂×C₄

C₂²×C₁₆

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle

C₂²

#1939

prime radical

C₂³

C₈×Q₈

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
58. #108	C₁₆	semidirect

H ≅ G/N

type

58. #108

C₁₆

semidirect

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

subgroup	index
#12015	2
C₂×#1367	2
#11410	2
C₈×#108	2
#7209	2

subgroup

index

Subgroups — #140409

Conjugacy classes of subgroups

Complements

Maximal subgroups