Group #129192 — subgroups

Subgroups — #129192

Order 512.

248 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 142 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	#12012	C₂	1	1	yes	maximal, characteristic
2	#11408	C₂	1	2	yes	maximal
3	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₈×#108	C₂	1	1	yes	maximal, characteristic
5	#7163	C₂	1	2	yes	maximal
6	#2162	C₂²	1	2	yes	—
7	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
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	C₄	1	2	yes	—
14	#1367	C₄	1	2	yes	—
15	#1312	—	2	2	no	—
16	#1311	—	2	2	no	—
17	C₈×C₁₆	C₄	1	2	yes	—
18	C₈×C₁₆	C₄	1	2	yes	—
19	#1278	C₄	1	2	yes	—
20	C₂²×C₁₆	—	2	1	no	—
21	C₂²×C₁₆	Q₈	1	2	yes	—
22	#431	C₂×C₄	1	2	yes	—
23	C₂×#108	C₈	1	2	yes	—
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	C₂×C₄	1	4	yes	—
29	C₂²⋊C₁₆	—	2	2	no	—
30	C₄×C₁₆	—	2	2	no	—
31	C₄×C₁₆	—	2	1	no	—
32	C₄×C₁₆	C₂×C₄	1	2	yes	—
33	C₄×C₁₆	C₂×C₄	1	2	yes	—
34	C₄×C₁₆	D₄	1	1	yes	characteristic
35	C₄×C₁₆	D₄	1	1	yes	characteristic
36	#341	C₈	1	2	yes	—
37	#335	C₈	1	2	yes	—
38	#334	C₈	1	2	yes	—
39	#322	—	2	2	no	—
40	C₈²	—	2	2	no	—
41	C₈²	C₂×C₄	1	1	yes	characteristic
42	C₈²	C₂×C₄	1	1	yes	characteristic
43	C₈²	C₂×C₄	1	1	yes	characteristic
44	C₈²	C₂×C₄	1	1	yes	characteristic
45	C₂×M₁₆	C₂×C₈	1	4	yes	—
46	C₂²×C₈	—	2	1	no	—
47	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₈	#41	1	1	yes	characteristic
51	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₁₆	—	2	4	no	—
53	C₂×C₁₆	—	2	2	no	—
54	C₂×C₁₆	—	2	4	no	—
55	C₂×C₁₆	—	2	4	no	—
56	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
57	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
58	#108	—	2	4	no	—
59	C₄⋊C₈	C₂×C₈	1	4	yes	—
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	C₄²	1	1	yes	characteristic
66	C₄×C₈	C₄²	1	1	yes	characteristic
67	C₄×C₈	C₄²	1	1	yes	characteristic
68	C₄×C₈	C₄²	1	1	yes	characteristic
69	C₄×C₈	C₂×C₈	1	1	yes	characteristic
70	C₄×C₈	C₂×C₈	1	1	yes	characteristic
71	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₄×C₈	C₂×D₄	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	4	no	—
79	M₁₆	—	2	4	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	4	no	—
87	C₂×C₈	—	2	4	no	—
88	C₂×C₈	C₄×C₈	1	1	yes	characteristic
89	C₂×C₈	C₄×C₈	1	1	yes	characteristic
90	C₂×C₈	C₄×C₈	1	1	yes	characteristic
91	C₂×C₈	C₄×C₈	1	1	yes	characteristic
92	C₂×C₈	M₃₂	1	4	yes	—
93	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₈	#117	1	1	yes	characteristic
95	C₂×C₈	C₄×D₄	1	1	yes	characteristic
96	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
97	C₄²	C₂×C₄²	1	1	yes	characteristic
98	C₄²	C₄×D₄	1	1	yes	characteristic
99	C₄²	C₂²×C₈	1	1	yes	characteristic
100	C₄²	#131	1	1	yes	characteristic
101	C₁₆	—	2	4	no	—
102	C₁₆	—	2	2	no	—
103	C₁₆	—	4	4	no	—
104	C₁₆	#108	1	4	yes	complemented
105	C₂³	C₈×Q₈	1	1	yes	characteristic
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
111	C₂×C₄	#405	1	1	yes	characteristic
112	C₂×C₄	C₈×D₄	1	1	yes	characteristic
113	C₂×C₄	C₈×D₄	1	1	yes	characteristic
114	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
115	C₂×C₄	C₂×M₃₂	1	2	yes	—
116	C₂×C₄	#504	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	2	2	no	—
119	C₈	—	2	1	no	—
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	2	1	no	—
123	C₈	—	4	4	no	—
124	C₈	—	4	4	no	—
125	C₈	#346	1	2	yes	—
126	C₈	#346	1	2	yes	—
127	C₈	C₂×#108	1	2	yes	—
128	C₈	#443	1	2	yes	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	C₈×C₄⋊C₄	1	1	yes	characteristic
132	C₂²	#2183	1	2	yes	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	#1772	1	1	yes	characteristic
136	C₄	C₄×#108	1	1	yes	characteristic
137	C₄	C₂×#346	1	1	yes	characteristic
138	C₄	#2108	1	1	yes	characteristic
139	C₄	#2169	1	2	yes	—
140	C₄	#2170	1	2	yes	—
141	C₂	—	2	2	no	—
142	C₂	C₈×#108	1	1	yes	characteristic
143	C₂	#12662	1	1	yes	characteristic
144	C₂	#12664	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12012

C₂

yes

maximal, characteristic

#11408

C₂

yes

maximal

C₂×C₈×C₁₆

C₂

yes

maximal, characteristic

C₈×#108

C₂

yes

maximal, characteristic

#7163

C₂

yes

maximal

#2162

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₄×#108

C₄

yes

characteristic

#1564

C₄

yes

characteristic

C₈×M₁₆

C₂²

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1370

C₄

yes

—

#1367

C₄

yes

—

#1312

—

#1311

—

C₈×C₁₆

C₄

yes

—

C₈×C₁₆

C₄

yes

—

#1278

C₄

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

Q₈

yes

—

#431

C₂×C₄

yes

—

C₂×#108

C₈

yes

—

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

C₂×C₄

yes

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

#341

C₈

yes

—

#335

C₈

yes

—

#334

C₈

yes

—

#322

—

C₈²

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

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

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

#108

—

C₄⋊C₈

C₂×C₈

yes

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

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

—

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

M₃₂

yes

—

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

#117

yes

characteristic

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₄²

C₂×C₄²

yes

characteristic

C₄²

C₄×D₄

yes

characteristic

C₄²

C₂²×C₈

yes

characteristic

100

C₄²

#131

yes

characteristic

101

C₁₆

—

102

C₁₆

—

103

C₁₆

—

104

C₁₆

#108

yes

complemented

105

C₂³

C₈×Q₈

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

110

C₂×C₄

C₂×C₄×C₈

yes

characteristic

111

C₂×C₄

#405

yes

characteristic

112

C₂×C₄

C₈×D₄

yes

characteristic

113

C₂×C₄

C₈×D₄

yes

characteristic

114

C₂×C₄

C₈×Q₈

yes

characteristic

115

C₂×C₄

C₂×M₃₂

yes

—

116

C₂×C₄

#504

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

#346

yes

—

126

C₈

#346

yes

—

127

C₈

C₂×#108

yes

—

128

C₈

#443

yes

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₈×C₄⋊C₄

yes

characteristic

132

C₂²

#2183

yes

—

133

C₄

—

134

C₄

—

135

C₄

#1772

yes

characteristic

136

C₄

C₄×#108

yes

characteristic

137

C₄

C₂×#346

yes

characteristic

138

C₄

#2108

yes

characteristic

139

C₄

#2169

yes

—

140

C₄

#2170

yes

—

141

C₂

—

142

C₂

C₈×#108

yes

characteristic

143

C₂

#12662

yes

characteristic

144

C₂

#12664

yes

characteristic

	subgroup	G/H
center	C₄×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₈×C₄⋊C₄
prime radical	C₂³	C₈×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄×C₈

C₂×D₄

commutator

C₂×C₄

C₂×C₄×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle

C₂²

C₈×C₄⋊C₄

prime radical

C₂³

C₈×Q₈

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

H ≅ G/N

type

104. C₁₆

#108

semidirect

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

subgroup	index
#12012	2
#11408	2
C₂×C₈×C₁₆	2
C₈×#108	2
#7163	2

subgroup

index

C₂×C₈×C₁₆

Subgroups — #129192

Conjugacy classes of subgroups

Complements

Maximal subgroups