Group #129142 — subgroups

Subgroups — #129142

Order 512.

258 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 138 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	#12670	C₂	1	1	yes	maximal, characteristic
2	#12664	C₂	1	2	yes	maximal
3	C₂×#1373	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
5	#7179	C₂	1	2	yes	maximal
6	C₂×#368	C₄	1	1	yes	characteristic
7	#2158	C₄	1	1	yes	characteristic
8	C₂×#364	C₂²	1	2	yes	—
9	#2108	—	2	2	no	—
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	#1772	C₂²	1	2	yes	—
13	C₂×C₈²	C₂²	1	1	yes	characteristic
14	#1382	C₄	1	2	yes	—
15	#1379	—	2	2	no	—
16	#1373	C₄	1	4	yes	complemented
17	C₈×C₁₆	C₄	1	4	yes	complemented
18	C₂×M₃₂	—	2	2	no	—
19	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₁₆	D₄	1	1	yes	characteristic
22	C₂²×C₁₆	D₄	1	1	yes	characteristic
23	C₂×#108	C₂×C₄	1	2	yes	—
24	#427	C₂×C₄	1	2	yes	—
25	#405	—	2	2	no	—
26	C₂×C₄×C₈	—	2	1	no	—
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	#368	C₈	1	4	yes	complemented
29	#367	C₈	1	2	yes	complemented
30	#366	C₈	1	2	yes	complemented
31	#364	—	2	2	no	—
32	#364	C₂×C₄	1	4	yes	—
33	#346	—	4	2	no	—
34	C₄×C₁₆	—	4	2	no	—
35	C₄×C₁₆	C₂×C₄	1	2	yes	—
36	C₄×C₁₆	C₂×C₄	1	2	yes	—
37	C₄×C₁₆	C₂×C₄	1	2	yes	—
38	C₄×C₁₆	Q₈	1	2	yes	—
39	#339	—	2	2	no	—
40	C₈²	—	2	1	no	—
41	C₈²	C₂×C₄	1	2	yes	—
42	C₂×M₁₆	—	2	2	no	—
43	C₂²×C₈	—	2	2	no	—
44	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
46	#117	—	2	2	no	—
47	C₂×C₄²	#41	1	1	yes	characteristic
48	M₃₂	—	2	4	no	—
49	M₃₂	—	4	2	no	—
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	—	2	1	no	—
53	C₂×C₁₆	—	2	1	no	—
54	C₂×C₁₆	—	4	2	no	—
55	C₂×C₁₆	C₄²	1	2	yes	—
56	C₂×C₁₆	C₄²	1	2	yes	—
57	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
58	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
59	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
60	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
61	C₂×C₁₆	C₂×C₈	1	2	yes	—
62	C₂×C₁₆	D₈	1	1	yes	characteristic
63	C₂×C₁₆	D₈	1	1	yes	characteristic
64	C₂×C₁₆	Q₁₆	1	2	yes	—
65	#108	—	2	2	no	—
66	#108	C₂×C₈	1	4	yes	—
67	#107	C₂×C₈	1	4	yes	—
68	#106	—	2	2	no	—
69	#97	—	4	2	no	—
70	C₄×C₈	—	2	2	no	—
71	C₄×C₈	—	2	2	no	—
72	C₄×C₈	—	4	2	no	—
73	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
74	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
76	C₄×C₈	#41	1	1	yes	characteristic
77	C₂²×C₄	—	2	1	no	—
78	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
79	M₁₆	—	2	4	no	—
80	M₁₆	—	4	2	no	—
81	C₂×C₈	—	2	4	no	—
82	C₂×C₈	—	2	8	no	—
83	C₂×C₈	—	4	2	no	—
84	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
85	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₈	C₄×D₄	1	1	yes	characteristic
87	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
88	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₈	#131	1	1	yes	characteristic
91	C₂×C₈	#131	1	1	yes	characteristic
92	C₂×C₈	C₂×D₈	1	1	yes	characteristic
93	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₈	#135	1	1	yes	characteristic
95	C₂×C₈	#135	1	1	yes	characteristic
96	C₄⋊C₄	—	2	4	no	—
97	C₂²⋊C₄	—	4	2	no	—
98	C₄²	—	2	1	no	—
99	C₄²	—	4	2	no	—
100	C₄²	#117	1	1	yes	characteristic
101	C₄²	C₄×Q₈	1	1	yes	characteristic
102	C₁₆	—	2	2	no	—
103	C₁₆	—	2	2	no	—
104	C₁₆	—	4	4	no	—
105	C₁₆	C₄×C₈	1	2	yes	—
106	C₁₆	C₄×C₈	1	2	yes	—
107	C₁₆	#107	1	2	yes	—
108	C₁₆	#107	1	2	yes	—
109	C₂³	#443	1	1	yes	characteristic
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	4	no	—
114	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄	C₈×D₄	1	1	yes	characteristic
116	C₂×C₄	C₈×D₄	1	1	yes	characteristic
117	C₂×C₄	C₄×D₈	1	1	yes	characteristic
118	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
119	C₂×C₄	#443	1	1	yes	characteristic
120	C₈	—	2	8	no	—
121	C₈	—	4	4	no	—
122	C₈	C₂×C₄×C₈	1	1	yes	characteristic
123	C₈	#405	1	1	yes	characteristic
124	C₈	C₂×#107	1	1	yes	characteristic
125	C₈	#427	1	1	yes	characteristic
126	C₈	C₄×D₈	1	1	yes	characteristic
127	C₈	C₄×Q₁₆	1	1	yes	characteristic
128	C₈	C₈×Q₈	1	1	yes	characteristic
129	C₈	C₈×Q₈	1	1	yes	characteristic
130	C₂²	—	2	2	no	—
131	C₂²	C₈×D₈	1	1	yes	characteristic
132	C₂²	C₈×Q₁₆	1	1	yes	characteristic
133	C₂²	#1772	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	2	2	no	—
136	C₄	—	8	4	no	—
137	C₄	C₈×D₈	1	1	yes	characteristic
138	C₄	C₈×Q₁₆	1	1	yes	characteristic
139	C₄	C₈×C₄⋊C₄	1	1	yes	characteristic
140	C₄	C₄×#107	1	1	yes	characteristic
141	C₂	—	2	1	no	—
142	C₂	—	2	1	no	—
143	C₂	C₈×#107	1	1	yes	characteristic
144	C₂	#12046	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#12670

C₂

yes

maximal, characteristic

#12664

C₂

yes

maximal

C₂×#1373

C₂

yes

maximal, characteristic

C₂×C₈×C₁₆

C₂

yes

maximal, characteristic

#7179

C₂

yes

maximal

C₂×#368

C₄

yes

characteristic

#2158

C₄

yes

characteristic

C₂×#364

C₂²

yes

—

#2108

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1772

C₂²

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1382

C₄

yes

—

#1379

—

#1373

C₄

yes

complemented

C₈×C₁₆

C₄

yes

complemented

C₂×M₃₂

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

C₂×#108

C₂×C₄

yes

—

#427

C₂×C₄

yes

—

#405

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

characteristic

#368

C₈

yes

complemented

#367

C₈

yes

complemented

#366

C₈

yes

complemented

#364

—

#364

C₂×C₄

yes

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

Q₈

yes

—

#339

—

C₈²

—

C₈²

C₂×C₄

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

—

C₂×C₄²

#41

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₂×C₈

yes

characteristic

C₂×C₁₆

C₂×C₈

yes

characteristic

C₂×C₁₆

C₂×C₈

yes

—

C₂×C₁₆

D₈

yes

characteristic

C₂×C₁₆

D₈

yes

characteristic

C₂×C₁₆

Q₁₆

yes

—

#108

—

#108

C₂×C₈

yes

—

#107

C₂×C₈

yes

—

#106

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄×D₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄²

yes

characteristic

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

C₂²×C₈

yes

characteristic

C₂×C₈

C₂²×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂×D₈

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

100

C₄²

#117

yes

characteristic

101

C₄²

C₄×Q₈

yes

characteristic

102

C₁₆

—

103

C₁₆

—

104

C₁₆

—

105

C₁₆

C₄×C₈

yes

—

106

C₁₆

C₄×C₈

yes

—

107

C₁₆

#107

yes

—

108

C₁₆

#107

yes

—

109

C₂³

#443

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

115

C₂×C₄

C₈×D₄

yes

characteristic

116

C₂×C₄

C₈×D₄

yes

characteristic

117

C₂×C₄

C₄×D₈

yes

characteristic

118

C₂×C₄

C₄×Q₁₆

yes

characteristic

119

C₂×C₄

#443

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

C₂×C₄×C₈

yes

characteristic

123

C₈

#405

yes

characteristic

124

C₈

C₂×#107

yes

characteristic

125

C₈

#427

yes

characteristic

126

C₈

C₄×D₈

yes

characteristic

127

C₈

C₄×Q₁₆

yes

characteristic

128

C₈

C₈×Q₈

yes

characteristic

129

C₈

C₈×Q₈

yes

characteristic

130

C₂²

—

131

C₂²

C₈×D₈

yes

characteristic

132

C₂²

C₈×Q₁₆

yes

characteristic

133

C₂²

#1772

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

C₈×D₈

yes

characteristic

138

C₄

C₈×Q₁₆

yes

characteristic

139

C₄

C₈×C₄⋊C₄

yes

characteristic

140

C₄

C₄×#107

yes

characteristic

141

C₂

—

142

C₂

—

143

C₂

C₈×#107

yes

characteristic

144

C₂

#12046

yes

—

	subgroup	G/H
center	C₂×C₁₆	D₈
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₈×D₈
prime radical	C₂³	#443
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₁₆

D₈

commutator

C₈

C₂×C₄×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle

C₂²

C₈×D₈

prime radical

C₂³

#443

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
16. #1373	C₄	semidirect
17. C₈×C₁₆	C₄	semidirect
28. #368	C₈	semidirect
29. #367	C₈	semidirect
30. #366	C₈	semidirect

H ≅ G/N

type

16. #1373

C₄

semidirect

17. C₈×C₁₆

C₄

semidirect

28. #368

C₈

semidirect

29. #367

C₈

semidirect

30. #366

C₈

semidirect

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

subgroup	index
#12670	2
#12664	2
C₂×#1373	2
C₂×C₈×C₁₆	2
#7179	2

subgroup

index

C₂×C₈×C₁₆

Subgroups — #129142

Conjugacy classes of subgroups

Complements

Maximal subgroups