Group #129134 — subgroups

Subgroups — C₁₆×#106 #129134

Order 512.

306 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 178 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	C₁₆×C₄⋊C₄	C₂	1	2	yes	maximal
2	C₂×#1367	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₈×#106	C₂	1	1	yes	maximal, characteristic
5	#7167	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₁₆	—	2	2	no	—
8	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	C₄×#106	C₄	1	1	yes	characteristic
11	C₈×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×#334	C₄	1	1	yes	characteristic
13	C₂×C₈²	C₂²	1	1	yes	characteristic
14	#1374	—	2	2	no	—
15	#1367	C₄	1	4	yes	complemented
16	C₈×C₁₆	C₄	1	4	yes	complemented
17	#1279	C₄	1	2	yes	—
18	C₂²×C₁₆	—	2	1	no	—
19	C₂²×C₁₆	—	2	2	no	—
20	C₂²×C₁₆	D₄	1	1	yes	characteristic
21	C₂²×C₁₆	D₄	1	1	yes	characteristic
22	C₂×#106	C₈	1	2	yes	—
23	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₈	—	2	2	no	—
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×C₄⋊C₄	C₂×C₄	1	2	yes	—
29	C₄⋊C₁₆	—	2	2	no	—
30	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
31	C₄×C₁₆	—	2	1	no	—
32	C₄×C₁₆	—	4	4	no	—
33	C₄×C₁₆	C₂×C₄	1	2	yes	—
34	C₄×C₁₆	C₂×C₄	1	2	yes	—
35	C₄×C₁₆	Q₈	1	2	yes	—
36	#340	C₈	1	2	yes	—
37	#336	—	2	2	no	—
38	#334	C₈	1	4	yes	—
39	C₈²	C₂×C₄	1	2	yes	—
40	C₈²	C₂×C₄	1	2	yes	—
41	C₂²×C₈	—	2	1	no	—
42	C₂²×C₈	—	2	2	no	—
43	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₈	#41	1	1	yes	characteristic
47	C₂×C₄⋊C₄	C₂×C₈	1	4	yes	—
48	C₂×C₄²	—	2	2	no	—
49	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₁₆	—	2	4	no	—
51	C₂×C₁₆	—	2	2	no	—
52	C₂×C₁₆	—	2	2	no	—
53	C₂×C₁₆	—	2	4	no	—
54	C₂×C₁₆	—	4	2	no	—
55	C₂×C₁₆	—	4	2	no	—
56	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
57	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
58	C₂×C₁₆	SD₁₆	1	4	yes	—
59	#106	C₁₆	1	8	yes	direct product factor
60	C₄⋊C₈	—	2	2	no	—
61	C₄⋊C₈	C₂×C₈	1	4	yes	—
62	C₄×C₈	—	2	1	no	—
63	C₄×C₈	—	4	2	no	—
64	C₄×C₈	—	4	2	no	—
65	C₄×C₈	C₄²	1	2	yes	—
66	C₄×C₈	C₄²	1	2	yes	—
67	C₄×C₈	C₂×C₈	1	2	yes	—
68	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
71	C₄×C₈	#41	1	1	yes	characteristic
72	#95	—	2	2	no	—
73	C₂²×C₄	—	2	4	no	—
74	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
75	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
76	C₂²×C₄	#131	1	1	yes	characteristic
77	C₂×C₈	—	2	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	4	no	—
82	C₂×C₈	—	4	2	no	—
83	C₂×C₈	—	4	2	no	—
84	C₂×C₈	C₄×C₈	1	2	yes	—
85	C₂×C₈	C₄×C₈	1	2	yes	—
86	C₂×C₈	C₂×C₁₆	1	4	yes	—
87	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₈	#117	1	1	yes	characteristic
89	C₂×C₈	C₄×D₄	1	1	yes	characteristic
90	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
91	C₂×C₈	C₂×SD₁₆	1	2	yes	—
92	C₂×C₈	#135	1	2	yes	—
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	C₂×C₁₆	1	8	yes	—
95	C₄²	—	4	2	no	—
96	C₄²	—	4	2	no	—
97	C₄²	C₂×C₄²	1	1	yes	characteristic
98	C₄²	C₄×Q₈	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	#106	1	4	yes	direct product factor
105	C₂³	C₈×D₄	1	1	yes	characteristic
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	8	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	—	4	4	no	—
110	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
112	C₂×C₄	#405	1	1	yes	characteristic
113	C₂×C₄	C₈×D₄	1	1	yes	characteristic
114	C₂×C₄	C₄×SD₁₆	1	2	yes	—
115	C₂×C₄	#443	1	2	yes	—
116	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
117	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
118	C₂×C₄	C₂²×C₁₆	1	2	yes	—
119	C₂×C₄	#504	1	2	yes	—
120	C₈	—	2	4	no	—
121	C₈	—	2	2	no	—
122	C₈	—	2	1	no	—
123	C₈	—	2	1	no	—
124	C₈	—	4	4	no	—
125	C₈	C₄×C₁₆	1	4	yes	complemented
126	C₈	C₂×#106	1	1	yes	characteristic
127	C₈	#427	1	1	yes	characteristic
128	C₈	C₄×SD₁₆	1	2	yes	—
129	C₂²	C₈×SD₁₆	1	2	yes	—
130	C₂²	C₈×C₄⋊C₄	1	1	yes	characteristic
131	C₂²	D₄×C₁₆	1	2	yes	—
132	C₂²	#2170	1	2	yes	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	8	no	—
136	C₄	C₈×SD₁₆	1	2	yes	—
137	C₄	#1772	1	1	yes	characteristic
138	C₄	C₄×#106	1	1	yes	characteristic
139	C₄	C₂×C₄×C₁₆	1	1	yes	characteristic
140	C₄	#2108	1	1	yes	characteristic
141	C₄	Q₈×C₁₆	1	2	yes	—
142	C₂	C₈×#106	1	1	yes	characteristic
143	C₂	C₁₆×SD₁₆	1	4	yes	—
144	C₂	C₁₆×C₄⋊C₄	1	1	yes	characteristic
145	C₂	#12664	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×C₄⋊C₄

C₂

yes

maximal

C₂×#1367

C₂

yes

maximal, characteristic

C₂×C₈×C₁₆

C₂

yes

maximal, characteristic

C₈×#106

C₂

yes

maximal, characteristic

#7167

C₂

yes

maximal

C₂×C₄⋊C₁₆

C₂²

yes

—

C₂×C₄×C₁₆

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₄×#106

C₄

yes

characteristic

C₈×C₄⋊C₄

C₂²

yes

—

C₂×#334

C₄

yes

characteristic

C₂×C₈²

C₂²

yes

characteristic

#1374

—

#1367

C₄

yes

complemented

C₈×C₁₆

C₄

yes

complemented

#1279

C₄

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

C₂×#106

C₈

yes

—

C₂×C₄⋊C₈

C₂×C₄

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄×C₄⋊C₄

C₂×C₄

yes

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×C₄

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

Q₈

yes

—

#340

C₈

yes

—

#336

—

#334

C₈

yes

—

C₈²

C₂×C₄

yes

—

C₈²

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄⋊C₄

C₂×C₈

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

SD₁₆

yes

—

#106

C₁₆

yes

direct product factor

C₄⋊C₈

—

C₄⋊C₈

C₂×C₈

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄²

yes

—

C₄×C₈

C₄²

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄

yes

characteristic

C₂²×C₄

C₂²×C₈

yes

characteristic

C₂²×C₄

#131

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×C₈

yes

—

C₂×C₈

C₄×C₈

yes

—

C₂×C₈

C₂×C₁₆

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

C₂×C₈

C₂×SD₁₆

yes

—

C₂×C₈

#135

yes

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₁₆

yes

—

C₄²

—

C₄²

—

C₄²

C₂×C₄²

yes

characteristic

C₄²

C₄×Q₈

yes

characteristic

C₄²

C₂²×C₈

yes

characteristic

100

C₄²

#131

yes

characteristic

101

C₁₆

—

102

C₁₆

—

103

C₁₆

—

104

C₁₆

#106

yes

direct product factor

105

C₂³

C₈×D₄

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

111

C₂×C₄

C₂×C₄×C₈

yes

characteristic

112

C₂×C₄

#405

yes

characteristic

113

C₂×C₄

C₈×D₄

yes

characteristic

114

C₂×C₄

C₄×SD₁₆

yes

—

115

C₂×C₄

#443

yes

—

116

C₂×C₄

C₈×Q₈

yes

characteristic

117

C₂×C₄

C₈×Q₈

yes

characteristic

118

C₂×C₄

C₂²×C₁₆

yes

—

119

C₂×C₄

#504

yes

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

C₄×C₁₆

yes

complemented

126

C₈

C₂×#106

yes

characteristic

127

C₈

#427

yes

characteristic

128

C₈

C₄×SD₁₆

yes

—

129

C₂²

C₈×SD₁₆

yes

—

130

C₂²

C₈×C₄⋊C₄

yes

characteristic

131

C₂²

D₄×C₁₆

yes

—

132

C₂²

#2170

yes

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

C₈×SD₁₆

yes

—

137

C₄

#1772

yes

characteristic

138

C₄

C₄×#106

yes

characteristic

139

C₄

C₂×C₄×C₁₆

yes

characteristic

140

C₄

#2108

yes

characteristic

141

C₄

Q₈×C₁₆

yes

—

142

C₂

C₈×#106

yes

characteristic

143

C₂

C₁₆×SD₁₆

yes

—

144

C₂

C₁₆×C₄⋊C₄

yes

characteristic

145

C₂

#12664

yes

characteristic

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

subgroup

G/H

center

C₂²×C₁₆

D₄

commutator

C₄

C₂×C₄×C₁₆

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle, prime radical

C₂³

C₈×D₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
15. #1367	C₄	semidirect
16. C₈×C₁₆	C₄	semidirect
59. #106	C₁₆	direct (N × H)
104. C₁₆	#106	direct (N × H)
125. C₈	C₄×C₁₆	semidirect

H ≅ G/N

type

15. #1367

C₄

semidirect

16. C₈×C₁₆

C₄

semidirect

59. #106

C₁₆

direct (N × H)

104. C₁₆

#106

direct (N × H)

125. C₈

C₄×C₁₆

semidirect

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

subgroup	index
C₁₆×C₄⋊C₄	2
C₂×#1367	2
C₂×C₈×C₁₆	2
C₈×#106	2
#7167	2

subgroup

index

C₁₆×C₄⋊C₄

C₂×#1367

C₂×C₈×C₁₆

C₈×#106

#7167

Subgroups — C₁₆×#106 #129134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C16×#106 #129134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×#106 #129134