Group #128508 — subgroups

Subgroups — #128508

Order 512.

294 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 136 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	C₂×Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#11493	C₂	1	1	yes	maximal, characteristic
3	#11408	C₂	1	2	yes	maximal
4	C₈×#101	C₂	1	1	yes	maximal, characteristic
5	#7209	C₂	1	2	yes	maximal
6	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
7	Q₈×C₁₆	—	2	2	no	—
8	Q₈×C₁₆	—	2	2	no	—
9	#2162	C₂²	1	2	yes	—
10	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
11	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₆	C₄	1	2	yes	—
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	C₄×#101	C₄	1	1	yes	characteristic
15	#1461	C₄	1	1	yes	characteristic
16	C₈×M₁₆	C₂²	1	2	yes	—
17	#1322	C₄	1	4	yes	—
18	#1312	—	2	2	no	—
19	#1311	—	2	2	no	—
20	#1283	C₄	1	2	yes	—
21	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₁₆	C₂×C₄	1	2	yes	—
24	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
25	C₈×Q₈	—	2	2	no	—
26	C₈×Q₈	—	2	2	no	—
27	#431	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×#101	C₈	1	2	yes	—
31	C₄×M₁₆	C₂×C₄	1	2	yes	—
32	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
34	C₄⋊C₁₆	—	2	4	no	—
35	C₄⋊C₁₆	—	2	2	no	—
36	C₄⋊C₁₆	—	2	2	no	—
37	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
38	C₄×C₁₆	—	2	2	no	—
39	C₄×C₁₆	—	2	2	no	—
40	C₄×C₁₆	—	2	2	no	—
41	C₄×C₁₆	D₄	1	2	yes	—
42	C₄×C₁₆	D₄	1	2	yes	—
43	#343	C₈	1	2	yes	—
44	#324	C₈	1	4	yes	—
45	#322	—	2	2	no	—
46	C₈²	—	2	2	no	—
47	C₂²×Q₈	C₂×C₈	1	1	yes	characteristic
48	C₂×M₁₆	C₂×C₈	1	4	yes	—
49	C₂²×C₈	C₄²	1	2	yes	—
50	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₄×Q₈	—	2	2	no	—
53	C₄×Q₈	—	2	2	no	—
54	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
55	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
56	C₂×C₄²	C₄²	1	2	yes	—
57	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₁₆	—	2	2	no	—
59	C₂×C₁₆	—	2	2	no	—
60	C₂×C₁₆	—	2	4	no	—
61	C₂×C₁₆	—	4	1	no	—
62	C₂×C₁₆	—	4	2	no	—
63	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
65	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
66	C₄⋊C₈	—	2	4	no	—
67	C₄⋊C₈	—	2	2	no	—
68	C₄⋊C₈	—	2	2	no	—
69	#101	—	2	4	no	—
70	C₂²⋊C₈	C₂×C₈	1	4	yes	—
71	#97	—	2	2	no	—
72	C₄×C₈	—	2	2	no	—
73	C₄×C₈	—	2	2	no	—
74	C₄×C₈	—	2	2	no	—
75	C₄×C₈	—	2	2	no	—
76	C₄×C₈	C₂×D₄	1	2	yes	—
77	C₄×C₈	#41	1	2	yes	—
78	C₂×Q₈	—	2	2	no	—
79	C₂×Q₈	—	2	2	no	—
80	C₂×Q₈	M₃₂	1	4	yes	—
81	C₂²×C₄	C₄×C₈	1	2	yes	—
82	C₂²×C₄	C₄×C₈	1	2	yes	—
83	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
84	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
85	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
86	M₁₆	—	2	4	no	—
87	M₁₆	—	2	4	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	2	4	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	—	4	2	no	—
95	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈	#117	1	1	yes	characteristic
97	C₂×C₈	C₄×D₄	1	2	yes	—
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	C₄×D₄	1	2	yes	—
103	C₄²	#131	1	2	yes	—
104	C₁₆	—	2	1	no	—
105	C₁₆	—	2	2	no	—
106	C₁₆	—	4	4	no	—
107	C₁₆	—	4	2	no	—
108	C₁₆	#101	1	2	yes	complemented
109	C₂³	C₂×C₄×C₈	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	#346	1	4	yes	—
119	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄	#405	1	1	yes	characteristic
121	C₂×C₄	C₈×D₄	1	2	yes	—
122	C₂×C₄	C₈×D₄	1	2	yes	—
123	C₂×C₄	C₂×M₃₂	1	2	yes	—
124	C₂×C₄	#504	1	2	yes	—
125	C₈	—	2	1	no	—
126	C₈	—	2	2	no	—
127	C₈	—	4	4	no	—
128	C₈	—	4	4	no	—
129	C₈	—	4	2	no	—
130	C₈	C₂×#101	1	1	yes	characteristic
131	C₈	#413	1	1	yes	characteristic
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	C₈×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²	C₂×#346	1	1	yes	characteristic
136	C₂²	#2108	1	1	yes	characteristic
137	C₄	—	2	4	no	—
138	C₄	—	2	2	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	2	no	—
141	C₄	C₄×#101	1	1	yes	characteristic
142	C₄	#1757	1	1	yes	characteristic
143	C₄	#2168	1	4	yes	—
144	C₂	—	2	1	no	—
145	C₂	—	2	1	no	—
146	C₂	C₈×#101	1	1	yes	characteristic
147	C₂	#12611	1	1	yes	characteristic
148	C₂	#12613	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×C₁₆

C₂

yes

maximal, characteristic

#11493

C₂

yes

maximal, characteristic

#11408

C₂

yes

maximal

C₈×#101

C₂

yes

maximal, characteristic

#7209

C₂

yes

maximal

C₂×C₈×Q₈

C₂²

yes

characteristic

Q₈×C₁₆

—

Q₈×C₁₆

—

#2162

C₂²

yes

—

C₂×C₄⋊C₁₆

C₄

yes

—

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₄×#101

C₄

yes

characteristic

#1461

C₄

yes

characteristic

C₈×M₁₆

C₂²

yes

—

#1322

C₄

yes

—

#1312

—

#1311

—

#1283

C₄

yes

—

C₂×C₄×Q₈

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

#431

C₂×C₄

yes

—

C₂×C₄⋊C₈

C₂×C₄

yes

—

C₂×C₄⋊C₈

C₂×C₄

yes

characteristic

C₂×#101

C₈

yes

—

C₄×M₁₆

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

C₂×C₄

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

—

C₄×C₁₆

D₄

yes

—

#343

C₈

yes

—

#324

C₈

yes

—

#322

—

C₈²

—

C₂²×Q₈

C₂×C₈

yes

characteristic

C₂×M₁₆

C₂×C₈

yes

—

C₂²×C₈

C₄²

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₂×C₈

yes

—

C₂×C₄⋊C₄

C₂×C₈

yes

characteristic

C₂×C₄²

C₄²

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#101

—

C₂²⋊C₈

C₂×C₈

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

M₃₂

yes

—

C₂²×C₄

C₄×C₈

yes

—

C₂²×C₄

C₄×C₈

yes

—

C₂²×C₄

C₂×C₄²

yes

characteristic

C₂²×C₄

C₂²×C₈

yes

characteristic

C₂²×C₄

C₂²×C₈

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

#117

yes

characteristic

C₂×C₈

C₄×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₄×D₄

yes

—

103

C₄²

#131

yes

—

104

C₁₆

—

105

C₁₆

—

106

C₁₆

—

107

C₁₆

—

108

C₁₆

#101

yes

complemented

109

C₂³

C₂×C₄×C₈

yes

characteristic

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

#346

yes

—

119

C₂×C₄

C₄×C₂²⋊C₄

yes

characteristic

120

C₂×C₄

#405

yes

characteristic

121

C₂×C₄

C₈×D₄

yes

—

122

C₂×C₄

C₈×D₄

yes

—

123

C₂×C₄

C₂×M₃₂

yes

—

124

C₂×C₄

#504

yes

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

C₂×#101

yes

characteristic

131

C₈

#413

yes

characteristic

132

C₂²

—

133

C₂²

—

134

C₂²

C₈×C₂²⋊C₄

yes

characteristic

135

C₂²

C₂×#346

yes

characteristic

136

C₂²

#2108

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

C₄×#101

yes

characteristic

142

C₄

#1757

yes

characteristic

143

C₄

#2168

yes

—

144

C₂

—

145

C₂

—

146

C₂

C₈×#101

yes

characteristic

147

C₂

#12611

yes

characteristic

148

C₂

#12613

yes

characteristic

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

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊C₄

commutator, prime radical

C₂³

C₂×C₄×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle

C₂²

C₈×C₂²⋊C₄

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
108. C₁₆	#101	semidirect

H ≅ G/N

type

108. C₁₆

#101

semidirect

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

subgroup	index
C₂×Q₈×C₁₆	2
#11493	2
#11408	2
C₈×#101	2
#7209	2

subgroup

index

C₂×Q₈×C₁₆

Subgroups — #128508

Conjugacy classes of subgroups

Complements

Maximal subgroups