Group #128504 — subgroups

Subgroups — #128504

Order 512.

317 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 122 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	#33434	C₂	1	1	yes	maximal, characteristic
2	#11490	C₂	1	1	yes	maximal, characteristic
3	C₁₆×M₁₆	C₂	1	2	yes	maximal, complemented
4	#8241	C₂	1	1	yes	maximal, characteristic
5	#7212	C₂	1	2	yes	maximal, complemented
6	C₂×#504	C₄	1	2	yes	—
7	#2931	C₂²	1	1	yes	characteristic
8	#2169	—	2	2	no	—
9	D₄×C₁₆	—	2	2	no	—
10	#2162	C₂²	1	2	yes	—
11	#2150	C₂²	1	1	yes	characteristic
12	#2113	C₄	1	2	yes	—
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	#1543	C₄	1	1	yes	characteristic
15	#1525	C₄	1	1	yes	characteristic
16	C₈×M₁₆	C₂²	1	2	yes	—
17	#1315	C₄	1	4	yes	—
18	#1312	—	2	2	no	—
19	C₈×C₁₆	—	2	2	no	—
20	#1299	C₄	1	2	yes	—
21	C₂×#131	C₂×C₄	1	2	yes	—
22	#532	C₂×C₄	1	1	yes	characteristic
23	#504	—	4	4	no	—
24	C₂×M₃₂	—	2	2	no	—
25	C₂×M₃₂	C₂×C₄	1	2	yes	—
26	C₂²×C₁₆	—	2	2	no	—
27	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
29	#436	—	2	2	no	—
30	C₈×D₄	—	2	2	no	—
31	#431	C₂×C₄	1	2	yes	—
32	#423	C₂×C₄	1	1	yes	characteristic
33	#408	C₂×C₄	1	2	yes	—
34	C₄×M₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
36	C₄⋊C₁₆	—	2	2	no	—
37	C₄⋊C₁₆	—	2	2	no	—
38	C₂²⋊C₁₆	—	2	4	no	—
39	C₂²⋊C₁₆	C₂×C₄	1	4	yes	—
40	C₄×C₁₆	—	2	2	no	—
41	C₄×C₁₆	D₄	1	2	yes	—
42	C₄×C₁₆	D₄	1	2	yes	—
43	#333	C₈	1	2	yes	—
44	#331	C₈	1	2	yes	—
45	#329	C₈	1	4	yes	—
46	#322	—	2	2	no	—
47	C₈²	—	2	2	no	—
48	C₂×#41	C₄²	1	2	yes	—
49	#131	—	4	2	no	—
50	#131	—	4	2	no	—
51	C₂×M₁₆	—	2	2	no	—
52	C₂×M₁₆	—	2	2	no	—
53	C₂×M₁₆	C₄²	1	2	yes	—
54	C₂²×C₈	—	2	2	no	—
55	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
57	C₄⋊Q₈	C₂×C₈	1	1	yes	characteristic
58	#127	C₂×C₈	1	1	yes	characteristic
59	#124	C₂×C₈	1	2	yes	—
60	#121	—	2	2	no	—
61	C₄×D₄	—	2	2	no	—
62	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
63	M₃₂	—	4	4	no	—
64	M₃₂	—	4	2	no	—
65	C₂×C₁₆	—	2	2	no	—
66	C₂×C₁₆	—	2	2	no	—
67	C₂×C₁₆	—	2	4	no	—
68	C₂×C₁₆	—	4	1	no	—
69	C₂×C₁₆	—	4	4	no	—
70	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
71	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
73	C₄⋊C₈	—	2	2	no	—
74	C₄⋊C₈	C₂×C₈	1	4	yes	—
75	C₂²⋊C₈	—	2	2	no	—
76	C₂²⋊C₈	—	2	4	no	—
77	#97	C₂×C₈	1	4	yes	—
78	C₄×C₈	—	2	2	no	—
79	C₄×C₈	—	2	2	no	—
80	C₄×C₈	C₂×D₄	1	2	yes	—
81	C₄×C₈	#41	1	2	yes	—
82	#41	—	4	2	no	—
83	#41	—	4	2	no	—
84	C₂×Q₈	C₄×C₈	1	2	yes	—
85	C₂×D₄	—	2	2	no	—
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	C₄×C₈	1	2	yes	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
90	C₂²×C₄	#131	1	2	yes	—
91	M₁₆	—	2	8	no	—
92	M₁₆	—	4	4	no	—
93	M₁₆	—	4	2	no	—
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	4	no	—
99	C₂×C₈	—	4	1	no	—
100	C₂×C₈	—	4	2	no	—
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₈	#117	1	1	yes	characteristic
104	C₂×C₈	C₄×D₄	1	2	yes	—
105	C₄⋊C₄	—	2	2	no	—
106	C₂²⋊C₄	—	2	4	no	—
107	C₄²	C₄×D₄	1	2	yes	—
108	C₄²	C₂²×C₈	1	1	yes	characteristic
109	C₄²	C₂²×C₈	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	2	2	no	—
112	C₁₆	—	4	4	no	—
113	C₁₆	—	4	2	no	—
114	C₁₆	#100	1	2	yes	—
115	C₂³	—	2	2	no	—
116	C₂³	#405	1	1	yes	characteristic
117	Q₈	—	4	2	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	4	no	—
120	D₄	—	4	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
129	C₂×C₄	C₈×D₄	1	2	yes	—
130	C₂×C₄	C₈×D₄	1	2	yes	—
131	C₈	—	2	8	no	—
132	C₈	—	2	1	no	—
133	C₈	—	2	2	no	—
134	C₈	—	4	4	no	—
135	C₈	—	4	2	no	—
136	C₈	C₂×#100	1	1	yes	characteristic
137	C₈	#413	1	1	yes	characteristic
138	C₂²	—	2	2	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	4	4	no	—
141	C₂²	—	4	2	no	—
142	C₂²	C₈×C₂²⋊C₄	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	2	1	no	—
145	C₄	—	4	2	no	—
146	C₄	C₄×#100	1	1	yes	characteristic
147	C₄	#1757	1	1	yes	characteristic
148	C₄	#2170	1	4	yes	—
149	C₂	—	4	1	no	—
150	C₂	—	8	2	no	—
151	C₂	C₈×#100	1	1	yes	characteristic
152	C₂	#12614	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33434

C₂

yes

maximal, characteristic

#11490

C₂

yes

maximal, characteristic

C₁₆×M₁₆

C₂

yes

maximal, complemented

#8241

C₂

yes

maximal, characteristic

#7212

C₂

yes

maximal, complemented

C₂×#504

C₄

yes

—

#2931

C₂²

yes

characteristic

#2169

—

D₄×C₁₆

—

#2162

C₂²

yes

—

#2150

C₂²

yes

characteristic

#2113

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1543

C₄

yes

characteristic

#1525

C₄

yes

characteristic

C₈×M₁₆

C₂²

yes

—

#1315

C₄

yes

—

#1312

—

C₈×C₁₆

—

#1299

C₄

yes

—

C₂×#131

C₂×C₄

yes

—

#532

C₂×C₄

yes

characteristic

#504

—

C₂×M₃₂

—

C₂×M₃₂

C₂×C₄

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

#436

—

C₈×D₄

—

#431

C₂×C₄

yes

—

#423

C₂×C₄

yes

characteristic

#408

C₂×C₄

yes

—

C₄×M₁₆

C₂×C₄

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

C₂×C₄

yes

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

—

C₄×C₁₆

D₄

yes

—

#333

C₈

yes

—

#331

C₈

yes

—

#329

C₈

yes

—

#322

—

C₈²

—

C₂×#41

C₄²

yes

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₄²

yes

—

C₂²×C₈

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₄⋊Q₈

C₂×C₈

yes

characteristic

#127

C₂×C₈

yes

characteristic

#124

C₂×C₈

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×C₈

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

C₂×C₈

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

#41

—

#41

—

C₂×Q₈

C₄×C₈

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₄×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄²

yes

characteristic

C₂²×C₄

#131

yes

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

103

C₂×C₈

#117

yes

characteristic

104

C₂×C₈

C₄×D₄

yes

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₄²

C₄×D₄

yes

—

108

C₄²

C₂²×C₈

yes

characteristic

109

C₄²

C₂²×C₈

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

113

C₁₆

—

114

C₁₆

#100

yes

—

115

C₂³

—

116

C₂³

#405

yes

characteristic

117

Q₈

—

118

D₄

—

119

D₄

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₄×C₂²⋊C₄

yes

characteristic

128

C₂×C₄

C₂×C₄×C₈

yes

characteristic

129

C₂×C₄

C₈×D₄

yes

—

130

C₂×C₄

C₈×D₄

yes

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

C₂×#100

yes

characteristic

137

C₈

#413

yes

characteristic

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₈×C₂²⋊C₄

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

C₄×#100

yes

characteristic

147

C₄

#1757

yes

characteristic

148

C₄

#2170

yes

—

149

C₂

—

150

C₂

—

151

C₂

C₈×#100

yes

characteristic

152

C₂

#12614

yes

—

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₈×C₂²⋊C₄
prime radical	#532	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₁₆

C₂²⋊C₄

commutator

C₂×C₄

C₂×C₄×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle

C₂²

C₈×C₂²⋊C₄

C₂×C₄

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
3. C₁₆×M₁₆	C₂	semidirect
5. #7212	C₂	semidirect

H ≅ G/N

type

3. C₁₆×M₁₆

C₂

semidirect

5. #7212

C₂

semidirect

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

subgroup	index
#33434	2
#11490	2
C₁₆×M₁₆	2
#8241	2
#7212	2

subgroup

index

C₁₆×M₁₆

Subgroups — #128504

Conjugacy classes of subgroups

Complements

Maximal subgroups