Group #129059 — subgroups

Subgroups — #129059

Order 512.

300 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 166 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	C₄×#364	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#11401	C₂	1	2	yes	maximal
4	#8266	C₂	1	1	yes	maximal, characteristic
5	#7141	C₂	1	2	yes	maximal
6	C₂×#364	C₄	1	2	yes	—
7	C₄×M₃₂	C₂²	1	2	yes	—
8	C₂×C₄×C₁₆	C₄	1	2	yes	—
9	C₂×C₄×C₁₆	C₄	1	2	yes	—
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	C₄²×C₈	C₂²	1	1	yes	characteristic
13	#1564	C₄	1	2	yes	—
14	#1450	C₂²	1	2	yes	—
15	#1376	C₄	1	2	yes	—
16	#1312	—	2	4	no	—
17	#1276	C₄	1	2	yes	—
18	C₂×M₃₂	C₂×C₄	1	4	yes	—
19	C₂²×C₁₆	C₂×C₄	1	2	yes	—
20	C₂²×C₁₆	C₂×C₄	1	2	yes	—
21	#431	C₂×C₄	1	4	yes	—
22	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
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	C₄³	C₂×C₄	1	1	yes	characteristic
29	#364	—	2	4	no	—
30	#346	—	2	2	no	—
31	C₄×C₁₆	—	2	4	no	—
32	C₄×C₁₆	—	2	4	no	—
33	C₄×C₁₆	—	2	2	no	—
34	C₄×C₁₆	—	2	1	no	—
35	C₄×C₁₆	—	2	1	no	—
36	C₄×C₁₆	D₄	1	2	yes	—
37	C₄×C₁₆	Q₈	1	2	yes	—
38	#335	C₈	1	4	yes	—
39	#334	C₈	1	4	yes	—
40	#322	—	2	2	no	—
41	C₈²	—	2	2	no	—
42	C₂²×C₈	C₄²	1	1	yes	characteristic
43	C₂²×C₈	C₄²	1	1	yes	characteristic
44	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄²	—	2	2	no	—
47	C₂×C₄²	C₄²	1	1	yes	characteristic
48	C₂×C₄²	C₄²	1	1	yes	characteristic
49	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
50	M₃₂	—	2	4	no	—
51	M₃₂	—	2	4	no	—
52	C₂×C₁₆	—	2	2	no	—
53	C₂×C₁₆	—	2	2	no	—
54	C₂×C₁₆	—	2	2	no	—
55	C₂×C₁₆	—	2	2	no	—
56	C₂×C₁₆	—	2	4	no	—
57	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
58	C₂×C₁₆	M₁₆	1	4	yes	—
59	C₄⋊C₈	C₂×C₈	1	8	yes	—
60	C₂²⋊C₈	—	2	4	no	—
61	C₄×C₈	—	2	2	no	—
62	C₄×C₈	—	2	2	no	—
63	C₄×C₈	—	2	4	no	—
64	C₄×C₈	C₂×C₈	1	2	yes	—
65	C₄×C₈	C₂×C₈	1	2	yes	—
66	C₄×C₈	M₁₆	1	4	yes	—
67	C₄×C₈	M₁₆	1	4	yes	—
68	C₄×C₈	C₂×D₄	1	1	yes	characteristic
69	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
70	C₄×C₈	#41	1	1	yes	characteristic
71	C₄×C₈	#41	1	1	yes	characteristic
72	C₂²×C₄	—	2	1	no	—
73	C₂²×C₄	—	2	1	no	—
74	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
75	C₂²×C₄	#131	1	2	yes	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	8	no	—
79	C₂×C₈	C₄×C₈	1	2	yes	—
80	C₂×C₈	C₄×C₈	1	1	yes	characteristic
81	C₂×C₈	C₄×C₈	1	1	yes	characteristic
82	C₂×C₈	#97	1	2	yes	—
83	C₂×C₈	#97	1	2	yes	—
84	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₈	#117	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	C₂×M₁₆	1	2	yes	—
89	C₂×C₈	C₂×M₁₆	1	2	yes	—
90	C₂×C₈	C₂×M₁₆	1	2	yes	—
91	C₂×C₈	#131	1	2	yes	—
92	C₄²	—	2	8	no	—
93	C₄²	—	2	1	no	—
94	C₄²	—	2	1	no	—
95	C₄²	#97	1	2	yes	—
96	C₄²	#97	1	2	yes	—
97	C₄²	C₄×D₄	1	1	yes	characteristic
98	C₄²	C₄×Q₈	1	1	yes	characteristic
99	C₄²	C₂²×C₈	1	2	yes	—
100	C₁₆	—	2	4	no	—
101	C₁₆	—	2	4	no	—
102	C₁₆	—	4	4	no	—
103	C₂³	#405	1	1	yes	characteristic
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
111	C₂×C₄	C₂×#97	1	1	yes	characteristic
112	C₂×C₄	C₂×#97	1	1	yes	characteristic
113	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
114	C₂×C₄	#405	1	1	yes	characteristic
115	C₂×C₄	C₈×D₄	1	2	yes	—
116	C₂×C₄	C₈×Q₈	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	8	no	—
119	C₈	#322	1	2	yes	—
120	C₈	#322	1	2	yes	—
121	C₈	#423	1	2	yes	—
122	C₈	#432	1	2	yes	—
123	C₈	#436	1	2	yes	—
124	C₈	#446	1	2	yes	—
125	C₂²	—	2	2	no	—
126	C₂²	#1451	1	1	yes	characteristic
127	C₂²	#1452	1	1	yes	characteristic
128	C₂²	C₈×C₄⋊C₄	1	1	yes	characteristic
129	C₄	—	2	4	no	—
130	C₄	—	2	4	no	—
131	C₄	—	2	2	no	—
132	C₄	C₂×#322	1	1	yes	characteristic
133	C₄	C₈×M₁₆	1	1	yes	characteristic
134	C₄	#1767	1	1	yes	characteristic
135	C₄	#1770	1	1	yes	characteristic
136	C₄	#2170	1	4	yes	—
137	C₂	—	2	1	no	—
138	C₂	—	2	1	no	—
139	C₂	#8261	1	1	yes	characteristic
140	C₂	#12664	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#364

C₂

yes

maximal, characteristic

C₄²×C₁₆

C₂

yes

maximal, characteristic

#11401

C₂

yes

maximal

#8266

C₂

yes

maximal, characteristic

#7141

C₂

yes

maximal

C₂×#364

C₄

yes

—

C₄×M₃₂

C₂²

yes

—

C₂×C₄×C₁₆

C₄

yes

—

C₂×C₄×C₁₆

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

#1564

C₄

yes

—

#1450

C₂²

yes

—

#1376

C₄

yes

—

#1312

—

#1276

C₄

yes

—

C₂×M₃₂

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

—

#431

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

#364

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

—

C₄×C₁₆

Q₈

yes

—

#335

C₈

yes

—

#334

C₈

yes

—

#322

—

C₈²

—

C₂²×C₈

C₄²

yes

characteristic

C₂²×C₈

C₄²

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₄²

yes

characteristic

C₂×C₄²

C₄²

yes

characteristic

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

M₁₆

yes

—

C₄⋊C₈

C₂×C₈

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄²

yes

characteristic

C₂²×C₄

#131

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×C₈

yes

—

C₂×C₈

C₄×C₈

yes

characteristic

C₂×C₈

C₄×C₈

yes

characteristic

C₂×C₈

#97

yes

—

C₂×C₈

#97

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×M₁₆

yes

—

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

#131

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#97

yes

—

C₄²

#97

yes

—

C₄²

C₄×D₄

yes

characteristic

C₄²

C₄×Q₈

yes

characteristic

C₄²

C₂²×C₈

yes

—

100

C₁₆

—

101

C₁₆

—

102

C₁₆

—

103

C₂³

#405

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

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

C₂×#97

yes

characteristic

112

C₂×C₄

C₂×#97

yes

characteristic

113

C₂×C₄

C₄×M₁₆

yes

characteristic

114

C₂×C₄

#405

yes

characteristic

115

C₂×C₄

C₈×D₄

yes

—

116

C₂×C₄

C₈×Q₈

yes

—

117

C₈

—

118

C₈

—

119

C₈

#322

yes

—

120

C₈

#322

yes

—

121

C₈

#423

yes

—

122

C₈

#432

yes

—

123

C₈

#436

yes

—

124

C₈

#446

yes

—

125

C₂²

—

126

C₂²

#1451

yes

characteristic

127

C₂²

#1452

yes

characteristic

128

C₂²

C₈×C₄⋊C₄

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

C₂×#322

yes

characteristic

133

C₄

C₈×M₁₆

yes

characteristic

134

C₄

#1767

yes

characteristic

135

C₄

#1770

yes

characteristic

136

C₄

#2170

yes

—

137

C₂

—

138

C₂

—

139

C₂

#8261

yes

characteristic

140

C₂

#12664

yes

—

	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	#405
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

C₂³

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₄×#364	2
C₄²×C₁₆	2
#11401	2
#8266	2
#7141	2

subgroup

index

C₄²×C₁₆

Subgroups — #129059

Conjugacy classes of subgroups

Maximal subgroups