Group #150077 — subgroups

Subgroups — #150077

Order 512.

232 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 112 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	#13627	C₂	1	1	yes	maximal, characteristic
3	C₄×M₆₄	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊C₃₂	C₂	1	4	yes	maximal, complemented
5	C₂×C₈×Q₈	C₄	1	1	yes	characteristic
6	C₂×M₆₄	—	2	1	no	—
7	C₂×M₆₄	—	2	1	no	—
8	Q₈×C₁₆	—	2	1	no	—
9	Q₈×C₁₆	—	2	1	no	—
10	Q₈×C₁₆	C₄	1	2	yes	—
11	Q₈×C₁₆	C₂²	1	2	yes	—
12	C₂×C₄⋊C₁₆	—	2	1	no	—
13	C₂×C₄⋊C₁₆	C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₆	—	2	1	no	—
15	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
16	C₄⋊C₃₂	—	2	1	no	—
17	C₄⋊C₃₂	C₂²	1	2	yes	—
18	#1397	—	2	1	no	—
19	C₄×C₃₂	C₂²	1	2	yes	—
20	C₂×C₄×Q₈	C₈	1	1	yes	characteristic
21	C₂²×C₁₆	—	2	1	no	—
22	C₂²×C₁₆	D₄	1	1	yes	characteristic
23	C₂²×C₁₆	D₄	1	1	yes	characteristic
24	C₈×Q₈	—	2	1	no	—
25	C₈×Q₈	—	2	1	no	—
26	C₈×Q₈	C₈	1	2	yes	—
27	C₈×Q₈	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₈	—	2	1	no	—
29	C₂×C₄⋊C₈	C₈	1	1	yes	characteristic
30	C₂×C₄×C₈	—	2	1	no	—
31	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	M₆₄	—	2	4	no	—
33	M₆₄	—	4	2	no	—
34	C₂×C₃₂	—	2	2	no	—
35	C₂×C₃₂	—	2	2	no	—
36	C₄⋊C₁₆	—	2	2	no	—
37	C₄⋊C₁₆	—	2	2	no	—
38	C₄⋊C₁₆	—	2	1	no	—
39	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
40	C₄×C₁₆	—	2	2	no	—
41	C₄×C₁₆	—	2	2	no	—
42	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
43	C₄×C₁₆	D₄	1	1	yes	characteristic
44	C₄×C₁₆	D₄	1	1	yes	characteristic
45	C₄×C₁₆	C₂³	1	1	yes	characteristic
46	C₂²×Q₈	C₁₆	1	1	yes	characteristic
47	C₂²×C₈	—	2	1	no	—
48	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₈	M₁₆	1	1	yes	characteristic
50	C₄×Q₈	—	2	1	no	—
51	C₄×Q₈	—	2	1	no	—
52	C₄×Q₈	C₁₆	1	2	yes	—
53	C₄×Q₈	C₂×C₈	1	2	yes	—
54	C₂×C₄⋊C₄	—	2	1	no	—
55	C₂×C₄⋊C₄	C₁₆	1	1	yes	characteristic
56	C₂×C₄²	—	2	1	no	—
57	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
58	C₂×C₁₆	—	2	2	no	—
59	C₂×C₁₆	—	2	2	no	—
60	C₂×C₁₆	—	4	2	no	—
61	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₁₆	SD₁₆	1	2	yes	—
64	C₂×C₁₆	Q₁₆	1	2	yes	—
65	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
66	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
67	C₄⋊C₈	—	2	2	no	—
68	C₄⋊C₈	—	2	2	no	—
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	C₂×C₈	1	2	yes	—
71	C₄×C₈	—	2	2	no	—
72	C₄×C₈	—	2	2	no	—
73	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₈	C₂×C₈	1	1	yes	characteristic
75	C₄×C₈	M₁₆	1	1	yes	characteristic
76	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₃₂	—	2	4	no	—
78	C₃₂	—	4	2	no	—
79	C₂×Q₈	—	2	4	no	—
80	C₂×Q₈	—	2	1	no	—
81	C₂×Q₈	C₂×C₁₆	1	2	yes	—
82	C₂²×C₄	—	2	1	no	—
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
85	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
86	C₂²×C₄	M₃₂	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	4	2	no	—
90	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
91	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
92	C₂×C₈	#103	1	2	yes	—
93	C₂×C₈	#104	1	2	yes	—
94	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	2	1	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	C₂×C₁₆	1	2	yes	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	C₂²⋊C₈	1	1	yes	characteristic
103	C₄²	C₂×C₁₆	1	1	yes	characteristic
104	C₄²	M₃₂	1	1	yes	characteristic
105	C₄²	C₂²×C₈	1	1	yes	characteristic
106	C₁₆	—	2	1	no	—
107	C₁₆	—	2	1	no	—
108	C₁₆	—	4	2	no	—
109	C₁₆	#103	1	2	yes	—
110	C₁₆	C₂×SD₁₆	1	1	yes	characteristic
111	C₁₆	C₂×Q₁₆	1	1	yes	characteristic
112	C₂³	C₂²⋊C₁₆	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	2	no	—
115	Q₈	M₆₄	1	4	yes	complemented
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	Q₈⋊C₈	1	2	yes	—
123	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
124	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
125	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
126	C₂×C₄	#350	1	2	yes	—
127	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
128	C₂×C₄	C₂²×C₁₆	1	1	yes	characteristic
129	C₂×C₄	C₂×M₃₂	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	—	2	1	no	—
132	C₈	—	4	2	no	—
133	C₈	Q₈⋊C₈	1	2	yes	—
134	C₈	C₂×#103	1	1	yes	characteristic
135	C₈	C₂×#104	1	1	yes	characteristic
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	Q₈⋊C₁₆	1	2	yes	—
139	C₂²	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
140	C₄	—	2	4	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	4	2	no	—
145	C₄	Q₈⋊C₁₆	1	2	yes	—
146	C₄	C₂×Q₈⋊C₈	1	1	yes	characteristic
147	C₄	C₂×#350	1	1	yes	characteristic
148	C₄	C₂×M₆₄	1	2	yes	—
149	C₂	—	2	2	no	—
150	C₂	C₂×Q₈⋊C₁₆	1	1	yes	characteristic
151	C₂	#13553	1	1	yes	characteristic
152	C₂	#13558	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×C₁₆

C₂

yes

maximal, characteristic

#13627

C₂

yes

maximal, characteristic

C₄×M₆₄

C₂

yes

maximal, characteristic

Q₈⋊C₃₂

C₂

yes

maximal, complemented

C₂×C₈×Q₈

C₄

yes

characteristic

C₂×M₆₄

—

C₂×M₆₄

—

Q₈×C₁₆

—

Q₈×C₁₆

—

Q₈×C₁₆

C₄

yes

—

Q₈×C₁₆

C₂²

yes

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

C₄

yes

characteristic

C₂×C₄×C₁₆

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₄⋊C₃₂

—

C₄⋊C₃₂

C₂²

yes

—

#1397

—

C₄×C₃₂

C₂²

yes

—

C₂×C₄×Q₈

C₈

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

C₈

yes

—

C₈×Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₈

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×C₄

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

C₂³

yes

characteristic

C₂²×Q₈

C₁₆

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

M₁₆

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₁₆

yes

—

C₄×Q₈

C₂×C₈

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₁₆

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×C₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

SD₁₆

yes

—

C₂×C₁₆

Q₁₆

yes

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₈

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₃₂

—

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₁₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₈

yes

characteristic

C₂²×C₄

C₂×C₁₆

yes

characteristic

C₂²×C₄

M₃₂

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₈

yes

characteristic

C₂×C₈

C₂²⋊C₈

yes

characteristic

C₂×C₈

#103

yes

—

C₂×C₈

#104

yes

—

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₁₆

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂²⋊C₈

yes

characteristic

103

C₄²

C₂×C₁₆

yes

characteristic

104

C₄²

M₃₂

yes

characteristic

105

C₄²

C₂²×C₈

yes

characteristic

106

C₁₆

—

107

C₁₆

—

108

C₁₆

—

109

C₁₆

#103

yes

—

110

C₁₆

C₂×SD₁₆

yes

characteristic

111

C₁₆

C₂×Q₁₆

yes

characteristic

112

C₂³

C₂²⋊C₁₆

yes

characteristic

113

Q₈

—

114

Q₈

—

115

Q₈

M₆₄

yes

complemented

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

Q₈⋊C₈

yes

—

123

C₂×C₄

C₂²⋊C₁₆

yes

characteristic

124

C₂×C₄

C₂²⋊C₁₆

yes

characteristic

125

C₂×C₄

C₂²⋊C₁₆

yes

characteristic

126

C₂×C₄

#350

yes

—

127

C₂×C₄

C₂×C₂²⋊C₈

yes

characteristic

128

C₂×C₄

C₂²×C₁₆

yes

characteristic

129

C₂×C₄

C₂×M₃₂

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

Q₈⋊C₈

yes

—

134

C₈

C₂×#103

yes

characteristic

135

C₈

C₂×#104

yes

characteristic

136

C₂²

—

137

C₂²

—

138

C₂²

Q₈⋊C₁₆

yes

—

139

C₂²

C₂×C₂²⋊C₁₆

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

Q₈⋊C₁₆

yes

—

146

C₄

C₂×Q₈⋊C₈

yes

characteristic

147

C₄

C₂×#350

yes

characteristic

148

C₄

C₂×M₆₄

yes

—

149

C₂

—

150

C₂

C₂×Q₈⋊C₁₆

yes

characteristic

151

C₂

#13553

yes

characteristic

152

C₂

#13558

yes

characteristic

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

subgroup

G/H

center

C₂×C₁₆

C₂×D₄

commutator

C₂×C₄

C₂²×C₁₆

Frattini, cosocle

C₄×C₁₆

C₂³

socle

C₂²

C₂×C₂²⋊C₁₆

prime radical

C₂³

C₂²⋊C₁₆

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
4. Q₈⋊C₃₂	C₂	semidirect
115. Q₈	M₆₄	semidirect

H ≅ G/N

type

4. Q₈⋊C₃₂

C₂

semidirect

115. Q₈

M₆₄

semidirect

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

subgroup	index
C₂×Q₈×C₁₆	2
#13627	2
C₄×M₆₄	2
Q₈⋊C₃₂	2

subgroup

index

C₂×Q₈×C₁₆

#13627

C₄×M₆₄

Q₈⋊C₃₂

Subgroups — #150077

Conjugacy classes of subgroups

Complements

Maximal subgroups