Group #7623950 — subgroups

Subgroups — C₂×#33962 #7623950

Order 512.

1370 conjugacy classes of subgroups; only the 562 normal subgroups are shown.

normal subgroups only

This group has 1370 conjugacy classes of subgroups — too many to draw. Only its 562 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×#3395	C₂	1	2	yes	maximal
2	#33962	C₂	1	16	yes	maximal, direct product factor
3	C₂×#2247	C₂	1	3	yes	maximal
4	C₂×#2209	C₂	1	6	yes	maximal
5	C₂×Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₃₂	C₂	1	3	yes	maximal
7	C₂×#557	—	2	2	no	—
8	C₂²×Q₃₂	C₂²	1	3	yes	—
9	C₂²×Q₃₂	C₂²	1	6	yes	—
10	#3395	—	2	8	no	—
11	#3395	C₂²	1	16	yes	—
12	C₂×#494	C₂²	1	6	yes	—
13	C₂×#462	—	2	6	no	—
14	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₁₆	C₂²	1	6	yes	—
16	#2247	C₂²	1	24	yes	—
17	#2209	C₂²	1	48	yes	—
18	Q₈×C₁₆	C₂²	1	8	yes	—
19	C₄×Q₃₂	C₂²	1	24	yes	—
20	C₂×#366	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₁₆	C₂²	1	3	yes	—
22	C₂×#358	C₂²	1	6	yes	—
23	C₂×C₄×C₁₆	C₂²	1	3	yes	—
24	C₂²×Q₁₆	—	2	6	no	—
25	C₂²×Q₁₆	C₂³	1	6	yes	—
26	#557	—	2	16	no	—
27	#557	—	4	8	no	—
28	C₂×C₄⋊Q₈	—	2	6	no	—
29	C₂×#125	—	4	6	no	—
30	C₂×C₄×Q₈	—	2	6	no	—
31	C₂×C₄×Q₈	—	4	2	no	—
32	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
33	C₂×Q₃₂	—	2	12	no	—
34	C₂×Q₃₂	—	4	12	no	—
35	C₂×Q₃₂	C₂³	1	12	yes	—
36	C₂×Q₃₂	C₂³	1	24	yes	—
37	C₂²×C₁₆	C₂³	1	3	yes	—
38	C₂²×C₁₆	C₂³	1	1	yes	characteristic
39	#494	—	2	12	no	—
40	#494	C₂³	1	24	yes	—
41	#462	—	2	48	no	—
42	C₈×Q₈	D₄	1	4	yes	—
43	C₈×Q₈	C₂³	1	4	yes	—
44	C₄×Q₁₆	—	2	12	no	—
45	C₄×Q₁₆	C₂³	1	24	yes	—
46	C₂×#107	C₂³	1	2	yes	—
47	C₂×C₄⋊C₈	D₄	1	3	yes	—
48	C₂×#103	—	2	6	no	—
49	C₂×C₄×C₈	C₂³	1	3	yes	—
50	#366	C₂³	1	4	yes	—
51	C₄⋊C₁₆	C₂³	1	12	yes	—
52	#358	C₂³	1	24	yes	—
53	C₄×C₁₆	C₂³	1	12	yes	—
54	C₂²×Q₈	—	2	6	no	—
55	C₂²×Q₈	D₈	1	1	yes	characteristic
56	C₂×Q₁₆	—	2	6	no	—
57	C₂×Q₁₆	—	2	24	no	—
58	C₂×Q₁₆	—	2	12	no	—
59	C₂×Q₁₆	—	2	12	no	—
60	C₂×Q₁₆	—	8	12	no	—
61	C₂×Q₁₆	C₂⁴	1	12	yes	—
62	C₂²×C₈	C₂×D₄	1	3	yes	—
63	C₂²×C₈	C₂⁴	1	1	yes	characteristic
64	C₄⋊Q₈	—	2	24	no	—
65	C₄⋊Q₈	—	4	12	no	—
66	#125	—	4	48	no	—
67	C₄×Q₈	—	2	24	no	—
68	C₄×Q₈	—	4	16	no	—
69	C₄×Q₈	—	4	12	no	—
70	C₄×Q₈	D₈	1	4	yes	—
71	C₄×Q₈	C₂×D₄	1	4	yes	—
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	—	4	6	no	—
74	C₂×C₄⋊C₄	—	4	6	no	—
75	C₂×C₄⋊C₄	—	4	6	no	—
76	C₂×C₄⋊C₄	D₈	1	3	yes	—
77	C₂×C₄²	—	4	6	no	—
78	C₂×C₄²	C₂×D₄	1	3	yes	—
79	Q₃₂	—	2	24	no	—
80	Q₃₂	—	4	24	no	—
81	C₂×C₁₆	—	2	6	no	—
82	C₂×C₁₆	#41	1	4	yes	—
83	C₂×C₁₆	C₂⁴	1	6	yes	—
84	C₂×C₁₆	C₂⁴	1	2	yes	—
85	#107	—	2	2	no	—
86	#107	C₂⁴	1	4	yes	—
87	C₄⋊C₈	C₂×D₄	1	12	yes	—
88	#103	—	2	24	no	—
89	C₄×C₈	C₂×D₄	1	6	yes	—
90	C₄×C₈	C₂⁴	1	6	yes	—
91	C₂×Q₈	—	2	12	no	—
92	C₂×Q₈	—	4	6	no	—
93	C₂×Q₈	—	4	12	no	—
94	C₂×Q₈	—	4	12	no	—
95	C₂×Q₈	Q₃₂	1	8	yes	complemented
96	C₂×Q₈	C₂×D₈	1	4	yes	—
97	C₂²×C₄	—	4	2	no	—
98	C₂²×C₄	—	4	6	no	—
99	C₂²×C₄	C₂×D₈	1	3	yes	—
100	C₂²×C₄	C₂×D₈	1	3	yes	—
101	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
102	Q₁₆	—	2	12	no	—
103	Q₁₆	—	2	12	no	—
104	Q₁₆	—	4	12	no	—
105	Q₁₆	—	8	24	no	—
106	C₂×C₈	—	2	6	no	—
107	C₂×C₈	C₂²×D₄	1	6	yes	—
108	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₈	C₂×#41	1	2	yes	—
110	C₂×C₈	H(5,𝔽₂)	1	2	yes	—
111	C₂×C₈	C₂⁵	1	1	yes	characteristic
112	C₄⋊C₄	—	2	4	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	4	24	no	—
115	C₄⋊C₄	—	4	24	no	—
116	C₄⋊C₄	—	4	24	no	—
117	C₄⋊C₄	C₂×D₈	1	12	yes	—
118	C₄²	—	4	24	no	—
119	C₄²	C₂×D₈	1	6	yes	—
120	C₄²	C₂²×D₄	1	6	yes	—
121	C₁₆	—	2	6	no	—
122	C₁₆	C₂×#41	1	4	yes	—
123	C₂³	C₂²×D₈	1	1	yes	characteristic
124	Q₈	—	4	12	no	—
125	Q₈	—	4	12	no	—
126	Q₈	—	8	12	no	—
127	Q₈	C₂×Q₃₂	1	16	yes	complemented
128	C₂×C₄	—	2	6	no	—
129	C₂×C₄	—	4	4	no	—
130	C₂×C₄	—	4	12	no	—
131	C₂×C₄	—	8	12	no	—
132	C₂×C₄	—	8	4	no	—
133	C₂×C₄	C₂×Q₃₂	1	12	yes	—
134	C₂×C₄	#550	1	2	yes	—
135	C₂×C₄	C₂²×D₈	1	6	yes	—
136	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
137	C₂×C₄	C₂²×D₈	1	6	yes	—
138	C₂×C₄	#576	1	2	yes	—
139	C₂×C₄	C₂³×D₄	1	1	yes	characteristic
140	C₈	—	2	6	no	—
141	C₈	#550	1	2	yes	—
142	C₈	C₂²×#41	1	1	yes	characteristic
143	C₈	C₂×H(5,𝔽₂)	1	1	yes	characteristic
144	C₂²	#3391	1	2	yes	—
145	C₂²	C₂²×Q₃₂	1	2	yes	—
146	C₂²	#3417	1	2	yes	—
147	C₂²	C₂³×D₈	1	1	yes	characteristic
148	C₄	—	2	6	no	—
149	C₄	—	8	4	no	—
150	C₄	—	8	12	no	—
151	C₄	#3391	1	2	yes	—
152	C₄	C₂²×Q₃₂	1	12	yes	—
153	C₄	C₂×#550	1	1	yes	characteristic
154	C₄	C₂×#576	1	1	yes	characteristic
155	C₂	#33962	1	4	yes	direct product factor
156	C₂	C₂×#3391	1	1	yes	characteristic
157	C₂	C₂³×Q₃₂	1	1	yes	characteristic
158	C₂	C₂×#3417	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#3395

C₂

yes

maximal

#33962

C₂

yes

maximal, direct product factor

C₂×#2247

C₂

yes

maximal

C₂×#2209

C₂

yes

maximal

C₂×Q₈×C₁₆

C₂

yes

maximal, characteristic

C₂×C₄×Q₃₂

C₂

yes

maximal

C₂×#557

—

C₂²×Q₃₂

C₂²

yes

—

C₂²×Q₃₂

C₂²

yes

—

#3395

—

#3395

C₂²

yes

—

C₂×#494

C₂²

yes

—

C₂×#462

—

C₂×C₈×Q₈

C₂²

yes

characteristic

C₂×C₄×Q₁₆

C₂²

yes

—

#2247

C₂²

yes

—

#2209

C₂²

yes

—

Q₈×C₁₆

C₂²

yes

—

C₄×Q₃₂

C₂²

yes

—

C₂×#366

C₂²

yes

characteristic

C₂×C₄⋊C₁₆

C₂²

yes

—

C₂×#358

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

—

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂³

yes

—

#557

—

#557

—

C₂×C₄⋊Q₈

—

C₂×#125

—

C₂×C₄×Q₈

—

C₂×C₄×Q₈

—

C₂×C₄×Q₈

D₄

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂³

yes

—

C₂×Q₃₂

C₂³

yes

—

C₂²×C₁₆

C₂³

yes

—

C₂²×C₁₆

C₂³

yes

characteristic

#494

—

#494

C₂³

yes

—

#462

—

C₈×Q₈

D₄

yes

—

C₈×Q₈

C₂³

yes

—

C₄×Q₁₆

—

C₄×Q₁₆

C₂³

yes

—

C₂×#107

C₂³

yes

—

C₂×C₄⋊C₈

D₄

yes

—

C₂×#103

—

C₂×C₄×C₈

C₂³

yes

—

#366

C₂³

yes

—

C₄⋊C₁₆

C₂³

yes

—

#358

C₂³

yes

—

C₄×C₁₆

C₂³

yes

—

C₂²×Q₈

—

C₂²×Q₈

D₈

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂⁴

yes

—

C₂²×C₈

C₂×D₄

yes

—

C₂²×C₈

C₂⁴

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

D₈

yes

—

C₄×Q₈

C₂×D₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₈

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

#41

yes

—

C₂×C₁₆

C₂⁴

yes

—

C₂×C₁₆

C₂⁴

yes

—

#107

—

#107

C₂⁴

yes

—

C₄⋊C₈

C₂×D₄

yes

—

#103

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂⁴

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Q₃₂

yes

complemented

C₂×Q₈

C₂×D₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

—

100

C₂²×C₄

C₂×D₈

yes

—

101

C₂²×C₄

C₂²×D₄

yes

characteristic

102

Q₁₆

—

103

Q₁₆

—

104

Q₁₆

—

105

Q₁₆

—

106

C₂×C₈

—

107

C₂×C₈

C₂²×D₄

yes

—

108

C₂×C₈

C₂²×D₄

yes

characteristic

109

C₂×C₈

C₂×#41

yes

—

110

C₂×C₈

H(5,𝔽₂)

yes

—

111

C₂×C₈

C₂⁵

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

C₂×D₈

yes

—

118

C₄²

—

119

C₄²

C₂×D₈

yes

—

120

C₄²

C₂²×D₄

yes

—

121

C₁₆

—

122

C₁₆

C₂×#41

yes

—

123

C₂³

C₂²×D₈

yes

characteristic

124

Q₈

—

125

Q₈

—

126

Q₈

—

127

Q₈

C₂×Q₃₂

yes

complemented

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂×Q₃₂

yes

—

134

C₂×C₄

#550

yes

—

135

C₂×C₄

C₂²×D₈

yes

—

136

C₂×C₄

C₂²×D₈

yes

characteristic

137

C₂×C₄

C₂²×D₈

yes

—

138

C₂×C₄

#576

yes

—

139

C₂×C₄

C₂³×D₄

yes

characteristic

140

C₈

—

141

C₈

#550

yes

—

142

C₈

C₂²×#41

yes

characteristic

143

C₈

C₂×H(5,𝔽₂)

yes

characteristic

144

C₂²

#3391

yes

—

145

C₂²

C₂²×Q₃₂

yes

—

146

C₂²

#3417

yes

—

147

C₂²

C₂³×D₈

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#3391

yes

—

152

C₄

C₂²×Q₃₂

yes

—

153

C₄

C₂×#550

yes

characteristic

154

C₄

C₂×#576

yes

characteristic

155

C₂

#33962

yes

direct product factor

156

C₂

C₂×#3391

yes

characteristic

157

C₂

C₂³×Q₃₂

yes

characteristic

158

C₂

C₂×#3417

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂²×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁵

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #33962	C₂	direct (N × H)
95. C₂×Q₈	Q₃₂	semidirect
127. Q₈	C₂×Q₃₂	semidirect
155. C₂	#33962	direct (N × H)

H ≅ G/N

type

2. #33962

C₂

direct (N × H)

95. C₂×Q₈

Q₃₂

semidirect

127. Q₈

C₂×Q₃₂

semidirect

155. C₂

#33962

direct (N × H)

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

subgroup	index
C₂×#3395	2
#33962	2
C₂×#2247	2
C₂×#2209	2
C₂×Q₈×C₁₆	2
C₂×C₄×Q₃₂	2

subgroup

index

C₂×Q₈×C₁₆

C₂×C₄×Q₃₂

Subgroups — C₂×#33962 #7623950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#33962 #7623950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#33962 #7623950