Group #9736 — subgroups

Subgroups — #9736

Order 256.

235 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 87 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×#487	C₂	1	1	yes	maximal, characteristic
2	#1988	C₂	1	1	yes	maximal, characteristic
3	#1981	C₂	1	1	yes	maximal, characteristic
4	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₂×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×#332	C₂	1	1	yes	maximal, characteristic
7	C₂×#332	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	#487	—	2	2	no	—
11	#487	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×#103	—	2	1	no	—
16	C₂×#103	—	2	1	no	—
17	C₂×#103	—	2	2	no	—
18	C₂×C₄×C₈	—	2	1	no	—
19	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	#340	—	2	2	no	—
22	#336	—	2	1	no	—
23	#334	—	2	2	no	—
24	#332	—	2	2	no	—
25	#332	—	2	2	no	—
26	C₂²×Q₈	—	2	1	no	—
27	C₂²×Q₈	—	2	1	no	—
28	C₂²×C₈	—	2	1	no	—
29	C₂²×C₈	—	2	1	no	—
30	C₂²×C₈	—	2	1	no	—
31	C₂²×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₈	D₄	1	1	yes	characteristic
33	C₄⋊Q₈	—	2	1	no	—
34	C₄⋊Q₈	—	2	1	no	—
35	C₄⋊Q₈	—	2	1	no	—
36	C₄⋊Q₈	—	2	1	no	—
37	C₄⋊Q₈	—	2	1	no	—
38	C₄⋊Q₈	—	2	1	no	—
39	C₄⋊Q₈	C₂×C₄	1	2	yes	—
40	C₄⋊Q₈	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄	—	2	1	no	—
42	C₂×C₄⋊C₄	—	2	1	no	—
43	C₂×C₄⋊C₄	—	2	1	no	—
44	C₂×C₄⋊C₄	—	2	1	no	—
45	C₂×C₄⋊C₄	D₄	1	2	yes	—
46	C₂×C₄²	—	2	1	no	—
47	C₂×C₄²	C₂³	1	1	yes	characteristic
48	C₄⋊C₈	—	2	1	no	—
49	C₄⋊C₈	—	2	1	no	—
50	C₄⋊C₈	—	2	1	no	—
51	C₄⋊C₈	—	2	1	no	—
52	C₄⋊C₈	—	2	1	no	—
53	C₄⋊C₈	D₄	1	2	yes	—
54	#103	—	2	4	no	—
55	#103	—	2	4	no	—
56	#103	—	4	4	no	—
57	C₄×C₈	—	4	2	no	—
58	C₄×C₈	C₂×C₄	1	2	yes	—
59	C₄×C₈	D₄	1	2	yes	—
60	#95	—	2	1	no	—
61	C₂×Q₈	—	2	2	no	—
62	C₂×Q₈	—	2	2	no	—
63	C₂×Q₈	—	4	1	no	—
64	C₂×Q₈	—	4	1	no	—
65	C₂×Q₈	—	4	2	no	—
66	C₂×Q₈	—	4	1	no	—
67	C₂×Q₈	—	4	1	no	—
68	C₂×Q₈	—	4	2	no	—
69	C₂×Q₈	—	4	1	no	—
70	C₂×Q₈	—	4	1	no	—
71	C₂²×C₄	—	2	2	no	—
72	C₂²×C₄	—	4	1	no	—
73	C₂²×C₄	—	4	1	no	—
74	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄	#41	1	1	yes	characteristic
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	4	1	no	—
83	C₂×C₈	—	4	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	4	1	no	—
86	C₂×C₈	—	4	2	no	—
87	C₂×C₈	—	4	2	no	—
88	C₂×C₈	C₂²⋊C₄	1	2	yes	—
89	C₂×C₈	C₂²⋊C₄	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	4	1	no	—
96	C₄⋊C₄	—	4	1	no	—
97	C₄⋊C₄	—	4	1	no	—
98	C₄⋊C₄	—	4	1	no	—
99	C₄⋊C₄	SD₁₆	1	4	yes	—
100	C₄²	—	4	1	no	—
101	C₄²	—	4	1	no	—
102	C₄²	C₂²×C₄	1	1	yes	characteristic
103	C₄²	C₂×D₄	1	1	yes	characteristic
104	C₄²	C₂×D₄	1	1	yes	characteristic
105	C₄²	#41	1	1	yes	characteristic
106	C₂³	#121	1	1	yes	characteristic
107	Q₈	—	4	2	no	—
108	Q₈	—	4	2	no	—
109	Q₈	—	4	4	no	—
110	Q₈	—	8	1	no	—
111	Q₈	—	8	1	no	—
112	Q₈	—	8	1	no	—
113	Q₈	—	8	1	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	8	1	no	—
121	C₂×C₄	—	8	1	no	—
122	C₂×C₄	—	8	2	no	—
123	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄	#121	1	1	yes	characteristic
127	C₂×C₄	#124	1	1	yes	characteristic
128	C₂×C₄	#127	1	1	yes	characteristic
129	C₂×C₄	C₂×SD₁₆	1	2	yes	—
130	C₂×C₄	#135	1	1	yes	characteristic
131	C₂×C₄	#135	1	1	yes	characteristic
132	C₂×C₄	#135	1	2	yes	—
133	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄	#137	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	2	2	no	—
137	C₈	—	4	2	no	—
138	C₈	—	8	1	no	—
139	C₈	—	8	1	no	—
140	C₂²	#390	1	1	yes	characteristic
141	C₂²	#463	1	1	yes	characteristic
142	C₂²	#464	1	1	yes	characteristic
143	C₂²	#465	1	2	yes	—
144	C₂²	#473	1	2	yes	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	4	4	no	—
148	C₄	—	8	2	no	—
149	C₄	—	8	2	no	—
150	C₄	#416	1	1	yes	characteristic
151	C₄	#417	1	1	yes	characteristic
152	C₄	C₄×SD₁₆	1	2	yes	—
153	C₄	#463	1	1	yes	characteristic
154	C₄	#464	1	1	yes	characteristic
155	C₄	#492	1	1	yes	characteristic
156	C₄	#495	1	1	yes	characteristic
157	C₂	#1679	1	2	yes	—
158	C₂	#1680	1	2	yes	—
159	C₂	#1968	1	1	yes	characteristic
160	C₂	#1979	1	1	yes	characteristic
161	C₂	#1983	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#487

C₂

yes

maximal, characteristic

#1988

C₂

yes

maximal, characteristic

#1981

C₂

yes

maximal, characteristic

C₈×C₄⋊C₄

C₂

yes

maximal, characteristic

C₂×#334

C₂

yes

maximal, characteristic

C₂×#332

C₂

yes

maximal, characteristic

C₂×#332

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₄⋊Q₈

C₂²

yes

characteristic

#487

—

#487

C₄

yes

complemented

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

—

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

—

—

—

—

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×C₄

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

—

#103

—

#103

—

#103

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

SD₁₆

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂²×C₄

yes

characteristic

103

C₄²

C₂×D₄

yes

characteristic

104

C₄²

C₂×D₄

yes

characteristic

105

C₄²

#41

yes

characteristic

106

C₂³

#121

yes

characteristic

107

Q₈

—

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

124

C₂×C₄

C₄×D₄

yes

characteristic

125

C₂×C₄

C₄×D₄

yes

characteristic

126

C₂×C₄

#121

yes

characteristic

127

C₂×C₄

#124

yes

characteristic

128

C₂×C₄

#127

yes

characteristic

129

C₂×C₄

C₂×SD₁₆

yes

—

130

C₂×C₄

#135

yes

characteristic

131

C₂×C₄

#135

yes

characteristic

132

C₂×C₄

#135

yes

—

133

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

134

C₂×C₄

#137

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₂²

#390

yes

characteristic

141

C₂²

#463

yes

characteristic

142

C₂²

#464

yes

characteristic

143

C₂²

#465

yes

—

144

C₂²

#473

yes

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#416

yes

characteristic

151

C₄

#417

yes

characteristic

152

C₄

C₄×SD₁₆

yes

—

153

C₄

#463

yes

characteristic

154

C₄

#464

yes

characteristic

155

C₄

#492

yes

characteristic

156

C₄

#495

yes

characteristic

157

C₂

#1679

yes

—

158

C₂

#1680

yes

—

159

C₂

#1968

yes

characteristic

160

C₂

#1979

yes

characteristic

161

C₂

#1983

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#121
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#121

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
11. #487	C₄	semidirect

H ≅ G/N

type

11. #487

C₄

semidirect

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

subgroup	index
C₂×#487	2
#1988	2
#1981	2
C₈×C₄⋊C₄	2
C₂×#334	2
C₂×#332	2

subgroup

index

C₈×C₄⋊C₄

Subgroups — #9736

Conjugacy classes of subgroups

Complements

Maximal subgroups