Group #9677 — subgroups

Subgroups — #9677

Order 256.

276 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 97 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×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#1767	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊C₈	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈	C₂²	1	2	yes	—
7	C₂×C₄⋊C₈	—	2	1	no	—
8	C₂×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×#97	—	2	1	no	—
10	C₂×C₄×C₈	C₂²	1	2	yes	—
11	#391	C₄	1	2	yes	—
12	#389	—	2	1	no	—
13	#389	—	2	1	no	—
14	#384	C₄	1	1	yes	characteristic
15	C₄×C₄⋊C₄	—	2	1	no	—
16	C₄×C₄⋊C₄	—	2	1	no	—
17	C₄³	C₂²	1	1	yes	characteristic
18	#336	—	2	1	no	—
19	#336	—	2	1	no	—
20	Q₈⋊C₈	—	2	8	no	—
21	C₂²×Q₈	C₂×C₄	1	2	yes	—
22	C₂²×C₈	—	2	2	no	—
23	C₂²×C₈	—	2	2	no	—
24	C₄⋊Q₈	—	2	4	no	—
25	C₄×Q₈	—	2	4	no	—
26	C₄×Q₈	—	2	4	no	—
27	C₂×C₄⋊C₄	—	2	1	no	—
28	C₂×C₄⋊C₄	—	2	1	no	—
29	C₂×C₄⋊C₄	—	2	1	no	—
30	C₂×C₄⋊C₄	—	2	1	no	—
31	C₂×C₄⋊C₄	—	2	2	no	—
32	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₂×C₄²	—	2	2	no	—
34	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²	D₄	1	1	yes	characteristic
37	C₂×C₄²	D₄	1	1	yes	characteristic
38	C₂×C₄²	D₄	1	1	yes	characteristic
39	C₂×C₄²	D₄	1	1	yes	characteristic
40	C₂×C₄²	C₂³	1	1	yes	characteristic
41	C₄⋊C₈	—	2	4	no	—
42	C₄⋊C₈	—	2	2	no	—
43	C₄⋊C₈	D₄	1	4	yes	—
44	#97	—	4	2	no	—
45	C₄×C₈	—	2	4	no	—
46	#95	—	2	2	no	—
47	#95	—	2	2	no	—
48	C₂×Q₈	—	2	2	no	—
49	C₂×Q₈	—	2	4	no	—
50	C₂×Q₈	—	2	4	no	—
51	C₂×Q₈	M₁₆	1	4	yes	—
52	C₂²×C₄	—	2	2	no	—
53	C₂²×C₄	—	2	2	no	—
54	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈	—	2	4	no	—
62	C₂×C₈	—	2	4	no	—
63	C₂×C₈	—	4	2	no	—
64	C₂×C₈	—	4	2	no	—
65	C₂×C₈	—	4	2	no	—
66	C₂×C₈	—	4	2	no	—
67	C₄⋊C₄	—	2	4	no	—
68	C₄⋊C₄	—	2	4	no	—
69	C₄⋊C₄	—	2	4	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄⋊C₄	—	4	2	no	—
72	C₄⋊C₄	—	4	2	no	—
73	C₄⋊C₄	—	4	2	no	—
74	C₄²	—	2	2	no	—
75	C₄²	—	2	2	no	—
76	C₄²	—	2	2	no	—
77	C₄²	—	2	2	no	—
78	C₄²	—	4	1	no	—
79	C₄²	—	4	4	no	—
80	C₄²	SD₁₆	1	2	yes	—
81	C₄²	Q₁₆	1	2	yes	—
82	C₄²	C₂×D₄	1	2	yes	—
83	C₄²	#41	1	2	yes	—
84	C₂³	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	Q₈	—	2	8	no	—
86	Q₈	—	4	4	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	#103	1	2	yes	—
105	C₂×C₄	#103	1	2	yes	—
106	C₂×C₄	#104	1	2	yes	—
107	C₂×C₄	C₄×D₄	1	2	yes	—
108	C₂×C₄	#121	1	2	yes	—
109	C₂×C₄	#124	1	1	yes	characteristic
110	C₂×C₄	#127	1	1	yes	characteristic
111	C₂×C₄	C₂×M₁₆	1	2	yes	—
112	C₂×C₄	#131	1	2	yes	—
113	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
114	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
115	C₂×C₄	#137	1	2	yes	—
116	C₈	—	4	4	no	—
117	C₈	—	4	4	no	—
118	C₂²	#390	1	1	yes	characteristic
119	C₂²	#407	1	1	yes	characteristic
120	C₂²	#408	1	1	yes	characteristic
121	C₂²	C₂×#103	1	1	yes	characteristic
122	C₂²	#419	1	1	yes	characteristic
123	C₂²	C₂×#104	1	1	yes	characteristic
124	C₂²	#421	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	2	2	no	—
128	C₄	—	2	2	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	4	no	—
131	C₄	—	4	1	no	—
132	C₄	—	8	2	no	—
133	C₄	#435	1	4	yes	—
134	C₄	#461	1	2	yes	—
135	C₄	#462	1	2	yes	—
136	C₂	#1487	1	1	yes	characteristic
137	C₂	#1491	1	1	yes	characteristic
138	C₂	#1493	1	1	yes	characteristic
139	C₂	#1497	1	1	yes	characteristic
140	C₂	#1955	1	1	yes	characteristic
141	C₂	#1980	1	1	yes	characteristic
142	C₂	#1981	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₄⋊Q₈

C₂

yes

maximal, characteristic

#1767

C₂

yes

maximal, characteristic

C₄×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂×Q₈⋊C₈

C₂

yes

maximal

C₂×C₄⋊Q₈

C₄

yes

characteristic

C₂×C₄×Q₈

C₂²

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×#97

—

C₂×C₄×C₈

C₂²

yes

—

#391

C₄

yes

—

#389

—

#389

—

#384

C₄

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₄³

C₂²

yes

characteristic

#336

—

#336

—

Q₈⋊C₈

—

C₂²×Q₈

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

—

#97

—

C₄×C₈

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

M₁₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

SD₁₆

yes

—

C₄²

Q₁₆

yes

—

C₄²

C₂×D₄

yes

—

C₄²

#41

yes

—

C₂³

C₂×C₂²⋊C₄

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

#103

yes

—

105

C₂×C₄

#103

yes

—

106

C₂×C₄

#104

yes

—

107

C₂×C₄

C₄×D₄

yes

—

108

C₂×C₄

#121

yes

—

109

C₂×C₄

#124

yes

characteristic

110

C₂×C₄

#127

yes

characteristic

111

C₂×C₄

C₂×M₁₆

yes

—

112

C₂×C₄

#131

yes

—

113

C₂×C₄

C₂×SD₁₆

yes

characteristic

114

C₂×C₄

C₂×Q₁₆

yes

characteristic

115

C₂×C₄

#137

yes

—

116

C₈

—

117

C₈

—

118

C₂²

#390

yes

characteristic

119

C₂²

#407

yes

characteristic

120

C₂²

#408

yes

characteristic

121

C₂²

C₂×#103

yes

characteristic

122

C₂²

#419

yes

characteristic

123

C₂²

C₂×#104

yes

characteristic

124

C₂²

#421

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#435

yes

—

134

C₄

#461

yes

—

135

C₄

#462

yes

—

136

C₂

#1487

yes

characteristic

137

C₂

#1491

yes

characteristic

138

C₂

#1493

yes

characteristic

139

C₂

#1497

yes

characteristic

140

C₂

#1955

yes

characteristic

141

C₂

#1980

yes

characteristic

142

C₂

#1981

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₄

commutator

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

socle, prime radical

C₂³

C₂×C₂²⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

subgroup	index
C₄×C₄⋊Q₈	2
#1767	2
C₄×C₄⋊C₈	2
C₂×Q₈⋊C₈	2

subgroup

index

C₄×C₄⋊Q₈

#1767

C₄×C₄⋊C₈

C₂×Q₈⋊C₈

Subgroups — #9677

Conjugacy classes of subgroups

Maximal subgroups