Group #9671 — subgroups

Subgroups — #9671

Order 256.

217 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 73 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×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#1564	C₂	1	1	yes	maximal, characteristic
3	#1479	C₂	1	2	yes	maximal
4	#1461	C₂	1	2	yes	maximal
5	#1452	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈	C₂²	1	2	yes	—
8	C₄×Q₁₆	—	2	1	no	—
9	C₄×Q₁₆	—	2	1	no	—
10	C₄×Q₁₆	—	2	1	no	—
11	C₄×Q₁₆	—	2	1	no	—
12	#431	C₂²	1	2	yes	—
13	#431	C₂²	1	1	yes	characteristic
14	#431	C₂²	1	1	yes	characteristic
15	C₂×#107	C₄	1	1	yes	characteristic
16	C₂×#103	C₄	1	2	yes	—
17	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
18	#335	—	2	1	no	—
19	#334	—	2	1	no	—
20	Q₈⋊C₈	—	2	4	no	—
21	#324	—	2	4	no	—
22	#322	—	2	1	no	—
23	#322	—	2	1	no	—
24	C₂²×Q₈	C₂×C₄	1	2	yes	—
25	C₂×Q₁₆	—	2	4	no	—
26	C₂×Q₁₆	—	2	1	no	—
27	C₂×Q₁₆	—	2	1	no	—
28	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
30	C₄×Q₈	—	2	2	no	—
31	C₄×Q₈	—	2	2	no	—
32	C₄×Q₈	—	2	2	no	—
33	C₄×Q₈	—	2	2	no	—
34	C₂×C₄⋊C₄	—	2	2	no	—
35	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₂×C₄²	—	2	2	no	—
37	C₂×C₄²	C₂³	1	1	yes	characteristic
38	#107	—	2	1	no	—
39	#107	—	2	1	no	—
40	C₄⋊C₈	—	2	2	no	—
41	C₄⋊C₈	—	2	1	no	—
42	C₄⋊C₈	D₄	1	2	yes	—
43	#103	—	2	2	no	—
44	#103	—	2	2	no	—
45	C₂²⋊C₈	—	2	1	no	—
46	C₂²⋊C₈	—	2	2	no	—
47	C₂²⋊C₈	D₄	1	2	yes	—
48	C₄×C₈	—	2	1	no	—
49	C₄×C₈	—	2	1	no	—
50	C₄×C₈	—	2	2	no	—
51	C₄×C₈	D₄	1	1	yes	characteristic
52	C₄×C₈	D₄	1	1	yes	characteristic
53	C₄×C₈	D₄	1	1	yes	characteristic
54	C₄×C₈	D₄	1	1	yes	characteristic
55	C₂×Q₈	—	2	2	no	—
56	C₂×Q₈	—	2	2	no	—
57	C₂×Q₈	—	2	2	no	—
58	C₂×Q₈	—	2	2	no	—
59	C₂×Q₈	—	2	2	no	—
60	C₂×Q₈	M₁₆	1	4	yes	—
61	C₂²×C₄	—	2	2	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄	#41	1	1	yes	characteristic
66	Q₁₆	—	2	4	no	—
67	Q₁₆	—	2	4	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	4	2	no	—
73	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	2	2	no	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄⋊C₄	—	4	2	no	—
81	C₄²	—	4	2	no	—
82	C₄²	—	4	2	no	—
83	C₄²	C₂×D₄	1	1	yes	characteristic
84	C₄²	C₂×D₄	1	1	yes	characteristic
85	C₄²	C₂×D₄	1	1	yes	characteristic
86	C₄²	#41	1	1	yes	characteristic
87	C₂³	C₄×D₄	1	1	yes	characteristic
88	Q₈	—	2	4	no	—
89	Q₈	—	2	4	no	—
90	Q₈	—	4	2	no	—
91	Q₈	—	4	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	8	2	no	—
99	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄	#121	1	1	yes	characteristic
102	C₂×C₄	#121	1	1	yes	characteristic
103	C₂×C₄	#124	1	1	yes	characteristic
104	C₂×C₄	#127	1	1	yes	characteristic
105	C₂×C₄	C₂×M₁₆	1	2	yes	—
106	C₂×C₄	#131	1	2	yes	—
107	C₈	—	2	1	no	—
108	C₈	—	2	1	no	—
109	C₈	—	2	1	no	—
110	C₈	—	2	1	no	—
111	C₈	—	4	2	no	—
112	C₈	—	4	2	no	—
113	C₈	—	8	2	no	—
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	#390	1	1	yes	characteristic
117	C₂²	#435	1	2	yes	—
118	C₄	—	2	1	no	—
119	C₄	—	2	1	no	—
120	C₄	—	2	1	no	—
121	C₄	—	4	4	no	—
122	C₄	—	8	2	no	—
123	C₄	#407	1	1	yes	characteristic
124	C₄	#408	1	1	yes	characteristic
125	C₄	#435	1	2	yes	—
126	C₄	#473	1	2	yes	—
127	C₂	—	2	2	no	—
128	C₂	#1955	1	1	yes	characteristic
129	C₂	#1976	1	1	yes	characteristic
130	C₂	#1979	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Q₁₆

C₂

yes

maximal, characteristic

#1564

C₂

yes

maximal, characteristic

#1479

C₂

yes

maximal

#1461

C₂

yes

maximal

#1452

C₂

yes

maximal, characteristic

C₂²×Q₁₆

C₄

yes

characteristic

C₂×C₄×Q₈

C₂²

yes

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₄×Q₁₆

—

#431

C₂²

yes

—

#431

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

C₂×#107

C₄

yes

characteristic

C₂×#103

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#335

—

#334

—

Q₈⋊C₈

—

#324

—

#322

—

#322

—

C₂²×Q₈

C₂×C₄

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄

yes

—

#103

—

#103

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

D₄

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

M₁₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

C₄×D₄

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

100

C₂×C₄

C₄×D₄

yes

characteristic

101

C₂×C₄

#121

yes

characteristic

102

C₂×C₄

#121

yes

characteristic

103

C₂×C₄

#124

yes

characteristic

104

C₂×C₄

#127

yes

characteristic

105

C₂×C₄

C₂×M₁₆

yes

—

106

C₂×C₄

#131

yes

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₂²

—

115

C₂²

—

116

C₂²

#390

yes

characteristic

117

C₂²

#435

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#407

yes

characteristic

124

C₄

#408

yes

characteristic

125

C₄

#435

yes

—

126

C₄

#473

yes

—

127

C₂

—

128

C₂

#1955

yes

characteristic

129

C₂

#1976

yes

characteristic

130

C₂

#1979

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#121
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#390
prime radical	C₂³	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

C₂×C₄

C₂²×C₄

C₂×C₄²

C₂³

socle

C₂²

#390

prime radical

C₂³

C₄×D₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ 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
C₂×C₄×Q₁₆	2
#1564	2
#1479	2
#1461	2
#1452	2

subgroup

index

C₂×C₄×Q₁₆

Subgroups — #9671

Conjugacy classes of subgroups

Maximal subgroups