Group #9656 — subgroups

Subgroups — #9656

Order 256.

251 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 251 conjugacy classes of subgroups — too many to draw. Only its 103 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	#1984	C₂	1	2	yes	maximal
3	C₂×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊C₈	C₂	1	2	yes	maximal
5	C₂×C₈²	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	2	no	—
9	C₄×Q₁₆	—	2	2	no	—
10	C₂×#107	C₄	1	1	yes	characteristic
11	C₂×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×#103	C₄	1	2	yes	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	#336	—	2	4	no	—
17	#335	—	2	2	no	—
18	Q₈⋊C₈	—	2	4	no	—
19	C₈²	—	2	2	no	—
20	C₂²×Q₈	C₂×C₄	1	2	yes	—
21	C₂×Q₁₆	—	2	2	no	—
22	C₂×Q₁₆	—	2	2	no	—
23	C₂×Q₁₆	C₈	1	4	yes	complemented
24	C₂²×C₈	—	2	1	no	—
25	C₂²×C₈	—	2	2	no	—
26	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈	D₄	1	2	yes	—
29	C₄×Q₈	—	2	2	no	—
30	C₄×Q₈	—	2	2	no	—
31	C₄×Q₈	—	2	2	no	—
32	C₄×Q₈	—	2	2	no	—
33	C₂×C₄⋊C₄	—	2	2	no	—
34	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₂×C₄²	—	2	2	no	—
36	C₂×C₄²	C₂³	1	1	yes	characteristic
37	#107	—	2	2	no	—
38	C₄⋊C₈	—	2	2	no	—
39	C₄⋊C₈	—	2	2	no	—
40	#103	—	2	4	no	—
41	C₄×C₈	—	2	1	no	—
42	C₄×C₈	—	2	1	no	—
43	C₄×C₈	—	2	1	no	—
44	C₄×C₈	D₄	1	2	yes	—
45	C₄×C₈	D₄	1	2	yes	—
46	C₄×C₈	D₄	1	2	yes	—
47	C₂×Q₈	—	2	2	no	—
48	C₂×Q₈	—	2	2	no	—
49	C₂×Q₈	—	2	2	no	—
50	C₂×Q₈	—	2	2	no	—
51	C₂×Q₈	—	2	2	no	—
52	C₂×Q₈	C₂×C₈	1	4	yes	—
53	C₂²×C₄	—	2	2	no	—
54	C₂²×C₄	—	2	2	no	—
55	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄	#41	1	1	yes	characteristic
58	Q₁₆	—	2	8	no	—
59	C₂×C₈	—	2	4	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	1	no	—
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	4	2	no	—
66	C₂×C₈	—	4	4	no	—
67	C₂×C₈	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈	C₂×C₈	1	2	yes	—
70	C₂×C₈	M₁₆	1	2	yes	—
71	C₂×C₈	Q₁₆	1	4	yes	complemented
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	4	2	no	—
75	C₄⋊C₄	—	4	2	no	—
76	C₄²	—	4	2	no	—
77	C₄²	—	4	2	no	—
78	C₄²	C₂×D₄	1	1	yes	characteristic
79	C₄²	C₂×D₄	1	1	yes	characteristic
80	C₄²	C₂×D₄	1	1	yes	characteristic
81	C₄²	#41	1	1	yes	characteristic
82	C₂³	C₄×D₄	1	1	yes	characteristic
83	Q₈	—	2	4	no	—
84	Q₈	—	2	4	no	—
85	Q₈	—	4	2	no	—
86	Q₈	—	4	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	4	4	no	—
93	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄	#121	1	1	yes	characteristic
96	C₂×C₄	#121	1	1	yes	characteristic
97	C₂×C₄	#124	1	1	yes	characteristic
98	C₂×C₄	#127	1	1	yes	characteristic
99	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄	#131	1	1	yes	characteristic
102	C₂×C₄	#131	1	1	yes	characteristic
103	C₂×C₄	C₂×Q₁₆	1	2	yes	—
104	C₂×C₄	#135	1	2	yes	—
105	C₈	—	2	4	no	—
106	C₈	—	2	2	no	—
107	C₈	—	2	4	no	—
108	C₈	—	8	2	no	—
109	C₈	C₂²⋊C₈	1	4	yes	—
110	C₂²	#390	1	1	yes	characteristic
111	C₂²	C₈×D₄	1	1	yes	characteristic
112	C₂²	#436	1	1	yes	characteristic
113	C₂²	C₄×Q₁₆	1	2	yes	—
114	C₂²	#443	1	2	yes	—
115	C₄	—	2	1	no	—
116	C₄	—	2	1	no	—
117	C₄	—	4	4	no	—
118	C₄	—	8	2	no	—
119	C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₄	#408	1	1	yes	characteristic
121	C₄	C₈×D₄	1	1	yes	characteristic
122	C₄	#436	1	1	yes	characteristic
123	C₄	#467	1	2	yes	—
124	C₄	#494	1	1	yes	characteristic
125	C₄	#495	1	1	yes	characteristic
126	C₂	C₈×Q₁₆	1	2	yes	—
127	C₂	#1591	1	2	yes	—
128	C₂	#1954	1	1	yes	characteristic
129	C₂	#1969	1	1	yes	characteristic
130	C₂	#1974	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Q₁₆

C₂

yes

maximal, characteristic

#1984

C₂

yes

maximal

C₂×#335

C₂

yes

maximal, characteristic

C₂×Q₈⋊C₈

C₂

yes

maximal

C₂×C₈²

C₂

yes

maximal, characteristic

C₂²×Q₁₆

C₄

yes

characteristic

C₂×C₄×Q₈

C₂²

yes

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₂×#107

C₄

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

C₂×#103

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#336

—

#335

—

Q₈⋊C₈

—

C₈²

—

C₂²×Q₈

C₂×C₄

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₈

yes

complemented

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

D₄

yes

—

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

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

Q₁₆

—

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

—

C₂×C₈

yes

—

C₂×C₈

M₁₆

yes

—

C₂×C₈

Q₁₆

yes

complemented

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×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²×C₈

yes

characteristic

100

C₂×C₄

C₂×M₁₆

yes

characteristic

101

C₂×C₄

#131

yes

characteristic

102

C₂×C₄

#131

yes

characteristic

103

C₂×C₄

C₂×Q₁₆

yes

—

104

C₂×C₄

#135

yes

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

C₂²⋊C₈

yes

—

110

C₂²

#390

yes

characteristic

111

C₂²

C₈×D₄

yes

characteristic

112

C₂²

#436

yes

characteristic

113

C₂²

C₄×Q₁₆

yes

—

114

C₂²

#443

yes

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

C₂×C₂²⋊C₈

yes

characteristic

120

C₄

#408

yes

characteristic

121

C₄

C₈×D₄

yes

characteristic

122

C₄

#436

yes

characteristic

123

C₄

#467

yes

—

124

C₄

#494

yes

characteristic

125

C₄

#495

yes

characteristic

126

C₂

C₈×Q₁₆

yes

—

127

C₂

#1591

yes

—

128

C₂

#1954

yes

characteristic

129

C₂

#1969

yes

characteristic

130

C₂

#1974

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₄

commutator

C₂×C₄

C₂²×C₈

Frattini, cosocle

C₂×C₄²

C₂³

socle, prime radical

C₂³

C₄×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
23. C₂×Q₁₆	C₈	semidirect
71. C₂×C₈	Q₁₆	semidirect

H ≅ G/N

type

23. C₂×Q₁₆

C₈

semidirect

71. C₂×C₈

Q₁₆

semidirect

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

subgroup	index
C₂×C₄×Q₁₆	2
#1984	2
C₂×#335	2
C₂×Q₈⋊C₈	2
C₂×C₈²	2

subgroup

index

C₂×C₄×Q₁₆

#1984

C₂×#335

C₂×Q₈⋊C₈

C₂×C₈²

Subgroups — #9656

Conjugacy classes of subgroups

Complements

Maximal subgroups