Group #9733 — subgroups

Subgroups — #9733

Order 256.

259 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 95 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×#494	C₂	1	1	yes	maximal, characteristic
2	#1988	C₂	1	2	yes	maximal
3	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂×#335	C₂	1	1	yes	maximal, characteristic
5	C₂×#332	C₂	1	2	yes	maximal
6	C₂²×Q₁₆	—	2	1	no	—
7	C₂²×Q₁₆	—	2	1	no	—
8	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
9	#494	—	2	2	no	—
10	#494	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈	—	2	1	no	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	#340	—	2	4	no	—
17	#336	—	2	1	no	—
18	#335	—	2	2	no	—
19	#332	—	2	4	no	—
20	C₂²×Q₈	—	2	2	no	—
21	C₂×Q₁₆	—	2	4	no	—
22	C₂×Q₁₆	—	2	4	no	—
23	C₂×Q₁₆	—	4	2	no	—
24	C₂×Q₁₆	—	4	2	no	—
25	C₂×Q₁₆	—	4	2	no	—
26	C₂×Q₁₆	—	4	2	no	—
27	C₂²×C₈	—	2	1	no	—
28	C₂²×C₈	—	2	2	no	—
29	C₂²×C₈	D₄	1	1	yes	characteristic
30	C₂²×C₈	D₄	1	1	yes	characteristic
31	C₄⋊Q₈	—	2	2	no	—
32	C₄⋊Q₈	—	2	2	no	—
33	C₄⋊Q₈	—	2	2	no	—
34	C₄⋊Q₈	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄	—	2	2	no	—
36	C₂×C₄⋊C₄	—	2	2	no	—
37	C₂×C₄⋊C₄	D₄	1	2	yes	—
38	C₂×C₄²	—	2	1	no	—
39	C₂×C₄²	C₂³	1	1	yes	characteristic
40	C₄⋊C₈	—	2	2	no	—
41	C₄⋊C₈	—	2	2	no	—
42	C₄⋊C₈	—	2	1	no	—
43	C₄⋊C₈	D₄	1	2	yes	—
44	C₄×C₈	—	4	2	no	—
45	C₄×C₈	C₂×C₄	1	2	yes	—
46	C₄×C₈	D₄	1	2	yes	—
47	#95	—	2	1	no	—
48	C₂×Q₈	—	2	4	no	—
49	C₂×Q₈	—	4	2	no	—
50	C₂×Q₈	—	4	2	no	—
51	C₂×Q₈	—	4	2	no	—
52	C₂×Q₈	—	4	2	no	—
53	C₂×Q₈	—	4	2	no	—
54	C₂²×C₄	—	2	2	no	—
55	C₂²×C₄	—	4	2	no	—
56	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄	#41	1	1	yes	characteristic
59	Q₁₆	—	4	8	no	—
60	Q₁₆	—	4	8	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	—	2	1	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	4	1	no	—
65	C₂×C₈	—	4	1	no	—
66	C₂×C₈	—	4	2	no	—
67	C₂×C₈	—	4	4	no	—
68	C₂×C₈	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈	D₈	1	2	yes	—
71	C₂×C₈	SD₁₆	1	2	yes	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	2	4	no	—
75	C₄⋊C₄	—	4	2	no	—
76	C₄⋊C₄	—	4	2	no	—
77	C₄⋊C₄	Q₁₆	1	4	yes	complemented
78	C₄²	—	4	1	no	—
79	C₄²	—	4	1	no	—
80	C₄²	C₂²×C₄	1	1	yes	characteristic
81	C₄²	C₂×D₄	1	1	yes	characteristic
82	C₄²	C₂×D₄	1	1	yes	characteristic
83	C₄²	#41	1	1	yes	characteristic
84	C₂³	#121	1	1	yes	characteristic
85	Q₈	—	4	4	no	—
86	Q₈	—	4	4	no	—
87	Q₈	—	8	2	no	—
88	Q₈	—	8	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	4	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	4	no	—
94	C₂×C₄	—	8	2	no	—
95	C₂×C₄	—	8	2	no	—
96	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄	#121	1	1	yes	characteristic
100	C₂×C₄	#124	1	1	yes	characteristic
101	C₂×C₄	#127	1	1	yes	characteristic
102	C₂×C₄	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄	C₂×Q₁₆	1	2	yes	—
105	C₂×C₄	#135	1	2	yes	—
106	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₈	—	2	2	no	—
109	C₈	—	4	2	no	—
110	C₈	—	8	2	no	—
111	C₈	#102	1	4	yes	—
112	C₂²	#390	1	1	yes	characteristic
113	C₂²	#459	1	1	yes	characteristic
114	C₂²	#460	1	1	yes	characteristic
115	C₂²	#467	1	2	yes	—
116	C₂²	#473	1	2	yes	—
117	C₄	—	2	1	no	—
118	C₄	—	2	1	no	—
119	C₄	—	4	4	no	—
120	C₄	—	8	4	no	—
121	C₄	C₂×#102	1	1	yes	characteristic
122	C₄	#418	1	1	yes	characteristic
123	C₄	C₄×Q₁₆	1	2	yes	—
124	C₄	#459	1	1	yes	characteristic
125	C₄	#460	1	1	yes	characteristic
126	C₄	#494	1	1	yes	characteristic
127	C₄	#495	1	1	yes	characteristic
128	C₂	#1669	1	2	yes	—
129	C₂	#1670	1	2	yes	—
130	C₂	#1969	1	1	yes	characteristic
131	C₂	#1979	1	1	yes	characteristic
132	C₂	#1982	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#494

C₂

yes

maximal, characteristic

#1988

C₂

yes

maximal

C₈×C₄⋊C₄

C₂

yes

maximal, characteristic

C₂×#335

C₂

yes

maximal, characteristic

C₂×#332

C₂

yes

maximal

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂²

yes

—

#494

—

#494

C₄

yes

complemented

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

—

—

—

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×C₄

yes

—

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

D₄

yes

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

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

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

D₈

yes

—

C₂×C₈

SD₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

Q₁₆

yes

complemented

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

C₄²

yes

C₂³

yes

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

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

#121

yes

characteristic

100

C₂×C₄

#124

yes

characteristic

101

C₂×C₄

#127

yes

characteristic

102

C₂×C₄

C₂×D₈

yes

characteristic

103

C₂×C₄

C₂×SD₁₆

yes

characteristic

104

C₂×C₄

C₂×Q₁₆

yes

—

105

C₂×C₄

#135

yes

—

106

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

107

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

#102

yes

—

112

C₂²

#390

yes

characteristic

113

C₂²

#459

yes

characteristic

114

C₂²

#460

yes

characteristic

115

C₂²

#467

yes

—

116

C₂²

#473

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₂×#102

yes

characteristic

122

C₄

#418

yes

characteristic

123

C₄

C₄×Q₁₆

yes

—

124

C₄

#459

yes

characteristic

125

C₄

#460

yes

characteristic

126

C₄

#494

yes

characteristic

127

C₄

#495

yes

characteristic

128

C₂

#1669

yes

—

129

C₂

#1670

yes

—

130

C₂

#1969

yes

characteristic

131

C₂

#1979

yes

characteristic

132

C₂

#1982

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
10. #494	C₄	semidirect
77. C₄⋊C₄	Q₁₆	semidirect

H ≅ G/N

type

10. #494

C₄

semidirect

77. C₄⋊C₄

Q₁₆

semidirect

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

subgroup	index
C₂×#494	2
#1988	2
C₈×C₄⋊C₄	2
C₂×#335	2
C₂×#332	2

subgroup

index

C₈×C₄⋊C₄

Subgroups — #9733

Conjugacy classes of subgroups

Complements

Maximal subgroups