Group #8659 — subgroups

Subgroups — #8659

Order 256.

197 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 197 conjugacy classes of subgroups — too many to draw. Only its 85 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×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#1848	C₂	1	1	yes	maximal, characteristic
3	#1564	C₂	1	1	yes	maximal, characteristic
4	#1487	C₂	1	2	yes	maximal
5	#1301	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
7	C₈×Q₈	—	2	1	no	—
8	C₈×Q₈	—	2	1	no	—
9	C₈×Q₈	C₄	1	2	yes	—
10	C₈×Q₈	C₄	1	2	yes	—
11	#431	C₂²	1	2	yes	—
12	C₂×#108	—	2	1	no	—
13	C₂×#108	—	2	1	no	—
14	#423	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈	—	2	1	no	—
16	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈	—	2	1	no	—
18	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
19	#335	—	2	1	no	—
20	#334	—	2	1	no	—
21	Q₈⋊C₈	—	2	4	no	—
22	C₂²×Q₈	Q₈	1	1	yes	characteristic
23	C₂×M₁₆	—	2	2	no	—
24	C₂²×C₈	—	2	1	no	—
25	C₂²×C₈	D₄	1	1	yes	characteristic
26	C₂²×C₈	D₄	1	1	yes	characteristic
27	C₄×Q₈	—	2	1	no	—
28	C₄×Q₈	—	2	1	no	—
29	C₄×Q₈	C₂×C₄	1	2	yes	—
30	C₄×Q₈	D₄	1	2	yes	—
31	C₂×C₄⋊C₄	—	2	1	no	—
32	C₂×C₄⋊C₄	Q₈	1	1	yes	characteristic
33	C₂×C₄²	—	2	1	no	—
34	C₂×C₄²	C₂³	1	1	yes	characteristic
35	#108	—	4	2	no	—
36	#108	—	4	2	no	—
37	C₄⋊C₈	—	2	2	no	—
38	C₄⋊C₈	—	2	2	no	—
39	C₄⋊C₈	—	2	1	no	—
40	C₄⋊C₈	—	2	2	no	—
41	C₄⋊C₈	—	2	2	no	—
42	C₄⋊C₈	—	2	2	no	—
43	C₄⋊C₈	C₂×C₄	1	2	yes	—
44	C₂²⋊C₈	—	2	2	no	—
45	C₄×C₈	—	2	2	no	—
46	C₄×C₈	—	2	2	no	—
47	C₄×C₈	—	2	2	no	—
48	C₄×C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈	D₄	1	1	yes	characteristic
51	C₄×C₈	D₄	1	1	yes	characteristic
52	C₂×Q₈	—	2	4	no	—
53	C₂×Q₈	—	2	1	no	—
54	C₂×Q₈	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄	—	2	1	no	—
56	C₂²×C₄	—	2	1	no	—
57	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄	#41	1	1	yes	characteristic
60	M₁₆	—	4	2	no	—
61	M₁₆	—	4	2	no	—
62	C₂×C₈	—	2	1	no	—
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	4	1	no	—
66	C₂×C₈	—	4	1	no	—
67	C₂×C₈	—	4	2	no	—
68	C₂×C₈	—	4	2	no	—
69	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈	SD₁₆	1	2	yes	—
74	C₂×C₈	Q₁₆	1	2	yes	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	1	no	—
78	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
79	C₄²	—	2	2	no	—
80	C₄²	—	2	2	no	—
81	C₄²	C₂²×C₄	1	1	yes	characteristic
82	C₄²	C₂×D₄	1	1	yes	characteristic
83	C₄²	C₂×D₄	1	1	yes	characteristic
84	C₄²	#41	1	1	yes	characteristic
85	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
86	Q₈	—	2	4	no	—
87	Q₈	—	2	2	no	—
88	Q₈	#108	1	4	yes	complemented
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄	#117	1	1	yes	characteristic
98	C₂×C₄	#121	1	1	yes	characteristic
99	C₂×C₄	#121	1	1	yes	characteristic
100	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄	#137	1	1	yes	characteristic
105	C₈	—	2	1	no	—
106	C₈	—	2	1	no	—
107	C₈	—	4	2	no	—
108	C₈	—	8	2	no	—
109	C₈	—	8	2	no	—
110	C₈	#103	1	2	yes	—
111	C₈	#103	1	2	yes	—
112	C₂²	—	2	1	no	—
113	C₂²	—	2	1	no	—
114	C₂²	#380	1	1	yes	characteristic
115	C₂²	#475	1	1	yes	characteristic
116	C₂²	#477	1	1	yes	characteristic
117	C₄	—	2	4	no	—
118	C₄	—	2	1	no	—
119	C₄	—	2	1	no	—
120	C₄	—	2	1	no	—
121	C₄	—	4	2	no	—
122	C₄	C₂×#103	1	1	yes	characteristic
123	C₄	#417	1	1	yes	characteristic
124	C₄	C₂×#108	1	2	yes	—
125	C₄	#460	1	1	yes	characteristic
126	C₄	#462	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	#1803	1	1	yes	characteristic
130	C₂	#1810	1	1	yes	characteristic
131	C₂	#1814	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Q₈

C₂

yes

maximal, characteristic

#1848

C₂

yes

maximal, characteristic

#1564

C₂

yes

maximal, characteristic

#1487

C₂

yes

maximal

#1301

C₂

yes

maximal

C₂×C₄×Q₈

C₂²

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

C₄

yes

—

C₈×Q₈

C₄

yes

—

#431

C₂²

yes

—

C₂×#108

—

C₂×#108

—

#423

C₂²

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

characteristic

#335

—

#334

—

Q₈⋊C₈

—

C₂²×Q₈

Q₈

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×C₄

yes

—

C₄×Q₈

D₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Q₈

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄⋊C₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

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

C₂×C₈

SD₁₆

yes

—

C₂×C₈

Q₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

C₂²⋊Q₈

yes

characteristic

Q₈

—

Q₈

—

Q₈

#108

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

C₂²⋊Q₈

yes

characteristic

101

C₂×C₄

C₂×SD₁₆

yes

characteristic

102

C₂×C₄

C₂×Q₁₆

yes

characteristic

103

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

104

C₂×C₄

#137

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

#103

yes

—

111

C₈

#103

yes

—

112

C₂²

—

113

C₂²

—

114

C₂²

#380

yes

characteristic

115

C₂²

#475

yes

characteristic

116

C₂²

#477

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

C₂×#103

yes

characteristic

123

C₄

#417

yes

characteristic

124

C₄

C₂×#108

yes

—

125

C₄

#460

yes

characteristic

126

C₄

#462

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

#1803

yes

characteristic

130

C₂

#1810

yes

characteristic

131

C₂

#1814

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#121
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#380
prime radical	C₂³	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#121

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

socle

C₂²

#380

prime radical

C₂³

C₂²⋊Q₈

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
88. Q₈	#108	semidirect

H ≅ G/N

type

88. Q₈

#108

semidirect

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

subgroup	index
C₂×C₈×Q₈	2
#1848	2
#1564	2
#1487	2
#1301	2

subgroup

index

C₂×C₈×Q₈

Subgroups — #8659

Conjugacy classes of subgroups

Complements

Maximal subgroups