Group #8643 — subgroups

Subgroups — #8643

Order 256.

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

normal subgroups only

This group has 227 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×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#1844	C₂	1	1	yes	maximal, characteristic
3	#1803	C₂	1	2	yes	maximal
4	C₂×#334	C₂	1	1	yes	maximal, characteristic
5	#1296	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈	C₂²	1	1	yes	characteristic
7	C₈×Q₈	—	2	2	no	—
8	C₈×Q₈	C₄	1	4	yes	complemented
9	C₂×#106	—	2	1	no	—
10	C₂×#106	—	2	1	no	—
11	C₂×C₄⋊C₈	—	2	1	no	—
12	C₂×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×#103	—	2	2	no	—
15	C₂×C₄×C₈	—	2	1	no	—
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	#384	C₂²	1	2	yes	—
18	#340	—	2	2	no	—
19	#334	—	2	2	no	—
20	C₂²×Q₈	D₄	1	1	yes	characteristic
21	C₂²×C₈	—	2	1	no	—
22	C₂²×C₈	—	2	2	no	—
23	C₂²×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₈	D₄	1	1	yes	characteristic
25	C₄×Q₈	—	2	1	no	—
26	C₄×Q₈	—	2	1	no	—
27	C₄×Q₈	C₂×C₄	1	2	yes	—
28	C₄×Q₈	Q₈	1	2	yes	—
29	C₂×C₄⋊C₄	—	2	1	no	—
30	C₂×C₄⋊C₄	—	2	2	no	—
31	C₂×C₄⋊C₄	—	2	2	no	—
32	C₂×C₄⋊C₄	—	2	2	no	—
33	C₂×C₄⋊C₄	D₄	1	1	yes	characteristic
34	C₂×C₄²	—	2	1	no	—
35	C₂×C₄²	C₂³	1	1	yes	characteristic
36	#106	—	4	2	no	—
37	#106	—	4	2	no	—
38	C₄⋊C₈	—	2	4	no	—
39	C₄⋊C₈	—	2	1	no	—
40	C₄⋊C₈	—	2	2	no	—
41	C₄⋊C₈	—	2	2	no	—
42	C₄⋊C₈	C₂×C₄	1	2	yes	—
43	#103	—	4	4	no	—
44	C₄×C₈	—	2	4	no	—
45	C₄×C₈	C₂×C₄	1	2	yes	—
46	C₄×C₈	D₄	1	2	yes	—
47	C₂×Q₈	—	2	1	no	—
48	C₂×Q₈	—	2	2	no	—
49	C₂×Q₈	C₄⋊C₄	1	2	yes	—
50	C₂×Q₈	SD₁₆	1	4	yes	—
51	C₂²×C₄	—	2	1	no	—
52	C₂²×C₄	—	2	1	no	—
53	C₂²×C₄	—	4	2	no	—
54	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄	#41	1	1	yes	characteristic
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	4	1	no	—
61	C₂×C₈	—	4	1	no	—
62	C₂×C₈	—	4	2	no	—
63	C₂×C₈	—	4	2	no	—
64	C₂×C₈	—	4	2	no	—
65	C₂×C₈	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈	SD₁₆	1	2	yes	—
68	C₂×C₈	Q₁₆	1	2	yes	—
69	C₄⋊C₄	—	2	2	no	—
70	C₄⋊C₄	—	2	1	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	4	2	no	—
73	C₄⋊C₄	—	4	2	no	—
74	C₄⋊C₄	—	4	2	no	—
75	C₄⋊C₄	—	4	2	no	—
76	C₄⋊C₄	—	8	2	no	—
77	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
78	C₄²	—	2	2	no	—
79	C₄²	—	2	2	no	—
80	C₄²	C₂²×C₄	1	1	yes	characteristic
81	C₄²	C₂×D₄	1	1	yes	characteristic
82	C₄²	C₂×Q₈	1	1	yes	characteristic
83	C₄²	#41	1	1	yes	characteristic
84	C₂³	#121	1	1	yes	characteristic
85	Q₈	—	2	2	no	—
86	Q₈	—	2	4	no	—
87	Q₈	#106	1	4	yes	complemented
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	8	2	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	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄	#117	1	1	yes	characteristic
99	C₂×C₄	#121	1	1	yes	characteristic
100	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄	C₂×SD₁₆	1	2	yes	—
104	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
105	C₂×C₄	#135	1	2	yes	—
106	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×C₄	#137	1	1	yes	characteristic
108	C₈	—	2	2	no	—
109	C₈	—	4	2	no	—
110	C₈	—	8	2	no	—
111	C₈	#103	1	4	yes	complemented
112	C₂²	#380	1	1	yes	characteristic
113	C₂²	#460	1	1	yes	characteristic
114	C₂²	#462	1	1	yes	characteristic
115	C₂²	#465	1	2	yes	—
116	C₂²	#471	1	2	yes	—
117	C₄	—	2	4	no	—
118	C₄	—	2	1	no	—
119	C₄	—	2	1	no	—
120	C₄	—	4	2	no	—
121	C₄	—	16	2	no	—
122	C₄	C₂×#103	1	1	yes	characteristic
123	C₄	#417	1	1	yes	characteristic
124	C₄	C₂×#106	1	1	yes	characteristic
125	C₄	#427	1	1	yes	characteristic
126	C₄	#465	1	2	yes	—
127	C₄	#475	1	1	yes	characteristic
128	C₄	#477	1	1	yes	characteristic
129	C₂	#1666	1	2	yes	—
130	C₂	#1672	1	2	yes	—
131	C₂	#1803	1	1	yes	characteristic
132	C₂	#1807	1	1	yes	characteristic
133	C₂	#1815	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Q₈

C₂

yes

maximal, characteristic

#1844

C₂

yes

maximal, characteristic

#1803

C₂

yes

maximal

C₂×#334

C₂

yes

maximal, characteristic

#1296

C₂

yes

maximal

C₂×C₄×Q₈

C₂²

yes

characteristic

C₈×Q₈

—

C₈×Q₈

C₄

yes

complemented

C₂×#106

—

C₂×#106

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

characteristic

#384

C₂²

yes

—

#340

—

#334

—

C₂²×Q₈

D₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×C₄

yes

—

C₄×Q₈

Q₈

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

#103

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₄⋊C₄

yes

—

C₂×Q₈

SD₁₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

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

SD₁₆

yes

—

C₂×C₈

Q₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×Q₈

yes

C₄²

yes

C₂³

yes

Q₈

—

Q₈

—

Q₈

#106

yes

complemented

C₂×C₄

—

C₂×C₄

—

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

100

C₂×C₄

C₂²⋊Q₈

yes

characteristic

101

C₂×C₄

C₂²⋊Q₈

yes

characteristic

102

C₂×C₄

C₂×SD₁₆

yes

characteristic

103

C₂×C₄

C₂×SD₁₆

yes

—

104

C₂×C₄

C₂×Q₁₆

yes

characteristic

105

C₂×C₄

#135

yes

—

106

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

107

C₂×C₄

#137

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

#103

yes

complemented

112

C₂²

#380

yes

characteristic

113

C₂²

#460

yes

characteristic

114

C₂²

#462

yes

characteristic

115

C₂²

#465

yes

—

116

C₂²

#471

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

C₂×#103

yes

characteristic

123

C₄

#417

yes

characteristic

124

C₄

C₂×#106

yes

characteristic

125

C₄

#427

yes

characteristic

126

C₄

#465

yes

—

127

C₄

#475

yes

characteristic

128

C₄

#477

yes

characteristic

129

C₂

#1666

yes

—

130

C₂

#1672

yes

—

131

C₂

#1803

yes

characteristic

132

C₂

#1807

yes

characteristic

133

C₂

#1815

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
8. C₈×Q₈	C₄	semidirect
87. Q₈	#106	semidirect
111. C₈	#103	semidirect

H ≅ G/N

type

8. C₈×Q₈

C₄

semidirect

87. Q₈

#106

semidirect

111. C₈

#103

semidirect

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

subgroup	index
C₂×C₈×Q₈	2
#1844	2
#1803	2
C₂×#334	2
#1296	2

subgroup

index

C₂×C₈×Q₈

Subgroups — #8643

Conjugacy classes of subgroups

Complements

Maximal subgroups