Group #19971 — subgroups

Subgroups — #19971

Order 256.

353 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 353 conjugacy classes of subgroups — too many to draw. Only its 111 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	#3334	C₂	1	2	yes	maximal, complemented
2	#3283	C₂	1	2	yes	maximal, complemented
3	#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	#1673	C₂	1	2	yes	maximal, complemented
5	#1669	C₂	1	2	yes	maximal, complemented
6	#1647	C₂	1	4	yes	maximal, complemented
7	C₈×D₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135	—	2	2	no	—
10	#547	C₂²	1	4	yes	—
11	#500	C₂²	1	2	yes	—
12	#500	C₂²	1	2	yes	—
13	#494	C₂²	1	2	yes	—
14	#480	—	2	4	no	—
15	#474	C₂²	1	4	yes	—
16	#467	—	2	2	no	—
17	#466	—	2	2	no	—
18	#461	C₂²	1	4	yes	—
19	#452	—	2	4	no	—
20	C₄×D₈	C₂²	1	1	yes	characteristic
21	C₄×D₈	C₂²	1	1	yes	characteristic
22	C₈×D₄	C₂²	1	2	yes	—
23	C₈×D₄	C₂²	1	2	yes	—
24	C₂×#107	—	2	2	no	—
25	#335	C₂²	1	1	yes	characteristic
26	#335	C₂²	1	1	yes	characteristic
27	#332	C₂²	1	4	yes	—
28	D₄⋊C₈	C₂²	1	2	yes	—
29	D₄⋊C₈	C₂²	1	2	yes	—
30	C₈²	C₂²	1	1	yes	characteristic
31	C₂×#41	—	2	4	no	—
32	#135	—	4	8	no	—
33	C₂×Q₁₆	—	2	2	no	—
34	C₂×Q₁₆	—	2	2	no	—
35	C₂×SD₁₆	—	2	4	no	—
36	C₂×D₈	D₄	1	1	yes	characteristic
37	C₂×D₈	D₄	1	1	yes	characteristic
38	C₂²×C₈	—	2	2	no	—
39	C₂²×C₈	—	2	2	no	—
40	C₄⋊Q₈	C₂³	1	4	yes	—
41	#123	—	4	4	no	—
42	C₂²⋊Q₈	—	2	4	no	—
43	#121	—	2	4	no	—
44	C₄×D₄	C₂³	1	2	yes	—
45	C₄×D₄	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄	—	2	4	no	—
47	#107	—	2	2	no	—
48	#107	—	2	2	no	—
49	#107	—	2	4	no	—
50	#107	—	4	2	no	—
51	#107	D₄	1	1	yes	characteristic
52	#107	D₄	1	1	yes	characteristic
53	C₄⋊C₈	C₂³	1	2	yes	—
54	C₄⋊C₈	C₂³	1	2	yes	—
55	#103	—	2	4	no	—
56	#102	—	2	4	no	—
57	#102	D₄	1	2	yes	—
58	#102	D₄	1	2	yes	—
59	C₂²⋊C₈	—	2	2	no	—
60	C₂²⋊C₈	—	2	2	no	—
61	C₄×C₈	C₂³	1	1	yes	characteristic
62	C₄×C₈	C₂³	1	1	yes	characteristic
63	C₄×C₈	C₂³	1	1	yes	characteristic
64	#41	—	4	8	no	—
65	#41	—	8	4	no	—
66	C₂×Q₈	—	2	4	no	—
67	C₂×D₄	—	4	4	no	—
68	C₂×D₄	C₂×D₄	1	2	yes	—
69	C₂×D₄	C₂×D₄	1	2	yes	—
70	C₂²×C₄	—	2	2	no	—
71	C₂²×C₄	—	2	2	no	—
72	C₂²×C₄	—	4	4	no	—
73	Q₁₆	—	4	4	no	—
74	Q₁₆	—	8	2	no	—
75	SD₁₆	—	4	8	no	—
76	D₈	—	2	2	no	—
77	D₈	D₈	1	4	yes	complemented
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	4	4	no	—
82	C₂×C₈	—	4	4	no	—
83	C₂×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈	C₂×D₄	1	1	yes	characteristic
87	C₄⋊C₄	—	2	4	no	—
88	C₄⋊C₄	—	2	4	no	—
89	C₄⋊C₄	—	4	4	no	—
90	C₄⋊C₄	C₂×D₄	1	2	yes	—
91	C₄⋊C₄	C₂×D₄	1	2	yes	—
92	C₂²⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₂²⋊C₄	—	4	4	no	—
95	C₄²	C₂⁴	1	1	yes	characteristic
96	C₂³	—	2	2	no	—
97	C₂³	—	2	2	no	—
98	Q₈	—	4	4	no	—
99	Q₈	—	4	4	no	—
100	D₄	—	2	2	no	—
101	D₄	—	2	2	no	—
102	D₄	—	2	2	no	—
103	D₄	—	8	8	no	—
104	D₄	C₂×D₈	1	4	yes	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	8	8	no	—
113	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
116	C₈	—	2	2	no	—
117	C₈	—	2	1	no	—
118	C₈	—	2	1	no	—
119	C₈	—	4	2	no	—
120	C₈	—	4	2	no	—
121	C₈	C₂×D₈	1	2	yes	—
122	C₈	#135	1	2	yes	complemented
123	C₂²	—	2	4	no	—
124	C₂²	—	4	2	no	—
125	C₂²	—	4	2	no	—
126	C₂²	—	4	2	no	—
127	C₂²	—	4	2	no	—
128	C₂²	—	4	2	no	—
129	C₂²	D₄²	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	4	2	no	—
132	C₄	—	4	2	no	—
133	C₄	—	8	4	no	—
134	C₄	C₂²×D₈	1	1	yes	characteristic
135	C₄	C₂×#135	1	1	yes	characteristic
136	C₄	#576	1	1	yes	characteristic
137	C₄	#578	1	1	yes	characteristic
138	C₂	—	4	4	no	—
139	C₂	—	8	2	no	—
140	C₂	D₄×D₈	1	1	yes	characteristic
141	C₂	#3283	1	1	yes	characteristic
142	C₂	#3293	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3334

C₂

yes

maximal, complemented

#3283

C₂

yes

maximal, complemented

#1713

C₂

yes

maximal, characteristic, complemented

#1673

C₂

yes

maximal, complemented

#1669

C₂

yes

maximal, complemented

#1647

C₂

yes

maximal, complemented

C₈×D₈

C₂

yes

maximal, characteristic, complemented

C₈×D₈

C₂

yes

maximal, characteristic, complemented

C₂×#135

—

#547

C₂²

yes

—

#500

C₂²

yes

—

#500

C₂²

yes

—

#494

C₂²

yes

—

#480

—

#474

C₂²

yes

—

#467

—

#466

—

#461

C₂²

yes

—

#452

—

C₄×D₈

C₂²

yes

characteristic

C₄×D₈

C₂²

yes

characteristic

C₈×D₄

C₂²

yes

—

C₈×D₄

C₂²

yes

—

C₂×#107

—

#335

C₂²

yes

characteristic

#335

C₂²

yes

characteristic

#332

C₂²

yes

—

D₄⋊C₈

C₂²

yes

—

D₄⋊C₈

C₂²

yes

—

C₈²

C₂²

yes

characteristic

C₂×#41

—

#135

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

D₄

yes

characteristic

C₂×D₈

D₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

C₂³

yes

—

#123

—

C₂²⋊Q₈

—

#121

—

C₄×D₄

C₂³

yes

—

C₄×D₄

C₂³

yes

—

C₂×C₄⋊C₄

—

—

—

—

—

D₄

yes

characteristic

#107

D₄

yes

characteristic

C₄⋊C₈

C₂³

yes

—

C₄⋊C₈

C₂³

yes

—

#103

—

#102

—

#102

D₄

yes

—

#102

D₄

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

yes

—

C₂×D₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

D₈

—

D₈

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄

yes

—

C₄⋊C₄

C₂×D₄

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂⁴

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

Q₈

—

100

D₄

—

101

D₄

—

102

D₄

—

103

D₄

—

104

D₄

C₂×D₈

yes

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²×D₄

yes

characteristic

114

C₂×C₄

C₂²×D₄

yes

characteristic

115

C₂×C₄

H(5,𝔽₂)

yes

characteristic

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

C₂×D₈

yes

—

122

C₈

#135

yes

complemented

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

D₄²

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

C₂²×D₈

yes

characteristic

135

C₄

C₂×#135

yes

characteristic

136

C₄

#576

yes

characteristic

137

C₄

#578

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

D₄×D₈

yes

characteristic

141

C₂

#3283

yes

characteristic

142

C₂

#3293

yes

characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

D₄²

commutator, Frattini, cosocle

C₄²

C₂⁴

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
1. #3334	C₂	semidirect
2. #3283	C₂	semidirect
3. #1713	C₂	semidirect
4. #1673	C₂	semidirect
5. #1669	C₂	semidirect
6. #1647	C₂	semidirect
7. C₈×D₈	C₂	semidirect
8. C₈×D₈	C₂	semidirect
77. D₈	D₈	semidirect
122. C₈	#135	semidirect

H ≅ G/N

type

1. #3334

C₂

semidirect

2. #3283

C₂

semidirect

3. #1713

C₂

semidirect

4. #1673

C₂

semidirect

5. #1669

C₂

semidirect

6. #1647

C₂

semidirect

7. C₈×D₈

C₂

semidirect

8. C₈×D₈

C₂

semidirect

77. D₈

D₈

semidirect

122. C₈

#135

semidirect

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

subgroup	index
#3334	2
#3283	2
#1713	2
#1673	2
#1669	2
#1647	2
C₈×D₈	2

subgroup

index

C₈×D₈

Subgroups — #19971

Conjugacy classes of subgroups

Complements

Maximal subgroups