Group #32872 — subgroups

Subgroups — #32872

Order 256.

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

normal subgroups only

This group has 241 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	#2103	C₂	1	2	yes	maximal, complemented
2	#2098	C₂	1	1	yes	maximal, characteristic
3	#2082	C₂	1	2	yes	maximal, complemented
4	#2070	C₂	1	2	yes	maximal, complemented
5	#2049	C₂	1	2	yes	maximal
6	#2047	C₂	1	2	yes	maximal, complemented
7	#2020	C₂	1	1	yes	maximal, characteristic
8	#1902	C₂	1	2	yes	maximal
9	#1858	C₂	1	1	yes	maximal, characteristic
10	C₂×#137	C₂²	1	2	yes	—
11	C₂²×Q₁₆	C₂²	1	1	yes	characteristic
12	#568	C₂²	1	1	yes	characteristic
13	#430	C₂²	1	2	yes	—
14	#430	C₂²	1	1	yes	characteristic
15	#430	C₂²	1	2	yes	—
16	#424	C₂²	1	2	yes	—
17	#421	C₂²	1	2	yes	—
18	#419	C₂²	1	2	yes	—
19	#419	C₂²	1	2	yes	—
20	#413	C₂²	1	1	yes	characteristic
21	C₂×#101	C₂²	1	2	yes	—
22	C₂×#101	C₂²	1	2	yes	—
23	C₂×#101	C₂²	1	2	yes	—
24	#408	C₂²	1	1	yes	characteristic
25	#344	C₂²	1	2	yes	—
26	#344	C₂²	1	2	yes	—
27	#341	C₂²	1	1	yes	characteristic
28	#341	C₂²	1	2	yes	—
29	#338	C₂²	1	2	yes	—
30	H(3,ℤ/4)	C₂²	1	1	yes	characteristic
31	C₂×#41	C₂³	1	1	yes	characteristic
32	C₂²×Q₈	C₂³	1	2	yes	—
33	#137	—	4	2	no	—
34	#137	—	4	2	no	—
35	C₂×Q₁₆	—	2	2	no	—
36	C₂×Q₁₆	—	4	1	no	—
37	C₂×Q₁₆	—	4	2	no	—
38	C₂×SD₁₆	—	2	2	no	—
39	#131	—	4	2	no	—
40	C₂×M₁₆	—	4	1	no	—
41	C₂×M₁₆	C₂³	1	2	yes	—
42	C₂×M₁₆	C₂³	1	2	yes	—
43	C₂×M₁₆	C₂³	1	1	yes	characteristic
44	C₂×M₁₆	C₂³	1	2	yes	—
45	C₂×M₁₆	C₂³	1	2	yes	—
46	C₂²×C₈	C₂³	1	1	yes	characteristic
47	#117	C₂³	1	2	yes	—
48	#108	—	4	1	no	—
49	#108	—	4	2	no	—
50	#108	—	4	2	no	—
51	C₄⋊C₈	—	4	2	no	—
52	#104	—	4	2	no	—
53	#103	—	4	2	no	—
54	#103	—	4	2	no	—
55	#101	—	2	1	no	—
56	#101	—	4	2	no	—
57	#101	—	4	2	no	—
58	#101	—	4	2	no	—
59	#100	—	2	1	no	—
60	C₂²⋊C₈	—	4	1	no	—
61	#41	—	2	2	no	—
62	#41	—	2	2	no	—
63	C₂×Q₈	—	2	2	no	—
64	C₂×Q₈	—	2	2	no	—
65	C₂×Q₈	—	2	2	no	—
66	C₂×Q₈	—	2	2	no	—
67	C₂×Q₈	—	2	2	no	—
68	C₂×Q₈	—	2	2	no	—
69	C₂×Q₈	#41	1	1	yes	characteristic
70	C₂×D₄	—	2	1	no	—
71	C₂×D₄	#41	1	1	yes	characteristic
72	C₂²×C₄	—	2	1	no	—
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	C₂⁴	1	1	yes	characteristic
75	Q₁₆	—	4	4	no	—
76	Q₁₆	—	4	4	no	—
77	SD₁₆	—	4	4	no	—
78	M₁₆	—	2	2	no	—
79	M₁₆	—	2	2	no	—
80	M₁₆	—	2	2	no	—
81	M₁₆	—	2	2	no	—
82	M₁₆	—	2	2	no	—
83	M₁₆	—	2	2	no	—
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	2	no	—
86	M₁₆	—	2	2	no	—
87	M₁₆	—	4	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	4	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₂²⋊C₄	—	2	2	no	—
98	C₄²	—	2	2	no	—
99	C₂³	—	2	1	no	—
100	C₂³	2₋¹⁺⁴	1	1	yes	characteristic
101	Q₈	—	2	2	no	—
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	4	2	no	—
105	Q₈	—	4	2	no	—
106	D₄	—	2	2	no	—
107	D₄	—	4	2	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	C₂×#41	1	1	yes	characteristic
116	C₂×C₄	H(5,𝔽₂)	1	2	yes	—
117	C₂×C₄	H(5,𝔽₂)	1	2	yes	—
118	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
119	C₈	—	4	2	no	—
120	C₈	—	4	2	no	—
121	C₈	—	4	1	no	—
122	C₈	—	4	2	no	—
123	C₈	—	4	1	no	—
124	C₈	—	4	2	no	—
125	C₂²	—	4	1	no	—
126	C₂²	—	4	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	#539	1	1	yes	characteristic
129	C₂²	#562	1	2	yes	—
130	C₄	—	4	1	no	—
131	C₄	—	8	2	no	—
132	C₄	—	8	2	no	—
133	C₄	#538	1	2	yes	—
134	C₄	#560	1	1	yes	characteristic
135	C₄	#560	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	—	2	1	no	—
138	C₂	—	8	1	no	—
139	C₂	#2826	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2103

C₂

yes

maximal, complemented

#2098

C₂

yes

maximal, characteristic

#2082

C₂

yes

maximal, complemented

#2070

C₂

yes

maximal, complemented

#2049

C₂

yes

maximal

#2047

C₂

yes

maximal, complemented

#2020

C₂

yes

maximal, characteristic

#1902

C₂

yes

maximal

#1858

C₂

yes

maximal, characteristic

C₂×#137

C₂²

yes

—

C₂²×Q₁₆

C₂²

yes

characteristic

#568

C₂²

yes

characteristic

#430

C₂²

yes

—

#430

C₂²

yes

characteristic

#430

C₂²

yes

—

#424

C₂²

yes

—

#421

C₂²

yes

—

#419

C₂²

yes

—

#419

C₂²

yes

—

#413

C₂²

yes

characteristic

C₂×#101

C₂²

yes

—

C₂×#101

C₂²

yes

—

C₂×#101

C₂²

yes

—

#408

C₂²

yes

characteristic

#344

C₂²

yes

—

#344

C₂²

yes

—

#341

C₂²

yes

characteristic

#341

C₂²

yes

—

#338

C₂²

yes

—

H(3,ℤ/4)

C₂²

yes

characteristic

C₂×#41

C₂³

yes

characteristic

C₂²×Q₈

C₂³

yes

—

#137

—

#137

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂³

yes

—

C₂×M₁₆

C₂³

yes

—

C₂×M₁₆

C₂³

yes

characteristic

C₂×M₁₆

C₂³

yes

—

C₂×M₁₆

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

#117

C₂³

yes

—

#108

—

#108

—

#108

—

C₄⋊C₈

—

—

—

—

—

—

—

—

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

#41

yes

characteristic

C₂×D₄

—

C₂×D₄

#41

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂⁴

yes

characteristic

Q₁₆

—

Q₁₆

—

SD₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

100

C₂³

2₋¹⁺⁴

yes

characteristic

101

Q₈

—

102

Q₈

—

103

Q₈

—

104

Q₈

—

105

Q₈

—

106

D₄

—

107

D₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₂×#41

yes

characteristic

116

C₂×C₄

H(5,𝔽₂)

yes

—

117

C₂×C₄

H(5,𝔽₂)

yes

—

118

C₂×C₄

H(5,𝔽₂)

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#539

yes

characteristic

129

C₂²

#562

yes

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#538

yes

—

134

C₄

#560

yes

characteristic

135

C₄

#560

yes

characteristic

136

C₂

—

137

C₂

—

138

C₂

—

139

C₂

#2826

yes

characteristic

	subgroup	G/H
center, socle	C₂	#2826
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂

#2826

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴

prime radical

C₂×#41

C₂³

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #2103	C₂	semidirect
3. #2082	C₂	semidirect
4. #2070	C₂	semidirect
6. #2047	C₂	semidirect

H ≅ G/N

type

1. #2103

C₂

semidirect

3. #2082

C₂

semidirect

4. #2070

C₂

semidirect

6. #2047

C₂

semidirect

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

subgroup	index
#2103	2
#2098	2
#2082	2
#2070	2
#2049	2
#2047	2
#2020	2
#1902	2
#1858	2

subgroup

index

Subgroups — #32872

Conjugacy classes of subgroups

Complements

Maximal subgroups