Group #17981 — subgroups

Subgroups — #17981

Order 256.

363 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 155 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	#3169	C₂	1	1	yes	maximal, characteristic
2	#2931	C₂	1	2	yes	maximal, complemented
3	#1586	C₂	1	8	yes	maximal, complemented
4	#1566	C₂	1	1	yes	maximal, characteristic, complemented
5	#1492	C₂	1	2	yes	maximal, complemented
6	C₂×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#131	—	2	2	no	—
8	#532	C₂²	1	2	yes	—
9	#500	C₄	1	1	yes	characteristic
10	#493	C₄	1	1	yes	characteristic
11	#486	C₄	1	2	yes	—
12	#466	C₄	1	4	yes	—
13	C₄×D₈	C₂²	1	4	yes	—
14	#436	—	2	2	no	—
15	#436	C₂²	1	4	yes	—
16	C₈×D₄	—	2	2	no	—
17	C₈×D₄	C₂²	1	4	yes	—
18	#423	C₂²	1	2	yes	—
19	#408	—	2	2	no	—
20	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
23	#334	C₂²	1	4	yes	—
24	D₄⋊C₈	C₂²	1	8	yes	—
25	#322	C₂²	1	4	yes	—
26	C₂×#41	—	2	2	no	—
27	C₂×D₈	C₂×C₄	1	4	yes	—
28	#131	—	4	8	no	—
29	C₂×M₁₆	—	2	2	no	—
30	C₂×M₁₆	—	2	4	no	—
31	C₂²×C₈	—	2	1	no	—
32	C₂²×C₈	—	2	4	no	—
33	C₂²×C₈	C₂×C₄	1	2	yes	—
34	C₂²×C₈	D₄	1	2	yes	—
35	C₄⋊Q₈	C₂×C₄	1	2	yes	—
36	#127	C₂×C₄	1	2	yes	—
37	#124	—	2	2	no	—
38	#121	—	2	2	no	—
39	#121	C₂×C₄	1	4	yes	—
40	C₄×D₄	—	2	2	no	—
41	C₄×D₄	C₂³	1	4	yes	—
42	C₂×C₄²	C₂³	1	1	yes	characteristic
43	#107	C₂×C₄	1	4	yes	—
44	C₄⋊C₈	—	2	2	no	—
45	C₄⋊C₈	C₂³	1	4	yes	—
46	#102	C₂×C₄	1	8	yes	—
47	C₂²⋊C₈	—	2	4	no	—
48	C₂²⋊C₈	—	2	4	no	—
49	C₄×C₈	—	2	1	no	—
50	C₄×C₈	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈	D₄	1	1	yes	characteristic
53	C₄×C₈	D₄	1	1	yes	characteristic
54	C₄×C₈	C₂³	1	2	yes	—
55	C₄×C₈	C₂³	1	2	yes	—
56	C₄×C₈	C₂³	1	2	yes	—
57	#41	—	4	4	no	—
58	#41	—	4	4	no	—
59	C₂×Q₈	—	2	2	no	—
60	C₂×D₄	—	2	4	no	—
61	C₂×D₄	—	2	2	no	—
62	C₂×D₄	—	2	2	no	—
63	C₂×D₄	C₂²×C₄	1	4	yes	—
64	C₂²×C₄	—	2	4	no	—
65	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄	#41	1	1	yes	characteristic
68	D₈	—	4	4	no	—
69	M₁₆	—	4	8	no	—
70	M₁₆	—	4	4	no	—
71	C₂×C₈	—	2	4	no	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	4	no	—
75	C₂×C₈	—	4	2	no	—
76	C₂×C₈	—	4	8	no	—
77	C₂×C₈	C₂²×C₄	1	2	yes	—
78	C₂×C₈	C₂²×C₄	1	2	yes	—
79	C₂×C₈	C₂×D₄	1	2	yes	—
80	C₂×C₈	C₂×D₄	1	2	yes	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	C₂²×C₄	1	4	yes	—
83	C₂²⋊C₄	—	2	4	no	—
84	C₂²⋊C₄	—	2	4	no	—
85	C₄²	C₂²×C₄	1	1	yes	characteristic
86	C₄²	C₂×D₄	1	1	yes	characteristic
87	C₄²	#41	1	1	yes	characteristic
88	C₄²	C₂⁴	1	1	yes	characteristic
89	C₂³	—	2	4	no	—
90	C₂³	C₄×D₄	1	1	yes	characteristic
91	Q₈	—	4	4	no	—
92	D₄	—	2	8	no	—
93	D₄	—	4	8	no	—
94	D₄	—	4	4	no	—
95	D₄	—	4	4	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	4	4	no	—
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄	AGL(1,ℤ/8)	1	4	yes	—
105	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄	C₂×#41	1	1	yes	characteristic
108	C₈	—	2	4	no	—
109	C₈	—	2	2	no	—
110	C₈	—	2	2	no	—
111	C₈	—	2	2	no	—
112	C₈	—	2	2	no	—
113	C₈	—	4	4	no	—
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	4	4	no	—
117	C₂²	—	4	4	no	—
118	C₂²	—	4	4	no	—
119	C₂²	#442	1	2	yes	—
120	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	4	4	no	—
124	C₄	#442	1	2	yes	—
125	C₄	#443	1	2	yes	—
126	C₄	#568	1	2	yes	—
127	C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	—	8	4	no	—
130	C₂	#2932	1	1	yes	characteristic
131	C₂	C₂×#442	1	1	yes	characteristic
132	C₂	C₂×#443	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3169

C₂

yes

maximal, characteristic

#2931

C₂

yes

maximal, complemented

#1586

C₂

yes

maximal, complemented

#1566

C₂

yes

maximal, characteristic, complemented

#1492

C₂

yes

maximal, complemented

C₂×#322

C₂

yes

maximal, characteristic, complemented

C₂×#131

—

#532

C₂²

yes

—

#500

C₄

yes

characteristic

#493

C₄

yes

characteristic

#486

C₄

yes

—

#466

C₄

yes

—

C₄×D₈

C₂²

yes

—

#436

—

#436

C₂²

yes

—

C₈×D₄

—

C₈×D₄

C₂²

yes

—

#423

C₂²

yes

—

#408

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#334

C₂²

yes

—

D₄⋊C₈

C₂²

yes

—

#322

C₂²

yes

—

C₂×#41

—

C₂×D₈

C₂×C₄

yes

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

#127

C₂×C₄

yes

—

#124

—

#121

—

#121

C₂×C₄

yes

—

C₄×D₄

—

C₄×D₄

C₂³

yes

—

C₂×C₄²

C₂³

yes

characteristic

#107

C₂×C₄

yes

—

C₄⋊C₈

—

C₄⋊C₈

C₂³

yes

—

#102

C₂×C₄

yes

—

C₂²⋊C₈

—

C₂²⋊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

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²×C₄

yes

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

D₈

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×C₄

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

characteristic

C₄²

C₂⁴

yes

characteristic

C₂³

—

C₂³

C₄×D₄

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄×D₄

yes

characteristic

102

C₂×C₄

C₄×D₄

yes

characteristic

103

C₂×C₄

C₄×D₄

yes

characteristic

104

C₂×C₄

AGL(1,ℤ/8)

yes

—

105

C₂×C₄

C₂³×C₄

yes

characteristic

106

C₂×C₄

C₂²×D₄

yes

characteristic

107

C₂×C₄

C₂×#41

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#442

yes

—

120

C₂²

C₂×C₄×D₄

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#442

yes

—

125

C₄

#443

yes

—

126

C₄

#568

yes

—

127

C₄

C₂×AGL(1,ℤ/8)

yes

—

128

C₂

—

129

C₂

—

130

C₂

#2932

yes

characteristic

131

C₂

C₂×#442

yes

characteristic

132

C₂

C₂×#443

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₄²	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

C₂²×D₄

commutator

C₂×C₄

C₂³×C₄

Frattini, cosocle

C₄²

C₂⁴

socle

C₂²

C₂×C₄×D₄

C₂

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
2. #2931	C₂	semidirect
3. #1586	C₂	semidirect
4. #1566	C₂	semidirect
5. #1492	C₂	semidirect
6. C₂×#322	C₂	semidirect

H ≅ G/N

type

2. #2931

C₂

semidirect

3. #1586

C₂

semidirect

4. #1566

C₂

semidirect

5. #1492

C₂

semidirect

6. C₂×#322

C₂

semidirect

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

subgroup	index
#3169	2
#2931	2
#1586	2
#1566	2
#1492	2
C₂×#322	2

subgroup

index

Subgroups — #17981

Conjugacy classes of subgroups

Complements

Maximal subgroups