Group #11581 — subgroups

Subgroups — C₂×D₄⋊C₁₆ #11581

Order 256.

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

normal subgroups only

This group has 235 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×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊C₁₆	C₂	1	4	yes	maximal, direct product factor
5	C₂³×C₈	—	2	1	no	—
6	C₂×C₄×D₄	C₄	1	1	yes	characteristic
7	C₂²×C₁₆	—	2	1	no	—
8	C₂²×C₁₆	—	2	1	no	—
9	C₈×D₄	—	2	1	no	—
10	C₈×D₄	—	2	1	no	—
11	C₈×D₄	C₄	1	2	yes	—
12	C₈×D₄	C₂²	1	2	yes	—
13	C₂×C₄⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₂²⋊C₈	—	2	1	no	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆	—	2	1	no	—
17	C₄⋊C₁₆	C₂²	1	2	yes	complemented
18	C₄×C₁₆	—	2	1	no	—
19	C₄×C₁₆	C₂²	1	2	yes	complemented
20	C₂²×D₄	C₈	1	1	yes	characteristic
21	C₂³×C₄	—	2	1	no	—
22	C₂²×C₈	—	2	1	no	—
23	C₂²×C₈	—	2	2	no	—
24	C₂²×C₈	—	2	2	no	—
25	C₂²×C₈	—	4	2	no	—
26	C₂²×C₈	—	4	2	no	—
27	C₂²×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₈	D₄	1	1	yes	characteristic
29	C₄×D₄	—	2	1	no	—
30	C₄×D₄	—	2	1	no	—
31	C₄×D₄	C₈	1	2	yes	—
32	C₄×D₄	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄	C₈	1	1	yes	characteristic
34	C₂×C₂²⋊C₄	—	2	1	no	—
35	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₆	—	2	4	no	—
37	C₂×C₁₆	—	2	2	no	—
38	C₂×C₁₆	—	2	2	no	—
39	C₂×C₁₆	—	4	2	no	—
40	C₄⋊C₈	—	2	1	no	—
41	C₄⋊C₈	C₂×C₄	1	2	yes	—
42	C₂²⋊C₈	—	2	2	no	—
43	C₂²⋊C₈	—	2	2	no	—
44	C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈	D₄	1	1	yes	characteristic
46	C₄×C₈	D₄	1	1	yes	characteristic
47	C₄×C₈	C₂³	1	1	yes	characteristic
48	C₂⁴	—	2	1	no	—
49	C₂×D₄	—	2	1	no	—
50	C₂×D₄	—	2	2	no	—
51	C₂×D₄	C₁₆	1	4	yes	complemented
52	C₂×D₄	C₂×C₈	1	2	yes	—
53	C₂²×C₄	—	2	1	no	—
54	C₂²×C₄	—	2	2	no	—
55	C₂²×C₄	—	2	2	no	—
56	C₂²×C₄	—	4	2	no	—
57	C₂²×C₄	—	4	2	no	—
58	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄	M₁₆	1	1	yes	characteristic
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	4	2	no	—
64	C₂×C₈	—	4	4	no	—
65	C₂×C₈	—	4	2	no	—
66	C₂×C₈	—	4	2	no	—
67	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈	D₈	1	2	yes	—
70	C₂×C₈	SD₁₆	1	2	yes	—
71	C₂×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈	C₂×D₄	1	1	yes	characteristic
73	C₄⋊C₄	—	2	1	no	—
74	C₄⋊C₄	C₂×C₈	1	2	yes	—
75	C₂²⋊C₄	—	2	2	no	—
76	C₂²⋊C₄	—	2	2	no	—
77	C₄²	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²	C₂×C₈	1	1	yes	characteristic
79	C₄²	M₁₆	1	1	yes	characteristic
80	C₄²	C₂²×C₄	1	1	yes	characteristic
81	C₁₆	—	2	4	no	—
82	C₁₆	—	4	2	no	—
83	C₂³	—	2	4	no	—
84	C₂³	—	2	2	no	—
85	C₂³	—	4	2	no	—
86	C₂³	—	4	2	no	—
87	C₂³	C₂²⋊C₈	1	1	yes	characteristic
88	D₄	—	2	4	no	—
89	D₄	—	2	2	no	—
90	D₄	C₂×C₁₆	1	4	yes	complemented
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	—	4	4	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
98	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
99	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
100	C₂×C₄	#102	1	2	yes	—
101	C₂×C₄	#104	1	2	yes	—
102	C₂×C₄	C₂×C₁₆	1	2	yes	—
103	C₂×C₄	M₃₂	1	2	yes	—
104	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	2	1	no	—
109	C₈	—	4	2	no	—
110	C₈	#102	1	2	yes	—
111	C₈	C₂×D₈	1	1	yes	characteristic
112	C₈	C₂×SD₁₆	1	1	yes	characteristic
113	C₂²	—	2	4	no	—
114	C₂²	—	4	8	no	—
115	C₂²	—	4	2	no	—
116	C₂²	—	4	2	no	—
117	C₂²	D₄⋊C₈	1	2	yes	—
118	C₂²	C₂²⋊C₁₆	1	2	yes	—
119	C₂²	#350	1	2	yes	—
120	C₂²	C₂×C₂²⋊C₈	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	4	2	no	—
124	C₄	D₄⋊C₈	1	2	yes	—
125	C₄	C₂²⋊C₁₆	1	2	yes	—
126	C₄	C₂×#102	1	1	yes	characteristic
127	C₄	C₂×#104	1	1	yes	characteristic
128	C₄	C₂²×C₁₆	1	1	yes	characteristic
129	C₄	C₂×M₃₂	1	1	yes	characteristic
130	C₂	—	4	4	no	—
131	C₂	D₄⋊C₁₆	1	4	yes	direct product factor
132	C₂	C₂×D₄⋊C₈	1	1	yes	characteristic
133	C₂	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
134	C₂	C₂×#350	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×D₄

C₂

yes

maximal, characteristic

C₂×C₄⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×C₄×C₁₆

C₂

yes

maximal, characteristic, complemented

D₄⋊C₁₆

C₂

yes

maximal, direct product factor

C₂³×C₈

—

C₂×C₄×D₄

C₄

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

—

C₈×D₄

—

C₈×D₄

—

C₈×D₄

C₄

yes

—

C₈×D₄

C₂²

yes

—

C₂×C₄⋊C₈

C₄

yes

characteristic

C₂×C₂²⋊C₈

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂²

yes

complemented

C₄×C₁₆

—

C₄×C₁₆

C₂²

yes

complemented

C₂²×D₄

C₈

yes

characteristic

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄×D₄

—

C₄×D₄

—

C₄×D₄

C₈

yes

—

C₄×D₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₈

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₁₆

yes

complemented

C₂×D₄

C₂×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

M₁₆

yes

characteristic

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

D₈

yes

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

C₂×C₈

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊C₄

yes

characteristic

C₄²

C₂×C₈

yes

characteristic

C₄²

M₁₆

yes

characteristic

C₄²

C₂²×C₄

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊C₈

yes

characteristic

D₄

—

D₄

—

D₄

C₂×C₁₆

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₈

yes

characteristic

C₂×C₄

C₂²⋊C₈

yes

characteristic

C₂×C₄

C₂²⋊C₈

yes

characteristic

100

C₂×C₄

#102

yes

—

101

C₂×C₄

#104

yes

—

102

C₂×C₄

C₂×C₁₆

yes

—

103

C₂×C₄

M₃₂

yes

—

104

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

105

C₂×C₄

C₂²×C₈

yes

characteristic

106

C₂×C₄

C₂×M₁₆

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

#102

yes

—

111

C₈

C₂×D₈

yes

characteristic

112

C₈

C₂×SD₁₆

yes

characteristic

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

D₄⋊C₈

yes

—

118

C₂²

C₂²⋊C₁₆

yes

—

119

C₂²

#350

yes

—

120

C₂²

C₂×C₂²⋊C₈

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

D₄⋊C₈

yes

—

125

C₄

C₂²⋊C₁₆

yes

—

126

C₄

C₂×#102

yes

characteristic

127

C₄

C₂×#104

yes

characteristic

128

C₄

C₂²×C₁₆

yes

characteristic

129

C₄

C₂×M₃₂

yes

characteristic

130

C₂

—

131

C₂

D₄⋊C₁₆

yes

direct product factor

132

C₂

C₂×D₄⋊C₈

yes

characteristic

133

C₂

C₂×C₂²⋊C₁₆

yes

characteristic

134

C₂

C₂×#350

yes

characteristic

	subgroup	G/H
center	C₂²×C₈	D₄
commutator	C₄	C₂²×C₁₆
Frattini, cosocle	C₄×C₈	C₂³
socle	C₂³	C₂²⋊C₈
prime radical	C₂²×D₄	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₈

D₄

commutator

C₄

C₂²×C₁₆

Frattini, cosocle

C₄×C₈

C₂³

socle

C₂³

C₂²⋊C₈

prime radical

C₂²×D₄

C₈

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. C₂×C₄⋊C₁₆	C₂	semidirect
3. C₂×C₄×C₁₆	C₂	semidirect
4. D₄⋊C₁₆	C₂	direct (N × H)
17. C₄⋊C₁₆	C₂²	semidirect
19. C₄×C₁₆	C₂²	semidirect
51. C₂×D₄	C₁₆	semidirect
90. D₄	C₂×C₁₆	semidirect
131. C₂	D₄⋊C₁₆	direct (N × H)

H ≅ G/N

type

2. C₂×C₄⋊C₁₆

C₂

semidirect

3. C₂×C₄×C₁₆

C₂

semidirect

4. D₄⋊C₁₆

C₂

direct (N × H)

17. C₄⋊C₁₆

C₂²

semidirect

19. C₄×C₁₆

C₂²

semidirect

51. C₂×D₄

C₁₆

semidirect

90. D₄

C₂×C₁₆

semidirect

131. C₂

D₄⋊C₁₆

direct (N × H)

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

subgroup	index
C₂×C₈×D₄	2
C₂×C₄⋊C₁₆	2
C₂×C₄×C₁₆	2
D₄⋊C₁₆	2

subgroup

index

C₂×C₈×D₄

C₂×C₄⋊C₁₆

C₂×C₄×C₁₆

D₄⋊C₁₆

Subgroups — C₂×D₄⋊C₁₆ #11581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×D4⋊C16 #11581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄⋊C₁₆ #11581