Group #19974 — subgroups

Subgroups — D₈×Q₁₆ #19974

Order 256.

345 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 119 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	Q₈×D₈	C₂	1	2	yes	maximal
2	D₄×Q₁₆	C₂	1	2	yes	maximal, complemented
3	#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	#1674	C₂	1	2	yes	maximal, complemented
5	#1669	C₂	1	2	yes	maximal, complemented
6	#1649	C₂	1	4	yes	maximal, complemented
7	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₈	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆	—	2	2	no	—
10	D₄×Q₈	C₂²	1	4	yes	—
11	#500	—	2	2	no	—
12	#500	C₂²	1	2	yes	—
13	#500	C₂²	1	2	yes	—
14	#494	C₂²	1	2	yes	—
15	#474	—	2	4	no	—
16	#474	C₂²	1	4	yes	—
17	#467	—	2	2	no	—
18	#462	C₂²	1	4	yes	—
19	#451	—	2	4	no	—
20	C₈×Q₈	C₂²	1	2	yes	—
21	C₄×Q₁₆	C₂²	1	1	yes	characteristic
22	C₄×D₈	—	2	2	no	—
23	C₄×D₈	C₂²	1	1	yes	characteristic
24	C₈×D₄	C₂²	1	2	yes	—
25	#335	C₂²	1	1	yes	characteristic
26	#335	C₂²	1	1	yes	characteristic
27	#332	C₂²	1	4	yes	—
28	Q₈⋊C₈	C₂²	1	2	yes	—
29	D₄⋊C₈	C₂²	1	2	yes	—
30	C₈²	C₂²	1	1	yes	characteristic
31	C₂²×Q₈	—	2	4	no	—
32	C₂×Q₁₆	—	2	4	no	—
33	C₂×Q₁₆	—	2	2	no	—
34	C₂×Q₁₆	—	2	2	no	—
35	C₂×Q₁₆	—	4	8	no	—
36	C₂×Q₁₆	D₄	1	1	yes	characteristic
37	C₂×D₈	D₄	1	1	yes	characteristic
38	C₂²×C₈	—	2	2	no	—
39	C₄⋊Q₈	—	2	4	no	—
40	C₄⋊Q₈	C₂³	1	4	yes	—
41	C₂²⋊Q₈	—	2	4	no	—
42	C₂²⋊Q₈	—	4	4	no	—
43	C₄×Q₈	C₂³	1	2	yes	—
44	C₄×D₄	—	2	4	no	—
45	C₄×D₄	C₂³	1	2	yes	—
46	#107	—	2	2	no	—
47	#107	—	2	2	no	—
48	#107	—	2	4	no	—
49	#107	—	4	2	no	—
50	#107	D₄	1	1	yes	characteristic
51	#107	D₄	1	1	yes	characteristic
52	C₄⋊C₈	—	2	2	no	—
53	C₄⋊C₈	C₂³	1	2	yes	—
54	C₄⋊C₈	C₂³	1	2	yes	—
55	#103	—	2	4	no	—
56	#103	D₄	1	2	yes	—
57	#102	—	2	4	no	—
58	#102	D₄	1	2	yes	—
59	C₂²⋊C₈	—	2	2	no	—
60	C₄×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	C₂×Q₈	—	2	4	no	—
65	C₂×Q₈	—	4	4	no	—
66	C₂×Q₈	—	4	8	no	—
67	C₂×Q₈	—	8	4	no	—
68	C₂×Q₈	C₂×D₄	1	2	yes	—
69	C₂×D₄	C₂×D₄	1	2	yes	—
70	C₂²×C₄	—	2	2	no	—
71	C₂²×C₄	—	4	4	no	—
72	Q₁₆	—	4	8	no	—
73	Q₁₆	—	4	4	no	—
74	Q₁₆	—	8	2	no	—
75	Q₁₆	D₈	1	4	yes	direct product factor
76	D₈	Q₁₆	1	4	yes	direct product factor
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	4	4	no	—
81	C₂×C₈	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈	C₂×D₄	1	1	yes	characteristic
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	2	4	no	—
87	C₄⋊C₄	—	2	4	no	—
88	C₄⋊C₄	—	4	4	no	—
89	C₄⋊C₄	C₂×D₄	1	2	yes	—
90	C₄⋊C₄	C₂×D₄	1	2	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₂²⋊C₄	—	4	4	no	—
93	C₄²	—	2	2	no	—
94	C₄²	C₂⁴	1	1	yes	characteristic
95	C₂³	—	2	2	no	—
96	Q₈	—	2	2	no	—
97	Q₈	—	4	4	no	—
98	Q₈	—	4	4	no	—
99	Q₈	—	8	8	no	—
100	Q₈	C₂×D₈	1	4	yes	—
101	D₄	—	2	2	no	—
102	D₄	C₂×Q₁₆	1	4	yes	complemented
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	—	8	8	no	—
110	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	2	1	no	—
115	C₈	—	2	1	no	—
116	C₈	—	4	2	no	—
117	C₈	—	4	2	no	—
118	C₈	C₂×D₈	1	2	yes	—
119	C₈	C₂×Q₁₆	1	2	yes	complemented
120	C₂²	—	2	4	no	—
121	C₂²	—	4	2	no	—
122	C₂²	—	4	2	no	—
123	C₂²	D₄²	1	1	yes	characteristic
124	C₄	—	2	4	no	—
125	C₄	—	2	1	no	—
126	C₄	—	4	2	no	—
127	C₄	—	4	2	no	—
128	C₄	—	8	4	no	—
129	C₄	C₂²×D₈	1	1	yes	characteristic
130	C₄	C₂²×Q₁₆	1	1	yes	characteristic
131	C₄	#576	1	1	yes	characteristic
132	C₄	#578	1	1	yes	characteristic
133	C₂	—	4	4	no	—
134	C₂	D₄×D₈	1	1	yes	characteristic
135	C₂	D₄×Q₁₆	1	1	yes	characteristic
136	C₂	#3293	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₈

C₂

yes

maximal

D₄×Q₁₆

C₂

yes

maximal, complemented

#1713

C₂

yes

maximal, characteristic, complemented

#1674

C₂

yes

maximal, complemented

#1669

C₂

yes

maximal, complemented

#1649

C₂

yes

maximal, complemented

C₈×Q₁₆

C₂

yes

maximal, characteristic, complemented

C₈×D₈

C₂

yes

maximal, characteristic

C₂²×Q₁₆

—

D₄×Q₈

C₂²

yes

—

#500

—

#500

C₂²

yes

—

#500

C₂²

yes

—

#494

C₂²

yes

—

#474

—

#474

C₂²

yes

—

#467

—

#462

C₂²

yes

—

#451

—

C₈×Q₈

C₂²

yes

—

C₄×Q₁₆

C₂²

yes

characteristic

C₄×D₈

—

C₄×D₈

C₂²

yes

characteristic

C₈×D₄

C₂²

yes

—

#335

C₂²

yes

characteristic

#335

C₂²

yes

characteristic

#332

C₂²

yes

—

Q₈⋊C₈

C₂²

yes

—

D₄⋊C₈

C₂²

yes

—

C₈²

C₂²

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

D₄

yes

characteristic

C₂×D₈

D₄

yes

characteristic

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂³

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

C₂³

yes

—

C₄×D₄

—

C₄×D₄

C₂³

yes

—

—

—

—

—

D₄

yes

characteristic

#107

D₄

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂³

yes

—

C₄⋊C₈

C₂³

yes

—

#103

—

#103

D₄

yes

—

#102

—

#102

D₄

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄

yes

—

C₂×D₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

D₈

yes

direct product factor

D₈

Q₁₆

yes

direct product factor

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

—

C₄⋊C₄

C₂×D₄

yes

—

C₄⋊C₄

C₂×D₄

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂⁴

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

100

Q₈

C₂×D₈

yes

—

101

D₄

—

102

D₄

C₂×Q₁₆

yes

complemented

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂²×D₄

yes

characteristic

111

C₂×C₄

C₂²×D₄

yes

characteristic

112

C₂×C₄

H(5,𝔽₂)

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

C₂×D₈

yes

—

119

C₈

C₂×Q₁₆

yes

complemented

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

D₄²

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

C₂²×D₈

yes

characteristic

130

C₄

C₂²×Q₁₆

yes

characteristic

131

C₄

#576

yes

characteristic

132

C₄

#578

yes

characteristic

133

C₂

—

134

C₂

D₄×D₈

yes

characteristic

135

C₂

D₄×Q₁₆

yes

characteristic

136

C₂

#3293

yes

characteristic

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×D₈	D₄
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

prime radical

C₂×D₈

D₄

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
2. D₄×Q₁₆	C₂	semidirect
3. #1713	C₂	semidirect
4. #1674	C₂	semidirect
5. #1669	C₂	semidirect
6. #1649	C₂	semidirect
7. C₈×Q₁₆	C₂	semidirect
75. Q₁₆	D₈	direct (N × H)
76. D₈	Q₁₆	direct (N × H)
102. D₄	C₂×Q₁₆	semidirect
119. C₈	C₂×Q₁₆	semidirect

H ≅ G/N

type

2. D₄×Q₁₆

C₂

semidirect

3. #1713

C₂

semidirect

4. #1674

C₂

semidirect

5. #1669

C₂

semidirect

6. #1649

C₂

semidirect

7. C₈×Q₁₆

C₂

semidirect

75. Q₁₆

D₈

direct (N × H)

76. D₈

Q₁₆

direct (N × H)

102. D₄

C₂×Q₁₆

semidirect

119. C₈

C₂×Q₁₆

semidirect

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

subgroup	index
Q₈×D₈	2
D₄×Q₁₆	2
#1713	2
#1674	2
#1669	2
#1649	2
C₈×Q₁₆	2
C₈×D₈	2

subgroup

index

Q₈×D₈

D₄×Q₁₆

C₈×Q₁₆

C₈×D₈

Subgroups — D₈×Q₁₆ #19974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D8×Q16 #19974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₈×Q₁₆ #19974