Group #17844 — subgroups

Subgroups — C₂×C₈×Q₁₆ #17844

Order 256.

367 conjugacy classes of subgroups; only the 223 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 223 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×Q₈	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₈×Q₁₆	C₂	1	8	yes	maximal, direct product factor
4	C₂×#335	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈⋊C₈	C₂	1	2	yes	maximal
6	C₂×C₈²	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈	C₂²	1	2	yes	—
9	C₈×Q₈	—	2	4	no	—
10	C₈×Q₈	C₂²	1	8	yes	—
11	C₄×Q₁₆	C₄	1	4	yes	—
12	C₄×Q₁₆	C₂²	1	4	yes	—
13	C₂×#107	C₄	1	1	yes	characteristic
14	C₂×C₄⋊C₈	—	2	2	no	—
15	C₂×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×#103	C₄	1	2	yes	—
17	C₂×C₄×C₈	—	2	2	no	—
18	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
21	#335	C₂²	1	4	yes	—
22	Q₈⋊C₈	C₂²	1	8	yes	—
23	C₈²	C₂²	1	4	yes	—
24	C₂²×Q₈	C₂×C₄	1	2	yes	—
25	C₂×Q₁₆	C₈	1	8	yes	direct product factor
26	C₂×Q₁₆	C₂×C₄	1	4	yes	—
27	C₂²×C₈	—	2	1	no	—
28	C₂²×C₈	—	2	2	no	—
29	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₈	D₄	1	1	yes	characteristic
33	C₄×Q₈	—	2	2	no	—
34	C₄×Q₈	—	2	2	no	—
35	C₄×Q₈	C₂×C₄	1	4	yes	—
36	C₄×Q₈	C₂³	1	4	yes	—
37	C₂×C₄⋊C₄	—	2	2	no	—
38	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₂×C₄²	—	2	2	no	—
40	C₂×C₄²	C₂³	1	1	yes	characteristic
41	#107	C₂×C₄	1	4	yes	—
42	C₄⋊C₈	—	2	8	no	—
43	C₄⋊C₈	—	2	2	no	—
44	C₄⋊C₈	C₂³	1	4	yes	—
45	#103	C₂×C₄	1	8	yes	—
46	C₄×C₈	—	2	8	no	—
47	C₄×C₈	—	2	1	no	—
48	C₄×C₈	C₂×C₄	1	2	yes	—
49	C₄×C₈	D₄	1	2	yes	—
50	C₄×C₈	C₂³	1	2	yes	—
51	C₄×C₈	C₂³	1	2	yes	—
52	C₄×C₈	C₂³	1	2	yes	—
53	C₂×Q₈	—	2	2	no	—
54	C₂×Q₈	—	2	4	no	—
55	C₂×Q₈	C₂×C₈	1	8	yes	—
56	C₂×Q₈	C₂²×C₄	1	4	yes	—
57	C₂²×C₄	—	2	2	no	—
58	C₂²×C₄	—	2	2	no	—
59	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄	#41	1	1	yes	characteristic
62	Q₁₆	C₂×C₈	1	16	yes	direct product factor
63	C₂×C₈	—	2	4	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	—	2	4	no	—
68	C₂×C₈	—	4	4	no	—
69	C₂×C₈	C₂×C₈	1	4	yes	—
70	C₂×C₈	Q₁₆	1	4	yes	direct product factor
71	C₂×C₈	C₂²×C₄	1	2	yes	—
72	C₂×C₈	C₂²×C₄	1	2	yes	—
73	C₂×C₈	C₂×D₄	1	2	yes	—
74	C₂×C₈	C₂×D₄	1	2	yes	—
75	C₄⋊C₄	—	2	4	no	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	4	no	—
78	C₄⋊C₄	C₂²×C₄	1	4	yes	—
79	C₄²	—	2	4	no	—
80	C₄²	—	2	4	no	—
81	C₄²	C₂²×C₄	1	1	yes	characteristic
82	C₄²	C₂×D₄	1	1	yes	characteristic
83	C₄²	#41	1	1	yes	characteristic
84	C₄²	C₂⁴	1	1	yes	characteristic
85	C₂³	C₄×D₄	1	1	yes	characteristic
86	Q₈	—	2	4	no	—
87	Q₈	—	2	8	no	—
88	Q₈	C₂²×C₈	1	8	yes	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	2	8	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	4	4	no	—
94	C₂×C₄	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄	C₂²×C₈	1	2	yes	—
98	C₂×C₄	#131	1	2	yes	—
99	C₂×C₄	C₂×Q₁₆	1	2	yes	—
100	C₂×C₄	#135	1	2	yes	—
101	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄	C₂×#41	1	1	yes	characteristic
104	C₈	—	2	4	no	—
105	C₈	—	2	2	no	—
106	C₈	—	2	2	no	—
107	C₈	—	4	4	no	—
108	C₈	C₂²×C₈	1	4	yes	—
109	C₈	C₂×Q₁₆	1	4	yes	direct product factor
110	C₂²	C₈×D₄	1	2	yes	—
111	C₂²	C₄×Q₁₆	1	2	yes	—
112	C₂²	#443	1	2	yes	—
113	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
114	C₄	—	2	8	no	—
115	C₄	—	2	1	no	—
116	C₄	—	2	1	no	—
117	C₄	—	4	4	no	—
118	C₄	C₈×D₄	1	2	yes	—
119	C₄	C₄×Q₁₆	1	2	yes	—
120	C₄	C₂³×C₈	1	1	yes	characteristic
121	C₄	C₂×#131	1	1	yes	characteristic
122	C₄	C₂²×Q₁₆	1	1	yes	characteristic
123	C₄	C₂×#135	1	1	yes	characteristic
124	C₂	C₈×Q₁₆	1	4	yes	direct product factor
125	C₂	C₂×C₈×D₄	1	1	yes	characteristic
126	C₂	C₂×C₄×Q₁₆	1	1	yes	characteristic
127	C₂	C₂×#443	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Q₈

C₂

yes

maximal

C₂×C₄×Q₁₆

C₂

yes

maximal, characteristic

C₈×Q₁₆

C₂

yes

maximal, direct product factor

C₂×#335

C₂

yes

maximal, characteristic

C₂×Q₈⋊C₈

C₂

yes

maximal

C₂×C₈²

C₂

yes

maximal, characteristic

C₂²×Q₁₆

C₄

yes

characteristic

C₂×C₄×Q₈

C₂²

yes

—

C₈×Q₈

—

C₈×Q₈

C₂²

yes

—

C₄×Q₁₆

C₄

yes

—

C₄×Q₁₆

C₂²

yes

—

C₂×#107

C₄

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×#103

C₄

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#335

C₂²

yes

—

Q₈⋊C₈

C₂²

yes

—

C₈²

C₂²

yes

—

C₂²×Q₈

C₂×C₄

yes

—

C₂×Q₁₆

C₈

yes

direct product factor

C₂×Q₁₆

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×C₄

yes

—

C₄×Q₈

C₂³

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

#107

C₂×C₄

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂³

yes

—

#103

C₂×C₄

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₈

yes

—

C₂×Q₈

C₂²×C₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

Q₁₆

C₂×C₈

yes

direct product factor

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

—

C₂×C₈

Q₁₆

yes

direct product factor

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×C₄

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

characteristic

C₄²

C₂⁴

yes

characteristic

C₂³

C₄×D₄

yes

characteristic

Q₈

—

Q₈

—

Q₈

C₂²×C₈

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

#131

yes

—

C₂×C₄

C₂×Q₁₆

yes

—

100

C₂×C₄

#135

yes

—

101

C₂×C₄

C₂³×C₄

yes

characteristic

102

C₂×C₄

C₂²×D₄

yes

characteristic

103

C₂×C₄

C₂×#41

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

C₂²×C₈

yes

—

109

C₈

C₂×Q₁₆

yes

direct product factor

110

C₂²

C₈×D₄

yes

—

111

C₂²

C₄×Q₁₆

yes

—

112

C₂²

#443

yes

—

113

C₂²

C₂×C₄×D₄

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

C₈×D₄

yes

—

119

C₄

C₄×Q₁₆

yes

—

120

C₄

C₂³×C₈

yes

characteristic

121

C₄

C₂×#131

yes

characteristic

122

C₄

C₂²×Q₁₆

yes

characteristic

123

C₄

C₂×#135

yes

characteristic

124

C₂

C₈×Q₁₆

yes

direct product factor

125

C₂

C₂×C₈×D₄

yes

characteristic

126

C₂

C₂×C₄×Q₁₆

yes

characteristic

127

C₂

C₂×#443

yes

characteristic

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

subgroup

G/H

center

C₂²×C₈

D₄

commutator

C₄

C₂³×C₈

Frattini, cosocle

C₄²

C₂⁴

socle, prime radical

C₂³

C₄×D₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
3. C₈×Q₁₆	C₂	direct (N × H)
25. C₂×Q₁₆	C₈	direct (N × H)
62. Q₁₆	C₂×C₈	direct (N × H)
70. C₂×C₈	Q₁₆	direct (N × H)
109. C₈	C₂×Q₁₆	direct (N × H)
124. C₂	C₈×Q₁₆	direct (N × H)

H ≅ G/N

type

3. C₈×Q₁₆

C₂

direct (N × H)

25. C₂×Q₁₆

C₈

direct (N × H)

62. Q₁₆

C₂×C₈

direct (N × H)

70. C₂×C₈

Q₁₆

direct (N × H)

109. C₈

C₂×Q₁₆

direct (N × H)

124. C₂

C₈×Q₁₆

direct (N × H)

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

subgroup	index
C₂×C₈×Q₈	2
C₂×C₄×Q₁₆	2
C₈×Q₁₆	2
C₂×#335	2
C₂×Q₈⋊C₈	2
C₂×C₈²	2

subgroup

index

C₂×C₈×Q₈

C₂×C₄×Q₁₆

C₈×Q₁₆

C₂×#335

C₂×Q₈⋊C₈

C₂×C₈²

Subgroups — C₂×C₈×Q₁₆ #17844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C8×Q16 #17844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×Q₁₆ #17844