Group #33429 — subgroups

Subgroups — C₂×D₄×C₁₆ #33429

Order 256.

403 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 255 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	C₂³×C₁₆	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₄	C₂	1	1	yes	maximal, characteristic
3	D₄×C₁₆	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊C₁₆	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₈	C₂²	1	2	yes	—
8	C₂×C₄×D₄	C₄	1	1	yes	characteristic
9	C₂²×C₁₆	—	2	8	no	—
10	C₂²×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆	C₂²	1	2	yes	—
12	C₂²×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆	C₂²	1	8	yes	complemented
14	C₈×D₄	C₄	1	4	yes	—
15	C₈×D₄	C₂²	1	4	yes	—
16	C₂×C₄⋊C₈	C₄	1	1	yes	characteristic
17	C₂×C₂²⋊C₈	C₄	1	2	yes	—
18	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
19	C₄⋊C₁₆	C₂²	1	4	yes	complemented
20	C₂²⋊C₁₆	C₂²	1	8	yes	complemented
21	C₄×C₁₆	C₂²	1	4	yes	complemented
22	C₂²×D₄	C₈	1	1	yes	characteristic
23	C₂³×C₄	C₂×C₄	1	2	yes	—
24	C₂²×C₈	—	2	4	no	—
25	C₂²×C₈	—	2	4	no	—
26	C₂²×C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₈	C₂×C₄	1	4	yes	—
28	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈	C₂³	1	4	yes	—
30	C₂²×C₈	C₂³	1	1	yes	characteristic
31	C₄×D₄	C₈	1	4	yes	—
32	C₄×D₄	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄	C₈	1	1	yes	characteristic
34	C₂×C₂²⋊C₄	C₈	1	2	yes	—
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	16	no	—
39	C₂×C₁₆	D₄	1	4	yes	direct product factor
40	C₂×C₁₆	C₂³	1	2	yes	—
41	C₂×C₁₆	C₂³	1	4	yes	—
42	C₂×C₁₆	C₂³	1	2	yes	—
43	C₄⋊C₈	C₂×C₄	1	4	yes	—
44	C₂²⋊C₈	C₂×C₄	1	8	yes	—
45	C₄×C₈	C₂×C₄	1	2	yes	—
46	C₄×C₈	C₂³	1	2	yes	—
47	C₂⁴	C₂×C₈	1	2	yes	—
48	C₂×D₄	C₁₆	1	8	yes	direct product factor
49	C₂×D₄	C₂×C₈	1	4	yes	—
50	C₂²×C₄	—	2	4	no	—
51	C₂²×C₄	—	2	4	no	—
52	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄	C₂×C₈	1	2	yes	—
54	C₂²×C₄	C₂×C₈	1	4	yes	—
55	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄	C₂²×C₄	1	4	yes	—
57	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈	—	2	4	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	2	8	no	—
61	C₂×C₈	—	2	4	no	—
62	C₂×C₈	—	2	4	no	—
63	C₂×C₈	C₂²×C₄	1	2	yes	—
64	C₂×C₈	C₂²×C₄	1	4	yes	—
65	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈	C₂×D₄	1	2	yes	—
67	C₂×C₈	#41	1	2	yes	—
68	C₂×C₈	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄	C₂×C₈	1	4	yes	—
70	C₂²⋊C₄	C₂×C₈	1	8	yes	—
71	C₄²	C₂×C₈	1	2	yes	—
72	C₄²	C₂²×C₄	1	2	yes	—
73	C₁₆	—	2	4	no	—
74	C₁₆	—	2	2	no	—
75	C₁₆	C₂×D₄	1	4	yes	direct product factor
76	C₂³	—	2	4	no	—
77	C₂³	—	2	4	no	—
78	C₂³	C₂×C₁₆	1	8	yes	complemented
79	C₂³	C₂²×C₈	1	4	yes	—
80	C₂³	C₂²×C₈	1	1	yes	characteristic
81	D₄	C₂×C₁₆	1	16	yes	direct product factor
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	4	no	—
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	2	8	no	—
87	C₂×C₄	C₂×C₁₆	1	4	yes	complemented
88	C₂×C₄	C₄×D₄	1	2	yes	—
89	C₂×C₄	C₂²×C₈	1	2	yes	—
90	C₂×C₄	C₂²×C₈	1	4	yes	—
91	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
92	C₂×C₄	C₂²×C₈	1	2	yes	—
93	C₂×C₄	#131	1	2	yes	—
94	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
95	C₈	—	2	4	no	—
96	C₈	—	2	2	no	—
97	C₈	C₄×D₄	1	2	yes	—
98	C₈	C₂²×D₄	1	1	yes	characteristic
99	C₈	C₂×#41	1	1	yes	characteristic
100	C₂²	—	2	4	no	—
101	C₂²	—	2	4	no	—
102	C₂²	—	2	16	no	—
103	C₂²	C₈×D₄	1	2	yes	—
104	C₂²	C₂²×C₁₆	1	2	yes	—
105	C₂²	C₂²×C₁₆	1	8	yes	complemented
106	C₂²	#504	1	2	yes	—
107	C₂²	C₂³×C₈	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	4	no	—
110	C₄	C₈×D₄	1	2	yes	—
111	C₄	C₂²×C₁₆	1	4	yes	complemented
112	C₄	C₂×C₄×D₄	1	1	yes	characteristic
113	C₄	C₂×#131	1	1	yes	characteristic
114	C₂	—	2	8	no	—
115	C₂	D₄×C₁₆	1	4	yes	direct product factor
116	C₂	C₂×C₈×D₄	1	1	yes	characteristic
117	C₂	C₂³×C₁₆	1	1	yes	characteristic
118	C₂	C₂×#504	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₁₆

C₂

yes

maximal, complemented

C₂×C₈×D₄

C₂

yes

maximal, characteristic

D₄×C₁₆

C₂

yes

maximal, direct product factor

C₂×C₄⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊C₁₆

C₂

yes

maximal, complemented

C₂×C₄×C₁₆

C₂

yes

maximal, characteristic, complemented

C₂³×C₈

C₂²

yes

—

C₂×C₄×D₄

C₄

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

—

C₂²×C₁₆

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

complemented

C₈×D₄

C₄

yes

—

C₈×D₄

C₂²

yes

—

C₂×C₄⋊C₈

C₄

yes

characteristic

C₂×C₂²⋊C₈

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄⋊C₁₆

C₂²

yes

complemented

C₂²⋊C₁₆

C₂²

yes

complemented

C₄×C₁₆

C₂²

yes

complemented

C₂²×D₄

C₈

yes

characteristic

C₂³×C₄

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₄×D₄

C₈

yes

—

C₄×D₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₈

yes

characteristic

C₂×C₂²⋊C₄

C₈

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₄

yes

direct product factor

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

—

C₂×C₁₆

C₂³

yes

—

C₄⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂³

yes

—

C₂⁴

C₂×C₈

yes

—

C₂×D₄

C₁₆

yes

direct product factor

C₂×D₄

C₂×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

yes

—

C₂²×C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

#41

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

C₂×C₈

yes

—

C₂²⋊C₄

C₂×C₈

yes

—

C₄²

C₂×C₈

yes

—

C₄²

C₂²×C₄

yes

—

C₁₆

—

C₁₆

—

C₁₆

C₂×D₄

yes

direct product factor

C₂³

—

C₂³

—

C₂³

C₂×C₁₆

yes

complemented

C₂³

C₂²×C₈

yes

—

C₂³

C₂²×C₈

yes

characteristic

D₄

C₂×C₁₆

yes

direct product factor

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₁₆

yes

complemented

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

#131

yes

—

C₂×C₄

C₂³×C₄

yes

characteristic

C₈

—

C₈

—

C₈

C₄×D₄

yes

—

C₈

C₂²×D₄

yes

characteristic

C₈

C₂×#41

yes

characteristic

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

C₈×D₄

yes

—

104

C₂²

C₂²×C₁₆

yes

—

105

C₂²

C₂²×C₁₆

yes

complemented

106

C₂²

#504

yes

—

107

C₂²

C₂³×C₈

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

C₈×D₄

yes

—

111

C₄

C₂²×C₁₆

yes

complemented

112

C₄

C₂×C₄×D₄

yes

characteristic

113

C₄

C₂×#131

yes

characteristic

114

C₂

—

115

C₂

D₄×C₁₆

yes

direct product factor

116

C₂

C₂×C₈×D₄

yes

characteristic

117

C₂

C₂³×C₁₆

yes

characteristic

118

C₂

C₂×#504

yes

characteristic

	subgroup	G/H
center	C₂²×C₁₆	C₂²
commutator	C₂	C₂³×C₁₆
Frattini, cosocle	C₂×C₈	C₂⁴
socle	C₂³	C₂²×C₈
prime radical	C₂²×D₄	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₁₆

C₂²

commutator

C₂

C₂³×C₁₆

Frattini, cosocle

C₂×C₈

C₂⁴

socle

C₂³

C₂²×C₈

prime radical

C₂²×D₄

C₈

LCS

G ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₆ ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. C₂³×C₁₆	C₂	semidirect
3. D₄×C₁₆	C₂	direct (N × H)
4. C₂×C₄⋊C₁₆	C₂	semidirect
5. C₂×C₂²⋊C₁₆	C₂	semidirect
6. C₂×C₄×C₁₆	C₂	semidirect
13. C₂²×C₁₆	C₂²	semidirect
19. C₄⋊C₁₆	C₂²	semidirect
20. C₂²⋊C₁₆	C₂²	semidirect
21. C₄×C₁₆	C₂²	semidirect
39. C₂×C₁₆	D₄	direct (N × H)
48. C₂×D₄	C₁₆	direct (N × H)
75. C₁₆	C₂×D₄	direct (N × H)
78. C₂³	C₂×C₁₆	semidirect
81. D₄	C₂×C₁₆	direct (N × H)
87. C₂×C₄	C₂×C₁₆	semidirect
105. C₂²	C₂²×C₁₆	semidirect
111. C₄	C₂²×C₁₆	semidirect
115. C₂	D₄×C₁₆	direct (N × H)

H ≅ G/N

type

1. C₂³×C₁₆

C₂

semidirect

3. D₄×C₁₆

C₂

direct (N × H)

4. C₂×C₄⋊C₁₆

C₂

semidirect

5. C₂×C₂²⋊C₁₆

C₂

semidirect

6. C₂×C₄×C₁₆

C₂

semidirect

13. C₂²×C₁₆

C₂²

semidirect

19. C₄⋊C₁₆

C₂²

semidirect

20. C₂²⋊C₁₆

C₂²

semidirect

21. C₄×C₁₆

C₂²

semidirect

39. C₂×C₁₆

D₄

direct (N × H)

48. C₂×D₄

C₁₆

direct (N × H)

75. C₁₆

C₂×D₄

direct (N × H)

78. C₂³

C₂×C₁₆

semidirect

81. D₄

C₂×C₁₆

direct (N × H)

87. C₂×C₄

C₂×C₁₆

semidirect

105. C₂²

C₂²×C₁₆

semidirect

111. C₄

C₂²×C₁₆

semidirect

115. 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
C₂³×C₁₆	2
C₂×C₈×D₄	2
D₄×C₁₆	2
C₂×C₄⋊C₁₆	2
C₂×C₂²⋊C₁₆	2
C₂×C₄×C₁₆	2

subgroup

index

C₂³×C₁₆

C₂×C₈×D₄

D₄×C₁₆

C₂×C₄⋊C₁₆

C₂×C₂²⋊C₁₆

C₂×C₄×C₁₆

Subgroups — C₂×D₄×C₁₆ #33429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×D4×C16 #33429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×C₁₆ #33429