Group #13145275 — subgroups

Subgroups — C₂×3₋¹⁺²×D₁₆ #13145275

Order 1728.

392 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 175 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×D₈×3₋¹⁺²	C₂	1	2	yes	maximal, complemented
2	3₋¹⁺²×D₁₆	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃²×D₁₆	C₃	1	1	yes	maximal, characteristic
5	C₂×C₉×D₁₆	C₃	1	3	yes	maximal, complemented
6	C₂×D₄×3₋¹⁺²	—	2	2	no	—
7	D₈×3₋¹⁺²	—	2	2	no	—
8	D₈×3₋¹⁺²	C₂²	1	4	yes	complemented
9	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
10	C₁₆×3₋¹⁺²	C₂²	1	2	yes	complemented
11	C₂×C₃²×D₈	C₂×C₃	1	2	yes	—
12	C₃²×D₁₆	C₂×C₃	1	4	yes	—
13	C₂×C₃²×C₁₆	C₂×C₃	1	1	yes	characteristic
14	C₂×C₉×D₈	C₂×C₃	1	6	yes	complemented
15	C₉×D₁₆	C₂×C₃	1	12	yes	complemented
16	C₂×C₉×C₁₆	C₂×C₃	1	3	yes	complemented
17	C₂³×3₋¹⁺²	—	4	2	no	—
18	D₄×3₋¹⁺²	—	2	4	no	—
19	D₄×3₋¹⁺²	—	4	2	no	—
20	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
21	C₈×3₋¹⁺²	D₄	1	1	yes	characteristic
22	C₈×3₋¹⁺²	C₂³	1	1	yes	characteristic
23	C₂×C₃×D₁₆	—	3	1	no	—
24	C₂×C₃×D₁₆	C₃²	1	1	yes	characteristic
25	C₂×C₃²×D₄	—	2	2	no	—
26	C₃²×D₈	—	2	2	no	—
27	C₃²×D₈	C₂²×C₃	1	4	yes	—
28	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
29	C₂×D₄×C₉	—	2	6	no	—
30	C₃²×C₁₆	C₂²×C₃	1	2	yes	—
31	C₉×D₈	—	2	6	no	—
32	C₉×D₈	C₂²×C₃	1	12	yes	complemented
33	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
34	C₉×C₁₆	C₂²×C₃	1	6	yes	complemented
35	C₂²×3₋¹⁺²	—	4	4	no	—
36	C₂²×3₋¹⁺²	—	8	4	no	—
37	C₂²×3₋¹⁺²	D₈	1	1	yes	characteristic
38	C₄×3₋¹⁺²	D₈	1	1	yes	characteristic
39	C₄×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
40	C₂×C₃×D₈	—	3	2	no	—
41	C₂×C₃×D₈	C₂×C₃²	1	2	yes	—
42	C₃×D₁₆	—	3	4	no	—
43	C₃×D₁₆	C₂×C₃²	1	4	yes	—
44	C₂×C₃×C₁₆	—	3	1	no	—
45	C₂×C₃×C₁₆	C₂×C₃²	1	1	yes	characteristic
46	C₂³×C₃²	—	4	2	no	—
47	C₃²×D₄	—	2	4	no	—
48	C₃²×D₄	—	4	2	no	—
49	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
50	C₂³×C₉	—	4	6	no	—
51	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
52	C₃²×C₈	C₂³×C₃	1	1	yes	characteristic
53	D₄×C₉	—	2	12	no	—
54	D₄×C₉	—	4	6	no	—
55	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
56	C₈×C₉	C₃×D₄	1	3	yes	—
57	C₈×C₉	C₂³×C₃	1	3	yes	—
58	C₂×D₁₆	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
59	C₂×3₋¹⁺²	—	8	4	no	—
60	C₂×3₋¹⁺²	D₁₆	1	2	yes	direct product factor
61	C₂×3₋¹⁺²	C₂×D₈	1	1	yes	characteristic
62	C₂×C₃×D₄	—	2	2	no	—
63	C₂×C₃×D₄	—	6	2	no	—
64	C₃×D₈	—	2	2	no	—
65	C₃×D₈	—	3	4	no	—
66	C₃×D₈	—	6	2	no	—
67	C₃×D₈	C₂²×C₃²	1	4	yes	—
68	C₂×C₃×C₈	—	3	1	no	—
69	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
70	C₃×C₁₆	—	3	2	no	—
71	C₃×C₁₆	C₂²×C₃²	1	2	yes	—
72	C₂²×C₃²	—	4	4	no	—
73	C₂²×C₃²	—	8	4	no	—
74	C₂²×C₃²	C₃×D₈	1	1	yes	characteristic
75	C₃²×C₄	C₃×D₈	1	1	yes	characteristic
76	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
77	C₂²×C₉	—	4	12	no	—
78	C₂²×C₉	—	8	12	no	—
79	C₂²×C₉	C₃×D₈	1	3	yes	—
80	C₄×C₉	C₃×D₈	1	3	yes	—
81	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
82	C₂×D₈	C₂×3₋¹⁺²	1	2	yes	complemented
83	D₁₆	C₂×3₋¹⁺²	1	4	yes	direct product factor
84	C₂×C₁₆	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
85	3₋¹⁺²	C₂×D₁₆	1	1	yes	characteristic, direct product factor, Sylow
86	C₂³×C₃	—	4	2	no	—
87	C₂³×C₃	—	12	2	no	—
88	C₃×D₄	—	2	4	no	—
89	C₃×D₄	—	4	2	no	—
90	C₃×D₄	—	6	4	no	—
91	C₃×D₄	—	12	2	no	—
92	C₂×C₃×C₄	—	3	1	no	—
93	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
94	C₃×C₈	—	3	1	no	—
95	C₃×C₈	—	3	1	no	—
96	C₃×C₈	C₃²×D₄	1	1	yes	characteristic
97	C₃×C₈	C₂³×C₃²	1	1	yes	characteristic
98	C₂×C₃²	—	8	4	no	—
99	C₂×C₃²	C₃×D₁₆	1	2	yes	—
100	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
101	C₂×C₉	—	8	12	no	—
102	C₂×C₉	C₃×D₁₆	1	6	yes	complemented
103	C₂×C₉	C₂×C₃×D₈	1	3	yes	—
104	C₂×D₄	—	2	2	no	—
105	D₈	—	2	2	no	—
106	D₈	C₂²×3₋¹⁺²	1	4	yes	complemented
107	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
108	C₁₆	C₂²×3₋¹⁺²	1	2	yes	complemented
109	C₂²×C₃	—	3	1	no	—
110	C₂²×C₃	—	4	4	no	—
111	C₂²×C₃	—	8	4	no	—
112	C₂²×C₃	—	12	4	no	—
113	C₂²×C₃	—	24	4	no	—
114	C₂²×C₃	C₃²×D₈	1	1	yes	characteristic
115	C₃×C₄	—	3	1	no	—
116	C₃×C₄	—	3	1	no	—
117	C₃×C₄	C₃²×D₈	1	1	yes	characteristic
118	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
119	C₃²	C₂×C₃×D₁₆	1	1	yes	characteristic
120	C₉	C₂×C₃×D₁₆	1	3	yes	complemented
121	C₂³	—	4	2	no	—
122	D₄	—	2	4	no	—
123	D₄	—	4	2	no	—
124	C₂×C₄	D₄×3₋¹⁺²	1	1	yes	characteristic
125	C₈	D₄×3₋¹⁺²	1	1	yes	characteristic
126	C₈	C₂³×3₋¹⁺²	1	1	yes	characteristic
127	C₂×C₃	—	3	2	no	—
128	C₂×C₃	—	3	1	no	—
129	C₂×C₃	—	8	4	no	—
130	C₂×C₃	—	24	4	no	—
131	C₂×C₃	C₃²×D₁₆	1	2	yes	—
132	C₂×C₃	C₂×C₃²×D₈	1	1	yes	characteristic
133	C₂²	—	4	4	no	—
134	C₂²	—	8	4	no	—
135	C₂²	D₈×3₋¹⁺²	1	1	yes	characteristic
136	C₄	D₈×3₋¹⁺²	1	1	yes	characteristic
137	C₄	C₂×D₄×3₋¹⁺²	1	1	yes	characteristic
138	C₃	—	3	1	no	—
139	C₃	C₂×C₃²×D₁₆	1	1	yes	characteristic
140	C₂	—	8	4	no	—
141	C₂	3₋¹⁺²×D₁₆	1	2	yes	direct product factor
142	C₂	C₂×D₈×3₋¹⁺²	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₈×3₋¹⁺²

C₂

yes

maximal, complemented

3₋¹⁺²×D₁₆

C₂

yes

maximal, direct product factor

C₂×C₁₆×3₋¹⁺²

C₂

yes

maximal, characteristic, complemented

C₂×C₃²×D₁₆

C₃

yes

maximal, characteristic

C₂×C₉×D₁₆

C₃

yes

maximal, complemented

C₂×D₄×3₋¹⁺²

—

D₈×3₋¹⁺²

—

D₈×3₋¹⁺²

C₂²

yes

complemented

C₂×C₈×3₋¹⁺²

C₂²

yes

characteristic

C₁₆×3₋¹⁺²

C₂²

yes

complemented

C₂×C₃²×D₈

C₂×C₃

yes

—

C₃²×D₁₆

C₂×C₃

yes

—

C₂×C₃²×C₁₆

C₂×C₃

yes

characteristic

C₂×C₉×D₈

C₂×C₃

yes

complemented

C₉×D₁₆

C₂×C₃

yes

complemented

C₂×C₉×C₁₆

C₂×C₃

yes

complemented

C₂³×3₋¹⁺²

—

D₄×3₋¹⁺²

—

D₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

D₄

yes

characteristic

C₈×3₋¹⁺²

D₄

yes

characteristic

C₈×3₋¹⁺²

C₂³

yes

characteristic

C₂×C₃×D₁₆

—

C₂×C₃×D₁₆

C₃²

yes

characteristic

C₂×C₃²×D₄

—

C₃²×D₈

—

C₃²×D₈

C₂²×C₃

yes

—

C₂×C₃²×C₈

C₂²×C₃

yes

characteristic

C₂×D₄×C₉

—

C₃²×C₁₆

C₂²×C₃

yes

—

C₉×D₈

—

C₉×D₈

C₂²×C₃

yes

complemented

C₂×C₈×C₉

C₂²×C₃

yes

—

C₉×C₁₆

C₂²×C₃

yes

complemented

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

D₈

yes

characteristic

C₄×3₋¹⁺²

D₈

yes

characteristic

C₄×3₋¹⁺²

C₂×D₄

yes

characteristic

C₂×C₃×D₈

—

C₂×C₃×D₈

C₂×C₃²

yes

—

C₃×D₁₆

—

C₃×D₁₆

C₂×C₃²

yes

—

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

C₂×C₃²

yes

characteristic

C₂³×C₃²

—

C₃²×D₄

—

C₃²×D₄

—

C₂×C₃²×C₄

C₃×D₄

yes

characteristic

C₂³×C₉

—

C₃²×C₈

C₃×D₄

yes

characteristic

C₃²×C₈

C₂³×C₃

yes

characteristic

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

C₃×D₄

yes

—

C₈×C₉

C₃×D₄

yes

—

C₈×C₉

C₂³×C₃

yes

—

C₂×D₁₆

3₋¹⁺²

yes

characteristic, direct product factor, Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

D₁₆

yes

direct product factor

C₂×3₋¹⁺²

C₂×D₈

yes

characteristic

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₃×D₈

—

C₃×D₈

—

C₃×D₈

—

C₃×D₈

C₂²×C₃²

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²×C₃²

yes

characteristic

C₃×C₁₆

—

C₃×C₁₆

C₂²×C₃²

yes

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₃×D₈

yes

characteristic

C₃²×C₄

C₃×D₈

yes

characteristic

C₃²×C₄

C₂×C₃×D₄

yes

characteristic

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₃×D₈

yes

—

C₄×C₉

C₃×D₈

yes

—

C₄×C₉

C₂×C₃×D₄

yes

—

C₂×D₈

C₂×3₋¹⁺²

yes

complemented

D₁₆

C₂×3₋¹⁺²

yes

direct product factor

C₂×C₁₆

C₂×3₋¹⁺²

yes

characteristic, complemented

3₋¹⁺²

C₂×D₁₆

yes

characteristic, direct product factor, Sylow

C₂³×C₃

—

C₂³×C₃

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₃²×D₄

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₃×C₈

C₃²×D₄

yes

characteristic

C₃×C₈

C₂³×C₃²

yes

characteristic

C₂×C₃²

—

C₂×C₃²

C₃×D₁₆

yes

—

100

C₂×C₃²

C₂×C₃×D₈

yes

characteristic

101

C₂×C₉

—

102

C₂×C₉

C₃×D₁₆

yes

complemented

103

C₂×C₉

C₂×C₃×D₈

yes

—

104

C₂×D₄

—

105

D₈

—

106

D₈

C₂²×3₋¹⁺²

yes

complemented

107

C₂×C₈

C₂²×3₋¹⁺²

yes

characteristic

108

C₁₆

C₂²×3₋¹⁺²

yes

complemented

109

C₂²×C₃

—

110

C₂²×C₃

—

111

C₂²×C₃

—

112

C₂²×C₃

—

113

C₂²×C₃

—

114

C₂²×C₃

C₃²×D₈

yes

characteristic

115

C₃×C₄

—

116

C₃×C₄

—

117

C₃×C₄

C₃²×D₈

yes

characteristic

118

C₃×C₄

C₂×C₃²×D₄

yes

characteristic

119

C₃²

C₂×C₃×D₁₆

yes

characteristic

120

C₉

C₂×C₃×D₁₆

yes

complemented

121

C₂³

—

122

D₄

—

123

D₄

—

124

C₂×C₄

D₄×3₋¹⁺²

yes

characteristic

125

C₈

D₄×3₋¹⁺²

yes

characteristic

126

C₈

C₂³×3₋¹⁺²

yes

characteristic

127

C₂×C₃

—

128

C₂×C₃

—

129

C₂×C₃

—

130

C₂×C₃

—

131

C₂×C₃

C₃²×D₁₆

yes

—

132

C₂×C₃

C₂×C₃²×D₈

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

D₈×3₋¹⁺²

yes

characteristic

136

C₄

D₈×3₋¹⁺²

yes

characteristic

137

C₄

C₂×D₄×3₋¹⁺²

yes

characteristic

138

C₃

—

139

C₃

C₂×C₃²×D₁₆

yes

characteristic

140

C₂

—

141

C₂

3₋¹⁺²×D₁₆

yes

direct product factor

142

C₂

C₂×D₈×3₋¹⁺²

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₃	C₃²×D₈
commutator, Frattini, cosocle	C₃×C₈	C₂³×C₃²
prime radical	C₂×C₃²×D₁₆	C₃
LCS	G ▷ C₃×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₄×3₋¹⁺² ◁ C₂×C₈×3₋¹⁺² ◁ G

subgroup

G/H

center, socle

C₂²×C₃

C₃²×D₈

commutator, Frattini, cosocle

C₃×C₈

C₂³×C₃²

prime radical

C₂×C₃²×D₁₆

C₃

LCS

G ▷ C₃×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₃ ◁ C₂×C₄×3₋¹⁺² ◁ C₂×C₈×3₋¹⁺² ◁ G

N	H ≅ G/N	type
1. C₂×D₈×3₋¹⁺²	C₂	semidirect
2. 3₋¹⁺²×D₁₆	C₂	direct (N × H)
3. C₂×C₁₆×3₋¹⁺²	C₂	semidirect
5. C₂×C₉×D₁₆	C₃	semidirect
8. D₈×3₋¹⁺²	C₂²	semidirect
10. C₁₆×3₋¹⁺²	C₂²	semidirect
14. C₂×C₉×D₈	C₂×C₃	semidirect
15. C₉×D₁₆	C₂×C₃	semidirect
16. C₂×C₉×C₁₆	C₂×C₃	semidirect
32. C₉×D₈	C₂²×C₃	semidirect
34. C₉×C₁₆	C₂²×C₃	semidirect
58. C₂×D₁₆	3₋¹⁺²	direct (N × H)
60. C₂×3₋¹⁺²	D₁₆	direct (N × H)
82. C₂×D₈	C₂×3₋¹⁺²	semidirect
83. D₁₆	C₂×3₋¹⁺²	direct (N × H)
84. C₂×C₁₆	C₂×3₋¹⁺²	semidirect
85. 3₋¹⁺²	C₂×D₁₆	direct (N × H)
102. C₂×C₉	C₃×D₁₆	semidirect
106. D₈	C₂²×3₋¹⁺²	semidirect
108. C₁₆	C₂²×3₋¹⁺²	semidirect
120. C₉	C₂×C₃×D₁₆	semidirect
141. C₂	3₋¹⁺²×D₁₆	direct (N × H)

H ≅ G/N

type

1. C₂×D₈×3₋¹⁺²

C₂

semidirect

2. 3₋¹⁺²×D₁₆

C₂

direct (N × H)

3. C₂×C₁₆×3₋¹⁺²

C₂

semidirect

5. C₂×C₉×D₁₆

C₃

semidirect

8. D₈×3₋¹⁺²

C₂²

semidirect

10. C₁₆×3₋¹⁺²

C₂²

semidirect

14. C₂×C₉×D₈

C₂×C₃

semidirect

15. C₉×D₁₆

C₂×C₃

semidirect

16. C₂×C₉×C₁₆

C₂×C₃

semidirect

32. C₉×D₈

C₂²×C₃

semidirect

34. C₉×C₁₆

C₂²×C₃

semidirect

58. C₂×D₁₆

3₋¹⁺²

direct (N × H)

60. C₂×3₋¹⁺²

D₁₆

direct (N × H)

82. C₂×D₈

C₂×3₋¹⁺²

semidirect

83. D₁₆

C₂×3₋¹⁺²

direct (N × H)

84. C₂×C₁₆

C₂×3₋¹⁺²

semidirect

85. 3₋¹⁺²

C₂×D₁₆

direct (N × H)

102. C₂×C₉

C₃×D₁₆

semidirect

106. D₈

C₂²×3₋¹⁺²

semidirect

108. C₁₆

C₂²×3₋¹⁺²

semidirect

120. C₉

C₂×C₃×D₁₆

semidirect

141. C₂

3₋¹⁺²×D₁₆

direct (N × H)

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

subgroup	index
C₂×D₈×3₋¹⁺²	2
3₋¹⁺²×D₁₆	2
C₂×C₁₆×3₋¹⁺²	2
C₂×C₃²×D₁₆	3
C₂×C₉×D₁₆	3

subgroup

index

C₂×D₈×3₋¹⁺²

3₋¹⁺²×D₁₆

C₂×C₁₆×3₋¹⁺²

C₂×C₃²×D₁₆

C₂×C₉×D₁₆

Subgroups — C₂×3₋¹⁺²×D₁₆ #13145275

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×3−1+2×D16 #13145275

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×3₋¹⁺²×D₁₆ #13145275