Group #13136050 — subgroups

Subgroups — 3₋¹⁺²×#349 #13136050

Order 1728.

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

normal subgroups only

This group has 232 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	C₂×M₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	3₋¹⁺²×M₃₂	C₂	1	2	yes	maximal, complemented
3	C₃²×#349	C₃	1	1	yes	maximal, characteristic
4	C₉×#349	C₃	1	3	yes	maximal, complemented
5	C₂²×C₄×3₋¹⁺²	C₄	1	1	yes	characteristic
6	M₁₆×3₋¹⁺²	—	2	2	no	—
7	C₂×C₈×3₋¹⁺²	C₄	1	1	yes	characteristic
8	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
9	C₁₆×3₋¹⁺²	—	2	2	no	—
10	C₂×C₃²×M₁₆	C₂×C₃	1	1	yes	characteristic
11	C₃²×M₃₂	C₂×C₃	1	2	yes	—
12	C₂×C₉×M₁₆	C₂×C₃	1	3	yes	—
13	C₉×M₃₂	C₂×C₃	1	6	yes	complemented
14	C₂³×3₋¹⁺²	C₈	1	1	yes	characteristic
15	C₂×C₄×3₋¹⁺²	—	2	1	no	—
16	C₂×C₄×3₋¹⁺²	—	2	1	no	—
17	C₂×C₄×3₋¹⁺²	C₈	1	1	yes	characteristic
18	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
19	C₈×3₋¹⁺²	—	2	1	no	—
20	C₈×3₋¹⁺²	D₄	1	2	yes	—
21	C₃×#349	—	3	1	no	—
22	C₃×#349	C₃²	1	1	yes	characteristic
23	C₂²×C₃²×C₄	C₃×C₄	1	1	yes	characteristic
24	C₃²×M₁₆	—	2	2	no	—
25	C₂×C₃²×C₈	C₃×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
27	C₂²×C₄×C₉	C₃×C₄	1	3	yes	—
28	C₃²×C₁₆	—	2	2	no	—
29	C₉×M₁₆	—	2	6	no	—
30	C₂×C₈×C₉	C₃×C₄	1	3	yes	—
31	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
32	C₉×C₁₆	—	2	6	no	—
33	C₂²×3₋¹⁺²	—	2	1	no	—
34	C₂²×3₋¹⁺²	—	4	1	no	—
35	C₂²×3₋¹⁺²	C₂×C₈	1	1	yes	characteristic
36	C₄×3₋¹⁺²	—	2	1	no	—
37	C₄×3₋¹⁺²	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×3₋¹⁺²	M₁₆	1	1	yes	characteristic
39	C₂×C₃×M₁₆	—	3	1	no	—
40	C₂×C₃×M₁₆	C₂×C₃²	1	1	yes	characteristic
41	C₃×M₃₂	—	3	2	no	—
42	C₃×M₃₂	C₂×C₃²	1	2	yes	—
43	C₂³×C₃²	C₃×C₈	1	1	yes	characteristic
44	C₂×C₃²×C₄	—	2	1	no	—
45	C₂×C₃²×C₄	—	2	1	no	—
46	C₂×C₃²×C₄	C₃×C₈	1	1	yes	characteristic
47	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
48	C₂³×C₉	C₃×C₈	1	3	yes	—
49	C₃²×C₈	—	2	1	no	—
50	C₃²×C₈	C₃×D₄	1	2	yes	—
51	C₂×C₄×C₉	—	2	3	no	—
52	C₂×C₄×C₉	—	2	3	no	—
53	C₂×C₄×C₉	C₃×C₈	1	3	yes	—
54	C₂×C₄×C₉	C₂×C₃×C₄	1	3	yes	—
55	C₈×C₉	—	2	3	no	—
56	C₈×C₉	C₃×D₄	1	6	yes	—
57	#349	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
58	C₂×3₋¹⁺²	—	2	1	no	—
59	C₂×3₋¹⁺²	—	4	1	no	—
60	C₂×3₋¹⁺²	C₂²⋊C₈	1	1	yes	characteristic
61	C₂²×C₃×C₄	—	3	1	no	—
62	C₂²×C₃×C₄	C₃²×C₄	1	1	yes	characteristic
63	C₃×M₁₆	—	2	2	no	—
64	C₃×M₁₆	—	6	2	no	—
65	C₂×C₃×C₈	—	3	1	no	—
66	C₂×C₃×C₈	—	3	1	no	—
67	C₂×C₃×C₈	C₃²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
69	C₃×C₁₆	—	2	2	no	—
70	C₃×C₁₆	—	6	2	no	—
71	C₂²×C₃²	—	2	1	no	—
72	C₂²×C₃²	—	4	1	no	—
73	C₂²×C₃²	C₂×C₃×C₈	1	1	yes	characteristic
74	C₃²×C₄	—	2	1	no	—
75	C₃²×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
76	C₃²×C₄	C₃×M₁₆	1	1	yes	characteristic
77	C₂²×C₉	—	2	3	no	—
78	C₂²×C₉	—	4	3	no	—
79	C₂²×C₉	C₂×C₃×C₈	1	3	yes	—
80	C₄×C₉	—	2	3	no	—
81	C₄×C₉	C₃×C₂²⋊C₄	1	3	yes	—
82	C₄×C₉	C₃×M₁₆	1	3	yes	—
83	C₂×M₁₆	C₂×3₋¹⁺²	1	1	yes	characteristic
84	M₃₂	C₂×3₋¹⁺²	1	2	yes	complemented
85	3₋¹⁺²	#349	1	1	yes	characteristic, direct product factor, Sylow
86	C₂³×C₃	—	3	1	no	—
87	C₂³×C₃	C₃²×C₈	1	1	yes	characteristic
88	C₂×C₃×C₄	—	2	1	no	—
89	C₂×C₃×C₄	—	2	1	no	—
90	C₂×C₃×C₄	—	3	1	no	—
91	C₂×C₃×C₄	—	3	1	no	—
92	C₂×C₃×C₄	—	6	1	no	—
93	C₂×C₃×C₄	—	6	1	no	—
94	C₂×C₃×C₄	C₃²×C₈	1	1	yes	characteristic
95	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
96	C₃×C₈	—	2	1	no	—
97	C₃×C₈	—	3	2	no	—
98	C₃×C₈	—	6	1	no	—
99	C₃×C₈	C₃²×D₄	1	2	yes	—
100	C₂×C₃²	—	2	1	no	—
101	C₂×C₃²	—	4	1	no	—
102	C₂×C₃²	C₃×C₂²⋊C₈	1	1	yes	characteristic
103	C₂×C₉	—	2	3	no	—
104	C₂×C₉	—	4	3	no	—
105	C₂×C₉	C₃×C₂²⋊C₈	1	3	yes	—
106	C₂²×C₄	C₄×3₋¹⁺²	1	1	yes	characteristic
107	M₁₆	—	2	2	no	—
108	C₂×C₈	C₄×3₋¹⁺²	1	1	yes	characteristic
109	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
110	C₁₆	—	2	2	no	—
111	C₂²×C₃	—	2	1	no	—
112	C₂²×C₃	—	3	1	no	—
113	C₂²×C₃	—	4	1	no	—
114	C₂²×C₃	—	6	1	no	—
115	C₂²×C₃	—	12	1	no	—
116	C₂²×C₃	C₂×C₃²×C₈	1	1	yes	characteristic
117	C₃×C₄	—	2	1	no	—
118	C₃×C₄	—	3	1	no	—
119	C₃×C₄	—	3	1	no	—
120	C₃×C₄	—	6	1	no	—
121	C₃×C₄	C₃²×C₂²⋊C₄	1	1	yes	characteristic
122	C₃×C₄	C₃²×M₁₆	1	1	yes	characteristic
123	C₃²	C₃×#349	1	1	yes	characteristic
124	C₉	C₃×#349	1	3	yes	complemented
125	C₂³	C₈×3₋¹⁺²	1	1	yes	characteristic
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	C₈×3₋¹⁺²	1	1	yes	characteristic
129	C₂×C₄	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	D₄×3₋¹⁺²	1	2	yes	—
132	C₂×C₃	—	2	1	no	—
133	C₂×C₃	—	3	1	no	—
134	C₂×C₃	—	4	1	no	—
135	C₂×C₃	—	6	1	no	—
136	C₂×C₃	—	12	1	no	—
137	C₂×C₃	C₃²×C₂²⋊C₈	1	1	yes	characteristic
138	C₂²	—	2	1	no	—
139	C₂²	—	4	1	no	—
140	C₂²	C₂×C₈×3₋¹⁺²	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	C₂²⋊C₄×3₋¹⁺²	1	1	yes	characteristic
143	C₄	M₁₆×3₋¹⁺²	1	1	yes	characteristic
144	C₃	—	3	1	no	—
145	C₃	C₃²×#349	1	1	yes	characteristic
146	C₂	—	2	1	no	—
147	C₂	—	4	1	no	—
148	C₂	3₋¹⁺²×C₂²⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₁₆×3₋¹⁺²

C₂

yes

maximal, characteristic

3₋¹⁺²×M₃₂

C₂

yes

maximal, complemented

C₃²×#349

C₃

yes

maximal, characteristic

C₉×#349

C₃

yes

maximal, complemented

C₂²×C₄×3₋¹⁺²

C₄

yes

characteristic

M₁₆×3₋¹⁺²

—

C₂×C₈×3₋¹⁺²

C₄

yes

characteristic

C₂×C₈×3₋¹⁺²

C₂²

yes

characteristic

C₁₆×3₋¹⁺²

—

C₂×C₃²×M₁₆

C₂×C₃

yes

characteristic

C₃²×M₃₂

C₂×C₃

yes

—

C₂×C₉×M₁₆

C₂×C₃

yes

—

C₉×M₃₂

C₂×C₃

yes

complemented

C₂³×3₋¹⁺²

C₈

yes

characteristic

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

C₈

yes

characteristic

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

characteristic

C₈×3₋¹⁺²

—

C₈×3₋¹⁺²

D₄

yes

—

C₃×#349

—

C₃×#349

C₃²

yes

characteristic

C₂²×C₃²×C₄

C₃×C₄

yes

characteristic

C₃²×M₁₆

—

C₂×C₃²×C₈

C₃×C₄

yes

characteristic

C₂×C₃²×C₈

C₂²×C₃

yes

characteristic

C₂²×C₄×C₉

C₃×C₄

yes

—

C₃²×C₁₆

—

C₉×M₁₆

—

C₂×C₈×C₉

C₃×C₄

yes

—

C₂×C₈×C₉

C₂²×C₃

yes

—

C₉×C₁₆

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

C₂×C₈

yes

characteristic

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

C₂²⋊C₄

yes

characteristic

C₄×3₋¹⁺²

M₁₆

yes

characteristic

C₂×C₃×M₁₆

—

C₂×C₃×M₁₆

C₂×C₃²

yes

characteristic

C₃×M₃₂

—

C₃×M₃₂

C₂×C₃²

yes

—

C₂³×C₃²

C₃×C₈

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₃×C₈

yes

characteristic

C₂×C₃²×C₄

C₂×C₃×C₄

yes

characteristic

C₂³×C₉

C₃×C₈

yes

—

C₃²×C₈

—

C₃²×C₈

C₃×D₄

yes

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₃×C₈

yes

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

—

C₈×C₉

—

C₈×C₉

C₃×D₄

yes

—

#349

3₋¹⁺²

yes

characteristic, direct product factor, Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₂²⋊C₈

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₃²×C₄

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₃²×C₄

yes

characteristic

C₂×C₃×C₈

C₂²×C₃²

yes

characteristic

C₃×C₁₆

—

C₃×C₁₆

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂×C₃×C₈

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₃×C₂²⋊C₄

yes

characteristic

C₃²×C₄

C₃×M₁₆

yes

characteristic

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂×C₃×C₈

yes

—

C₄×C₉

—

C₄×C₉

C₃×C₂²⋊C₄

yes

—

C₄×C₉

C₃×M₁₆

yes

—

C₂×M₁₆

C₂×3₋¹⁺²

yes

characteristic

M₃₂

C₂×3₋¹⁺²

yes

complemented

3₋¹⁺²

#349

yes

characteristic, direct product factor, Sylow

C₂³×C₃

—

C₂³×C₃

C₃²×C₈

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₃²×C₈

yes

characteristic

C₂×C₃×C₄

C₂×C₃²×C₄

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

C₃²×D₄

yes

—

100

C₂×C₃²

—

101

C₂×C₃²

—

102

C₂×C₃²

C₃×C₂²⋊C₈

yes

characteristic

103

C₂×C₉

—

104

C₂×C₉

—

105

C₂×C₉

C₃×C₂²⋊C₈

yes

—

106

C₂²×C₄

C₄×3₋¹⁺²

yes

characteristic

107

M₁₆

—

108

C₂×C₈

C₄×3₋¹⁺²

yes

characteristic

109

C₂×C₈

C₂²×3₋¹⁺²

yes

characteristic

110

C₁₆

—

111

C₂²×C₃

—

112

C₂²×C₃

—

113

C₂²×C₃

—

114

C₂²×C₃

—

115

C₂²×C₃

—

116

C₂²×C₃

C₂×C₃²×C₈

yes

characteristic

117

C₃×C₄

—

118

C₃×C₄

—

119

C₃×C₄

—

120

C₃×C₄

—

121

C₃×C₄

C₃²×C₂²⋊C₄

yes

characteristic

122

C₃×C₄

C₃²×M₁₆

yes

characteristic

123

C₃²

C₃×#349

yes

characteristic

124

C₉

C₃×#349

yes

complemented

125

C₂³

C₈×3₋¹⁺²

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₈×3₋¹⁺²

yes

characteristic

129

C₂×C₄

C₂×C₄×3₋¹⁺²

yes

characteristic

130

C₈

—

131

C₈

D₄×3₋¹⁺²

yes

—

132

C₂×C₃

—

133

C₂×C₃

—

134

C₂×C₃

—

135

C₂×C₃

—

136

C₂×C₃

—

137

C₂×C₃

C₃²×C₂²⋊C₈

yes

characteristic

138

C₂²

—

139

C₂²

—

140

C₂²

C₂×C₈×3₋¹⁺²

yes

characteristic

141

C₄

—

142

C₄

C₂²⋊C₄×3₋¹⁺²

yes

characteristic

143

C₄

M₁₆×3₋¹⁺²

yes

characteristic

144

C₃

—

145

C₃

C₃²×#349

yes

characteristic

146

C₂

—

147

C₂

—

148

C₂

3₋¹⁺²×C₂²⋊C₈

yes

characteristic

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

subgroup

G/H

center

C₃×C₄

C₃²×C₂²⋊C₄

commutator

C₂²×C₃

C₂×C₃²×C₈

Frattini, cosocle

C₂×C₃×C₈

C₂²×C₃²

socle

C₂×C₃

C₃²×C₂²⋊C₈

prime radical

C₂³×C₃²

C₃×C₈

LCS

G ▷ C₂²×C₃ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₃×C₄ ◁ C₂×C₈×3₋¹⁺² ◁ G

N	H ≅ G/N	type
2. 3₋¹⁺²×M₃₂	C₂	semidirect
4. C₉×#349	C₃	semidirect
13. C₉×M₃₂	C₂×C₃	semidirect
57. #349	3₋¹⁺²	direct (N × H)
84. M₃₂	C₂×3₋¹⁺²	semidirect
85. 3₋¹⁺²	#349	direct (N × H)
124. C₉	C₃×#349	semidirect

H ≅ G/N

type

2. 3₋¹⁺²×M₃₂

C₂

semidirect

4. C₉×#349

C₃

semidirect

13. C₉×M₃₂

C₂×C₃

semidirect

57. #349

3₋¹⁺²

direct (N × H)

84. M₃₂

C₂×3₋¹⁺²

semidirect

85. 3₋¹⁺²

#349

direct (N × H)

124. C₉

C₃×#349

semidirect

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

subgroup	index
C₂×M₁₆×3₋¹⁺²	2
3₋¹⁺²×M₃₂	2
C₃²×#349	3
C₉×#349	3

subgroup

index

C₂×M₁₆×3₋¹⁺²

3₋¹⁺²×M₃₂

C₃²×#349

C₉×#349

Subgroups — 3₋¹⁺²×#349 #13136050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3−1+2×#349 #13136050

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#349 #13136050