Group #13135073 — subgroups

Subgroups — #13135073

Order 1728.

170 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 55 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	#11720552	C₂	1	1	yes	maximal, characteristic
2	#63646⋊C₃	C₂	1	1	yes	maximal, characteristic
3	C₂×(C₃⋊(C₃⋊C₁₆))⋊C₃	C₂	1	1	yes	maximal, characteristic
4	#10589144	—	3	1	no	maximal
5	#10589143	—	3	1	no	maximal
6	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	#88538	—	2	1	no	—
8	#3748⋊C₃	—	2	1	no	—
9	(C₃⋊(C₃⋊C₁₆))⋊C₃	C₄	1	2	yes	—
10	#63646	D₃	1	1	yes	characteristic
11	C₂×C₃⋊(C₃⋊C₁₆)	D₃	1	1	yes	characteristic
12	C₃×#819	—	3	1	no	—
13	C₃×#819	—	3	1	no	—
14	C₃×#819	—	3	1	no	—
15	C₃×#819	—	6	1	no	—
16	C₂×C₃×C₃⋊C₁₆	—	3	1	no	—
17	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
18	C₈×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
19	C₈×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
20	#6180	—	4	1	no	—
21	(C₃⋊(C₃⋊C₈))⋊C₃	—	4	1	no	—
22	#4506	—	9	1	no	—
23	#4493	—	9	1	no	—
24	#4493	—	9	1	no	—
25	C₂×C₃²×C₈	—	2	1	no	—
26	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
27	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
28	#3748	—	2	1	no	—
29	C₃×#208	—	6	2	no	—
30	C₃×#208	—	6	1	no	—
31	C₃×#208	—	6	1	no	—
32	C₃×#208	—	6	1	no	—
33	C₃⋊(C₃⋊C₁₆)	Dic₃	1	2	yes	—
34	C₃×C₃⋊C₁₆	—	3	2	no	—
35	C₂²×H(3,𝔽₃)	Q₁₆	1	1	yes	characteristic
36	C₄×H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
37	C₄×H(3,𝔽₃)	D₈	1	1	yes	characteristic
38	C₂×C₃×C₁₆	—	9	1	no	—
39	C₃×#108	—	9	1	no	—
40	C₃×#108	—	9	1	no	—
41	#819	—	3	1	no	—
42	#819	—	3	1	no	—
43	#819	—	3	1	no	—
44	#819	—	6	1	no	—
45	#819	—	9	1	no	—
46	C₂×C₃⋊C₁₆	—	3	1	no	—
47	C₂×C₃⋊C₁₆	—	9	1	no	—
48	C₂×C₃²×C₄	—	2	1	no	—
49	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
50	C₂×C₃²×C₄	#67	1	1	yes	characteristic
51	C₃²×C₈	—	2	1	no	—
52	C₃²×C₈	—	2	1	no	—
53	C₃²×C₈	Dic₆	1	1	yes	characteristic
54	C₃²×C₈	Dic₆	1	1	yes	characteristic
55	C₃²×C₈	C₄×D₃	1	1	yes	characteristic
56	C₃²×C₈	C₂×Dic₃	1	1	yes	characteristic
57	C₃⋊(C₃⋊C₈)	—	4	1	no	—
58	C₃×C₃⋊C₈	—	12	2	no	—
59	C₃×C₃⋊C₈	—	12	1	no	—
60	C₃×C₃⋊C₈	—	12	1	no	—
61	C₃×C₃⋊C₈	—	12	1	no	—
62	#368	—	27	1	no	Sylow
63	C₂×H(3,𝔽₃)	—	2	1	no	—
64	C₂×H(3,𝔽₃)	#107	1	1	yes	characteristic
65	C₃×M₁₆	—	18	1	no	—
66	C₃×M₁₆	—	18	1	no	—
67	C₂×C₃×C₈	—	3	1	no	—
68	C₂×C₃×C₈	—	3	1	no	—
69	C₂×C₃×C₈	—	6	1	no	—
70	C₂×C₃×C₈	D₃²	1	1	yes	characteristic
71	#208	—	6	1	no	—
72	#208	—	6	1	no	—
73	#208	—	6	2	no	—
74	#208	—	6	1	no	—
75	#208	—	18	1	no	—
76	C₃×C₁₆	—	9	2	no	—
77	C₃⋊C₁₆	—	3	2	no	—
78	C₃⋊C₁₆	—	9	2	no	—
79	C₂²×C₃²	—	2	1	no	—
80	C₂²×C₃²	Dic₁₂	1	1	yes	characteristic
81	C₂²×C₃²	#216	1	1	yes	characteristic
82	C₃²×C₄	—	2	1	no	—
83	C₃²×C₄	—	2	1	no	—
84	C₃²×C₄	D₂₄	1	1	yes	characteristic
85	C₃²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
86	C₃²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
87	C₃²×C₄	#213	1	1	yes	characteristic
88	C₂×C₁₆	—	9	1	no	—
89	#108	—	9	1	no	—
90	#108	—	9	1	no	—
91	H(3,𝔽₃)	#368	1	1	yes	characteristic, complemented, Sylow
92	C₂×C₃×C₄	—	3	1	no	—
93	C₂×C₃×C₄	—	3	1	no	—
94	C₂×C₃×C₄	—	6	1	no	—
95	C₂×C₃×C₄	#626	1	1	yes	characteristic
96	C₃×C₈	—	3	1	no	—
97	C₃×C₈	—	3	1	no	—
98	C₃×C₈	—	3	1	no	—
99	C₃×C₈	—	3	1	no	—
100	C₃×C₈	—	6	1	no	—
101	C₃×C₈	—	6	1	no	—
102	C₃×C₈	—	36	1	no	—
103	C₃×C₈	—	36	1	no	—
104	C₃×C₈	D₃×Dic₃	1	1	yes	characteristic
105	C₃×C₈	#627	1	1	yes	characteristic
106	C₃⋊C₈	—	12	1	no	—
107	C₃⋊C₈	—	12	1	no	—
108	C₃⋊C₈	—	12	2	no	—
109	C₃⋊C₈	—	12	1	no	—
110	C₃⋊C₈	—	36	1	no	—
111	C₂×C₃²	—	2	1	no	—
112	C₂×C₃²	—	2	1	no	—
113	C₂×C₃²	—	2	1	no	—
114	C₂×C₃²	—	4	1	no	—
115	C₂×C₃²	#807	1	1	yes	characteristic
116	C₂×C₃²	#818	1	1	yes	characteristic
117	M₁₆	—	18	1	no	—
118	M₁₆	—	18	1	no	—
119	C₂×C₈	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
120	C₁₆	—	9	2	no	—
121	C₂²×C₃	—	3	1	no	—
122	C₂²×C₃	—	3	1	no	—
123	C₂²×C₃	—	6	1	no	—
124	C₂²×C₃	#3719	1	1	yes	characteristic
125	C₃×C₄	—	3	1	no	—
126	C₃×C₄	—	3	1	no	—
127	C₃×C₄	—	3	1	no	—
128	C₃×C₄	—	3	1	no	—
129	C₃×C₄	—	6	1	no	—
130	C₃×C₄	—	6	1	no	—
131	C₃×C₄	#3714	1	1	yes	characteristic
132	C₃×C₄	Dic₃⋊Dic₃	1	1	yes	characteristic
133	C₃²	—	2	1	no	—
134	C₃²	#4493	1	1	yes	characteristic, complemented
135	C₃²	#4506	1	1	yes	characteristic, complemented
136	C₂×C₄	#6200	1	1	yes	characteristic
137	C₈	—	36	1	no	—
138	C₈	—	36	1	no	—
139	C₈	#6196	1	1	yes	characteristic
140	C₈	((C₃⋊D₃)⋊C₃)⋊C₄	1	1	yes	characteristic
141	C₂×C₃	—	2	1	no	—
142	C₂×C₃	—	3	1	no	—
143	C₂×C₃	—	3	1	no	—
144	C₂×C₃	—	6	1	no	—
145	C₂×C₃	—	6	1	no	—
146	C₂×C₃	—	6	1	no	—
147	C₂×C₃	—	12	1	no	—
148	C₂×C₃	#63575	1	1	yes	characteristic
149	C₂²	#88439	1	1	yes	characteristic
150	C₄	#88436	1	1	yes	characteristic
151	C₄	#88448	1	1	yes	characteristic
152	C₃	—	3	1	no	—
153	C₃	—	3	1	no	—
154	C₃	—	6	1	no	—
155	C₃	#10589143	1	1	yes	characteristic
156	C₂	—	2	1	no	—
157	C₂	#11720306	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11720552

C₂

yes

maximal, characteristic

#63646⋊C₃

C₂

yes

maximal, characteristic

C₂×(C₃⋊(C₃⋊C₁₆))⋊C₃

C₂

yes

maximal, characteristic

—

—

C₂×C₈×H(3,𝔽₃)

C₂²

yes

characteristic

#88538

—

#3748⋊C₃

—

(C₃⋊(C₃⋊C₁₆))⋊C₃

C₄

yes

—

#63646

D₃

yes

characteristic

C₂×C₃⋊(C₃⋊C₁₆)

D₃

yes

—

—

—

—

C₂×C₃×C₃⋊C₁₆

—

C₂×C₄×H(3,𝔽₃)

D₄

yes

characteristic

C₈×H(3,𝔽₃)

C₂×C₄

yes

characteristic

C₈×H(3,𝔽₃)

Q₈

yes

characteristic

#6180

—

(C₃⋊(C₃⋊C₈))⋊C₃

—

#4506

—

#4493

—

#4493

—

C₂×C₃²×C₈

—

C₂×C₃²×C₈

C₂×D₃

yes

characteristic

C₂×C₃²×C₈

C₂×D₃

yes

—

—

—

—

—

C₃⋊(C₃⋊C₁₆)

Dic₃

yes

—

C₃×C₃⋊C₁₆

—

C₂²×H(3,𝔽₃)

Q₁₆

yes

characteristic

C₄×H(3,𝔽₃)

C₄⋊C₄

yes

characteristic

C₄×H(3,𝔽₃)

D₈

yes

characteristic

C₂×C₃×C₁₆

—

—

—

—

—

—

—

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

D₁₂

yes

characteristic

C₂×C₃²×C₄

#67

yes

characteristic

C₃²×C₈

—

C₃²×C₈

—

C₃²×C₈

Dic₆

yes

characteristic

C₃²×C₈

Dic₆

yes

characteristic

C₃²×C₈

C₄×D₃

yes

characteristic

C₃²×C₈

C₂×Dic₃

yes

characteristic

C₃⋊(C₃⋊C₈)

—

C₃×C₃⋊C₈

—

C₃×C₃⋊C₈

—

C₃×C₃⋊C₈

—

C₃×C₃⋊C₈

—

#368

—

Sylow

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

#107

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

D₃²

yes

—

—

—

—

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

—

C₂²×C₃²

Dic₁₂

yes

characteristic

C₂²×C₃²

#216

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

D₂₄

yes

characteristic

C₃²×C₄

Dic₃⋊C₄

yes

characteristic

C₃²×C₄

C₄⋊Dic₃

yes

characteristic

C₃²×C₄

#213

yes

characteristic

C₂×C₁₆

—

—

—

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#626

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

100

C₃×C₈

—

101

C₃×C₈

—

102

C₃×C₈

—

103

C₃×C₈

—

104

C₃×C₈

D₃×Dic₃

yes

characteristic

105

C₃×C₈

#627

yes

characteristic

106

C₃⋊C₈

—

107

C₃⋊C₈

—

108

C₃⋊C₈

—

109

C₃⋊C₈

—

110

C₃⋊C₈

—

111

C₂×C₃²

—

112

C₂×C₃²

—

113

C₂×C₃²

—

114

C₂×C₃²

—

115

C₂×C₃²

#807

yes

characteristic

116

C₂×C₃²

#818

yes

characteristic

117

M₁₆

—

118

M₁₆

—

119

C₂×C₈

((C₃⋊D₃)⋊C₃)⋊C₂

yes

characteristic

120

C₁₆

—

121

C₂²×C₃

—

122

C₂²×C₃

—

123

C₂²×C₃

—

124

C₂²×C₃

#3719

yes

characteristic

125

C₃×C₄

—

126

C₃×C₄

—

127

C₃×C₄

—

128

C₃×C₄

—

129

C₃×C₄

—

130

C₃×C₄

—

131

C₃×C₄

#3714

yes

characteristic

132

C₃×C₄

Dic₃⋊Dic₃

yes

characteristic

133

C₃²

—

134

C₃²

#4493

yes

characteristic, complemented

135

C₃²

#4506

yes

characteristic, complemented

136

C₂×C₄

#6200

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

#6196

yes

characteristic

140

C₈

((C₃⋊D₃)⋊C₃)⋊C₄

yes

characteristic

141

C₂×C₃

—

142

C₂×C₃

—

143

C₂×C₃

—

144

C₂×C₃

—

145

C₂×C₃

—

146

C₂×C₃

—

147

C₂×C₃

—

148

C₂×C₃

#63575

yes

characteristic

149

C₂²

#88439

yes

characteristic

150

C₄

#88436

yes

characteristic

151

C₄

#88448

yes

characteristic

152

C₃

—

153

C₃

—

154

C₃

—

155

C₃

#10589143

yes

characteristic

156

C₂

—

157

C₂

#11720306

yes

characteristic

	subgroup	G/H
center	C₄	#88436
commutator	C₈×H(3,𝔽₃)	C₂×C₄
Frattini	C₂×C₃×C₈	D₃²
Fitting, cosocle	C₂×C₈×H(3,𝔽₃)	C₂²
socle	C₂×C₃	#63575
prime radical	C₂²×H(3,𝔽₃)	Q₁₆
hypercenter	C₂×C₈	((C₃⋊D₃)⋊C₃)⋊C₂
hypocenter	H(3,𝔽₃)	#368
LCS	G ▷ C₈×H(3,𝔽₃) ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

subgroup

G/H

center

C₄

#88436

commutator

C₈×H(3,𝔽₃)

C₂×C₄

Frattini

C₂×C₃×C₈

D₃²

Fitting, cosocle

C₂×C₈×H(3,𝔽₃)

C₂²

socle

C₂×C₃

#63575

prime radical

C₂²×H(3,𝔽₃)

Q₁₆

hypercenter

C₂×C₈

((C₃⋊D₃)⋊C₃)⋊C₂

G ▷ C₈×H(3,𝔽₃) ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

N	H ≅ G/N	type
91. H(3,𝔽₃)	#368	semidirect
134. C₃²	#4493	semidirect
135. C₃²	#4506	semidirect

H ≅ G/N

type

91. H(3,𝔽₃)

#368

semidirect

134. C₃²

#4493

semidirect

135. C₃²

#4506

semidirect

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

subgroup	index
#11720552	2
#63646⋊C₃	2
C₂×(C₃⋊(C₃⋊C₁₆))⋊C₃	2
#10589144	3
#10589143	3

subgroup

index

C₂×(C₃⋊(C₃⋊C₁₆))⋊C₃

Subgroups — #13135073

Conjugacy classes of subgroups

Complements

Maximal subgroups