Group #13147103 — subgroups

Subgroups — #13147103

Order 1728.

206 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 48 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	C₄×C₉⋊SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	Dic₉⋊SL(2,𝔽₃)	C₂	1	2	yes	maximal
3	#5570⋊C₃	—	3	1	no	maximal
4	#10589672	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×C₉⋊SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₄⋊(3₋¹⁺²⋊C₄)	—	4	1	no	maximal
7	C₃×C₄×SL(2,𝔽₃)	D₃	1	1	yes	characteristic
8	Dic₃×SL(2,𝔽₃)	—	3	2	no	—
9	C₄×Q₈×C₉	C₂×C₃	1	1	yes	characteristic
10	Q₈×Dic₉	C₂×C₃	1	2	yes	—
11	#63452	—	3	2	no	—
12	#63431	—	3	1	no	—
13	C₄×Dic₁₈	—	3	1	no	—
14	C₄×Q₈⋊C₉	—	2	1	no	—
15	C₂×C₄×3₋¹⁺²	—	4	1	no	—
16	C₂×3₋¹⁺²⋊C₄	—	4	2	no	—
17	C₉⋊SL(2,𝔽₃)	Q₈	1	1	yes	characteristic, complemented
18	#5570	—	3	1	no	—
19	#557⋊C₃	—	9	1	no	—
20	C₂×C₃×SL(2,𝔽₃)	C₂×D₃	1	1	yes	characteristic
21	C₃×C₄⋊Dic₃	—	12	1	no	—
22	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
23	C₂×Dic₁₈	—	3	1	no	—
24	C₂×Q₈⋊C₉	—	2	1	no	—
25	C₉×C₄⋊C₄	—	3	1	no	—
26	C₄²×C₉	—	3	1	no	—
27	C₄⋊Dic₉	—	3	2	no	—
28	C₄⋊Dic₉	—	3	2	no	—
29	C₄⋊Dic₉	A₄	1	1	yes	characteristic
30	Dic₉⋊C₄	—	3	2	no	—
31	C₄×Dic₉	—	3	2	no	—
32	C₂²×3₋¹⁺²	—	4	1	no	—
33	C₄×3₋¹⁺²	—	4	2	no	—
34	3₋¹⁺²⋊C₄	—	8	2	no	—
35	C₃×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
36	Q₈×Dic₃	—	3	2	no	—
37	#890	—	9	2	no	—
38	#869	—	9	1	no	—
39	C₄×Dic₆	—	9	1	no	—
40	C₄×SL(2,𝔽₃)	—	3	1	no	—
41	C₄×SL(2,𝔽₃)	—	9	2	no	—
42	C₂×C₃²×C₄	—	4	1	no	—
43	C₂×C₃×Dic₃	—	12	2	no	—
44	C₃×SL(2,𝔽₃)	Dic₆	1	1	yes	characteristic
45	Q₈×C₉	—	3	1	no	—
46	Q₈×C₉	C₃×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₉	—	3	1	no	—
48	C₂×C₄×C₉	—	3	1	no	—
49	C₂×C₄×C₉	—	8	1	no	—
50	C₂×C₄×C₉	C₂×A₄	1	1	yes	characteristic
51	C₂×Dic₉	—	3	2	no	—
52	C₂×Dic₉	C₂×A₄	1	2	yes	—
53	Dic₁₈	—	6	2	no	—
54	Q₈⋊C₉	—	2	1	no	—
55	#557	—	9	1	no	Sylow
56	C₂×3₋¹⁺²	—	4	1	no	—
57	C₂×3₋¹⁺²	—	4	1	no	—
58	C₂×3₋¹⁺²	—	4	1	no	—
59	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
60	C₂×Dic₆	—	9	1	no	—
61	C₂×SL(2,𝔽₃)	—	3	1	no	—
62	C₃×C₄⋊C₄	—	3	1	no	—
63	C₃×C₄⋊C₄	—	36	1	no	—
64	C₃×C₄²	—	3	1	no	—
65	C₄⋊Dic₃	—	3	1	no	—
66	C₄⋊Dic₃	—	9	2	no	—
67	C₄⋊Dic₃	—	9	2	no	—
68	Dic₃⋊C₄	—	9	2	no	—
69	C₄×Dic₃	—	9	2	no	—
70	C₂²×C₃²	—	4	1	no	—
71	C₃²×C₄	—	4	2	no	—
72	C₃×Dic₃	—	24	2	no	—
73	C₂²×C₉	—	8	1	no	—
74	C₂²×C₉	C₂²×A₄	1	1	yes	characteristic
75	C₄×C₉	—	3	1	no	—
76	C₄×C₉	—	3	1	no	—
77	C₄×C₉	—	6	1	no	—
78	C₄×C₉	—	8	2	no	—
79	C₄×C₉	#41⋊C₃	1	2	yes	—
80	Dic₉	—	2	2	no	—
81	Dic₉	—	6	2	no	—
82	C₄⋊Q₈	—	27	1	no	—
83	#125	—	27	2	no	—
84	C₄×Q₈	—	9	2	no	—
85	C₄×Q₈	—	27	1	no	—
86	C₄×Q₈	AGL(1,ℤ/9)	1	1	yes	characteristic
87	3₋¹⁺²	—	4	1	no	Sylow
88	C₃×Q₈	—	3	1	no	—
89	C₃×Q₈	C₃×Dic₆	1	1	yes	characteristic
90	C₂×C₃×C₄	—	3	1	no	—
91	C₂×C₃×C₄	—	3	1	no	—
92	C₂×C₃×C₄	—	12	1	no	—
93	C₂×C₃×C₄	—	36	2	no	—
94	C₂×C₃×C₄	D₃×A₄	1	1	yes	characteristic
95	C₂×Dic₃	—	3	2	no	—
96	C₂×Dic₃	—	9	2	no	—
97	Dic₆	—	18	2	no	—
98	SL(2,𝔽₃)	—	3	1	no	—
99	C₂×C₃²	—	4	1	no	—
100	C₂×C₃²	—	4	1	no	—
101	C₂×C₃²	—	4	1	no	—
102	C₂×C₉	—	8	1	no	—
103	C₂×C₉	—	8	1	no	—
104	C₂×C₉	—	8	1	no	—
105	C₂×C₉	Q₈×A₄	1	1	yes	characteristic
106	C₂×C₉	C₂×#41⋊C₃	1	1	yes	characteristic
107	C₂×C₉	2₋¹⁺⁴⋊C₃	1	1	yes	characteristic
108	C₂×Q₈	—	27	1	no	—
109	C₂×Q₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
110	C₄⋊C₄	—	3	1	no	—
111	C₄⋊C₄	—	9	1	no	—
112	C₄⋊C₄	—	27	2	no	—
113	C₄⋊C₄	—	27	2	no	—
114	C₄⋊C₄	—	27	2	no	—
115	C₄²	—	3	1	no	—
116	C₄²	—	27	2	no	—
117	C₂²×C₃	—	12	1	no	—
118	C₂²×C₃	C₂×D₃×A₄	1	1	yes	characteristic
119	C₃×C₄	—	3	1	no	—
120	C₃×C₄	—	3	1	no	—
121	C₃×C₄	—	6	1	no	—
122	C₃×C₄	—	12	2	no	—
123	C₃×C₄	—	72	2	no	—
124	C₃×C₄	C₃⋊(#41⋊C₃)	1	2	yes	—
125	Dic₃	—	6	2	no	—
126	Dic₃	—	18	2	no	—
127	C₃²	—	4	1	no	—
128	C₉	—	8	1	no	—
129	C₉	#557⋊C₃	1	1	yes	characteristic, complemented
130	Q₈	—	3	1	no	—
131	Q₈	—	54	2	no	—
132	Q₈	3₋¹⁺²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄	—	3	1	no	—
134	C₂×C₄	—	3	1	no	—
135	C₂×C₄	—	9	2	no	—
136	C₂×C₄	—	27	2	no	—
137	C₂×C₄	D₉⋊A₄	1	1	yes	characteristic
138	C₂×C₃	—	12	1	no	—
139	C₂×C₃	—	12	1	no	—
140	C₂×C₃	—	12	1	no	—
141	C₂×C₃	A₄×Dic₆	1	1	yes	characteristic
142	C₂×C₃	C₂×C₃⋊(#41⋊C₃)	1	1	yes	characteristic
143	C₂×C₃	#1010⋊C₃	1	1	yes	characteristic
144	C₂²	C₂×D₉⋊A₄	1	1	yes	characteristic
145	C₄	—	3	1	no	—
146	C₄	—	3	1	no	—
147	C₄	—	6	1	no	—
148	C₄	—	18	2	no	—
149	C₄	—	54	2	no	—
150	C₄	#88614	1	2	yes	—
151	C₃	—	12	1	no	—
152	C₃	#5570⋊C₃	1	1	yes	characteristic
153	C₂	Dic₁₈⋊A₄	1	1	yes	characteristic
154	C₂	C₂×#88614	1	1	yes	characteristic
155	C₂	#11722414	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₉⋊SL(2,𝔽₃)

C₂

yes

maximal, characteristic

Dic₉⋊SL(2,𝔽₃)

C₂

yes

—

C₃

yes

maximal, characteristic, complemented

C₂×C₉⋊SL(2,𝔽₃)

C₂²

yes

characteristic

C₄⋊(3₋¹⁺²⋊C₄)

—

maximal

C₃×C₄×SL(2,𝔽₃)

D₃

yes

characteristic

Dic₃×SL(2,𝔽₃)

—

C₄×Q₈×C₉

C₂×C₃

yes

characteristic

Q₈×Dic₉

C₂×C₃

yes

—

#63452

—

#63431

—

C₄×Dic₁₈

—

C₄×Q₈⋊C₉

—

C₂×C₄×3₋¹⁺²

—

C₂×3₋¹⁺²⋊C₄

—

C₉⋊SL(2,𝔽₃)

Q₈

yes

characteristic, complemented

#5570

—

#557⋊C₃

—

C₂×C₃×SL(2,𝔽₃)

C₂×D₃

yes

characteristic

C₃×C₄⋊Dic₃

—

C₂×Q₈×C₉

C₂²×C₃

yes

characteristic

C₂×Dic₁₈

—

C₂×Q₈⋊C₉

—

C₉×C₄⋊C₄

—

C₄²×C₉

—

C₄⋊Dic₉

—

C₄⋊Dic₉

—

C₄⋊Dic₉

A₄

yes

characteristic

Dic₉⋊C₄

—

C₄×Dic₉

—

C₂²×3₋¹⁺²

—

C₄×3₋¹⁺²

—

3₋¹⁺²⋊C₄

—

C₃×C₄×Q₈

C₃×D₃

yes

characteristic

Q₈×Dic₃

—

#890

—

#869

—

C₄×Dic₆

—

C₄×SL(2,𝔽₃)

—

C₄×SL(2,𝔽₃)

—

C₂×C₃²×C₄

—

C₂×C₃×Dic₃

—

C₃×SL(2,𝔽₃)

Dic₆

yes

characteristic

Q₈×C₉

—

Q₈×C₉

C₃×Q₈

yes

characteristic

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×A₄

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

C₂×A₄

yes

—

Dic₁₈

—

Q₈⋊C₉

—

#557

—

Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×C₃×Q₈

C₂×C₃×D₃

yes

characteristic

C₂×Dic₆

—

C₂×SL(2,𝔽₃)

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂²×C₃²

—

C₃²×C₄

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

C₂²×A₄

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

#41⋊C₃

yes

—

Dic₉

—

Dic₉

—

C₄⋊Q₈

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

AGL(1,ℤ/9)

yes

characteristic

3₋¹⁺²

—

Sylow

C₃×Q₈

—

C₃×Q₈

C₃×Dic₆

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

D₃×A₄

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

SL(2,𝔽₃)

—

C₂×C₃²

—

100

C₂×C₃²

—

101

C₂×C₃²

—

102

C₂×C₉

—

103

C₂×C₉

—

104

C₂×C₉

—

105

C₂×C₉

Q₈×A₄

yes

characteristic

106

C₂×C₉

C₂×#41⋊C₃

yes

characteristic

107

C₂×C₉

2₋¹⁺⁴⋊C₃

yes

characteristic

108

C₂×Q₈

—

109

C₂×Q₈

C₂×AGL(1,ℤ/9)

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₂²×C₃

—

118

C₂²×C₃

C₂×D₃×A₄

yes

characteristic

119

C₃×C₄

—

120

C₃×C₄

—

121

C₃×C₄

—

122

C₃×C₄

—

123

C₃×C₄

—

124

C₃×C₄

C₃⋊(#41⋊C₃)

yes

—

125

Dic₃

—

126

Dic₃

—

127

C₃²

—

128

C₉

—

129

C₉

#557⋊C₃

yes

characteristic, complemented

130

Q₈

—

131

Q₈

—

132

Q₈

3₋¹⁺²⋊Q₈

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

D₉⋊A₄

yes

characteristic

138

C₂×C₃

—

139

C₂×C₃

—

140

C₂×C₃

—

141

C₂×C₃

A₄×Dic₆

yes

characteristic

142

C₂×C₃

C₂×C₃⋊(#41⋊C₃)

yes

characteristic

143

C₂×C₃

#1010⋊C₃

yes

characteristic

144

C₂²

C₂×D₉⋊A₄

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#88614

yes

—

151

C₃

—

152

C₃

#5570⋊C₃

yes

characteristic

153

C₂

Dic₁₈⋊A₄

yes

characteristic

154

C₂

C₂×#88614

yes

characteristic

155

C₂

#11722414

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₉⋊A₄
commutator, cosocle	C₂×Q₈×C₉	C₂²×C₃
Frattini, socle	C₂²×C₃	C₂×D₃×A₄
Fitting	C₄×Q₈×C₉	C₂×C₃
prime radical	C₂×C₃×SL(2,𝔽₃)	C₂×D₃
hypercenter	C₂×C₄	D₉⋊A₄
hypocenter	Q₈×C₉	C₃×Q₈
LCS	G ▷ C₂×Q₈×C₉ ▷ Q₈×C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄

subgroup

G/H

center

C₂²

C₂×D₉⋊A₄

commutator, cosocle

C₂×Q₈×C₉

C₂²×C₃

Frattini, socle

C₂²×C₃

C₂×D₃×A₄

Fitting

C₄×Q₈×C₉

C₂×C₃

prime radical

C₂×C₃×SL(2,𝔽₃)

C₂×D₃

hypercenter

C₂×C₄

D₉⋊A₄

hypocenter

Q₈×C₉

C₃×Q₈

LCS

G ▷ C₂×Q₈×C₉ ▷ Q₈×C₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄

N	H ≅ G/N	type
4. #10589672	C₃	semidirect
17. C₉⋊SL(2,𝔽₃)	Q₈	semidirect
129. C₉	#557⋊C₃	semidirect

H ≅ G/N

type

4. #10589672

C₃

semidirect

17. C₉⋊SL(2,𝔽₃)

Q₈

semidirect

129. C₉

#557⋊C₃

semidirect

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

subgroup	index
C₄×C₉⋊SL(2,𝔽₃)	2
Dic₉⋊SL(2,𝔽₃)	2
#5570⋊C₃	3
#10589672	3
C₄⋊(3₋¹⁺²⋊C₄)	4

subgroup

index

C₄×C₉⋊SL(2,𝔽₃)

Dic₉⋊SL(2,𝔽₃)

#5570⋊C₃

#10589672

C₄⋊(3₋¹⁺²⋊C₄)

Subgroups — #13147103

Conjugacy classes of subgroups

Complements

Maximal subgroups