Group #13135633 — subgroups

Subgroups — C₃×#10588223 #13135633

Order 1728.

260 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 102 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	C₃×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊(C₉⋊C₈)	C₂	1	2	yes	maximal
3	C₃×#4546	—	3	1	no	maximal
4	#10588223	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂×C₃×Q₈×C₉	C₄	1	2	yes	—
6	C₃×C₉×C₄⋊C₄	—	2	2	no	—
7	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₉⋊C₈	—	2	2	no	—
9	C₃²×C₄⋊Q₈	D₃	1	1	yes	characteristic
10	C₃×C₄⋊(C₃⋊C₈)	—	3	2	no	—
11	C₉×C₄⋊Q₈	—	2	1	no	—
12	C₉×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
13	C₄⋊(C₉⋊C₈)	C₂×C₃	1	2	yes	—
14	C₃×Q₈×C₉	—	4	2	no	—
15	C₂×C₃×C₄×C₉	—	2	2	no	—
16	C₂×C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
18	C₃×C₉⋊C₈	—	4	2	no	—
19	C₃×#333	—	9	1	no	—
20	#4546	—	3	1	no	—
21	C₂×C₃²×Q₈	Dic₃	1	2	yes	—
22	C₃²×C₄⋊C₄	—	2	2	no	—
23	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
24	C₂×C₃×C₃⋊C₈	—	6	2	no	—
25	C₂×Q₈×C₉	—	2	2	no	—
26	C₂×Q₈×C₉	C₃×C₄	1	2	yes	—
27	C₉×C₄⋊C₄	—	2	4	no	—
28	C₉×C₄⋊C₄	—	2	2	no	—
29	C₄²×C₉	—	2	1	no	—
30	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
31	C₂×C₉⋊C₈	—	2	2	no	—
32	C₂²×C₃×C₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₃×C₄×C₉	—	2	1	no	—
34	C₃×C₄×C₉	—	4	2	no	—
35	C₃×C₄×C₉	SD₁₆	1	2	yes	—
36	C₃×C₄×C₉	Q₁₆	1	2	yes	—
37	C₃×C₄⋊Q₈	—	2	1	no	—
38	C₃×C₄⋊Q₈	D₉	1	1	yes	characteristic
39	C₃×C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic
40	C₃×C₄⋊C₈	—	9	2	no	—
41	C₄⋊(C₃⋊C₈)	—	3	2	no	—
42	C₃²×Q₈	—	4	2	no	—
43	C₂×C₃²×C₄	—	2	2	no	—
44	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
45	C₂×C₃²×C₄	#67	1	2	yes	—
46	C₃×C₃⋊C₈	—	12	2	no	—
47	Q₈×C₉	—	4	2	no	—
48	Q₈×C₉	—	4	4	no	—
49	C₂×C₄×C₉	—	2	4	no	—
50	C₂×C₄×C₉	—	2	2	no	—
51	C₂×C₄×C₉	—	2	1	no	—
52	C₂×C₄×C₉	—	2	2	no	—
53	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
54	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
55	C₉⋊C₈	—	4	2	no	—
56	#333	—	9	1	no	Sylow
57	C₂×C₃×C₉	#100	1	1	yes	characteristic
58	C₂×C₃×C₉	#103	1	2	yes	—
59	C₂×C₃×Q₈	—	2	2	no	—
60	C₂×C₃×Q₈	Dic₉	1	2	yes	—
61	C₂×C₃×Q₈	C₃×Dic₃	1	2	yes	—
62	C₂×C₃×C₈	—	18	2	no	—
63	C₃×C₄⋊C₄	—	2	4	no	—
64	C₃×C₄⋊C₄	—	2	2	no	—
65	C₃×C₄⋊C₄	—	2	2	no	—
66	C₃×C₄²	—	2	1	no	—
67	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
68	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
69	C₂×C₃⋊C₈	—	6	2	no	—
70	C₂²×C₃²	#217	1	1	yes	characteristic
71	C₃²×C₄	—	2	1	no	—
72	C₃²×C₄	—	4	2	no	—
73	C₃²×C₄	#215	1	2	yes	—
74	C₃²×C₄	#216	1	2	yes	—
75	C₂²×C₉	—	2	1	no	—
76	C₂²×C₉	C₃×C₂²⋊C₄	1	1	yes	characteristic
77	C₄×C₉	—	2	2	no	—
78	C₄×C₉	—	2	1	no	—
79	C₄×C₉	—	2	2	no	—
80	C₄×C₉	—	2	2	no	—
81	C₄×C₉	—	4	4	no	—
82	C₄×C₉	—	4	2	no	—
83	C₄×C₉	C₃×SD₁₆	1	2	yes	—
84	C₄×C₉	C₃×Q₁₆	1	2	yes	—
85	C₄⋊Q₈	C₃×D₉	1	1	yes	characteristic
86	C₄⋊C₈	—	9	2	no	—
87	C₃×C₉	#333	1	1	yes	characteristic, complemented, Sylow
88	C₃×Q₈	—	4	2	no	—
89	C₃×Q₈	—	4	2	no	—
90	C₃×Q₈	—	4	4	no	—
91	C₂×C₃×C₄	—	2	4	no	—
92	C₂×C₃×C₄	—	2	2	no	—
93	C₂×C₃×C₄	—	2	2	no	—
94	C₂×C₃×C₄	—	2	1	no	—
95	C₂×C₃×C₄	—	2	2	no	—
96	C₂×C₃×C₄	C₂×Dic₉	1	1	yes	characteristic
97	C₂×C₃×C₄	#611	1	2	yes	—
98	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
99	C₂×C₃×C₄	C₃×#67	1	2	yes	—
100	C₃×C₈	—	36	2	no	—
101	C₃⋊C₈	—	12	2	no	—
102	C₂×C₃²	#833	1	1	yes	characteristic
103	C₂×C₃²	#835	1	2	yes	—
104	C₂×C₉	—	2	1	no	—
105	C₂×C₉	—	2	2	no	—
106	C₂×C₉	C₃×#100	1	1	yes	characteristic
107	C₂×C₉	C₃×#103	1	2	yes	—
108	C₂×Q₈	C₃×Dic₉	1	2	yes	—
109	C₂×C₈	—	18	2	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
112	C₂²×C₃	—	2	1	no	—
113	C₂²×C₃	#3676	1	1	yes	characteristic
114	C₂²×C₃	C₃×#217	1	1	yes	characteristic
115	C₃×C₄	—	2	2	no	—
116	C₃×C₄	—	2	1	no	—
117	C₃×C₄	—	2	1	no	—
118	C₃×C₄	—	2	2	no	—
119	C₃×C₄	—	2	2	no	—
120	C₃×C₄	—	4	4	no	—
121	C₃×C₄	—	4	2	no	—
122	C₃×C₄	—	4	2	no	—
123	C₃×C₄	#3674	1	2	yes	—
124	C₃×C₄	#3675	1	2	yes	—
125	C₃×C₄	C₃×#215	1	2	yes	—
126	C₃×C₄	C₃×#216	1	2	yes	—
127	C₃²	#4546	1	1	yes	characteristic
128	C₉	—	2	1	no	—
129	C₉	C₃×#333	1	1	yes	characteristic, complemented
130	Q₈	—	4	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	C₂×C₃×Dic₉	1	1	yes	characteristic
133	C₂×C₄	C₃×#611	1	2	yes	—
134	C₈	—	36	2	no	—
135	C₂×C₃	—	2	1	no	—
136	C₂×C₃	—	2	2	no	—
137	C₂×C₃	#63391	1	1	yes	characteristic
138	C₂×C₃	#63395	1	2	yes	—
139	C₂×C₃	C₃×#833	1	1	yes	characteristic
140	C₂×C₃	C₃×#835	1	2	yes	—
141	C₂²	C₃×#3676	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	4	2	no	—
144	C₄	C₃×#3674	1	2	yes	—
145	C₄	C₃×#3675	1	2	yes	—
146	C₃	—	2	1	no	—
147	C₃	#10588223	1	1	yes	characteristic, direct product factor
148	C₃	C₃×#4546	1	1	yes	characteristic
149	C₂	C₃×#63391	1	1	yes	characteristic
150	C₂	C₃×#63395	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₃×C₄⋊(C₉⋊C₈)

C₂

yes

—

C₃

yes

maximal, characteristic, direct product factor

C₂×C₃×Q₈×C₉

C₄

yes

—

C₃×C₉×C₄⋊C₄

—

C₃×C₄²×C₉

C₂²

yes

characteristic

C₂×C₃×C₉⋊C₈

—

C₃²×C₄⋊Q₈

D₃

yes

characteristic

C₃×C₄⋊(C₃⋊C₈)

—

C₉×C₄⋊Q₈

—

C₉×C₄⋊Q₈

C₂×C₃

yes

characteristic

C₄⋊(C₉⋊C₈)

C₂×C₃

yes

—

C₃×Q₈×C₉

—

C₂×C₃×C₄×C₉

—

C₂×C₃×C₄×C₉

C₂×C₄

yes

characteristic

C₂×C₃×C₄×C₉

D₄

yes

—

C₃×C₉⋊C₈

—

C₃×#333

—

#4546

—

C₂×C₃²×Q₈

Dic₃

yes

—

C₃²×C₄⋊C₄

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₂×C₃×C₃⋊C₈

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

C₃×C₄

yes

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₄²×C₉

—

C₄²×C₉

C₂²×C₃

yes

characteristic

C₂×C₉⋊C₈

—

C₂²×C₃×C₉

C₂²⋊C₄

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

SD₁₆

yes

—

C₃×C₄×C₉

Q₁₆

yes

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

D₉

yes

characteristic

C₃×C₄⋊Q₈

C₃×D₃

yes

characteristic

C₃×C₄⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×Dic₃

yes

characteristic

C₂×C₃²×C₄

#67

yes

—

C₃×C₃⋊C₈

—

Q₈×C₉

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

characteristic

C₂×C₄×C₉

C₃×D₄

yes

—

C₉⋊C₈

—

—

C₂×C₃×C₉

yes

C₂×C₃×C₉

yes

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

Dic₉

yes

—

C₂×C₃×Q₈

C₃×Dic₃

yes

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×D₉

yes

characteristic

C₃×C₄²

C₂×C₃×D₃

yes

characteristic

C₂×C₃⋊C₈

—

C₂²×C₃²

#217

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

#215

yes

—

C₃²×C₄

#216

yes

—

C₂²×C₉

—

C₂²×C₉

C₃×C₂²⋊C₄

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₃×SD₁₆

yes

—

C₄×C₉

C₃×Q₁₆

yes

—

C₄⋊Q₈

C₃×D₉

yes

characteristic

C₄⋊C₈

—

C₃×C₉

#333

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×Dic₉

yes

characteristic

C₂×C₃×C₄

#611

yes

—

C₂×C₃×C₄

C₂×C₃×Dic₃

yes

characteristic

C₂×C₃×C₄

C₃×#67

yes

—

100

C₃×C₈

—

101

C₃⋊C₈

—

102

C₂×C₃²

#833

yes

characteristic

103

C₂×C₃²

#835

yes

—

104

C₂×C₉

—

105

C₂×C₉

—

106

C₂×C₉

C₃×#100

yes

characteristic

107

C₂×C₉

C₃×#103

yes

—

108

C₂×Q₈

C₃×Dic₉

yes

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₄²

C₂×C₃×D₉

yes

characteristic

112

C₂²×C₃

—

113

C₂²×C₃

#3676

yes

characteristic

114

C₂²×C₃

C₃×#217

yes

characteristic

115

C₃×C₄

—

116

C₃×C₄

—

117

C₃×C₄

—

118

C₃×C₄

—

119

C₃×C₄

—

120

C₃×C₄

—

121

C₃×C₄

—

122

C₃×C₄

—

123

C₃×C₄

#3674

yes

—

124

C₃×C₄

#3675

yes

—

125

C₃×C₄

C₃×#215

yes

—

126

C₃×C₄

C₃×#216

yes

—

127

C₃²

#4546

yes

characteristic

128

C₉

—

129

C₉

C₃×#333

yes

characteristic, complemented

130

Q₈

—

131

C₂×C₄

—

132

C₂×C₄

C₂×C₃×Dic₉

yes

characteristic

133

C₂×C₄

C₃×#611

yes

—

134

C₈

—

135

C₂×C₃

—

136

C₂×C₃

—

137

C₂×C₃

#63391

yes

characteristic

138

C₂×C₃

#63395

yes

—

139

C₂×C₃

C₃×#833

yes

characteristic

140

C₂×C₃

C₃×#835

yes

—

141

C₂²

C₃×#3676

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

C₃×#3674

yes

—

145

C₄

C₃×#3675

yes

—

146

C₃

—

147

C₃

#10588223

yes

characteristic, direct product factor

148

C₃

C₃×#4546

yes

characteristic

149

C₂

C₃×#63391

yes

characteristic

150

C₂

C₃×#63395

yes

—

	subgroup	G/H
center	C₂²×C₃	#3676
commutator	C₂×C₄×C₉	C₂×C₃×C₄
Frattini	C₃×C₄²	C₂×C₃×D₃
Fitting	C₃×C₉×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₃²	#217
cosocle	C₄²×C₉	C₂²×C₃
hypercenter	C₃×C₄⋊Q₈	D₉
hypocenter	C₉	C₃×#333
LCS	G ▷ C₂×C₄×C₉ ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄⋊Q₈

subgroup

G/H

center

C₂²×C₃

#3676

commutator

C₂×C₄×C₉

C₂×C₃×C₄

Frattini

C₃×C₄²

C₂×C₃×D₃

Fitting

C₃×C₉×C₄⋊Q₈

C₂

socle, prime radical

C₂²×C₃²

#217

cosocle

C₄²×C₉

C₂²×C₃

hypercenter

C₃×C₄⋊Q₈

D₉

hypocenter

C₉

C₃×#333

LCS

G ▷ C₂×C₄×C₉ ▷ C₂²×C₉ ▷ C₉

UCS

C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄⋊Q₈

N	H ≅ G/N	type
4. #10588223	C₃	direct (N × H)
87. C₃×C₉	#333	semidirect
129. C₉	C₃×#333	semidirect
147. C₃	#10588223	direct (N × H)

H ≅ G/N

type

4. #10588223

C₃

direct (N × H)

87. C₃×C₉

#333

semidirect

129. C₉

C₃×#333

semidirect

147. C₃

#10588223

direct (N × H)

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

subgroup	index
C₃×C₉×C₄⋊Q₈	2
C₃×C₄⋊(C₉⋊C₈)	2
C₃×#4546	3
#10588223	3

subgroup

index

C₃×C₉×C₄⋊Q₈

C₃×C₄⋊(C₉⋊C₈)

C₃×#4546

#10588223

Subgroups — C₃×#10588223 #13135633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#10588223 #13135633

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588223 #13135633