Group #13135990 — subgroups

Subgroups — 3₋¹⁺²×#329 #13135990

Order 1728.

280 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 105 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	3₋¹⁺²×#124	C₂	1	1	yes	maximal, characteristic
2	3₋¹⁺²×#97	C₂	1	2	yes	maximal, complemented
3	C₃²×#329	C₃	1	1	yes	maximal, characteristic
4	C₉×#329	C₃	1	3	yes	maximal, complemented
5	C₂×Q₈×3₋¹⁺²	C₄	1	1	yes	characteristic
6	C₂×D₄×3₋¹⁺²	C₄	1	1	yes	characteristic
7	C₂×C₈×3₋¹⁺²	—	2	2	no	—
8	C₂²⋊C₄×3₋¹⁺²	—	2	2	no	—
9	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
10	C₃²×#124	C₂×C₃	1	1	yes	characteristic
11	C₃²×#97	C₂×C₃	1	2	yes	—
12	C₉×#124	C₂×C₃	1	3	yes	—
13	C₉×#97	C₂×C₃	1	6	yes	complemented
14	C₂³×3₋¹⁺²	—	2	1	no	—
15	Q₈×3₋¹⁺²	—	4	1	no	—
16	D₄×3₋¹⁺²	—	4	1	no	—
17	C₂×C₄×3₋¹⁺²	—	2	1	no	—
18	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
20	C₈×3₋¹⁺²	—	2	4	no	—
21	C₃×#329	—	3	1	no	—
22	C₃×#329	C₃²	1	1	yes	characteristic
23	C₂×C₃²×Q₈	C₃×C₄	1	1	yes	characteristic
24	C₂×C₃²×D₄	C₃×C₄	1	1	yes	characteristic
25	C₂×C₃²×C₈	—	2	2	no	—
26	C₃²×C₂²⋊C₄	—	2	2	no	—
27	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
28	C₂×Q₈×C₉	C₃×C₄	1	3	yes	—
29	C₂×D₄×C₉	C₃×C₄	1	3	yes	—
30	C₂×C₈×C₉	—	2	6	no	—
31	C₉×C₂²⋊C₄	—	2	6	no	—
32	C₄²×C₉	C₂²×C₃	1	3	yes	—
33	C₂²×3₋¹⁺²	—	4	1	no	—
34	C₂²×3₋¹⁺²	—	4	2	no	—
35	C₂²×3₋¹⁺²	C₂²⋊C₄	1	1	yes	characteristic
36	C₄×3₋¹⁺²	—	2	1	no	—
37	C₄×3₋¹⁺²	—	2	2	no	—
38	C₄×3₋¹⁺²	—	4	1	no	—
39	C₃×#124	—	3	1	no	—
40	C₃×#124	C₂×C₃²	1	1	yes	characteristic
41	C₃×#97	—	3	2	no	—
42	C₃×#97	C₂×C₃²	1	2	yes	—
43	C₂³×C₃²	—	2	1	no	—
44	C₃²×Q₈	—	4	1	no	—
45	C₃²×D₄	—	4	1	no	—
46	C₂×C₃²×C₄	—	2	1	no	—
47	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
48	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
49	C₂³×C₉	—	2	3	no	—
50	C₃²×C₈	—	2	4	no	—
51	Q₈×C₉	—	4	3	no	—
52	D₄×C₉	—	4	3	no	—
53	C₂×C₄×C₉	—	2	3	no	—
54	C₂×C₄×C₉	C₂×C₃×C₄	1	3	yes	—
55	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
56	C₈×C₉	—	2	12	no	—
57	#329	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
58	C₂×3₋¹⁺²	—	8	1	no	—
59	C₂×3₋¹⁺²	#100	1	1	yes	characteristic
60	C₂×3₋¹⁺²	#104	1	2	yes	—
61	C₂×C₃×Q₈	—	3	1	no	—
62	C₂×C₃×Q₈	C₃²×C₄	1	1	yes	characteristic
63	C₂×C₃×D₄	—	3	1	no	—
64	C₂×C₃×D₄	C₃²×C₄	1	1	yes	characteristic
65	C₂×C₃×C₈	—	2	2	no	—
66	C₂×C₃×C₈	—	6	2	no	—
67	C₃×C₂²⋊C₄	—	2	2	no	—
68	C₃×C₂²⋊C₄	—	6	2	no	—
69	C₃×C₄²	—	3	1	no	—
70	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
71	C₂²×C₃²	—	4	1	no	—
72	C₂²×C₃²	—	4	2	no	—
73	C₂²×C₃²	C₃×C₂²⋊C₄	1	1	yes	characteristic
74	C₃²×C₄	—	2	1	no	—
75	C₃²×C₄	—	2	2	no	—
76	C₃²×C₄	—	4	1	no	—
77	C₂²×C₉	—	4	3	no	—
78	C₂²×C₉	—	4	6	no	—
79	C₂²×C₉	C₃×C₂²⋊C₄	1	3	yes	—
80	C₄×C₉	—	2	3	no	—
81	C₄×C₉	—	2	6	no	—
82	C₄×C₉	—	4	3	no	—
83	#124	C₂×3₋¹⁺²	1	1	yes	characteristic
84	#97	C₂×3₋¹⁺²	1	2	yes	complemented
85	3₋¹⁺²	#329	1	1	yes	characteristic, direct product factor, Sylow
86	C₂³×C₃	—	2	1	no	—
87	C₂³×C₃	—	6	1	no	—
88	C₃×Q₈	—	4	1	no	—
89	C₃×Q₈	—	12	1	no	—
90	C₃×D₄	—	4	1	no	—
91	C₃×D₄	—	12	1	no	—
92	C₂×C₃×C₄	—	2	1	no	—
93	C₂×C₃×C₄	—	3	1	no	—
94	C₂×C₃×C₄	—	3	2	no	—
95	C₂×C₃×C₄	—	6	1	no	—
96	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
97	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
98	C₃×C₈	—	2	4	no	—
99	C₃×C₈	—	6	4	no	—
100	C₂×C₃²	—	8	1	no	—
101	C₂×C₃²	C₃×#100	1	1	yes	characteristic
102	C₂×C₃²	C₃×#104	1	2	yes	—
103	C₂×C₉	—	8	3	no	—
104	C₂×C₉	C₃×#100	1	3	yes	—
105	C₂×C₉	C₃×#104	1	6	yes	—
106	C₂×Q₈	C₄×3₋¹⁺²	1	1	yes	characteristic
107	C₂×D₄	C₄×3₋¹⁺²	1	1	yes	characteristic
108	C₂×C₈	—	2	2	no	—
109	C₂²⋊C₄	—	2	2	no	—
110	C₄²	C₂²×3₋¹⁺²	1	1	yes	characteristic
111	C₂²×C₃	—	3	1	no	—
112	C₂²×C₃	—	4	1	no	—
113	C₂²×C₃	—	4	2	no	—
114	C₂²×C₃	—	12	1	no	—
115	C₂²×C₃	—	12	2	no	—
116	C₂²×C₃	C₃²×C₂²⋊C₄	1	1	yes	characteristic
117	C₃×C₄	—	2	1	no	—
118	C₃×C₄	—	2	2	no	—
119	C₃×C₄	—	4	1	no	—
120	C₃×C₄	—	6	2	no	—
121	C₃×C₄	—	6	1	no	—
122	C₃×C₄	—	12	1	no	—
123	C₃²	C₃×#329	1	1	yes	characteristic
124	C₉	C₃×#329	1	3	yes	complemented
125	C₂³	—	2	1	no	—
126	Q₈	—	4	1	no	—
127	D₄	—	4	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
130	C₂×C₄	D₄×3₋¹⁺²	1	2	yes	—
131	C₈	—	2	4	no	—
132	C₂×C₃	—	3	1	no	—
133	C₂×C₃	—	3	2	no	—
134	C₂×C₃	—	8	1	no	—
135	C₂×C₃	—	24	1	no	—
136	C₂×C₃	C₃²×#100	1	1	yes	characteristic
137	C₂×C₃	C₃²×#104	1	2	yes	—
138	C₂²	—	4	1	no	—
139	C₂²	—	4	2	no	—
140	C₂²	C₂²⋊C₄×3₋¹⁺²	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	2	2	no	—
143	C₄	—	4	1	no	—
144	C₃	—	3	1	no	—
145	C₃	C₃²×#329	1	1	yes	characteristic
146	C₂	—	8	1	no	—
147	C₂	3₋¹⁺²×#100	1	1	yes	characteristic
148	C₂	3₋¹⁺²×#104	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²×#124

C₂

yes

maximal, characteristic

3₋¹⁺²×#97

C₂

yes

maximal, complemented

C₃²×#329

C₃

yes

maximal, characteristic

C₉×#329

C₃

yes

maximal, complemented

C₂×Q₈×3₋¹⁺²

C₄

yes

characteristic

C₂×D₄×3₋¹⁺²

C₄

yes

characteristic

C₂×C₈×3₋¹⁺²

—

C₂²⋊C₄×3₋¹⁺²

—

C₄²×3₋¹⁺²

C₂²

yes

characteristic

C₃²×#124

C₂×C₃

yes

characteristic

C₃²×#97

C₂×C₃

yes

—

C₉×#124

C₂×C₃

yes

—

C₉×#97

C₂×C₃

yes

complemented

C₂³×3₋¹⁺²

—

Q₈×3₋¹⁺²

—

D₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

characteristic

C₂×C₄×3₋¹⁺²

D₄

yes

—

C₈×3₋¹⁺²

—

C₃×#329

—

C₃×#329

C₃²

yes

characteristic

C₂×C₃²×Q₈

C₃×C₄

yes

characteristic

C₂×C₃²×D₄

C₃×C₄

yes

characteristic

C₂×C₃²×C₈

—

C₃²×C₂²⋊C₄

—

C₃²×C₄²

C₂²×C₃

yes

characteristic

C₂×Q₈×C₉

C₃×C₄

yes

—

C₂×D₄×C₉

C₃×C₄

yes

—

C₂×C₈×C₉

—

C₉×C₂²⋊C₄

—

C₄²×C₉

C₂²×C₃

yes

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

C₂²⋊C₄

yes

characteristic

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₃×#124

—

C₃×#124

C₂×C₃²

yes

characteristic

C₃×#97

—

C₃×#97

C₂×C₃²

yes

—

C₂³×C₃²

—

C₃²×Q₈

—

C₃²×D₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×C₃×C₄

yes

characteristic

C₂×C₃²×C₄

C₃×D₄

yes

—

C₂³×C₉

—

C₃²×C₈

—

Q₈×C₉

—

D₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

—

C₂×C₄×C₉

C₃×D₄

yes

—

C₈×C₉

—

#329

3₋¹⁺²

yes

characteristic, direct product factor, Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

#100

yes

characteristic

C₂×3₋¹⁺²

#104

yes

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₃²×C₄

yes

characteristic

C₂×C₃×D₄

—

C₂×C₃×D₄

C₃²×C₄

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₂²⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂²×C₃²

yes

characteristic

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₃×C₂²⋊C₄

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₃×C₂²⋊C₄

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

#124

C₂×3₋¹⁺²

yes

characteristic

#97

C₂×3₋¹⁺²

yes

complemented

3₋¹⁺²

#329

yes

characteristic, direct product factor, Sylow

C₂³×C₃

—

C₂³×C₃

—

C₃×Q₈

—

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃²×C₄

yes

characteristic

C₂×C₃×C₄

C₃²×D₄

yes

—

C₃×C₈

—

C₃×C₈

—

100

C₂×C₃²

—

101

C₂×C₃²

C₃×#100

yes

characteristic

102

C₂×C₃²

C₃×#104

yes

—

103

C₂×C₉

—

104

C₂×C₉

C₃×#100

yes

—

105

C₂×C₉

C₃×#104

yes

—

106

C₂×Q₈

C₄×3₋¹⁺²

yes

characteristic

107

C₂×D₄

C₄×3₋¹⁺²

yes

characteristic

108

C₂×C₈

—

109

C₂²⋊C₄

—

110

C₄²

C₂²×3₋¹⁺²

yes

characteristic

111

C₂²×C₃

—

112

C₂²×C₃

—

113

C₂²×C₃

—

114

C₂²×C₃

—

115

C₂²×C₃

—

116

C₂²×C₃

C₃²×C₂²⋊C₄

yes

characteristic

117

C₃×C₄

—

118

C₃×C₄

—

119

C₃×C₄

—

120

C₃×C₄

—

121

C₃×C₄

—

122

C₃×C₄

—

123

C₃²

C₃×#329

yes

characteristic

124

C₉

C₃×#329

yes

complemented

125

C₂³

—

126

Q₈

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₄×3₋¹⁺²

yes

characteristic

130

C₂×C₄

D₄×3₋¹⁺²

yes

—

131

C₈

—

132

C₂×C₃

—

133

C₂×C₃

—

134

C₂×C₃

—

135

C₂×C₃

—

136

C₂×C₃

C₃²×#100

yes

characteristic

137

C₂×C₃

C₃²×#104

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

C₂²⋊C₄×3₋¹⁺²

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₃

—

145

C₃

C₃²×#329

yes

characteristic

146

C₂

—

147

C₂

3₋¹⁺²×#100

yes

characteristic

148

C₂

3₋¹⁺²×#104

yes

—

	subgroup	G/H
center, socle	C₂²×C₃	C₃²×C₂²⋊C₄
commutator	C₂×C₃×C₄	C₂×C₃²×C₄
Frattini, cosocle	C₃×C₄²	C₂²×C₃²
prime radical	C₂×C₃²×D₄	C₃×C₄
LCS	G ▷ C₂×C₃×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₄²×3₋¹⁺² ◁ G

subgroup

G/H

center, socle

C₂²×C₃

C₃²×C₂²⋊C₄

commutator

C₂×C₃×C₄

C₂×C₃²×C₄

Frattini, cosocle

C₃×C₄²

C₂²×C₃²

prime radical

C₂×C₃²×D₄

C₃×C₄

LCS

G ▷ C₂×C₃×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₃ ◁ C₄²×3₋¹⁺² ◁ G

N	H ≅ G/N	type
2. 3₋¹⁺²×#97	C₂	semidirect
4. C₉×#329	C₃	semidirect
13. C₉×#97	C₂×C₃	semidirect
57. #329	3₋¹⁺²	direct (N × H)
84. #97	C₂×3₋¹⁺²	semidirect
85. 3₋¹⁺²	#329	direct (N × H)
124. C₉	C₃×#329	semidirect

H ≅ G/N

type

2. 3₋¹⁺²×#97

C₂

semidirect

4. C₉×#329

C₃

semidirect

13. C₉×#97

C₂×C₃

semidirect

57. #329

3₋¹⁺²

direct (N × H)

84. #97

C₂×3₋¹⁺²

semidirect

85. 3₋¹⁺²

#329

direct (N × H)

124. C₉

C₃×#329

semidirect

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

subgroup	index
3₋¹⁺²×#124	2
3₋¹⁺²×#97	2
C₃²×#329	3
C₉×#329	3

subgroup

index

3₋¹⁺²×#124

3₋¹⁺²×#97

C₃²×#329

C₉×#329

Subgroups — 3₋¹⁺²×#329 #13135990

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3−1+2×#329 #13135990

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#329 #13135990