Group #13135996 — subgroups

Subgroups — 3₋¹⁺²×#331 #13135996

Order 1728.

336 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 133 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×#127	C₂	1	1	yes	maximal, characteristic
2	3₋¹⁺²×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₃²×#331	C₃	1	1	yes	maximal, characteristic
4	C₉×#331	C₃	1	3	yes	maximal, complemented
5	C₂×D₄×3₋¹⁺²	—	2	2	no	—
6	C₂×D₄×3₋¹⁺²	C₄	1	2	yes	—
7	C₂×C₈×3₋¹⁺²	—	2	2	no	—
8	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
9	C₃²×#127	C₂×C₃	1	1	yes	characteristic
10	C₃²×C₄⋊C₈	C₂×C₃	1	2	yes	—
11	C₉×#127	C₂×C₃	1	3	yes	—
12	C₉×C₄⋊C₈	C₂×C₃	1	6	yes	complemented
13	C₂³×3₋¹⁺²	—	2	2	no	—
14	D₄×3₋¹⁺²	—	4	2	no	—
15	D₄×3₋¹⁺²	—	4	2	no	—
16	D₄×3₋¹⁺²	—	4	2	no	—
17	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
19	C₈×3₋¹⁺²	—	4	2	no	—
20	C₃×#331	—	3	1	no	—
21	C₃×#331	C₃²	1	1	yes	characteristic
22	C₂×C₃²×D₄	—	2	2	no	—
23	C₂×C₃²×D₄	C₃×C₄	1	2	yes	—
24	C₂×C₃²×C₈	—	2	2	no	—
25	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
26	C₂×D₄×C₉	—	2	6	no	—
27	C₂×D₄×C₉	C₃×C₄	1	6	yes	—
28	C₂×C₈×C₉	—	2	6	no	—
29	C₄²×C₉	C₂²×C₃	1	3	yes	—
30	C₂²×3₋¹⁺²	—	4	2	no	—
31	C₂²×3₋¹⁺²	—	4	4	no	—
32	C₂²×3₋¹⁺²	C₂²⋊C₄	1	1	yes	characteristic
33	C₄×3₋¹⁺²	—	2	1	no	—
34	C₄×3₋¹⁺²	D₈	1	2	yes	—
35	C₄×3₋¹⁺²	SD₁₆	1	2	yes	—
36	C₃×#127	—	3	1	no	—
37	C₃×#127	C₂×C₃²	1	1	yes	characteristic
38	C₃×C₄⋊C₈	—	3	2	no	—
39	C₃×C₄⋊C₈	C₂×C₃²	1	2	yes	—
40	C₂³×C₃²	—	2	2	no	—
41	C₃²×D₄	—	4	2	no	—
42	C₃²×D₄	—	4	2	no	—
43	C₃²×D₄	—	4	2	no	—
44	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
45	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
46	C₂³×C₉	—	2	6	no	—
47	C₃²×C₈	—	4	2	no	—
48	D₄×C₉	—	4	6	no	—
49	D₄×C₉	—	4	6	no	—
50	D₄×C₉	—	4	6	no	—
51	C₂×C₄×C₉	C₂×C₃×C₄	1	3	yes	—
52	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
53	C₈×C₉	—	4	6	no	—
54	#331	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
55	C₂×3₋¹⁺²	—	8	2	no	—
56	C₂×3₋¹⁺²	#100	1	1	yes	characteristic
57	C₂×3₋¹⁺²	#102	1	2	yes	—
58	C₂×C₃×D₄	—	2	2	no	—
59	C₂×C₃×D₄	—	3	2	no	—
60	C₂×C₃×D₄	—	6	2	no	—
61	C₂×C₃×D₄	C₃²×C₄	1	2	yes	—
62	C₂×C₃×C₈	—	2	2	no	—
63	C₂×C₃×C₈	—	6	2	no	—
64	C₃×C₄²	—	3	1	no	—
65	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
66	C₂²×C₃²	—	4	2	no	—
67	C₂²×C₃²	—	4	4	no	—
68	C₂²×C₃²	C₃×C₂²⋊C₄	1	1	yes	characteristic
69	C₃²×C₄	—	2	1	no	—
70	C₃²×C₄	C₃×D₈	1	2	yes	—
71	C₃²×C₄	C₃×SD₁₆	1	2	yes	—
72	C₂²×C₉	—	4	6	no	—
73	C₂²×C₉	—	4	12	no	—
74	C₂²×C₉	C₃×C₂²⋊C₄	1	3	yes	—
75	C₄×C₉	—	2	3	no	—
76	C₄×C₉	C₃×D₈	1	6	yes	—
77	C₄×C₉	C₃×SD₁₆	1	6	yes	—
78	#127	C₂×3₋¹⁺²	1	1	yes	characteristic
79	C₄⋊C₈	C₂×3₋¹⁺²	1	2	yes	complemented
80	3₋¹⁺²	#331	1	1	yes	characteristic, direct product factor, Sylow
81	C₂³×C₃	—	2	2	no	—
82	C₂³×C₃	—	6	2	no	—
83	C₃×D₄	—	4	2	no	—
84	C₃×D₄	—	4	2	no	—
85	C₃×D₄	—	4	2	no	—
86	C₃×D₄	—	12	2	no	—
87	C₃×D₄	—	12	2	no	—
88	C₃×D₄	—	12	2	no	—
89	C₂×C₃×C₄	—	3	1	no	—
90	C₂×C₃×C₄	—	3	2	no	—
91	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
92	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
93	C₃×C₈	—	4	2	no	—
94	C₃×C₈	—	12	2	no	—
95	C₂×C₃²	—	8	2	no	—
96	C₂×C₃²	C₃×#100	1	1	yes	characteristic
97	C₂×C₃²	C₃×#102	1	2	yes	—
98	C₂×C₉	—	8	6	no	—
99	C₂×C₉	C₃×#100	1	3	yes	—
100	C₂×C₉	C₃×#102	1	6	yes	—
101	C₂×D₄	—	2	2	no	—
102	C₂×D₄	C₄×3₋¹⁺²	1	2	yes	—
103	C₂×C₈	—	2	2	no	—
104	C₄²	C₂²×3₋¹⁺²	1	1	yes	characteristic
105	C₂²×C₃	—	3	1	no	—
106	C₂²×C₃	—	4	2	no	—
107	C₂²×C₃	—	4	4	no	—
108	C₂²×C₃	—	12	4	no	—
109	C₂²×C₃	—	12	2	no	—
110	C₂²×C₃	C₃²×C₂²⋊C₄	1	1	yes	characteristic
111	C₃×C₄	—	2	1	no	—
112	C₃×C₄	—	3	2	no	—
113	C₃×C₄	—	3	2	no	—
114	C₃×C₄	—	6	1	no	—
115	C₃×C₄	C₃²×D₈	1	2	yes	—
116	C₃×C₄	C₃²×SD₁₆	1	2	yes	—
117	C₃²	C₃×#331	1	1	yes	characteristic
118	C₉	C₃×#331	1	3	yes	complemented
119	C₂³	—	2	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	C₂×C₄	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
124	C₂×C₄	D₄×3₋¹⁺²	1	2	yes	—
125	C₈	—	4	2	no	—
126	C₂×C₃	—	3	1	no	—
127	C₂×C₃	—	3	2	no	—
128	C₂×C₃	—	8	2	no	—
129	C₂×C₃	—	24	2	no	—
130	C₂×C₃	C₃²×#100	1	1	yes	characteristic
131	C₂×C₃	C₃²×#102	1	2	yes	—
132	C₂²	—	4	2	no	—
133	C₂²	—	4	4	no	—
134	C₂²	C₂²⋊C₄×3₋¹⁺²	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	D₈×3₋¹⁺²	1	2	yes	—
137	C₄	SD₁₆×3₋¹⁺²	1	2	yes	—
138	C₃	—	3	1	no	—
139	C₃	C₃²×#331	1	1	yes	characteristic
140	C₂	—	8	2	no	—
141	C₂	3₋¹⁺²×#100	1	1	yes	characteristic
142	C₂	3₋¹⁺²×#102	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²×#127

C₂

yes

maximal, characteristic

3₋¹⁺²×C₄⋊C₈

C₂

yes

maximal, complemented

C₃²×#331

C₃

yes

maximal, characteristic

C₉×#331

C₃

yes

maximal, complemented

C₂×D₄×3₋¹⁺²

—

C₂×D₄×3₋¹⁺²

C₄

yes

—

C₂×C₈×3₋¹⁺²

—

C₄²×3₋¹⁺²

C₂²

yes

characteristic

C₃²×#127

C₂×C₃

yes

characteristic

C₃²×C₄⋊C₈

C₂×C₃

yes

—

C₉×#127

C₂×C₃

yes

—

C₉×C₄⋊C₈

C₂×C₃

yes

complemented

C₂³×3₋¹⁺²

—

D₄×3₋¹⁺²

—

D₄×3₋¹⁺²

—

D₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

characteristic

C₂×C₄×3₋¹⁺²

D₄

yes

—

C₈×3₋¹⁺²

—

C₃×#331

—

C₃×#331

C₃²

yes

characteristic

C₂×C₃²×D₄

—

C₂×C₃²×D₄

C₃×C₄

yes

—

C₂×C₃²×C₈

—

C₃²×C₄²

C₂²×C₃

yes

characteristic

C₂×D₄×C₉

—

C₂×D₄×C₉

C₃×C₄

yes

—

C₂×C₈×C₉

—

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

D₈

yes

—

C₄×3₋¹⁺²

SD₁₆

yes

—

C₃×#127

—

C₃×#127

C₂×C₃²

yes

characteristic

C₃×C₄⋊C₈

—

C₃×C₄⋊C₈

C₂×C₃²

yes

—

C₂³×C₃²

—

C₃²×D₄

—

C₃²×D₄

—

C₃²×D₄

—

C₂×C₃²×C₄

C₂×C₃×C₄

yes

characteristic

C₂×C₃²×C₄

C₃×D₄

yes

—

C₂³×C₉

—

C₃²×C₈

—

D₄×C₉

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

—

C₂×C₄×C₉

C₃×D₄

yes

—

C₈×C₉

—

#331

3₋¹⁺²

yes

characteristic, direct product factor, Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

#100

yes

characteristic

C₂×3₋¹⁺²

#102

yes

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

C₃²×C₄

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₃×C₄²

C₂²×C₃²

yes

characteristic

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₃×C₂²⋊C₄

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₃×D₈

yes

—

C₃²×C₄

C₃×SD₁₆

yes

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₃×C₂²⋊C₄

yes

—

C₄×C₉

—

C₄×C₉

C₃×D₈

yes

—

C₄×C₉

C₃×SD₁₆

yes

—

#127

C₂×3₋¹⁺²

yes

characteristic

C₄⋊C₈

C₂×3₋¹⁺²

yes

complemented

3₋¹⁺²

#331

yes

characteristic, direct product factor, Sylow

C₂³×C₃

—

C₂³×C₃

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃²×C₄

yes

characteristic

C₂×C₃×C₄

C₃²×D₄

yes

—

C₃×C₈

—

C₃×C₈

—

C₂×C₃²

—

C₂×C₃²

C₃×#100

yes

characteristic

C₂×C₃²

C₃×#102

yes

—

C₂×C₉

—

C₂×C₉

C₃×#100

yes

—

100

C₂×C₉

C₃×#102

yes

—

101

C₂×D₄

—

102

C₂×D₄

C₄×3₋¹⁺²

yes

—

103

C₂×C₈

—

104

C₄²

C₂²×3₋¹⁺²

yes

characteristic

105

C₂²×C₃

—

106

C₂²×C₃

—

107

C₂²×C₃

—

108

C₂²×C₃

—

109

C₂²×C₃

—

110

C₂²×C₃

C₃²×C₂²⋊C₄

yes

characteristic

111

C₃×C₄

—

112

C₃×C₄

—

113

C₃×C₄

—

114

C₃×C₄

—

115

C₃×C₄

C₃²×D₈

yes

—

116

C₃×C₄

C₃²×SD₁₆

yes

—

117

C₃²

C₃×#331

yes

characteristic

118

C₉

C₃×#331

yes

complemented

119

C₂³

—

120

D₄

—

121

D₄

—

122

D₄

—

123

C₂×C₄

C₂×C₄×3₋¹⁺²

yes

characteristic

124

C₂×C₄

D₄×3₋¹⁺²

yes

—

125

C₈

—

126

C₂×C₃

—

127

C₂×C₃

—

128

C₂×C₃

—

129

C₂×C₃

—

130

C₂×C₃

C₃²×#100

yes

characteristic

131

C₂×C₃

C₃²×#102

yes

—

132

C₂²

—

133

C₂²

—

134

C₂²

C₂²⋊C₄×3₋¹⁺²

yes

characteristic

135

C₄

—

136

C₄

D₈×3₋¹⁺²

yes

—

137

C₄

SD₁₆×3₋¹⁺²

yes

—

138

C₃

—

139

C₃

C₃²×#331

yes

characteristic

140

C₂

—

141

C₂

3₋¹⁺²×#100

yes

characteristic

142

C₂

3₋¹⁺²×#102

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	C₃²×#127	C₂×C₃
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

C₃²×#127

C₂×C₃

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×C₄⋊C₈	C₂	semidirect
4. C₉×#331	C₃	semidirect
12. C₉×C₄⋊C₈	C₂×C₃	semidirect
54. #331	3₋¹⁺²	direct (N × H)
79. C₄⋊C₈	C₂×3₋¹⁺²	semidirect
80. 3₋¹⁺²	#331	direct (N × H)
118. C₉	C₃×#331	semidirect

H ≅ G/N

type

2. 3₋¹⁺²×C₄⋊C₈

C₂

semidirect

4. C₉×#331

C₃

semidirect

12. C₉×C₄⋊C₈

C₂×C₃

semidirect

54. #331

3₋¹⁺²

direct (N × H)

79. C₄⋊C₈

C₂×3₋¹⁺²

semidirect

80. 3₋¹⁺²

#331

direct (N × H)

118. C₉

C₃×#331

semidirect

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

subgroup	index
3₋¹⁺²×#127	2
3₋¹⁺²×C₄⋊C₈	2
C₃²×#331	3
C₉×#331	3

subgroup

index

3₋¹⁺²×#127

3₋¹⁺²×C₄⋊C₈

C₃²×#331

C₉×#331

Subgroups — 3₋¹⁺²×#331 #13135996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3−1+2×#331 #13135996

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#331 #13135996