Group #13047878 — subgroups

Subgroups — 3₋¹⁺²×#216 #13047878

Order 1296.

210 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 77 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×Q₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	Dic₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
3	C₃⋊C₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
4	C₃²×#216	C₃	1	1	yes	maximal, characteristic
5	Q₁₆×3₋¹⁺²	—	3	1	no	maximal
6	C₉×#216	C₃	1	3	yes	maximal, complemented
7	C₃×C₄×3₋¹⁺²	—	2	1	no	—
8	C₃×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
9	Dic₃×3₋¹⁺²	—	2	1	no	—
10	C₃³×Q₈	C₂×C₃	1	1	yes	characteristic
11	C₃²×Dic₆	C₂×C₃	1	1	yes	characteristic
12	C₃²×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
13	Q₈×3₋¹⁺²	—	2	1	no	—
14	Q₈×3₋¹⁺²	—	2	3	no	—
15	Q₈×3₋¹⁺²	—	3	1	no	—
16	Q₈×3₋¹⁺²	D₃	1	1	yes	characteristic
17	C₃×Q₈×C₉	C₂×C₃	1	3	yes	—
18	C₉×Dic₆	C₂×C₃	1	3	yes	—
19	C₈×3₋¹⁺²	—	3	1	no	—
20	C₉×C₃⋊C₈	C₂×C₃	1	3	yes	—
21	C₂×C₃×3₋¹⁺²	D₄	1	1	yes	characteristic
22	C₃²×Q₁₆	—	3	1	no	—
23	C₃×#216	—	3	1	no	—
24	C₃×#216	C₃²	1	1	yes	characteristic
25	C₉×Q₁₆	—	3	3	no	—
26	C₃³×C₄	—	2	1	no	—
27	C₃³×C₄	C₂²×C₃	1	1	yes	characteristic
28	C₃²×Dic₃	—	2	1	no	—
29	C₄×3₋¹⁺²	—	2	2	no	—
30	C₄×3₋¹⁺²	—	2	6	no	—
31	C₄×3₋¹⁺²	—	2	1	no	—
32	C₄×3₋¹⁺²	—	2	3	no	—
33	C₄×3₋¹⁺²	—	2	1	no	—
34	C₄×3₋¹⁺²	—	6	1	no	—
35	C₄×3₋¹⁺²	C₂×D₃	1	1	yes	characteristic
36	C₃×C₄×C₉	—	2	3	no	—
37	C₃×C₄×C₉	C₂²×C₃	1	3	yes	—
38	C₉×Dic₃	—	2	3	no	—
39	C₃×3₋¹⁺²	Q₁₆	1	1	yes	characteristic, complemented, Sylow
40	C₃²×Q₈	—	2	1	no	—
41	C₃²×Q₈	—	3	1	no	—
42	C₃²×Q₈	—	3	1	no	—
43	C₃²×Q₈	—	6	1	no	—
44	C₃²×Q₈	C₃×D₃	1	1	yes	characteristic
45	C₃²×Q₈	C₂×C₃²	1	1	yes	characteristic
46	C₃×Dic₆	—	3	1	no	—
47	C₃×Dic₆	C₂×C₃²	1	1	yes	characteristic
48	C₃²×C₈	—	3	1	no	—
49	C₃×C₃⋊C₈	—	3	1	no	—
50	C₃×C₃⋊C₈	C₂×C₃²	1	1	yes	characteristic
51	Q₈×C₉	—	2	3	no	—
52	Q₈×C₉	—	3	3	no	—
53	Q₈×C₉	C₃×D₃	1	3	yes	—
54	C₈×C₉	—	3	3	no	—
55	C₂×C₃³	C₃×D₄	1	1	yes	characteristic
56	C₂×3₋¹⁺²	—	2	1	no	—
57	C₂×3₋¹⁺²	—	2	3	no	—
58	C₂×3₋¹⁺²	#67	1	1	yes	characteristic
59	C₂×C₃×C₉	C₃×D₄	1	3	yes	—
60	C₃×Q₁₆	—	3	1	no	—
61	C₃×Q₁₆	—	9	1	no	—
62	#216	3₋¹⁺²	1	1	yes	characteristic, direct product factor
63	C₃²×C₄	—	2	1	no	—
64	C₃²×C₄	—	2	2	no	—
65	C₃²×C₄	—	2	1	no	—
66	C₃²×C₄	—	2	1	no	—
67	C₃²×C₄	—	3	1	no	—
68	C₃²×C₄	—	6	1	no	—
69	C₃²×C₄	—	6	1	no	—
70	C₃²×C₄	—	6	2	no	—
71	C₃²×C₄	—	6	1	no	—
72	C₃²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
73	C₃²×C₄	C₂²×C₃²	1	1	yes	characteristic
74	C₃×Dic₃	—	2	1	no	—
75	C₃×Dic₃	—	6	1	no	—
76	C₄×C₉	—	2	6	no	—
77	C₄×C₉	—	2	3	no	—
78	C₄×C₉	—	2	3	no	—
79	C₄×C₉	—	6	3	no	—
80	C₄×C₉	C₂×C₃×D₃	1	3	yes	—
81	C₃³	C₃×Q₁₆	1	1	yes	characteristic
82	3₋¹⁺²	—	2	1	no	—
83	3₋¹⁺²	—	2	3	no	—
84	3₋¹⁺²	#216	1	1	yes	characteristic, direct product factor
85	C₃×C₉	C₃×Q₁₆	1	3	yes	complemented
86	C₃×Q₈	—	2	1	no	—
87	C₃×Q₈	—	3	1	no	—
88	C₃×Q₈	—	3	1	no	—
89	C₃×Q₈	—	6	1	no	—
90	C₃×Q₈	—	9	1	no	—
91	C₃×Q₈	C₂×3₋¹⁺²	1	1	yes	characteristic
92	C₃×Q₈	C₃²×D₃	1	1	yes	characteristic
93	Dic₆	C₂×3₋¹⁺²	1	1	yes	characteristic
94	C₃×C₈	—	3	1	no	—
95	C₃×C₈	—	9	1	no	—
96	C₃⋊C₈	C₂×3₋¹⁺²	1	1	yes	characteristic
97	C₂×C₃²	—	2	1	no	—
98	C₂×C₃²	—	3	1	no	—
99	C₂×C₃²	—	6	1	no	—
100	C₂×C₃²	C₃×#67	1	1	yes	characteristic
101	C₂×C₃²	C₃²×D₄	1	1	yes	characteristic
102	C₂×C₉	—	2	3	no	—
103	C₂×C₉	C₃×#67	1	3	yes	—
104	Q₁₆	—	3	1	no	Sylow
105	C₃×C₄	—	2	2	no	—
106	C₃×C₄	—	2	1	no	—
107	C₃×C₄	—	2	1	no	—
108	C₃×C₄	—	2	1	no	—
109	C₃×C₄	—	3	1	no	—
110	C₃×C₄	—	6	1	no	—
111	C₃×C₄	—	6	2	no	—
112	C₃×C₄	—	6	1	no	—
113	C₃×C₄	—	6	1	no	—
114	C₃×C₄	—	18	1	no	—
115	C₃×C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
116	C₃×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
117	Dic₃	—	2	1	no	—
118	C₃²	—	2	1	no	—
119	C₃²	—	3	1	no	—
120	C₃²	—	6	1	no	—
121	C₃²	C₃×#216	1	1	yes	characteristic
122	C₃²	C₃²×Q₁₆	1	1	yes	characteristic
123	C₉	—	2	3	no	—
124	C₉	C₃×#216	1	3	yes	complemented
125	Q₈	—	3	1	no	—
126	Q₈	D₃×3₋¹⁺²	1	1	yes	characteristic
127	C₈	—	3	1	no	—
128	C₂×C₃	—	2	1	no	—
129	C₂×C₃	—	3	1	no	—
130	C₂×C₃	—	6	1	no	—
131	C₂×C₃	D₄×3₋¹⁺²	1	1	yes	characteristic
132	C₂×C₃	C₃²×#67	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	6	1	no	—
135	C₄	C₂×D₃×3₋¹⁺²	1	1	yes	characteristic
136	C₃	—	2	1	no	—
137	C₃	—	3	1	no	—
138	C₃	—	6	1	no	—
139	C₃	Q₁₆×3₋¹⁺²	1	1	yes	characteristic, complemented
140	C₃	C₃²×#216	1	1	yes	characteristic
141	C₂	#67×3₋¹⁺²	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×Q₈×3₋¹⁺²

C₂

yes

maximal, characteristic

Dic₆×3₋¹⁺²

C₂

yes

maximal, characteristic

C₃⋊C₈×3₋¹⁺²

C₂

yes

maximal, characteristic

C₃²×#216

C₃

yes

maximal, characteristic

Q₁₆×3₋¹⁺²

—

maximal

C₉×#216

C₃

yes

maximal, complemented

C₃×C₄×3₋¹⁺²

—

C₃×C₄×3₋¹⁺²

C₂²

yes

characteristic

Dic₃×3₋¹⁺²

—

C₃³×Q₈

C₂×C₃

yes

characteristic

C₃²×Dic₆

C₂×C₃

yes

characteristic

C₃²×C₃⋊C₈

C₂×C₃

yes

characteristic

Q₈×3₋¹⁺²

—

Q₈×3₋¹⁺²

—

Q₈×3₋¹⁺²

—

Q₈×3₋¹⁺²

D₃

yes

characteristic

C₃×Q₈×C₉

C₂×C₃

yes

—

C₉×Dic₆

C₂×C₃

yes

—

C₈×3₋¹⁺²

—

C₉×C₃⋊C₈

C₂×C₃

yes

—

C₂×C₃×3₋¹⁺²

D₄

yes

characteristic

C₃²×Q₁₆

—

C₃×#216

—

C₃×#216

C₃²

yes

characteristic

C₉×Q₁₆

—

C₃³×C₄

—

C₃³×C₄

C₂²×C₃

yes

characteristic

C₃²×Dic₃

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

C₂×D₃

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

C₂²×C₃

yes

—

C₉×Dic₃

—

C₃×3₋¹⁺²

Q₁₆

yes

characteristic, complemented, Sylow

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

C₃×D₃

yes

characteristic

C₃²×Q₈

C₂×C₃²

yes

characteristic

C₃×Dic₆

—

C₃×Dic₆

C₂×C₃²

yes

characteristic

C₃²×C₈

—

C₃×C₃⋊C₈

—

C₃×C₃⋊C₈

C₂×C₃²

yes

characteristic

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

C₃×D₃

yes

—

C₈×C₉

—

C₂×C₃³

C₃×D₄

yes

characteristic

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

#67

yes

characteristic

C₂×C₃×C₉

C₃×D₄

yes

—

C₃×Q₁₆

—

C₃×Q₁₆

—

#216

3₋¹⁺²

yes

characteristic, direct product factor

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×C₃×D₃

yes

characteristic

C₃²×C₄

C₂²×C₃²

yes

characteristic

C₃×Dic₃

—

C₃×Dic₃

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂×C₃×D₃

yes

—

C₃³

C₃×Q₁₆

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

#216

yes

characteristic, direct product factor

C₃×C₉

C₃×Q₁₆

yes

complemented

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

C₂×3₋¹⁺²

yes

characteristic

C₃×Q₈

C₃²×D₃

yes

characteristic

Dic₆

C₂×3₋¹⁺²

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

C₂×3₋¹⁺²

yes

characteristic

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

100

C₂×C₃²

C₃×#67

yes

characteristic

101

C₂×C₃²

C₃²×D₄

yes

characteristic

102

C₂×C₉

—

103

C₂×C₉

C₃×#67

yes

—

104

Q₁₆

—

Sylow

105

C₃×C₄

—

106

C₃×C₄

—

107

C₃×C₄

—

108

C₃×C₄

—

109

C₃×C₄

—

110

C₃×C₄

—

111

C₃×C₄

—

112

C₃×C₄

—

113

C₃×C₄

—

114

C₃×C₄

—

115

C₃×C₄

C₂²×3₋¹⁺²

yes

characteristic

116

C₃×C₄

C₂×C₃²×D₃

yes

characteristic

117

Dic₃

—

118

C₃²

—

119

C₃²

—

120

C₃²

—

121

C₃²

C₃×#216

yes

characteristic

122

C₃²

C₃²×Q₁₆

yes

characteristic

123

C₉

—

124

C₉

C₃×#216

yes

complemented

125

Q₈

—

126

Q₈

D₃×3₋¹⁺²

yes

characteristic

127

C₈

—

128

C₂×C₃

—

129

C₂×C₃

—

130

C₂×C₃

—

131

C₂×C₃

D₄×3₋¹⁺²

yes

characteristic

132

C₂×C₃

C₃²×#67

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

C₂×D₃×3₋¹⁺²

yes

characteristic

136

C₃

—

137

C₃

—

138

C₃

—

139

C₃

Q₁₆×3₋¹⁺²

yes

characteristic, complemented

140

C₃

C₃²×#216

yes

characteristic

141

C₂

#67×3₋¹⁺²

yes

characteristic

	subgroup	G/H
center	C₂×C₃	C₃²×#67
commutator, cosocle	C₃²×C₄	C₂²×C₃²
Frattini	C₃×C₄	C₂×C₃²×D₃
Fitting	C₃×Q₈×3₋¹⁺²	C₂
socle	C₂×C₃²	C₃²×D₄
prime radical	C₂×C₃³	C₃×D₄
hypercenter	Q₈×3₋¹⁺²	D₃
hypocenter	C₃	Q₁₆×3₋¹⁺²
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃ ◁ C₄×3₋¹⁺² ◁ Q₈×3₋¹⁺²

subgroup

G/H

center

C₂×C₃

C₃²×#67

commutator, cosocle

C₃²×C₄

C₂²×C₃²

Frattini

C₃×C₄

C₂×C₃²×D₃

Fitting

C₃×Q₈×3₋¹⁺²

C₂

socle

C₂×C₃²

C₃²×D₄

prime radical

C₂×C₃³

C₃×D₄

hypercenter

Q₈×3₋¹⁺²

D₃

hypocenter

C₃

Q₁₆×3₋¹⁺²

LCS

G ▷ C₃²×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₃ ◁ C₄×3₋¹⁺² ◁ Q₈×3₋¹⁺²

N	H ≅ G/N	type
6. C₉×#216	C₃	semidirect
39. C₃×3₋¹⁺²	Q₁₆	semidirect
62. #216	3₋¹⁺²	direct (N × H)
84. 3₋¹⁺²	#216	direct (N × H)
85. C₃×C₉	C₃×Q₁₆	semidirect
124. C₉	C₃×#216	semidirect
139. C₃	Q₁₆×3₋¹⁺²	semidirect

H ≅ G/N

type

6. C₉×#216

C₃

semidirect

39. C₃×3₋¹⁺²

Q₁₆

semidirect

62. #216

3₋¹⁺²

direct (N × H)

84. 3₋¹⁺²

#216

direct (N × H)

85. C₃×C₉

C₃×Q₁₆

semidirect

124. C₉

C₃×#216

semidirect

139. C₃

Q₁₆×3₋¹⁺²

semidirect

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

subgroup	index
C₃×Q₈×3₋¹⁺²	2
Dic₆×3₋¹⁺²	2
C₃⋊C₈×3₋¹⁺²	2
C₃²×#216	3
Q₁₆×3₋¹⁺²	3
C₉×#216	3

subgroup

index

C₃×Q₈×3₋¹⁺²

Dic₆×3₋¹⁺²

C₃⋊C₈×3₋¹⁺²

C₃²×#216

Q₁₆×3₋¹⁺²

C₉×#216

Subgroups — 3₋¹⁺²×#216 #13047878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3−1+2×#216 #13047878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#216 #13047878