Group #14518923 — subgroups

Subgroups — 3₋¹⁺²×#634 #14518923

Order 1944.

648 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 147 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	C₃²×C₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	3₋¹⁺²×C₃⋊Dic₃	C₂	1	2	yes	maximal
3	C₃²×#634	C₃	1	1	yes	maximal, characteristic
4	C₉×#634	C₃	1	3	yes	maximal, complemented
5	Dic₆×3₋¹⁺²	—	3	4	no	maximal
6	C₂×C₃²×3₋¹⁺²	C₂²	1	1	yes	characteristic
7	C₃⁴×C₄	C₂×C₃	1	1	yes	characteristic
8	C₃²×C₃⋊Dic₃	C₂×C₃	1	2	yes	—
9	C₃×C₄×3₋¹⁺²	—	2	4	no	—
10	C₃×C₄×3₋¹⁺²	—	2	12	no	—
11	C₃×C₄×3₋¹⁺²	D₃	1	4	yes	—
12	C₃²×C₄×C₉	C₂×C₃	1	3	yes	—
13	C₉×C₃⋊Dic₃	C₂×C₃	1	6	yes	—
14	Dic₃×3₋¹⁺²	—	3	8	no	—
15	C₃²×3₋¹⁺²	Q₈	1	1	yes	characteristic, complemented, Sylow
16	C₃×#634	—	3	1	no	—
17	C₃×#634	C₃²	1	1	yes	characteristic
18	C₃²×Dic₆	—	3	4	no	—
19	Q₈×3₋¹⁺²	—	9	1	no	—
20	C₉×Dic₆	—	3	12	no	—
21	C₂×C₃⁴	C₂²×C₃	1	1	yes	characteristic
22	C₂×C₃×3₋¹⁺²	—	2	4	no	—
23	C₂×C₃×3₋¹⁺²	—	2	12	no	—
24	C₂×C₃×3₋¹⁺²	C₂×D₃	1	4	yes	—
25	C₂×C₃²×C₉	C₂²×C₃	1	3	yes	—
26	C₃³×C₄	—	2	4	no	—
27	C₃³×C₄	—	3	1	no	—
28	C₃³×C₄	—	6	4	no	—
29	C₃³×C₄	C₃×D₃	1	4	yes	—
30	C₃³×C₄	C₂×C₃²	1	1	yes	characteristic
31	C₃×C₃⋊Dic₃	—	3	2	no	—
32	C₃×C₃⋊Dic₃	C₂×C₃²	1	2	yes	—
33	C₃²×Dic₃	—	3	8	no	—
34	C₄×3₋¹⁺²	—	2	24	no	—
35	C₄×3₋¹⁺²	—	2	4	no	—
36	C₄×3₋¹⁺²	—	2	12	no	—
37	C₄×3₋¹⁺²	—	9	2	no	—
38	C₄×3₋¹⁺²	C₃⋊D₃	1	1	yes	characteristic
39	C₃×C₄×C₉	—	2	12	no	—
40	C₃×C₄×C₉	C₃×D₃	1	12	yes	—
41	C₉×Dic₃	—	3	24	no	—
42	C₃⁴	C₃×Q₈	1	1	yes	characteristic
43	C₃×3₋¹⁺²	—	2	4	no	—
44	C₃×3₋¹⁺²	—	2	12	no	—
45	C₃×3₋¹⁺²	Dic₆	1	4	yes	complemented
46	C₃²×C₉	C₃×Q₈	1	3	yes	complemented
47	C₃²×Q₈	—	9	1	no	—
48	#634	3₋¹⁺²	1	1	yes	characteristic, direct product factor
49	C₃×Dic₆	—	3	4	no	—
50	C₃×Dic₆	—	9	4	no	—
51	Q₈×C₉	—	9	3	no	—
52	C₂×C₃³	—	2	4	no	—
53	C₂×C₃³	—	3	1	no	—
54	C₂×C₃³	—	6	4	no	—
55	C₂×C₃³	C₂×C₃×D₃	1	4	yes	—
56	C₂×C₃³	C₂²×C₃²	1	1	yes	characteristic
57	C₂×3₋¹⁺²	—	2	24	no	—
58	C₂×3₋¹⁺²	—	2	4	no	—
59	C₂×3₋¹⁺²	—	2	12	no	—
60	C₂×3₋¹⁺²	C₂×C₃⋊D₃	1	1	yes	characteristic
61	C₂×C₃×C₉	—	2	12	no	—
62	C₂×C₃×C₉	C₂×C₃×D₃	1	12	yes	—
63	C₃²×C₄	—	2	4	no	—
64	C₃²×C₄	—	2	4	no	—
65	C₃²×C₄	—	3	4	no	—
66	C₃²×C₄	—	6	4	no	—
67	C₃²×C₄	—	6	8	no	—
68	C₃²×C₄	—	6	4	no	—
69	C₃²×C₄	—	9	2	no	—
70	C₃²×C₄	C₂×3₋¹⁺²	1	1	yes	characteristic
71	C₃²×C₄	C₃²×D₃	1	4	yes	—
72	C₃²×C₄	C₃×C₃⋊D₃	1	1	yes	characteristic
73	C₃⋊Dic₃	C₂×3₋¹⁺²	1	2	yes	—
74	C₃×Dic₃	—	3	8	no	—
75	C₃×Dic₃	—	9	8	no	—
76	C₄×C₉	—	2	12	no	—
77	C₄×C₉	—	9	6	no	—
78	C₄×C₉	C₃×C₃⋊D₃	1	3	yes	—
79	C₃³	—	2	4	no	—
80	C₃³	—	3	1	no	—
81	C₃³	—	6	4	no	—
82	C₃³	C₃×Dic₆	1	4	yes	—
83	C₃³	C₃²×Q₈	1	1	yes	characteristic
84	3₋¹⁺²	—	2	24	no	—
85	3₋¹⁺²	—	2	4	no	—
86	3₋¹⁺²	—	2	12	no	—
87	3₋¹⁺²	#634	1	1	yes	characteristic, direct product factor
88	C₃×C₉	—	2	12	no	—
89	C₃×C₉	C₃×Dic₆	1	12	yes	complemented
90	C₃×Q₈	—	9	1	no	—
91	C₃×Q₈	—	27	1	no	—
92	Dic₆	—	3	4	no	—
93	C₂×C₃²	—	2	4	no	—
94	C₂×C₃²	—	2	4	no	—
95	C₂×C₃²	—	3	4	no	—
96	C₂×C₃²	—	6	4	no	—
97	C₂×C₃²	—	6	8	no	—
98	C₂×C₃²	—	6	4	no	—
99	C₂×C₃²	C₂²×3₋¹⁺²	1	1	yes	characteristic
100	C₂×C₃²	C₂×C₃²×D₃	1	4	yes	—
101	C₂×C₃²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
102	C₂×C₉	—	2	12	no	—
103	C₂×C₉	C₂×C₃×C₃⋊D₃	1	3	yes	—
104	C₃×C₄	—	2	4	no	—
105	C₃×C₄	—	3	1	no	—
106	C₃×C₄	—	6	4	no	—
107	C₃×C₄	—	9	2	no	—
108	C₃×C₄	—	27	2	no	—
109	C₃×C₄	D₃×3₋¹⁺²	1	4	yes	—
110	C₃×C₄	C₃²×C₃⋊D₃	1	1	yes	characteristic
111	Dic₃	—	3	8	no	—
112	C₃²	—	2	4	no	—
113	C₃²	—	2	4	no	—
114	C₃²	—	3	4	no	—
115	C₃²	—	6	4	no	—
116	C₃²	—	6	8	no	—
117	C₃²	—	6	4	no	—
118	C₃²	Q₈×3₋¹⁺²	1	1	yes	characteristic, complemented
119	C₃²	C₃²×Dic₆	1	4	yes	—
120	C₃²	C₃×#634	1	1	yes	characteristic
121	C₉	—	2	12	no	—
122	C₉	C₃×#634	1	3	yes	complemented
123	Q₈	—	9	1	no	Sylow
124	C₂×C₃	—	2	4	no	—
125	C₂×C₃	—	3	1	no	—
126	C₂×C₃	—	6	4	no	—
127	C₂×C₃	C₂×D₃×3₋¹⁺²	1	4	yes	—
128	C₂×C₃	C₂×C₃²×C₃⋊D₃	1	1	yes	characteristic
129	C₄	—	9	2	no	—
130	C₄	C₃⋊D₃×3₋¹⁺²	1	1	yes	characteristic
131	C₃	—	2	4	no	—
132	C₃	—	3	1	no	—
133	C₃	—	6	4	no	—
134	C₃	Dic₆×3₋¹⁺²	1	4	yes	complemented
135	C₃	C₃²×#634	1	1	yes	characteristic
136	C₂	C₂×C₃⋊D₃×3₋¹⁺²	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₄×3₋¹⁺²

C₂

yes

maximal, characteristic

3₋¹⁺²×C₃⋊Dic₃

C₂

yes

maximal

C₃²×#634

C₃

yes

maximal, characteristic

C₉×#634

C₃

yes

maximal, complemented

Dic₆×3₋¹⁺²

—

maximal

C₂×C₃²×3₋¹⁺²

C₂²

yes

characteristic

C₃⁴×C₄

C₂×C₃

yes

characteristic

C₃²×C₃⋊Dic₃

C₂×C₃

yes

—

C₃×C₄×3₋¹⁺²

—

C₃×C₄×3₋¹⁺²

—

C₃×C₄×3₋¹⁺²

D₃

yes

—

C₃²×C₄×C₉

C₂×C₃

yes

—

C₉×C₃⋊Dic₃

C₂×C₃

yes

—

Dic₃×3₋¹⁺²

—

C₃²×3₋¹⁺²

Q₈

yes

characteristic, complemented, Sylow

C₃×#634

—

C₃×#634

C₃²

yes

characteristic

C₃²×Dic₆

—

Q₈×3₋¹⁺²

—

C₉×Dic₆

—

C₂×C₃⁴

C₂²×C₃

yes

characteristic

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

C₂×D₃

yes

—

C₂×C₃²×C₉

C₂²×C₃

yes

—

C₃³×C₄

—

C₃³×C₄

—

C₃³×C₄

—

C₃³×C₄

C₃×D₃

yes

—

C₃³×C₄

C₂×C₃²

yes

characteristic

C₃×C₃⋊Dic₃

—

C₃×C₃⋊Dic₃

C₂×C₃²

yes

—

C₃²×Dic₃

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

C₃⋊D₃

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

C₃×D₃

yes

—

C₉×Dic₃

—

C₃⁴

C₃×Q₈

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

Dic₆

yes

complemented

C₃²×C₉

C₃×Q₈

yes

complemented

C₃²×Q₈

—

#634

3₋¹⁺²

yes

characteristic, direct product factor

C₃×Dic₆

—

C₃×Dic₆

—

Q₈×C₉

—

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

C₂×C₃×D₃

yes

—

C₂×C₃³

C₂²×C₃²

yes

characteristic

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₂×C₃⋊D₃

yes

characteristic

C₂×C₃×C₉

—

C₂×C₃×C₉

C₂×C₃×D₃

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×3₋¹⁺²

yes

characteristic

C₃²×C₄

C₃²×D₃

yes

—

C₃²×C₄

C₃×C₃⋊D₃

yes

characteristic

C₃⋊Dic₃

C₂×3₋¹⁺²

yes

—

C₃×Dic₃

—

C₃×Dic₃

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₃×C₃⋊D₃

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×Dic₆

yes

—

C₃³

C₃²×Q₈

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

#634

yes

characteristic, direct product factor

C₃×C₉

—

C₃×C₉

C₃×Dic₆

yes

complemented

C₃×Q₈

—

C₃×Q₈

—

Dic₆

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₂²×3₋¹⁺²

yes

characteristic

100

C₂×C₃²

C₂×C₃²×D₃

yes

—

101

C₂×C₃²

C₂×C₃×C₃⋊D₃

yes

characteristic

102

C₂×C₉

—

103

C₂×C₉

C₂×C₃×C₃⋊D₃

yes

—

104

C₃×C₄

—

105

C₃×C₄

—

106

C₃×C₄

—

107

C₃×C₄

—

108

C₃×C₄

—

109

C₃×C₄

D₃×3₋¹⁺²

yes

—

110

C₃×C₄

C₃²×C₃⋊D₃

yes

characteristic

111

Dic₃

—

112

C₃²

—

113

C₃²

—

114

C₃²

—

115

C₃²

—

116

C₃²

—

117

C₃²

—

118

C₃²

Q₈×3₋¹⁺²

yes

characteristic, complemented

119

C₃²

C₃²×Dic₆

yes

—

120

C₃²

C₃×#634

yes

characteristic

121

C₉

—

122

C₉

C₃×#634

yes

complemented

123

Q₈

—

Sylow

124

C₂×C₃

—

125

C₂×C₃

—

126

C₂×C₃

—

127

C₂×C₃

C₂×D₃×3₋¹⁺²

yes

—

128

C₂×C₃

C₂×C₃²×C₃⋊D₃

yes

characteristic

129

C₄

—

130

C₄

C₃⋊D₃×3₋¹⁺²

yes

characteristic

131

C₃

—

132

C₃

—

133

C₃

—

134

C₃

Dic₆×3₋¹⁺²

yes

complemented

135

C₃

C₃²×#634

yes

characteristic

136

C₂

C₂×C₃⋊D₃×3₋¹⁺²

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₃	C₂×C₃²×C₃⋊D₃
commutator, socle, cosocle	C₂×C₃³	C₂²×C₃²
Fitting	C₃²×C₄×3₋¹⁺²	C₂
prime radical	C₂×C₃⁴	C₂²×C₃
hypercenter	C₄×3₋¹⁺²	C₃⋊D₃
hypocenter	C₃²	Q₈×3₋¹⁺²
LCS	G ▷ C₂×C₃³ ▷ C₃²
UCS	C₁ ◁ C₂×C₃ ◁ C₄×3₋¹⁺²

subgroup

G/H

center, Frattini

C₂×C₃

C₂×C₃²×C₃⋊D₃

commutator, socle, cosocle

C₂×C₃³

C₂²×C₃²

Fitting

C₃²×C₄×3₋¹⁺²

C₂

prime radical

C₂×C₃⁴

C₂²×C₃

hypercenter

C₄×3₋¹⁺²

C₃⋊D₃

hypocenter

C₃²

Q₈×3₋¹⁺²

LCS

G ▷ C₂×C₃³ ▷ C₃²

UCS

C₁ ◁ C₂×C₃ ◁ C₄×3₋¹⁺²

N	H ≅ G/N	type
4. C₉×#634	C₃	semidirect
15. C₃²×3₋¹⁺²	Q₈	semidirect
45. C₃×3₋¹⁺²	Dic₆	semidirect
46. C₃²×C₉	C₃×Q₈	semidirect
48. #634	3₋¹⁺²	direct (N × H)
87. 3₋¹⁺²	#634	direct (N × H)
89. C₃×C₉	C₃×Dic₆	semidirect
118. C₃²	Q₈×3₋¹⁺²	semidirect
122. C₉	C₃×#634	semidirect
134. C₃	Dic₆×3₋¹⁺²	semidirect

H ≅ G/N

type

4. C₉×#634

C₃

semidirect

15. C₃²×3₋¹⁺²

Q₈

semidirect

45. C₃×3₋¹⁺²

Dic₆

semidirect

46. C₃²×C₉

C₃×Q₈

semidirect

48. #634

3₋¹⁺²

direct (N × H)

87. 3₋¹⁺²

#634

direct (N × H)

89. C₃×C₉

C₃×Dic₆

semidirect

118. C₃²

Q₈×3₋¹⁺²

semidirect

122. C₉

C₃×#634

semidirect

134. C₃

Dic₆×3₋¹⁺²

semidirect

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

subgroup	index
C₃²×C₄×3₋¹⁺²	2
3₋¹⁺²×C₃⋊Dic₃	2
C₃²×#634	3
C₉×#634	3
Dic₆×3₋¹⁺²	3

subgroup

index

C₃²×C₄×3₋¹⁺²

3₋¹⁺²×C₃⋊Dic₃

C₃²×#634

C₉×#634

Dic₆×3₋¹⁺²

Subgroups — 3₋¹⁺²×#634 #14518923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3−1+2×#634 #14518923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#634 #14518923