Group #13163473 — subgroups

Subgroups — #13163473

Order 1728.

1344 conjugacy classes of subgroups; only the 461 normal subgroups are shown.

normal subgroups only

This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 461 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	#11723910	C₂	1	1	yes	maximal, characteristic
2	C₄×#6308	C₂	1	1	yes	maximal, characteristic
3	C₃³×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11722188	C₂	1	2	yes	maximal
5	#11722186	C₂	1	2	yes	maximal
6	#10592549	—	3	13	no	maximal
7	C₂×#6308	—	2	1	no	—
8	C₂×#6308	C₂²	1	1	yes	characteristic
9	C₂×C₃³×C₈	C₂²	1	2	yes	—
10	C₃³×C₄²	C₂²	1	1	yes	characteristic
11	C₄⋊(C₃⋊(C₃⋊Dic₃))	—	2	1	no	—
12	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	1	yes	characteristic
13	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	2	yes	—
14	#88857	—	2	1	no	—
15	C₄×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
16	#64078	—	3	13	no	—
17	C₄×#634	—	3	13	no	—
18	C₃²×C₄⋊C₈	D₃	1	13	yes	—
19	#63645	—	3	26	no	—
20	#63643	—	3	26	no	—
21	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
22	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
23	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
24	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
25	C₂×C₃⋊(C₃⋊Dic₃)	—	2	1	no	—
26	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
27	#6308	—	2	1	no	—
28	#6308	—	4	2	no	—
29	#6308	Q₈	1	2	yes	complemented
30	C₃³×C₈	—	2	2	no	—
31	#4850	—	9	13	no	—
32	C₂×#634	—	3	13	no	—
33	C₂×#634	—	6	13	no	—
34	C₂×C₃²×C₈	C₂×D₃	1	26	yes	—
35	C₃²×C₄²	C₂×D₃	1	13	yes	—
36	C₄⋊(C₃⋊Dic₃)	—	3	13	no	—
37	C₄⋊(C₃⋊Dic₃)	—	3	26	no	—
38	C₄⋊(C₃⋊Dic₃)	—	6	13	no	—
39	#3750	—	6	13	no	—
40	C₄×C₃⋊Dic₃	—	6	13	no	—
41	C₂²×C₃³	C₂×D₄	1	1	yes	characteristic
42	C₃³×C₄	—	2	1	no	—
43	C₃³×C₄	Q₁₆	1	2	yes	—
44	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
45	C₃³×C₄	#41	1	1	yes	characteristic
46	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
47	C₃⋊(C₃⋊Dic₃)	—	4	2	no	—
48	C₃⋊(C₃⋊Dic₃)	—	4	1	no	—
49	#869	—	9	13	no	—
50	C₄×Dic₆	—	9	13	no	—
51	C₃×C₄⋊C₈	C₃⋊D₃	1	13	yes	—
52	#818	—	9	26	no	—
53	#816	—	9	26	no	—
54	C₂×C₃²×C₄	D₁₂	1	13	yes	—
55	C₂×C₃²×C₄	D₁₂	1	13	yes	—
56	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
57	C₂×C₃⋊Dic₃	—	6	13	no	—
58	C₂×C₃⋊Dic₃	—	6	13	no	—
59	C₂×C₃⋊Dic₃	—	6	26	no	—
60	#634	—	3	26	no	—
61	#634	—	6	13	no	—
62	#634	—	12	26	no	—
63	C₃²×C₈	—	2	26	no	—
64	#477	—	27	1	no	Sylow
65	C₂×C₃³	C₂²⋊Q₈	1	1	yes	characteristic
66	C₂×C₃³	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×C₃³	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂×Dic₆	—	9	13	no	—
69	C₂×Dic₆	—	18	13	no	—
70	C₂×C₃×C₈	C₂×C₃⋊D₃	1	26	yes	—
71	C₃×C₄²	C₂×C₃⋊D₃	1	13	yes	—
72	C₄⋊Dic₃	—	9	26	no	—
73	C₄⋊Dic₃	—	9	13	no	—
74	C₄⋊Dic₃	—	18	13	no	—
75	Dic₃⋊C₄	—	18	13	no	—
76	C₄×Dic₃	—	18	13	no	—
77	C₂²×C₃²	C₂×D₁₂	1	13	yes	—
78	C₃²×C₄	—	2	13	no	—
79	C₃²×C₄	Dic₁₂	1	26	yes	—
80	C₃²×C₄	#237	1	13	yes	—
81	C₃²×C₄	D₃×Q₈	1	13	yes	—
82	C₃⋊Dic₃	—	6	26	no	—
83	C₃⋊Dic₃	—	12	26	no	—
84	C₃⋊Dic₃	—	12	13	no	—
85	C₄⋊Q₈	—	27	1	no	—
86	C₄×Q₈	—	27	1	no	—
87	#107	—	27	2	no	—
88	C₄⋊C₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
89	#103	—	27	2	no	—
90	C₃³	#477	1	1	yes	characteristic, complemented, Sylow
91	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	13	yes	—
92	C₂×C₃×C₄	C₄⋊(C₃⋊D₃)	1	13	yes	—
93	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
94	C₂×Dic₃	—	18	13	no	—
95	C₂×Dic₃	—	18	13	no	—
96	C₂×Dic₃	—	18	26	no	—
97	Dic₆	—	9	26	no	—
98	Dic₆	—	18	13	no	—
99	Dic₆	—	36	26	no	—
100	C₃×C₈	—	2	26	no	—
101	C₂×C₃²	#897	1	13	yes	—
102	C₂×C₃²	C₂×Dic₁₂	1	13	yes	—
103	C₂×C₃²	#908	1	13	yes	—
104	C₂×Q₈	—	27	1	no	—
105	C₂×Q₈	—	54	1	no	—
106	C₂×C₈	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
107	C₄⋊C₄	—	27	2	no	—
108	C₄⋊C₄	—	27	1	no	—
109	C₄⋊C₄	—	54	1	no	—
110	C₄⋊C₄	—	54	1	no	—
111	C₄²	—	54	1	no	—
112	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
113	C₂²×C₃	C₂×C₄⋊(C₃⋊D₃)	1	13	yes	—
114	C₃×C₄	—	2	13	no	—
115	C₃×C₄	#3746	1	26	yes	—
116	C₃×C₄	#3830	1	13	yes	—
117	C₃×C₄	Q₈×C₃⋊D₃	1	13	yes	—
118	Dic₃	—	18	26	no	—
119	Dic₃	—	36	26	no	—
120	Dic₃	—	36	13	no	—
121	C₃²	#4850	1	13	yes	complemented
122	Q₈	—	27	2	no	—
123	Q₈	—	54	1	no	—
124	Q₈	—	108	2	no	—
125	C₂×C₄	—	54	1	no	—
126	C₂×C₄	—	54	1	no	—
127	C₂×C₄	—	54	2	no	—
128	C₂×C₄	C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
129	C₂×C₄	C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
130	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₂×C₃	#64106	1	13	yes	—
133	C₂×C₃	C₂×#3746	1	13	yes	—
134	C₂×C₃	#64117	1	13	yes	—
135	C₂²	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
136	C₄	—	2	1	no	—
137	C₄	—	54	2	no	—
138	C₄	—	108	2	no	—
139	C₄	—	108	1	no	—
140	C₄	#88853	1	2	yes	—
141	C₄	#89078	1	1	yes	characteristic
142	C₄	Q₈×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
143	C₃	#10592549	1	13	yes	complemented
144	C₂	#11723938	1	1	yes	characteristic
145	C₂	C₂×#88853	1	1	yes	characteristic
146	C₂	#11723949	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11723910

C₂

yes

maximal, characteristic

C₄×#6308

C₂

yes

maximal, characteristic

C₃³×C₄⋊C₈

C₂

yes

maximal, characteristic

#11722188

C₂

yes

maximal

#11722186

C₂

yes

—

—

C₂²

yes

characteristic

C₂×C₃³×C₈

C₂²

yes

—

C₃³×C₄²

C₂²

yes

characteristic

C₄⋊(C₃⋊(C₃⋊Dic₃))

—

C₄⋊(C₃⋊(C₃⋊Dic₃))

C₂²

yes

characteristic

C₄⋊(C₃⋊(C₃⋊Dic₃))

C₂²

yes

—

#88857

—

C₄×C₃⋊(C₃⋊Dic₃)

—

#64078

—

C₄×#634

—

C₃²×C₄⋊C₈

D₃

yes

—

#63645

—

#63643

—

C₂×C₃³×C₄

D₄

yes

characteristic

C₂×C₃³×C₄

D₄

yes

characteristic

C₂×C₃³×C₄

C₂³

yes

characteristic

C₂×C₃⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊Dic₃)

—

#6308

—

#6308

—

#6308

Q₈

yes

complemented

C₃³×C₈

—

#4850

—

C₂×#634

—

C₂×#634

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

—

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

#3750

—

C₄×C₃⋊Dic₃

—

C₂²×C₃³

C₂×D₄

yes

characteristic

C₃³×C₄

—

C₃³×C₄

Q₁₆

yes

—

C₃³×C₄

C₂×Q₈

yes

characteristic

C₃³×C₄

#41

yes

characteristic

C₃⋊(C₃⋊Dic₃)

—

C₃⋊(C₃⋊Dic₃)

—

C₃⋊(C₃⋊Dic₃)

—

#869

—

C₄×Dic₆

—

C₃×C₄⋊C₈

C₃⋊D₃

yes

—

#818

—

#816

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

D₁₂

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

#634

—

#634

—

#634

—

C₃²×C₈

—

#477

—

Sylow

C₂×C₃³

C₂²⋊Q₈

yes

characteristic

C₂×C₃³

C₂×Q₁₆

yes

characteristic

C₂×C₃³

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

—

C₃×C₄²

C₂×C₃⋊D₃

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂²×C₃²

C₂×D₁₂

yes

—

C₃²×C₄

—

C₃²×C₄

Dic₁₂

yes

—

C₃²×C₄

#237

yes

—

C₃²×C₄

D₃×Q₈

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₄⋊Q₈

—

C₄×Q₈

—

#107

—

C₄⋊C₈

C₃⋊(C₃⋊D₃)

yes

characteristic

#103

—

C₃³

#477

yes

characteristic, complemented, Sylow

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

—

C₂×C₃×C₄

C₄⋊(C₃⋊D₃)

yes

—

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

Dic₆

—

100

C₃×C₈

—

101

C₂×C₃²

#897

yes

—

102

C₂×C₃²

C₂×Dic₁₂

yes

—

103

C₂×C₃²

#908

yes

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×C₈

C₂×C₃⋊(C₃⋊D₃)

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄²

—

112

C₄²

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

113

C₂²×C₃

C₂×C₄⋊(C₃⋊D₃)

yes

—

114

C₃×C₄

—

115

C₃×C₄

#3746

yes

—

116

C₃×C₄

#3830

yes

—

117

C₃×C₄

Q₈×C₃⋊D₃

yes

—

118

Dic₃

—

119

Dic₃

—

120

Dic₃

—

121

C₃²

#4850

yes

complemented

122

Q₈

—

123

Q₈

—

124

Q₈

—

108

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₄⋊(C₃⋊(C₃⋊D₃))

yes

characteristic

129

C₂×C₄

C₄⋊(C₃⋊(C₃⋊D₃))

yes

characteristic

130

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

131

C₈

—

132

C₂×C₃

#64106

yes

—

133

C₂×C₃

C₂×#3746

yes

—

134

C₂×C₃

#64117

yes

—

135

C₂²

C₂×C₄⋊(C₃⋊(C₃⋊D₃))

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

108

—

139

C₄

—

108

—

140

C₄

#88853

yes

—

141

C₄

#89078

yes

characteristic

142

C₄

Q₈×C₃⋊(C₃⋊D₃)

yes

characteristic

143

C₃

#10592549

yes

complemented

144

C₂

#11723938

yes

characteristic

145

C₂

C₂×#88853

yes

characteristic

146

C₂

#11723949

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊(C₃⋊(C₃⋊D₃))
commutator, cosocle	C₂×C₃³×C₄	C₂³
Frattini	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₃³×C₄⋊C₈	C₂
socle, prime radical	C₂²×C₃³	C₂×D₄
hypercenter	C₄⋊C₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#477
LCS	G ▷ C₂×C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊(C₃⋊(C₃⋊D₃))

commutator, cosocle

C₂×C₃³×C₄

C₂³

Frattini

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

Fitting

C₃³×C₄⋊C₈

C₂

socle, prime radical

C₂²×C₃³

C₂×D₄

hypercenter

C₄⋊C₈

C₃⋊(C₃⋊D₃)

hypocenter

C₃³

#477

LCS

G ▷ C₂×C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
29. #6308	Q₈	semidirect
90. C₃³	#477	semidirect
121. C₃²	#4850	semidirect
143. C₃	#10592549	semidirect

H ≅ G/N

type

29. #6308

Q₈

semidirect

90. C₃³

#477

semidirect

121. C₃²

#4850

semidirect

143. C₃

#10592549

semidirect

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

subgroup	index
#11723910	2
C₄×#6308	2
C₃³×C₄⋊C₈	2
#11722188	2
#11722186	2
#10592549	3

subgroup

index

C₃³×C₄⋊C₈

Subgroups — #13163473

Conjugacy classes of subgroups

Complements

Maximal subgroups