Group #10592758 — subgroups

Subgroups — #10592758

Order 576.

390 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 99 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×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#3756	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#3755	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#3754	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#3753	C₂	1	1	yes	maximal, characteristic
6	#64084	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	#5059	—	3	4	no	maximal
9	C₂×C₃²×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
12	C₂×#634	C₂²	1	1	yes	characteristic
13	C₃²×C₂²⋊C₄	—	2	2	no	—
14	C₃²×C₄²	C₂²	1	1	yes	characteristic
15	#3756	—	2	2	no	—
16	#3755	—	2	2	no	—
17	#3754	—	2	2	no	—
18	#3753	—	2	2	no	—
19	#3752	—	2	2	no	—
20	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
21	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
22	C₃×#124	D₃	1	4	yes	complemented
23	C₂×#216	—	3	4	no	—
24	C₂×#215	—	3	4	no	—
25	C₂×#214	—	3	4	no	—
26	C₂×#213	—	3	4	no	—
27	#875	—	3	4	no	—
28	C₄×C₃⋊C₈	—	3	4	no	—
29	C₂³×C₃²	—	2	1	no	—
30	C₂²×C₃⋊D₃	—	2	1	no	—
31	C₃²×Q₈	—	2	1	no	—
32	C₃²×Q₈	—	2	1	no	—
33	C₃²×D₄	—	2	1	no	—
34	C₃²×D₄	—	2	1	no	—
35	C₂×C₃²×C₄	—	2	1	no	—
36	C₂×C₃²×C₄	D₄	1	2	yes	—
37	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
38	C₂×C₃⋊Dic₃	—	2	1	no	—
39	C₄⋊(C₃⋊D₃)	—	2	1	no	—
40	C₄⋊(C₃⋊D₃)	—	2	1	no	—
41	#634	—	2	1	no	—
42	#634	—	2	1	no	—
43	C₃⋊(C₃⋊C₈)	D₄	1	2	yes	—
44	C₃⋊(C₃⋊C₈)	D₄	1	2	yes	—
45	#495	—	9	1	no	Sylow
46	C₂×C₃×Q₈	C₂×D₃	1	4	yes	—
47	C₂×C₃×D₄	C₂×D₃	1	4	yes	—
48	C₂×D₁₂	—	3	4	no	—
49	C₂×Dic₆	—	3	4	no	—
50	C₃×C₂²⋊C₄	—	2	8	no	—
51	C₃×C₄²	C₂×D₃	1	4	yes	—
52	#216	—	6	8	no	—
53	#215	—	6	8	no	—
54	#214	—	6	8	no	—
55	#213	—	6	8	no	—
56	(C₂×D₃)⋊C₄	—	6	8	no	—
57	C₂×C₃⋊C₈	—	3	4	no	—
58	C₂×C₃⋊C₈	—	3	4	no	—
59	C₂²×C₃²	—	4	1	no	—
60	C₂²×C₃²	—	4	2	no	—
61	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃⋊D₃	—	4	1	no	—
63	C₂×C₃⋊D₃	—	4	2	no	—
64	C₃²×C₄	—	2	2	no	—
65	C₃²×C₄	—	4	1	no	—
66	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
67	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
68	C₃⋊Dic₃	—	4	1	no	—
69	C₂×Q₁₆	—	9	1	no	—
70	C₂×SD₁₆	—	9	1	no	—
71	C₂×SD₁₆	—	9	1	no	—
72	C₂×D₈	—	9	1	no	—
73	#124	—	9	1	no	—
74	#124	C₃⋊D₃	1	1	yes	characteristic, complemented
75	C₄×C₈	—	9	1	no	—
76	C₂³×C₃	—	2	4	no	—
77	C₂²×D₃	—	6	4	no	—
78	C₃×Q₈	—	2	4	no	—
79	C₃×Q₈	—	2	4	no	—
80	C₃×D₄	—	2	4	no	—
81	C₃×D₄	—	2	4	no	—
82	C₂×C₃×C₄	—	2	4	no	—
83	C₂×C₃×C₄	#67	1	8	yes	—
84	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
85	C₂×Dic₃	—	6	4	no	—
86	D₁₂	—	6	4	no	—
87	D₁₂	—	6	4	no	—
88	Dic₆	—	6	4	no	—
89	Dic₆	—	6	4	no	—
90	C₃⋊C₈	—	3	8	no	—
91	C₃⋊C₈	—	3	8	no	—
92	C₂×C₃²	—	8	1	no	—
93	C₂×C₃²	#127	1	1	yes	characteristic
94	C₂×C₃²	#135	1	2	yes	—
95	C₃⋊D₃	—	8	1	no	—
96	C₂×Q₈	—	9	1	no	—
97	C₂×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
98	C₂×D₄	—	9	1	no	—
99	C₂×D₄	C₂×C₃⋊D₃	1	1	yes	characteristic
100	Q₁₆	—	18	2	no	—
101	SD₁₆	—	18	2	no	—
102	SD₁₆	—	18	2	no	—
103	D₈	—	18	2	no	—
104	C₂×C₈	—	9	1	no	—
105	C₂×C₈	—	9	1	no	—
106	C₂²⋊C₄	—	2	2	no	—
107	C₂²⋊C₄	—	18	2	no	—
108	C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
109	C₂²×C₃	—	4	4	no	—
110	C₂²×C₃	—	4	8	no	—
111	C₂²×C₃	C₂×#67	1	4	yes	—
112	C₂×D₃	—	12	4	no	—
113	C₂×D₃	—	12	8	no	—
114	C₃×C₄	—	2	8	no	—
115	C₃×C₄	—	4	4	no	—
116	C₃×C₄	D₃×D₄	1	4	yes	—
117	C₃×C₄	D₃×D₄	1	4	yes	—
118	Dic₃	—	12	4	no	—
119	C₃²	#495	1	1	yes	characteristic, complemented, Sylow
120	C₂³	—	2	1	no	—
121	C₂³	—	18	1	no	—
122	Q₈	—	2	1	no	—
123	Q₈	—	2	1	no	—
124	Q₈	—	18	1	no	—
125	Q₈	—	18	1	no	—
126	D₄	—	2	1	no	—
127	D₄	—	2	1	no	—
128	D₄	—	18	1	no	—
129	D₄	—	18	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	18	1	no	—
132	C₂×C₄	#638	1	2	yes	—
133	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
134	C₈	—	9	2	no	—
135	C₈	—	9	2	no	—
136	C₂×C₃	—	8	4	no	—
137	C₂×C₃	#940	1	4	yes	—
138	C₂×C₃	#950	1	8	yes	—
139	D₃	—	24	4	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	2	no	—
142	C₂²	—	36	1	no	—
143	C₂²	—	36	2	no	—
144	C₂²	C₂×#638	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	4	1	no	—
147	C₄	—	36	1	no	—
148	C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
149	C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
150	C₃	#5059	1	4	yes	complemented
151	C₂	—	8	1	no	—
152	C₂	—	72	1	no	—
153	C₂	#64149	1	1	yes	characteristic
154	C₂	#64159	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#124

C₂

yes

maximal, characteristic, complemented

C₂×#3756

C₂

yes

maximal, characteristic, complemented

C₂×#3755

C₂

yes

maximal, characteristic, complemented

C₂×#3754

C₂

yes

maximal, characteristic, complemented

C₂×#3753

C₂

yes

maximal, characteristic

#64084

C₂

yes

maximal, characteristic, complemented

C₄×C₃⋊(C₃⋊C₈)

C₂

yes

maximal, characteristic, complemented

#5059

—

maximal

C₂×C₃²×Q₈

C₂²

yes

characteristic

C₂×C₃²×D₄

C₂²

yes

characteristic

C₂×C₄⋊(C₃⋊D₃)

C₂²

yes

characteristic

C₂×#634

C₂²

yes

characteristic

C₃²×C₂²⋊C₄

—

C₃²×C₄²

C₂²

yes

—

—

—

—

—

C₂×C₃⋊(C₃⋊C₈)

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

C₂²

yes

characteristic

C₃×#124

D₃

yes

—

—

—

—

—

C₄×C₃⋊C₈

—

C₂³×C₃²

—

C₂²×C₃⋊D₃

—

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×D₄

—

C₃²×D₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

D₄

yes

—

C₂×C₃²×C₄

C₂³

yes

characteristic

C₂×C₃⋊Dic₃

—

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

#634

—

#634

—

C₃⋊(C₃⋊C₈)

D₄

yes

—

C₃⋊(C₃⋊C₈)

D₄

yes

—

#495

—

Sylow

C₂×C₃×Q₈

C₂×D₃

yes

—

C₂×C₃×D₄

C₂×D₃

yes

—

C₂×D₁₂

—

C₂×Dic₆

—

C₃×C₂²⋊C₄

—

C₃×C₄²

C₂×D₃

yes

—

—

—

—

—

(C₂×D₃)⋊C₄

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂×D₄

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×D₄

yes

characteristic

C₃²×C₄

C₂×D₄

yes

characteristic

C₃⋊Dic₃

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

#124

—

#124

C₃⋊D₃

yes

characteristic, complemented

C₄×C₈

—

C₂³×C₃

—

C₂²×D₃

—

C₃×Q₈

—

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#67

yes

—

C₂×C₃×C₄

C₂²×D₃

yes

—

C₂×Dic₃

—

D₁₂

—

D₁₂

—

Dic₆

—

Dic₆

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

#127

yes

characteristic

C₂×C₃²

#135

yes

—

C₃⋊D₃

—

C₂×Q₈

—

C₂×Q₈

C₂×C₃⋊D₃

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂×C₃⋊D₃

yes

characteristic

100

Q₁₆

—

101

SD₁₆

—

102

SD₁₆

—

103

D₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

—

108

C₄²

C₂×C₃⋊D₃

yes

characteristic

109

C₂²×C₃

—

110

C₂²×C₃

—

111

C₂²×C₃

C₂×#67

yes

—

112

C₂×D₃

—

113

C₂×D₃

—

114

C₃×C₄

—

115

C₃×C₄

—

116

C₃×C₄

D₃×D₄

yes

—

117

C₃×C₄

D₃×D₄

yes

—

118

Dic₃

—

119

C₃²

#495

yes

characteristic, complemented, Sylow

120

C₂³

—

121

C₂³

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

D₄

—

127

D₄

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

#638

yes

—

133

C₂×C₄

C₂²×C₃⋊D₃

yes

characteristic

134

C₈

—

135

C₈

—

136

C₂×C₃

—

137

C₂×C₃

#940

yes

—

138

C₂×C₃

#950

yes

—

139

D₃

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

C₂×#638

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

D₄×C₃⋊D₃

yes

characteristic

149

C₄

D₄×C₃⋊D₃

yes

characteristic

150

C₃

#5059

yes

complemented

151

C₂

—

152

C₂

—

153

C₂

#64149

yes

characteristic

154

C₂

#64159

yes

—

	subgroup	G/H
center	C₂²	C₂×#638
commutator, cosocle	C₂×C₃²×C₄	C₂³
Frattini	C₂×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×#124	C₂
socle	C₂²×C₃²	C₂×D₄
prime radical	C₂×#3753	C₂
hypercenter	#124	C₃⋊D₃
hypocenter	C₃²	#495
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #124

subgroup

G/H

center

C₂²

C₂×#638

commutator, cosocle

C₂×C₃²×C₄

C₂³

Frattini

C₂×C₄

C₂²×C₃⋊D₃

Fitting

C₃²×#124

C₂

socle

C₂²×C₃²

C₂×D₄

C₂

C₃⋊D₃

C₃²

G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #124

N	H ≅ G/N	type
1. C₃²×#124	C₂	semidirect
2. C₂×#3756	C₂	semidirect
3. C₂×#3755	C₂	semidirect
4. C₂×#3754	C₂	semidirect
6. #64084	C₂	semidirect
7. C₄×C₃⋊(C₃⋊C₈)	C₂	semidirect
22. C₃×#124	D₃	semidirect
74. #124	C₃⋊D₃	semidirect
119. C₃²	#495	semidirect
150. C₃	#5059	semidirect

H ≅ G/N

type

1. C₃²×#124

C₂

semidirect

2. C₂×#3756

C₂

semidirect

3. C₂×#3755

C₂

semidirect

4. C₂×#3754

C₂

semidirect

6. #64084

C₂

semidirect

7. C₄×C₃⋊(C₃⋊C₈)

C₂

semidirect

22. C₃×#124

D₃

semidirect

74. #124

C₃⋊D₃

semidirect

119. C₃²

#495

semidirect

150. C₃

#5059

semidirect

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

subgroup	index
C₃²×#124	2
C₂×#3756	2
C₂×#3755	2
C₂×#3754	2
C₂×#3753	2
#64084	2
C₄×C₃⋊(C₃⋊C₈)	2
#5059	3

subgroup

index

C₃²×#124

C₄×C₃⋊(C₃⋊C₈)

Subgroups — #10592758

Conjugacy classes of subgroups

Complements

Maximal subgroups