Group #135672 — subgroups

Subgroups — #135672

Order 512.

238 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 110 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	#12591	C₂	1	1	yes	maximal, characteristic
2	#8807	C₂	1	1	yes	maximal, characteristic
3	#8805	C₂	1	1	yes	maximal, characteristic
4	#7156	C₂	1	4	yes	maximal
5	C₂×#346	C₂²	1	2	yes	—
6	#1842	—	2	1	no	—
7	#1831	—	2	1	no	—
8	C₄²×C₈	C₂²	1	1	yes	characteristic
9	C₂×#335	—	2	1	no	—
10	C₂×#334	—	2	1	no	—
11	C₂×#334	C₂²	1	2	yes	—
12	C₂×#334	C₂²	1	2	yes	—
13	#1374	C₄	1	2	yes	—
14	#1374	C₄	1	2	yes	—
15	C₂²×C₁₆	C₂×C₄	1	4	yes	—
16	C₂×C₄⋊C₈	—	2	2	no	—
17	C₂×C₄⋊C₈	—	2	2	no	—
18	C₂×C₄⋊C₈	—	4	2	no	—
19	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₈	D₄	1	1	yes	characteristic
22	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
23	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
25	C₄³	D₄	1	1	yes	characteristic
26	#346	—	2	2	no	—
27	#346	—	2	2	no	—
28	#336	—	4	2	no	—
29	#335	—	4	2	no	—
30	#334	—	2	4	no	—
31	#334	—	2	4	no	—
32	#334	—	4	2	no	—
33	C₂²×C₈	—	4	2	no	—
34	C₂²×C₈	—	4	2	no	—
35	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄²	SD₁₆	1	2	yes	—
42	C₂×C₄²	SD₁₆	1	2	yes	—
43	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
44	C₂×C₁₆	—	2	4	no	—
45	C₂×C₁₆	—	2	4	no	—
46	C₂×C₁₆	—	2	4	no	—
47	C₄⋊C₈	—	4	2	no	—
48	C₄⋊C₈	—	4	2	no	—
49	C₄⋊C₈	—	4	2	no	—
50	C₄⋊C₈	—	4	2	no	—
51	C₄⋊C₈	—	8	4	no	—
52	C₄×C₈	—	2	2	no	—
53	C₄×C₈	—	2	2	no	—
54	C₄×C₈	—	4	1	no	—
55	C₄×C₈	—	4	1	no	—
56	C₄×C₈	D₈	1	2	yes	—
57	C₄×C₈	Q₁₆	1	2	yes	—
58	C₄×C₈	#41	1	2	yes	—
59	C₄×C₈	#41	1	2	yes	—
60	C₂²×C₄	#106	1	2	yes	—
61	C₂²×C₄	#106	1	2	yes	—
62	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
64	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	—	2	1	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	8	4	no	—
72	C₂×C₈	—	8	4	no	—
73	C₂×C₈	—	8	2	no	—
74	C₂×C₈	—	8	2	no	—
75	C₂×C₈	#107	1	2	yes	—
76	C₂×C₈	#107	1	2	yes	—
77	C₂×C₈	#117	1	2	yes	—
78	C₂×C₈	#125	1	2	yes	—
79	C₂×C₈	#131	1	2	yes	—
80	C₂×C₈	#131	1	2	yes	—
81	C₄²	—	2	4	no	—
82	C₄²	—	2	1	no	—
83	C₄²	—	2	1	no	—
84	C₄²	—	2	1	no	—
85	C₄²	—	2	1	no	—
86	C₄²	—	4	2	no	—
87	C₄²	#124	1	1	yes	characteristic
88	C₄²	#124	1	1	yes	characteristic
89	C₄²	C₂×D₈	1	1	yes	characteristic
90	C₄²	C₂×Q₁₆	1	1	yes	characteristic
91	C₁₆	—	4	4	no	—
92	C₂³	C₂×#106	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	#365	1	4	yes	—
101	C₂×C₄	#382	1	1	yes	characteristic
102	C₂×C₄	#424	1	1	yes	characteristic
103	C₂×C₄	#424	1	1	yes	characteristic
104	C₂×C₄	C₂×#107	1	1	yes	characteristic
105	C₂×C₄	#430	1	1	yes	characteristic
106	C₂×C₄	#430	1	1	yes	characteristic
107	C₂×C₄	#486	1	1	yes	characteristic
108	C₂×C₄	#486	1	1	yes	characteristic
109	C₂×C₄	#487	1	1	yes	characteristic
110	C₂×C₄	#487	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	2	2	no	—
113	C₈	—	2	2	no	—
114	C₈	—	16	2	no	—
115	C₈	—	16	2	no	—
116	C₈	#433	1	4	yes	—
117	C₂²	#1570	1	1	yes	characteristic
118	C₂²	#1570	1	1	yes	characteristic
119	C₂²	#1834	1	1	yes	characteristic
120	C₂²	C₂×#365	1	2	yes	—
121	C₂²	#2155	1	2	yes	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	4	no	—
127	C₄	—	4	2	no	—
128	C₄	#1571	1	1	yes	characteristic
129	C₄	#1571	1	1	yes	characteristic
130	C₄	#1832	1	1	yes	characteristic
131	C₄	#1838	1	1	yes	characteristic
132	C₂	#8787	1	1	yes	characteristic
133	C₂	#12028	1	2	yes	—
134	C₂	#12029	1	2	yes	—
135	C₂	#12886	1	1	yes	characteristic
136	C₂	#12887	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12591

C₂

yes

maximal, characteristic

#8807

C₂

yes

maximal, characteristic

#8805

C₂

yes

maximal, characteristic

#7156

C₂

yes

maximal

C₂×#346

C₂²

yes

—

#1842

—

#1831

—

C₄²×C₈

C₂²

yes

—

—

C₂²

yes

—

C₂×#334

C₂²

yes

—

#1374

C₄

yes

—

#1374

C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

D₄

yes

—

—

—

—

—

—

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

SD₁₆

yes

—

C₂×C₄²

SD₁₆

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

#41

yes

—

C₄×C₈

#41

yes

—

C₂²×C₄

#106

yes

—

C₂²×C₄

#106

yes

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#107

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#117

yes

—

C₂×C₈

#125

yes

—

C₂×C₈

#131

yes

—

C₂×C₈

#131

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

yes

C₄²

yes

C₄²

C₂×D₈

yes

characteristic

C₄²

C₂×Q₁₆

yes

characteristic

C₁₆

—

C₂³

C₂×#106

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#365

yes

—

101

C₂×C₄

#382

yes

characteristic

102

C₂×C₄

#424

yes

characteristic

103

C₂×C₄

#424

yes

characteristic

104

C₂×C₄

C₂×#107

yes

characteristic

105

C₂×C₄

#430

yes

characteristic

106

C₂×C₄

#430

yes

characteristic

107

C₂×C₄

#486

yes

characteristic

108

C₂×C₄

#486

yes

characteristic

109

C₂×C₄

#487

yes

characteristic

110

C₂×C₄

#487

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

#433

yes

—

117

C₂²

#1570

yes

characteristic

118

C₂²

#1570

yes

characteristic

119

C₂²

#1834

yes

characteristic

120

C₂²

C₂×#365

yes

—

121

C₂²

#2155

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#1571

yes

characteristic

129

C₄

#1571

yes

characteristic

130

C₄

#1832

yes

characteristic

131

C₄

#1838

yes

characteristic

132

C₂

#8787

yes

characteristic

133

C₂

#12028

yes

—

134

C₂

#12029

yes

—

135

C₂

#12886

yes

characteristic

136

C₂

#12887

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#106
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×#106

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#12591	2
#8807	2
#8805	2
#7156	2

subgroup

index

Subgroups — #135672

Conjugacy classes of subgroups

Maximal subgroups