Group #135733 — subgroups

Subgroups — #135733

Order 512.

260 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 134 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	#12883	C₂	1	2	yes	maximal
2	#12004	C₂	1	2	yes	maximal
3	#11996	C₂	1	2	yes	maximal
4	#7841	C₂	1	1	yes	maximal, characteristic
5	#2150	C₂²	1	4	yes	—
6	#1744	C₄	1	1	yes	characteristic
7	#1744	C₄	1	1	yes	characteristic
8	C₄²×C₈	C₂²	1	1	yes	characteristic
9	C₂×#322	C₄	1	1	yes	characteristic
10	C₂×#322	C₄	1	1	yes	characteristic
11	C₂×C₈²	C₂²	1	1	yes	characteristic
12	C₂×C₈²	C₂²	1	1	yes	characteristic
13	#1376	—	2	4	no	—
14	#1370	—	2	4	no	—
15	#1367	—	2	4	no	—
16	C₂×M₃₂	—	2	4	no	—
17	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₈	D₄	1	2	yes	—
23	C₂×C₄×C₈	Q₈	1	2	yes	—
24	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
25	C₄³	C₂×C₄	1	1	yes	characteristic
26	C₄⋊C₁₆	—	2	4	no	—
27	C₄⋊C₁₆	—	2	4	no	—
28	#336	—	2	2	no	—
29	#336	—	2	2	no	—
30	#322	—	2	2	no	—
31	#322	C₈	1	4	yes	—
32	C₈²	D₄	1	2	yes	—
33	C₈²	D₄	1	2	yes	—
34	C₈²	Q₈	1	2	yes	—
35	C₈²	Q₈	1	2	yes	—
36	C₂²×C₈	—	2	1	no	—
37	C₂²×C₈	—	2	1	no	—
38	C₂²×C₈	C₄⋊C₄	1	2	yes	—
39	C₂²×C₈	M₁₆	1	2	yes	—
40	C₂²×C₈	M₁₆	1	2	yes	—
41	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄²	—	2	1	no	—
44	C₂×C₄²	—	2	1	no	—
45	C₂×C₄²	C₄⋊C₄	1	2	yes	—
46	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
47	M₃₂	—	4	8	no	—
48	C₂×C₁₆	—	4	4	no	—
49	C₄×C₈	—	2	4	no	—
50	C₄×C₈	—	2	4	no	—
51	C₄×C₈	—	2	1	no	—
52	C₄×C₈	—	2	1	no	—
53	C₄×C₈	C₄⋊C₄	1	2	yes	—
54	C₄×C₈	C₄⋊C₄	1	2	yes	—
55	C₄×C₈	C₄⋊C₄	1	2	yes	—
56	C₄×C₈	C₄⋊C₄	1	2	yes	—
57	C₄×C₈	C₂×C₈	1	2	yes	—
58	C₄×C₈	C₂×C₈	1	2	yes	—
59	C₄×C₈	C₂×C₈	1	2	yes	—
60	C₄×C₈	M₁₆	1	2	yes	—
61	C₄×C₈	C₂×D₄	1	1	yes	characteristic
62	C₄×C₈	C₂×D₄	1	1	yes	characteristic
63	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
64	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄	—	2	2	no	—
66	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
68	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	4	2	no	—
75	C₂×C₈	—	4	2	no	—
76	C₂×C₈	—	4	2	no	—
77	C₂×C₈	—	4	2	no	—
78	C₂×C₈	C₄⋊C₈	1	4	yes	—
79	C₂×C₈	C₄⋊C₈	1	4	yes	—
80	C₂×C₈	C₄⋊C₈	1	4	yes	—
81	C₂×C₈	#127	1	1	yes	characteristic
82	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
84	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₈	#135	1	4	yes	—
86	C₄²	—	2	2	no	—
87	C₄²	—	4	2	no	—
88	C₄²	—	4	2	no	—
89	C₄²	#108	1	4	yes	—
90	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₄²	C₂²×C₈	1	1	yes	characteristic
93	C₄²	C₂×M₁₆	1	1	yes	characteristic
94	C₁₆	—	8	4	no	—
95	C₂³	#423	1	1	yes	characteristic
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	#384	1	1	yes	characteristic
102	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
103	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
105	C₂×C₄	#423	1	1	yes	characteristic
106	C₂×C₄	#423	1	1	yes	characteristic
107	C₂×C₄	#427	1	2	yes	—
108	C₂×C₄	C₂×#108	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₈	—	2	1	no	—
111	C₈	—	2	1	no	—
112	C₈	—	2	4	no	—
113	C₈	—	2	4	no	—
114	C₈	—	4	2	no	—
115	C₈	—	4	2	no	—
116	C₈	#495	1	2	yes	—
117	C₈	#499	1	2	yes	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	#1568	1	2	yes	—
121	C₂²	#1842	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	4	4	no	—
125	C₄	#1564	1	2	yes	—
126	C₄	#1846	1	1	yes	characteristic
127	C₄	#1848	1	1	yes	characteristic
128	C₄	#2153	1	2	yes	—
129	C₄	#2153	1	2	yes	—
130	C₂	—	2	2	no	—
131	C₂	#8807	1	1	yes	characteristic
132	C₂	#12901	1	1	yes	characteristic
133	C₂	#12901	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12883

C₂

yes

maximal

#12004

C₂

yes

maximal

#11996

C₂

yes

maximal

#7841

C₂

yes

maximal, characteristic

#2150

C₂²

yes

—

#1744

C₄

yes

characteristic

#1744

C₄

yes

characteristic

C₄²×C₈

C₂²

yes

characteristic

C₂×#322

C₄

yes

characteristic

C₂×#322

C₄

yes

characteristic

C₂×C₈²

C₂²

yes

characteristic

C₂×C₈²

C₂²

yes

—

—

—

C₂×M₃₂

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

Q₈

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

C₈

yes

—

C₈²

D₄

yes

—

C₈²

D₄

yes

—

C₈²

Q₈

yes

—

C₈²

Q₈

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

M₁₆

yes

—

C₂²×C₈

M₁₆

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₄⋊C₄

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂×M₁₆

yes

characteristic

C₂²×C₄

C₂×M₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

#127

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

#135

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#108

yes

—

C₄²

C₂×C₄⋊C₄

yes

characteristic

C₄²

C₂×C₄⋊C₄

yes

characteristic

C₄²

C₂²×C₈

yes

characteristic

C₄²

C₂×M₁₆

yes

characteristic

C₁₆

—

C₂³

#423

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

#384

yes

characteristic

102

C₂×C₄

C₂×C₄⋊C₈

yes

characteristic

103

C₂×C₄

C₂×C₄⋊C₈

yes

characteristic

104

C₂×C₄

C₂×C₄⋊C₈

yes

characteristic

105

C₂×C₄

#423

yes

characteristic

106

C₂×C₄

#423

yes

characteristic

107

C₂×C₄

#427

yes

—

108

C₂×C₄

C₂×#108

yes

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

#495

yes

—

117

C₈

#499

yes

—

118

C₂²

—

119

C₂²

—

120

C₂²

#1568

yes

—

121

C₂²

#1842

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#1564

yes

—

126

C₄

#1846

yes

characteristic

127

C₄

#1848

yes

characteristic

128

C₄

#2153

yes

—

129

C₄

#2153

yes

—

130

C₂

—

131

C₂

#8807

yes

characteristic

132

C₂

#12901

yes

characteristic

133

C₂

#12901

yes

characteristic

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1842
prime radical	C₂³	#423
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

#127

commutator

C₄²

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₈ ◁ C₂×C₄×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
#12883	2
#12004	2
#11996	2
#7841	2

subgroup

index

Subgroups — #135733

Conjugacy classes of subgroups

Maximal subgroups