Group #134849 — subgroups

Subgroups — #134849

Order 512.

277 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 90 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	#33434	C₂	1	1	yes	maximal, characteristic
2	#12023	C₂	1	2	yes	maximal, complemented
3	#8638	C₂	1	2	yes	maximal
4	#7245	C₂	1	2	yes	maximal, complemented
5	C₂×#504	C₄	1	2	yes	—
6	#2931	C₂²	1	1	yes	characteristic
7	#2169	—	2	2	no	—
8	D₄×C₁₆	—	2	2	no	—
9	#2150	C₂²	1	1	yes	characteristic
10	#2113	C₄	1	2	yes	—
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	#1567	C₂²	1	4	yes	—
13	#1560	—	2	2	no	—
14	#1372	—	2	2	no	—
15	#1372	—	2	2	no	—
16	#1309	—	2	2	no	—
17	C₂×#131	C₂×C₄	1	2	yes	—
18	#532	D₄	1	1	yes	characteristic
19	#504	—	4	4	no	—
20	C₂×M₃₂	—	2	2	no	—
21	C₂×M₃₂	C₂×C₄	1	2	yes	—
22	C₂²×C₁₆	—	2	2	no	—
23	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
25	#436	—	2	2	no	—
26	C₈×D₄	—	2	2	no	—
27	#423	—	2	4	no	—
28	#423	D₄	1	1	yes	characteristic
29	#408	Q₈	1	2	yes	—
30	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
31	C₄⋊C₁₆	—	2	2	no	—
32	C₂²⋊C₁₆	—	2	4	no	—
33	C₄×C₁₆	D₄	1	2	yes	—
34	C₄×C₁₆	D₄	1	2	yes	—
35	#335	—	2	4	no	—
36	#335	—	2	4	no	—
37	#333	—	4	2	no	—
38	#331	—	4	2	no	—
39	C₂×#41	C₄⋊C₄	1	2	yes	—
40	#131	—	4	2	no	—
41	#131	—	4	2	no	—
42	C₂×M₁₆	—	2	2	no	—
43	C₂×M₁₆	—	4	4	no	—
44	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
45	C₂²×C₈	—	2	2	no	—
46	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
48	C₄⋊Q₈	D₈	1	1	yes	characteristic
49	#127	D₈	1	1	yes	characteristic
50	#124	Q₁₆	1	2	yes	—
51	#121	—	2	2	no	—
52	C₄×D₄	—	2	2	no	—
53	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
54	M₃₂	—	4	4	no	—
55	M₃₂	—	4	2	no	—
56	C₂×C₁₆	—	2	2	no	—
57	C₂×C₁₆	—	2	2	no	—
58	C₂×C₁₆	—	4	1	no	—
59	C₂×C₁₆	—	4	4	no	—
60	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
61	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	4	4	no	—
65	C₄⋊C₈	—	4	4	no	—
66	C₂²⋊C₈	—	2	4	no	—
67	C₄×C₈	C₂×D₄	1	2	yes	—
68	C₄×C₈	#41	1	2	yes	—
69	#41	—	4	2	no	—
70	#41	—	4	2	no	—
71	C₂×Q₈	#107	1	2	yes	—
72	C₂×D₄	—	2	2	no	—
73	C₂×D₄	—	2	2	no	—
74	C₂×D₄	#107	1	2	yes	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄	#135	1	2	yes	—
78	M₁₆	—	4	4	no	—
79	M₁₆	—	4	2	no	—
80	M₁₆	—	16	4	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	4	2	no	—
86	C₂×C₈	—	4	2	no	—
87	C₂×C₈	—	8	4	no	—
88	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈	#117	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
91	C₄⋊C₄	—	2	2	no	—
92	C₂²⋊C₄	—	2	4	no	—
93	C₄²	#121	1	2	yes	—
94	C₄²	C₂×D₈	1	1	yes	characteristic
95	C₄²	C₂×Q₁₆	1	1	yes	characteristic
96	C₁₆	—	2	1	no	—
97	C₁₆	—	2	2	no	—
98	C₁₆	—	4	2	no	—
99	C₁₆	#100	1	2	yes	—
100	C₂³	—	2	2	no	—
101	C₂³	#427	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	4	no	—
105	D₄	—	4	4	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	#380	1	1	yes	characteristic
113	C₂×C₄	C₂×#107	1	1	yes	characteristic
114	C₂×C₄	#466	1	2	yes	—
115	C₂×C₄	#467	1	2	yes	—
116	C₈	—	2	1	no	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	16	4	no	—
120	C₈	C₂×#100	1	1	yes	characteristic
121	C₈	#413	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	4	4	no	—
125	C₂²	—	4	2	no	—
126	C₂²	#1808	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	1	no	—
129	C₄	—	4	2	no	—
130	C₄	#1800	1	1	yes	characteristic
131	C₄	#1813	1	1	yes	characteristic
132	C₄	#2223	1	4	yes	—
133	C₂	—	4	1	no	—
134	C₂	—	8	2	no	—
135	C₂	#8616	1	1	yes	characteristic
136	C₂	#12843	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33434

C₂

yes

maximal, characteristic

#12023

C₂

yes

maximal, complemented

#8638

C₂

yes

maximal

#7245

C₂

yes

maximal, complemented

C₂×#504

C₄

yes

—

#2931

C₂²

yes

characteristic

#2169

—

D₄×C₁₆

—

#2150

C₂²

yes

characteristic

#2113

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1567

C₂²

yes

—

—

—

—

—

C₂×C₄

yes

—

#532

D₄

yes

characteristic

#504

—

C₂×M₃₂

—

C₂×M₃₂

C₂×C₄

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

#436

—

C₈×D₄

—

#423

—

#423

D₄

yes

characteristic

#408

Q₈

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

D₄

yes

—

C₄×C₁₆

D₄

yes

—

—

—

—

—

C₂×#41

C₄⋊C₄

yes

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₄⋊C₄

yes

—

C₂²×C₈

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₄⋊Q₈

D₈

yes

characteristic

#127

D₈

yes

characteristic

#124

Q₁₆

yes

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

#41

—

#41

—

C₂×Q₈

#107

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

#107

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

#135

yes

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

#117

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

#121

yes

—

C₄²

C₂×D₈

yes

characteristic

C₄²

C₂×Q₁₆

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

#100

yes

—

100

C₂³

—

101

C₂³

#427

yes

characteristic

102

Q₈

—

103

D₄

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

#380

yes

characteristic

113

C₂×C₄

C₂×#107

yes

characteristic

114

C₂×C₄

#466

yes

—

115

C₂×C₄

#467

yes

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

C₂×#100

yes

characteristic

121

C₈

#413

yes

characteristic

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

#1808

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#1800

yes

characteristic

131

C₄

#1813

yes

characteristic

132

C₄

#2223

yes

—

133

C₂

—

134

C₂

—

135

C₂

#8616

yes

characteristic

136

C₂

#12843

yes

—

	subgroup	G/H
center, socle	C₂²	#1808
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1808

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

#532

D₄

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #12023	C₂	semidirect
4. #7245	C₂	semidirect

H ≅ G/N

type

2. #12023

C₂

semidirect

4. #7245

C₂

semidirect

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

subgroup	index
#33434	2
#12023	2
#8638	2
#7245	2

subgroup

index

Subgroups — #134849

Conjugacy classes of subgroups

Complements

Maximal subgroups