Group #134855 — subgroups

Subgroups — #134855

Order 512.

244 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 92 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	C₂×Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#12024	C₂	1	2	yes	maximal
3	#8623	C₂	1	2	yes	maximal
4	#7242	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
6	Q₈×C₁₆	—	2	2	no	—
7	Q₈×C₁₆	—	2	2	no	—
8	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
9	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₄	1	2	yes	—
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	#1568	C₂²	1	4	yes	—
13	#1467	—	2	2	no	—
14	#1372	—	2	2	no	—
15	#1371	—	2	2	no	—
16	#1290	—	2	2	no	—
17	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
18	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₁₆	C₂×C₄	1	2	yes	—
20	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₈×Q₈	—	2	2	no	—
22	C₈×Q₈	—	2	2	no	—
23	#432	—	2	4	no	—
24	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
25	C₂×C₄⋊C₈	Q₈	1	2	yes	—
26	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	C₄⋊C₁₆	—	2	4	no	—
29	C₄⋊C₁₆	—	2	2	no	—
30	C₄×C₁₆	—	2	2	no	—
31	C₄×C₁₆	—	2	2	no	—
32	C₄×C₁₆	D₄	1	2	yes	—
33	C₄×C₁₆	D₄	1	2	yes	—
34	#335	—	2	4	no	—
35	#334	—	2	4	no	—
36	#324	—	4	4	no	—
37	C₂²×Q₈	Q₁₆	1	1	yes	characteristic
38	C₂²×C₈	C₄⋊C₄	1	2	yes	—
39	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
41	C₄×Q₈	—	2	2	no	—
42	C₄×Q₈	—	2	2	no	—
43	C₂×C₄⋊C₄	D₈	1	2	yes	—
44	C₂×C₄⋊C₄	Q₁₆	1	1	yes	characteristic
45	C₂×C₄²	C₄⋊C₄	1	2	yes	—
46	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
47	C₂×C₁₆	—	2	2	no	—
48	C₂×C₁₆	—	2	2	no	—
49	C₂×C₁₆	—	4	1	no	—
50	C₂×C₁₆	—	4	2	no	—
51	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
53	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
54	C₄⋊C₈	—	2	4	no	—
55	C₄⋊C₈	—	2	2	no	—
56	C₄⋊C₈	—	4	4	no	—
57	C₂²⋊C₈	—	4	4	no	—
58	#97	—	4	4	no	—
59	C₄×C₈	—	2	2	no	—
60	C₄×C₈	—	2	2	no	—
61	C₄×C₈	C₂×D₄	1	2	yes	—
62	C₄×C₈	#41	1	2	yes	—
63	C₂×Q₈	—	2	4	no	—
64	C₂×Q₈	—	2	2	no	—
65	C₂²×C₄	#107	1	2	yes	—
66	C₂²×C₄	#107	1	2	yes	—
67	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
69	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	4	1	no	—
74	C₂×C₈	—	4	2	no	—
75	C₂×C₈	—	8	4	no	—
76	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈	#117	1	1	yes	characteristic
78	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄²	—	2	2	no	—
82	C₄²	—	2	2	no	—
83	C₄²	#121	1	2	yes	—
84	C₄²	#135	1	2	yes	—
85	C₁₆	—	2	1	no	—
86	C₁₆	—	2	2	no	—
87	C₁₆	—	4	2	no	—
88	C₁₆	#101	1	2	yes	—
89	C₂³	C₂×#107	1	1	yes	characteristic
90	Q₈	—	2	8	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	#368	1	4	yes	—
100	C₂×C₄	#380	1	1	yes	characteristic
101	C₂×C₄	#427	1	1	yes	characteristic
102	C₂×C₄	#466	1	2	yes	—
103	C₂×C₄	#467	1	2	yes	—
104	C₈	—	2	1	no	—
105	C₈	—	2	2	no	—
106	C₈	—	4	2	no	—
107	C₈	—	16	4	no	—
108	C₈	C₂×#101	1	1	yes	characteristic
109	C₈	#413	1	1	yes	characteristic
110	C₂²	—	2	1	no	—
111	C₂²	—	2	1	no	—
112	C₂²	#1808	1	1	yes	characteristic
113	C₂²	C₂×#368	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	2	1	no	—
118	C₄	—	4	2	no	—
119	C₄	#1801	1	1	yes	characteristic
120	C₄	#1813	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	#8617	1	1	yes	characteristic
123	C₂	#12837	1	1	yes	characteristic
124	C₂	#12839	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×C₁₆

C₂

yes

maximal, characteristic

#12024

C₂

yes

maximal

#8623

C₂

yes

maximal

#7242

C₂

yes

maximal

C₂×C₈×Q₈

C₂²

yes

characteristic

Q₈×C₁₆

—

Q₈×C₁₆

—

C₂×C₄⋊C₁₆

C₄

yes

—

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1568

C₂²

yes

—

—

—

—

—

C₂×C₄×Q₈

D₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

#432

—

C₂×C₄⋊C₈

D₄

yes

characteristic

C₂×C₄⋊C₈

Q₈

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

—

C₄×C₁₆

D₄

yes

—

#335

—

#334

—

#324

—

C₂²×Q₈

Q₁₆

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

D₈

yes

—

C₂×C₄⋊C₄

Q₁₆

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

#107

yes

—

C₂²×C₄

#107

yes

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

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

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

#121

yes

—

C₄²

#135

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

#101

yes

—

C₂³

C₂×#107

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#368

yes

—

100

C₂×C₄

#380

yes

characteristic

101

C₂×C₄

#427

yes

characteristic

102

C₂×C₄

#466

yes

—

103

C₂×C₄

#467

yes

—

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

C₂×#101

yes

characteristic

109

C₈

#413

yes

characteristic

110

C₂²

—

111

C₂²

—

112

C₂²

#1808

yes

characteristic

113

C₂²

C₂×#368

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#1801

yes

characteristic

120

C₄

#1813

yes

characteristic

121

C₂

—

122

C₂

#8617

yes

characteristic

123

C₂

#12837

yes

characteristic

124

C₂

#12839

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1808
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#107
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ 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

C₂³

C₂×#107

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ 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
C₂×Q₈×C₁₆	2
#12024	2
#8623	2
#7242	2

subgroup

index

C₂×Q₈×C₁₆

#12024

#8623

#7242

Subgroups — #134855

Conjugacy classes of subgroups

Maximal subgroups