Group #512473 — subgroups

Subgroups — #512473

Order 512.

446 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 194 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	#33257	C₂	1	2	yes	maximal
2	#13166	C₂	1	4	yes	maximal
3	#13165	C₂	1	4	yes	maximal
4	#12907	C₂	1	2	yes	maximal
5	#12889	C₂	1	2	yes	maximal
6	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
7	#2864	—	2	2	no	—
8	C₂×#367	C₂²	1	4	yes	—
9	C₂×#366	C₂²	1	4	yes	—
10	C₂×C₄×C₁₆	C₂²	1	2	yes	—
11	C₂×C₄×C₁₆	C₂²	1	2	yes	—
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	#1942	C₂²	1	8	yes	—
15	#1941	—	2	4	no	—
16	#1846	C₂²	1	4	yes	—
17	#1835	—	2	2	no	—
18	C₄²×C₈	C₂²	1	1	yes	characteristic
19	#1380	C₂²	1	8	yes	—
20	C₂²×C₁₆	C₂³	1	2	yes	—
21	C₂²×C₁₆	C₂³	1	1	yes	characteristic
22	C₂²×C₁₆	C₂³	1	1	yes	characteristic
23	C₂×#107	C₂³	1	8	yes	—
24	C₂×#106	—	2	4	no	—
25	C₂×C₄×C₈	D₄	1	2	yes	—
26	C₂×C₄×C₈	D₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂³	1	2	yes	—
28	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
29	#389	—	2	8	no	—
30	#387	—	4	8	no	—
31	#382	—	2	4	no	—
32	C₄³	D₄	1	1	yes	characteristic
33	#367	—	2	8	no	—
34	#366	—	2	8	no	—
35	C₄×C₁₆	—	2	4	no	—
36	C₄×C₁₆	—	2	4	no	—
37	C₄×C₁₆	Q₈	1	4	yes	—
38	C₄×C₁₆	Q₈	1	4	yes	—
39	#340	—	2	8	no	—
40	C₂²×C₈	C₂×D₄	1	2	yes	—
41	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₈	C₂⁴	1	1	yes	characteristic
43	C₂×C₄⋊C₄	—	2	8	no	—
44	C₂×C₄⋊C₄	—	4	8	no	—
45	C₂×C₄²	D₈	1	4	yes	—
46	C₂×C₄²	C₂×D₄	1	2	yes	—
47	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
48	C₂×C₁₆	—	2	2	no	—
49	C₂×C₁₆	—	2	2	no	—
50	C₂×C₁₆	C₂×Q₈	1	4	yes	—
51	C₂×C₁₆	C₂×Q₈	1	2	yes	—
52	C₂×C₁₆	C₂×Q₈	1	2	yes	—
53	C₂×C₁₆	#41	1	4	yes	—
54	C₂×C₁₆	#41	1	4	yes	—
55	#107	—	2	8	no	—
56	#107	—	2	8	no	—
57	#106	—	4	8	no	—
58	C₄×C₈	—	2	4	no	—
59	C₄×C₈	—	2	2	no	—
60	C₄×C₈	—	2	2	no	—
61	C₄×C₈	Q₁₆	1	4	yes	—
62	C₄×C₈	C₂×Q₈	1	2	yes	—
63	C₄×C₈	C₂×Q₈	1	2	yes	—
64	#95	—	4	8	no	—
65	#95	—	4	8	no	—
66	C₂²×C₄	—	4	8	no	—
67	C₂²×C₄	C₂×D₈	1	2	yes	—
68	C₂²×C₄	C₂×D₈	1	4	yes	—
69	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	C₄⋊Q₈	1	2	yes	—
73	C₂×C₈	C₂×Q₁₆	1	4	yes	—
74	C₂×C₈	#135	1	4	yes	—
75	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
76	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
77	C₂×C₈	C₂×#41	1	2	yes	—
78	C₂×C₈	C₂×#41	1	2	yes	—
79	C₄⋊C₄	—	4	8	no	—
80	C₄⋊C₄	—	4	8	no	—
81	C₄⋊C₄	—	8	16	no	—
82	C₄²	—	2	2	no	—
83	C₄²	—	2	2	no	—
84	C₄²	—	2	8	no	—
85	C₄²	C₄⋊Q₈	1	2	yes	—
86	C₄²	C₂×Q₁₆	1	2	yes	—
87	C₁₆	—	2	4	no	—
88	C₁₆	#125	1	4	yes	—
89	C₁₆	#125	1	4	yes	—
90	C₂³	C₂²×D₈	1	1	yes	characteristic
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	8	16	no	—
95	C₂×C₄	—	8	8	no	—
96	C₂×C₄	#500	1	4	yes	—
97	C₂×C₄	#508	1	8	yes	—
98	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₄	#532	1	2	yes	—
100	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
101	C₂×C₄	C₂×#135	1	2	yes	—
102	C₈	—	2	4	no	—
103	C₈	#499	1	4	yes	—
104	C₈	C₂×#125	1	1	yes	characteristic
105	C₈	C₂×#125	1	1	yes	characteristic
106	C₈	#533	1	2	yes	—
107	C₂²	C₂×#500	1	1	yes	characteristic
108	C₂²	#3169	1	2	yes	—
109	C₂²	C₂×#508	1	4	yes	—
110	C₄	—	2	4	no	—
111	C₄	—	2	8	no	—
112	C₄	—	16	8	no	—
113	C₄	#2859	1	1	yes	characteristic
114	C₄	C₂×#499	1	1	yes	characteristic
115	C₄	#3170	1	2	yes	—
116	C₂	#33258	1	1	yes	characteristic
117	C₂	#33726	1	2	yes	—
118	C₂	#33735	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33257

C₂

yes

maximal

#13166

C₂

yes

maximal

#13165

C₂

yes

maximal

#12907

C₂

yes

maximal

#12889

C₂

yes

maximal

C₄²×C₁₆

C₂

yes

maximal, characteristic

#2864

—

C₂×#367

C₂²

yes

—

C₂×#366

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1942

C₂²

yes

—

#1941

—

#1846

C₂²

yes

—

#1835

—

C₄²×C₈

C₂²

yes

characteristic

#1380

C₂²

yes

—

C₂²×C₁₆

C₂³

yes

—

C₂²×C₁₆

C₂³

yes

characteristic

C₂²×C₁₆

C₂³

yes

characteristic

C₂×#107

C₂³

yes

—

C₂×#106

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

C₂×C₄×C₈

C₂³

yes

—

—

—

C₄³

D₄

yes

characteristic

#367

—

#366

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

Q₈

yes

—

C₄×C₁₆

Q₈

yes

—

#340

—

C₂²×C₈

C₂×D₄

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂⁴

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₈

yes

—

C₂×C₄²

C₂×D₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

#41

yes

—

C₂×C₁₆

#41

yes

—

#107

—

#107

—

#106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

C₂×Q₈

yes

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

—

C₂²×C₄

C₂×D₈

yes

—

C₂²×C₄

C₂²×D₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊Q₈

yes

—

C₂×C₈

C₂×Q₁₆

yes

—

C₂×C₈

#135

yes

—

C₂×C₈

C₂²×Q₈

yes

characteristic

C₂×C₈

C₂²×Q₈

yes

characteristic

C₂×C₈

C₂×#41

yes

—

C₂×C₈

C₂×#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄⋊Q₈

yes

—

C₄²

C₂×Q₁₆

yes

—

C₁₆

—

C₁₆

#125

yes

—

C₁₆

#125

yes

—

C₂³

C₂²×D₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#500

yes

—

C₂×C₄

#508

yes

—

C₂×C₄

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₄

#532

yes

—

100

C₂×C₄

C₂²×Q₁₆

yes

characteristic

101

C₂×C₄

C₂×#135

yes

—

102

C₈

—

103

C₈

#499

yes

—

104

C₈

C₂×#125

yes

characteristic

105

C₈

C₂×#125

yes

characteristic

106

C₈

#533

yes

—

107

C₂²

C₂×#500

yes

characteristic

108

C₂²

#3169

yes

—

109

C₂²

C₂×#508

yes

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#2859

yes

characteristic

114

C₄

C₂×#499

yes

characteristic

115

C₄

#3170

yes

—

116

C₂

#33258

yes

characteristic

117

C₂

#33726

yes

—

118

C₂

#33735

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂²×D₈

commutator, Frattini, cosocle

C₂²×C₈

C₂⁴

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#33257	2
#13166	2
#13165	2
#12907	2
#12889	2
C₄²×C₁₆	2

subgroup

index

C₄²×C₁₆

Subgroups — #512473

Conjugacy classes of subgroups

Maximal subgroups