Group #512474 — subgroups

Subgroups — #512474

Order 512.

482 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 218 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	#33258	C₂	1	2	yes	maximal
2	#13166	C₂	1	8	yes	maximal
3	#12906	C₂	1	2	yes	maximal
4	#12888	C₂	1	2	yes	maximal
5	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
6	#2859	—	2	2	no	—
7	C₂×#367	C₂²	1	8	yes	—
8	C₂×C₄×C₁₆	C₂²	1	4	yes	—
9	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	#1942	—	2	4	no	—
12	#1942	C₂²	1	8	yes	—
13	#1845	C₂²	1	4	yes	—
14	#1834	—	2	2	no	—
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	#1380	C₂²	1	8	yes	—
17	C₂×#125	—	4	2	no	—
18	C₂²×C₁₆	C₂³	1	2	yes	—
19	C₂²×C₁₆	C₂³	1	2	yes	—
20	C₂×#107	—	2	4	no	—
21	C₂×#107	C₂³	1	8	yes	—
22	C₂×C₄×C₈	D₄	1	2	yes	—
23	C₂×C₄×C₈	D₄	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂³	1	2	yes	—
25	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
26	#389	—	2	8	no	—
27	#389	—	4	4	no	—
28	#384	—	2	4	no	—
29	#382	—	4	2	no	—
30	C₄³	D₄	1	1	yes	characteristic
31	#367	—	2	16	no	—
32	C₄×C₁₆	—	2	8	no	—
33	C₄×C₁₆	Q₈	1	4	yes	—
34	C₄×C₁₆	Q₈	1	4	yes	—
35	#340	—	2	8	no	—
36	C₂²×C₈	C₂×D₄	1	2	yes	—
37	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₈	C₂⁴	1	1	yes	characteristic
39	#125	—	8	8	no	—
40	C₂×C₄⋊C₄	—	2	8	no	—
41	C₂×C₄⋊C₄	—	4	4	no	—
42	C₂×C₄⋊C₄	—	4	8	no	—
43	C₂×C₄⋊C₄	—	4	4	no	—
44	C₂×C₄²	D₈	1	2	yes	—
45	C₂×C₄²	D₈	1	2	yes	—
46	C₂×C₄²	C₂×D₄	1	2	yes	—
47	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
48	C₂×C₁₆	—	2	4	no	—
49	C₂×C₁₆	C₂×Q₈	1	4	yes	—
50	C₂×C₁₆	C₂×Q₈	1	4	yes	—
51	C₂×C₁₆	#41	1	8	yes	—
52	#107	—	2	16	no	—
53	#107	—	4	8	no	—
54	C₄×C₈	—	2	4	no	—
55	C₄×C₈	—	2	4	no	—
56	C₄×C₈	Q₁₆	1	4	yes	—
57	C₄×C₈	C₂×Q₈	1	2	yes	—
58	C₄×C₈	C₂×Q₈	1	2	yes	—
59	#95	—	4	8	no	—
60	C₂²×C₄	—	4	8	no	—
61	C₂²×C₄	C₂×D₈	1	4	yes	—
62	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
64	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	C₄⋊Q₈	1	2	yes	—
68	C₂×C₈	C₂×Q₁₆	1	4	yes	—
69	C₂×C₈	#135	1	4	yes	—
70	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
71	C₂×C₈	C₂²×Q₈	1	1	yes	characteristic
72	C₂×C₈	C₂×#41	1	4	yes	—
73	C₄⋊C₄	—	4	8	no	—
74	C₄⋊C₄	—	4	8	no	—
75	C₄⋊C₄	—	8	16	no	—
76	C₄⋊C₄	—	8	8	no	—
77	C₄⋊C₄	—	8	8	no	—
78	C₄²	—	2	2	no	—
79	C₄²	—	2	2	no	—
80	C₄²	—	2	4	no	—
81	C₄²	SD₃₂	1	8	yes	—
82	C₄²	C₄⋊Q₈	1	2	yes	—
83	C₄²	C₂×Q₁₆	1	2	yes	—
84	C₁₆	—	2	4	no	—
85	C₁₆	#125	1	8	yes	complemented
86	C₂³	C₂²×D₈	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	8	8	no	—
90	C₂×C₄	—	8	8	no	—
91	C₂×C₄	—	8	8	no	—
92	C₂×C₄	#500	1	2	yes	—
93	C₂×C₄	#500	1	2	yes	—
94	C₂×C₄	C₂×SD₃₂	1	8	yes	—
95	C₂×C₄	C₂×SD₃₂	1	4	yes	—
96	C₂×C₄	#508	1	4	yes	—
97	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₄	#532	1	2	yes	—
99	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
100	C₂×C₄	C₂×#135	1	2	yes	—
101	C₈	—	2	4	no	—
102	C₈	#499	1	4	yes	—
103	C₈	C₂×#125	1	2	yes	—
104	C₈	#533	1	2	yes	—
105	C₂²	C₂×#500	1	1	yes	characteristic
106	C₂²	#3169	1	2	yes	—
107	C₂²	C₂²×SD₃₂	1	2	yes	—
108	C₂²	C₂×#508	1	2	yes	—
109	C₄	—	2	8	no	—
110	C₄	—	16	8	no	—
111	C₄	#2253	1	4	yes	—
112	C₄	#2254	1	4	yes	—
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	C₂×#2253	1	1	yes	characteristic
118	C₂	C₂×#2254	1	1	yes	characteristic
119	C₂	#33736	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33258

C₂

yes

maximal

#13166

C₂

yes

maximal

#12906

C₂

yes

maximal

#12888

C₂

yes

maximal

C₄²×C₁₆

C₂

yes

maximal, characteristic

#2859

—

C₂×#367

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1942

—

#1942

C₂²

yes

—

#1845

C₂²

yes

—

#1834

—

C₄²×C₈

C₂²

yes

characteristic

#1380

C₂²

yes

—

C₂×#125

—

C₂²×C₁₆

C₂³

yes

—

C₂²×C₁₆

C₂³

yes

—

C₂×#107

—

C₂×#107

C₂³

yes

—

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

—

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

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₈

yes

—

C₂×C₄²

D₈

yes

—

C₂×C₄²

C₂×D₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

#41

yes

—

#107

—

#107

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

C₂×Q₈

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

—

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

SD₃₂

yes

—

C₄²

C₄⋊Q₈

yes

—

C₄²

C₂×Q₁₆

yes

—

C₁₆

—

C₁₆

#125

yes

complemented

C₂³

C₂²×D₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#500

yes

—

C₂×C₄

#500

yes

—

C₂×C₄

C₂×SD₃₂

yes

—

C₂×C₄

C₂×SD₃₂

yes

—

C₂×C₄

#508

yes

—

C₂×C₄

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₄

#532

yes

—

C₂×C₄

C₂²×Q₁₆

yes

characteristic

100

C₂×C₄

C₂×#135

yes

—

101

C₈

—

102

C₈

#499

yes

—

103

C₈

C₂×#125

yes

—

104

C₈

#533

yes

—

105

C₂²

C₂×#500

yes

characteristic

106

C₂²

#3169

yes

—

107

C₂²

C₂²×SD₃₂

yes

—

108

C₂²

C₂×#508

yes

—

109

C₄

—

110

C₄

—

111

C₄

#2253

yes

—

112

C₄

#2254

yes

—

113

C₄

#2859

yes

characteristic

114

C₄

C₂×#499

yes

characteristic

115

C₄

#3170

yes

—

116

C₂

#33258

yes

characteristic

117

C₂

C₂×#2253

yes

characteristic

118

C₂

C₂×#2254

yes

characteristic

119

C₂

#33736

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

N	H ≅ G/N	type
85. C₁₆	#125	semidirect

H ≅ G/N

type

85. C₁₆

#125

semidirect

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

subgroup	index
#33258	2
#13166	2
#12906	2
#12888	2
C₄²×C₁₆	2

subgroup

index

C₄²×C₁₆

Subgroups — #512474

Conjugacy classes of subgroups

Complements

Maximal subgroups