Group #512508 — subgroups

Subgroups — #512508

Order 512.

610 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 282 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	#33312	C₂	1	2	yes	maximal
2	#13282	C₂	1	8	yes	maximal
3	#12888	C₂	1	4	yes	maximal
4	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂×#500	—	2	2	no	—
6	C₂×#500	C₂²	1	4	yes	—
7	#2868	—	2	2	no	—
8	C₂×C₄×C₁₆	C₂²	1	4	yes	—
9	C₂×C₄×C₁₆	C₂²	1	2	yes	—
10	#1845	—	2	2	no	—
11	#1845	—	2	2	no	—
12	#1845	C₂²	1	4	yes	—
13	#1845	C₂²	1	4	yes	—
14	C₄²×C₈	C₂²	1	1	yes	characteristic
15	#1380	C₂²	1	16	yes	—
16	C₂×C₄⋊Q₈	—	2	4	no	—
17	C₂×C₄⋊Q₈	—	4	2	no	—
18	C₂×C₄⋊Q₈	—	4	2	no	—
19	C₂²×C₁₆	C₂³	1	4	yes	—
20	#500	—	2	16	no	—
21	#500	—	4	8	no	—
22	C₂×#107	—	2	4	no	—
23	C₂×#107	—	2	4	no	—
24	C₂×#107	—	2	4	no	—
25	C₂×#107	C₂³	1	8	yes	—
26	C₂×C₄×C₈	D₄	1	2	yes	—
27	C₂×C₄×C₈	D₄	1	1	yes	characteristic
28	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂³	1	2	yes	—
30	#384	—	2	4	no	—
31	#384	—	2	4	no	—
32	#384	—	4	2	no	—
33	#384	—	4	2	no	—
34	C₄³	D₄	1	1	yes	characteristic
35	C₄×C₁₆	—	2	8	no	—
36	C₄×C₁₆	Q₈	1	8	yes	complemented
37	C₂²×Q₈	—	4	4	no	—
38	C₂²×C₈	C₂×D₄	1	2	yes	—
39	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₈	C₂⁴	1	1	yes	characteristic
41	C₄⋊Q₈	—	4	8	no	—
42	C₄⋊Q₈	—	4	8	no	—
43	C₄⋊Q₈	—	8	8	no	—
44	C₄⋊Q₈	—	8	8	no	—
45	C₂×C₄⋊C₄	—	2	8	no	—
46	C₂×C₄⋊C₄	—	4	4	no	—
47	C₂×C₄⋊C₄	—	4	4	no	—
48	C₂×C₄⋊C₄	—	4	4	no	—
49	C₂×C₄⋊C₄	—	4	4	no	—
50	C₂×C₄⋊C₄	—	4	4	no	—
51	C₂×C₄²	D₈	1	1	yes	characteristic
52	C₂×C₄²	D₈	1	2	yes	—
53	C₂×C₄²	D₈	1	1	yes	characteristic
54	C₂×C₄²	C₂×D₄	1	2	yes	—
55	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
56	C₂×C₁₆	—	2	8	no	—
57	C₂×C₁₆	C₂×Q₈	1	8	yes	—
58	#107	—	2	16	no	—
59	#107	—	4	8	no	—
60	#107	—	4	8	no	—
61	#107	—	4	8	no	—
62	C₄×C₈	SD₁₆	1	8	yes	—
63	C₄×C₈	Q₁₆	1	4	yes	—
64	C₄×C₈	C₂×Q₈	1	4	yes	—
65	C₄×C₈	#41	1	8	yes	—
66	C₂×Q₈	—	8	8	no	—
67	C₂×Q₈	—	8	8	no	—
68	C₂×Q₈	—	8	8	no	—
69	C₂²×C₄	—	4	8	no	—
70	C₂²×C₄	C₂×D₈	1	2	yes	—
71	C₂²×C₄	C₂×D₈	1	2	yes	—
72	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
73	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
74	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
75	C₂×C₈	#124	1	4	yes	—
76	C₂×C₈	C₄⋊Q₈	1	2	yes	—
77	C₂×C₈	C₂×SD₁₆	1	4	yes	—
78	C₂×C₈	C₂×SD₁₆	1	4	yes	—
79	C₂×C₈	C₂×Q₁₆	1	4	yes	—
80	C₂×C₈	C₂²×Q₈	1	2	yes	—
81	C₂×C₈	C₂×#41	1	4	yes	—
82	C₄⋊C₄	—	4	8	no	—
83	C₄⋊C₄	—	4	8	no	—
84	C₄⋊C₄	—	8	8	no	—
85	C₄⋊C₄	—	8	8	no	—
86	C₄⋊C₄	—	8	8	no	—
87	C₄⋊C₄	—	8	8	no	—
88	C₄⋊C₄	—	8	8	no	—
89	C₄²	D₁₆	1	4	yes	—
90	C₄²	SD₃₂	1	8	yes	—
91	C₄²	Q₃₂	1	4	yes	—
92	C₄²	#124	1	4	yes	—
93	C₄²	C₄⋊Q₈	1	2	yes	—
94	C₄²	C₂×SD₁₆	1	4	yes	—
95	C₄²	C₂×Q₁₆	1	2	yes	—
96	C₁₆	—	2	8	no	—
97	C₂³	C₂²×D₈	1	1	yes	characteristic
98	Q₈	—	16	16	no	—
99	C₂×C₄	—	8	8	no	—
100	C₂×C₄	—	8	8	no	—
101	C₂×C₄	—	8	8	no	—
102	C₂×C₄	#486	1	4	yes	—
103	C₂×C₄	#486	1	4	yes	—
104	C₂×C₄	#500	1	2	yes	—
105	C₂×C₄	#500	1	2	yes	—
106	C₂×C₄	C₂×D₁₆	1	4	yes	—
107	C₂×C₄	C₂×D₁₆	1	2	yes	—
108	C₂×C₄	C₂×SD₃₂	1	4	yes	—
109	C₂×C₄	C₂×SD₃₂	1	4	yes	—
110	C₂×C₄	C₂×SD₃₂	1	4	yes	—
111	C₂×C₄	C₂×Q₃₂	1	4	yes	—
112	C₂×C₄	C₂×Q₃₂	1	2	yes	—
113	C₂×C₄	C₂×#124	1	2	yes	—
114	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
115	C₂×C₄	C₂²×SD₁₆	1	2	yes	—
116	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
117	C₈	#487	1	8	yes	—
118	C₈	#498	1	4	yes	—
119	C₈	#533	1	4	yes	—
120	C₂²	C₂×#486	1	2	yes	—
121	C₂²	C₂×#500	1	1	yes	characteristic
122	C₂²	C₂²×D₁₆	1	1	yes	characteristic
123	C₂²	C₂²×SD₃₂	1	2	yes	—
124	C₂²	C₂²×Q₃₂	1	1	yes	characteristic
125	C₄	—	16	8	no	—
126	C₄	#2240	1	8	yes	—
127	C₄	#2241	1	8	yes	—
128	C₄	#2252	1	4	yes	—
129	C₄	#2254	1	4	yes	—
130	C₄	#2862	1	1	yes	characteristic
131	C₄	C₂×#487	1	2	yes	—
132	C₄	C₂×#498	1	1	yes	characteristic
133	C₂	#33289	1	1	yes	characteristic
134	C₂	C₂×#2240	1	2	yes	—
135	C₂	C₂×#2241	1	2	yes	—
136	C₂	C₂×#2252	1	1	yes	characteristic
137	C₂	C₂×#2254	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33312

C₂

yes

maximal

#13282

C₂

yes

maximal

#12888

C₂

yes

maximal

C₄²×C₁₆

C₂

yes

maximal, characteristic

C₂×#500

—

C₂×#500

C₂²

yes

—

#2868

—

C₂×C₄×C₁₆

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

—

#1845

—

#1845

—

#1845

C₂²

yes

—

#1845

C₂²

yes

—

C₄²×C₈

C₂²

yes

characteristic

#1380

C₂²

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

—

C₂³

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

C₄³

D₄

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

Q₈

yes

complemented

C₂²×Q₈

—

C₂²×C₈

C₂×D₄

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂⁴

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

D₈

yes

—

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

C₂×D₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×Q₈

yes

—

—

—

—

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

—

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

#124

yes

—

C₂×C₈

C₄⋊Q₈

yes

—

C₂×C₈

C₂×SD₁₆

yes

—

C₂×C₈

C₂×SD₁₆

yes

—

C₂×C₈

C₂×Q₁₆

yes

—

C₂×C₈

C₂²×Q₈

yes

—

C₂×C₈

C₂×#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₁₆

yes

—

C₄²

SD₃₂

yes

—

C₄²

Q₃₂

yes

—

C₄²

#124

yes

—

C₄²

C₄⋊Q₈

yes

—

C₄²

C₂×SD₁₆

yes

—

C₄²

C₂×Q₁₆

yes

—

C₁₆

—

C₂³

C₂²×D₈

yes

characteristic

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

#486

yes

—

103

C₂×C₄

#486

yes

—

104

C₂×C₄

#500

yes

—

105

C₂×C₄

#500

yes

—

106

C₂×C₄

C₂×D₁₆

yes

—

107

C₂×C₄

C₂×D₁₆

yes

—

108

C₂×C₄

C₂×SD₃₂

yes

—

109

C₂×C₄

C₂×SD₃₂

yes

—

110

C₂×C₄

C₂×SD₃₂

yes

—

111

C₂×C₄

C₂×Q₃₂

yes

—

112

C₂×C₄

C₂×Q₃₂

yes

—

113

C₂×C₄

C₂×#124

yes

—

114

C₂×C₄

C₂×C₄⋊Q₈

yes

characteristic

115

C₂×C₄

C₂²×SD₁₆

yes

—

116

C₂×C₄

C₂²×Q₁₆

yes

characteristic

117

C₈

#487

yes

—

118

C₈

#498

yes

—

119

C₈

#533

yes

—

120

C₂²

C₂×#486

yes

—

121

C₂²

C₂×#500

yes

characteristic

122

C₂²

C₂²×D₁₆

yes

characteristic

123

C₂²

C₂²×SD₃₂

yes

—

124

C₂²

C₂²×Q₃₂

yes

characteristic

125

C₄

—

126

C₄

#2240

yes

—

127

C₄

#2241

yes

—

128

C₄

#2252

yes

—

129

C₄

#2254

yes

—

130

C₄

#2862

yes

characteristic

131

C₄

C₂×#487

yes

—

132

C₄

C₂×#498

yes

characteristic

133

C₂

#33289

yes

characteristic

134

C₂

C₂×#2240

yes

—

135

C₂

C₂×#2241

yes

—

136

C₂

C₂×#2252

yes

characteristic

137

C₂

C₂×#2254

yes

characteristic

	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
36. C₄×C₁₆	Q₈	semidirect

H ≅ G/N

type

36. C₄×C₁₆

Q₈

semidirect

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

subgroup	index
#33312	2
#13282	2
#12888	2
C₄²×C₁₆	2

subgroup

index

#33312

#13282

#12888

C₄²×C₁₆

Subgroups — #512508

Conjugacy classes of subgroups

Complements

Maximal subgroups