Group #498643 — subgroups

Subgroups — #498643

Order 512.

380 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 198 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	#21735	C₂	1	1	yes	maximal, characteristic
2	#13270	C₂	1	4	yes	maximal
3	#13164	C₂	1	4	yes	maximal
4	#12667	C₂	1	4	yes	maximal
5	#12656	C₂	1	2	yes	maximal
6	#3162	C₄	1	1	yes	characteristic
7	#2960	C₂²	1	2	yes	—
8	#2153	C₂²	1	4	yes	—
9	C₂×#364	C₂²	1	2	yes	—
10	C₂×#364	C₂²	1	2	yes	—
11	#2110	C₂²	1	4	yes	—
12	C₄×M₃₂	C₂²	1	4	yes	—
13	#1995	C₄	1	2	yes	—
14	#1946	C₄	1	4	yes	—
15	#1849	C₄	1	1	yes	characteristic
16	#1772	C₂²	1	4	yes	—
17	#1726	C₂²	1	1	yes	characteristic
18	#1378	C₂²	1	8	yes	—
19	#1376	C₂²	1	4	yes	—
20	#533	C₂×C₄	1	2	yes	—
21	C₂×M₃₂	C₂³	1	4	yes	—
22	C₂×M₃₂	C₂³	1	4	yes	—
23	#501	—	2	4	no	—
24	#446	—	4	2	no	—
25	C₈×Q₈	—	4	2	no	—
26	C₂×#108	C₂×C₄	1	4	yes	—
27	#427	C₂×C₄	1	4	yes	—
28	#424	C₂×C₄	1	4	yes	—
29	#423	C₂×C₄	1	2	yes	—
30	#405	C₂³	1	4	yes	—
31	C₂×#97	C₂×C₄	1	2	yes	—
32	C₂×#97	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₈	C₂³	1	2	yes	—
35	#376	C₂×C₄	1	1	yes	characteristic
36	#364	—	2	4	no	—
37	#364	—	4	4	no	—
38	#364	Q₈	1	8	yes	complemented
39	#347	—	4	4	no	—
40	#346	—	2	4	no	—
41	C₄×C₁₆	—	2	4	no	—
42	#339	C₂×C₄	1	8	yes	—
43	C₂×M₁₆	C₂²×C₄	1	4	yes	—
44	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₈	C₂²×C₄	1	2	yes	—
46	C₂²×C₈	C₂⁴	1	1	yes	characteristic
47	C₄⋊Q₈	—	2	2	no	—
48	#125	—	2	2	no	—
49	C₂²⋊Q₈	—	4	2	no	—
50	C₄×Q₈	—	4	2	no	—
51	#117	C₂²×C₄	1	4	yes	—
52	C₂×C₄²	C₂²×C₄	1	2	yes	—
53	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
54	M₃₂	—	2	4	no	—
55	M₃₂	—	2	4	no	—
56	M₃₂	—	2	4	no	—
57	M₃₂	C₂×Q₈	1	8	yes	—
58	C₂×C₁₆	—	2	4	no	—
59	C₂×C₁₆	—	2	4	no	—
60	#108	—	2	8	no	—
61	#107	—	2	4	no	—
62	#106	—	2	4	no	—
63	C₄⋊C₈	—	4	2	no	—
64	C₄⋊C₈	—	4	4	no	—
65	#97	—	2	4	no	—
66	#97	—	2	4	no	—
67	#97	M₁₆	1	4	yes	—
68	C₄×C₈	—	2	4	no	—
69	C₄×C₈	—	2	1	no	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	C₂×Q₈	1	4	yes	—
72	C₄×C₈	#41	1	4	yes	—
73	#95	—	4	1	no	—
74	C₂×Q₈	—	4	2	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	—	2	1	no	—
77	C₂²×C₄	C₂³×C₄	1	1	yes	characteristic
78	M₁₆	—	2	8	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	C₄×Q₈	1	4	yes	—
83	C₂×C₈	C₂×M₁₆	1	4	yes	—
84	C₂×C₈	#131	1	4	yes	—
85	C₂×C₈	C₂²×Q₈	1	2	yes	—
86	C₂×C₈	C₂×#41	1	2	yes	—
87	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
88	C₂×C₈	H(5,𝔽₂)	1	1	yes	characteristic
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	—	4	4	no	—
91	C₄⋊C₄	—	4	2	no	—
92	C₂²⋊C₄	—	2	4	no	—
93	C₄²	—	2	1	no	—
94	C₄²	—	2	2	no	—
95	C₄²	—	2	4	no	—
96	C₄²	C₄×Q₈	1	4	yes	—
97	C₄²	C₂×M₁₆	1	2	yes	—
98	C₁₆	—	2	8	no	—
99	C₁₆	—	4	4	no	—
100	C₂³	#568	1	1	yes	characteristic
101	Q₈	—	8	4	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	—	8	2	no	—
106	C₂×C₄	—	8	1	no	—
107	C₂×C₄	C₂×C₄×Q₈	1	2	yes	—
108	C₂×C₄	#518	1	1	yes	characteristic
109	C₂×C₄	C₂²×M₁₆	1	1	yes	characteristic
110	C₂×C₄	C₂×#131	1	2	yes	—
111	C₈	—	4	1	no	—
112	C₈	—	4	4	no	—
113	C₈	#446	1	8	yes	—
114	C₈	#554	1	2	yes	—
115	C₈	#555	1	1	yes	characteristic
116	C₈	Q₈²	1	1	yes	characteristic
117	C₂²	—	4	1	no	—
118	C₂²	#2962	1	2	yes	—
119	C₂²	#2975	1	1	yes	characteristic
120	C₄	—	4	4	no	—
121	C₄	—	4	2	no	—
122	C₄	—	8	4	no	—
123	C₄	#2369	1	1	yes	characteristic
124	C₄	C₂×#446	1	2	yes	—
125	C₄	#2979	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	2	2	no	—
128	C₂	#23250	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21735

C₂

yes

maximal, characteristic

#13270

C₂

yes

maximal

#13164

C₂

yes

maximal

#12667

C₂

yes

maximal

#12656

C₂

yes

maximal

#3162

C₄

yes

characteristic

#2960

C₂²

yes

—

#2153

C₂²

yes

—

C₂×#364

C₂²

yes

—

C₂×#364

C₂²

yes

—

#2110

C₂²

yes

—

C₄×M₃₂

C₂²

yes

—

#1995

C₄

yes

—

#1946

C₄

yes

—

#1849

C₄

yes

characteristic

#1772

C₂²

yes

—

#1726

C₂²

yes

characteristic

#1378

C₂²

yes

—

#1376

C₂²

yes

—

#533

C₂×C₄

yes

—

C₂×M₃₂

C₂³

yes

—

C₂×M₃₂

C₂³

yes

—

#501

—

#446

—

C₈×Q₈

—

C₂×#108

C₂×C₄

yes

—

#427

C₂×C₄

yes

—

#424

C₂×C₄

yes

—

#423

C₂×C₄

yes

—

#405

C₂³

yes

—

C₂×#97

C₂×C₄

yes

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

#376

C₂×C₄

yes

—

—

Q₈

yes

complemented

#347

—

#346

—

C₄×C₁₆

—

#339

C₂×C₄

yes

—

C₂×M₁₆

C₂²×C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

—

C₂²×C₈

C₂⁴

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂²×C₄

yes

—

C₂×C₄²

C₂²×C₄

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

C₂×Q₈

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

#107

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

—

#97

M₁₆

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

#95

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂³×C₄

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×Q₈

yes

—

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

#131

yes

—

C₂×C₈

C₂²×Q₈

yes

—

C₂×C₈

C₂×#41

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×Q₈

yes

—

C₄²

C₂×M₁₆

yes

—

C₁₆

—

C₁₆

—

100

C₂³

#568

yes

characteristic

101

Q₈

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×C₄×Q₈

yes

—

108

C₂×C₄

#518

yes

characteristic

109

C₂×C₄

C₂²×M₁₆

yes

characteristic

110

C₂×C₄

C₂×#131

yes

—

111

C₈

—

112

C₈

—

113

C₈

#446

yes

—

114

C₈

#554

yes

—

115

C₈

#555

yes

characteristic

116

C₈

Q₈²

yes

characteristic

117

C₂²

—

118

C₂²

#2962

yes

—

119

C₂²

#2975

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#2369

yes

characteristic

124

C₄

C₂×#446

yes

—

125

C₄

#2979

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

#23250

yes

characteristic

	subgroup	G/H
center	C₈	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23250
prime radical	C₂³	#568
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₈

#555

commutator

C₂²×C₄

C₂³×C₄

Frattini, cosocle

C₂²×C₈

C₂⁴

C₂

C₂³

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
38. #364	Q₈	semidirect

H ≅ G/N

type

38. #364

Q₈

semidirect

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

subgroup	index
#21735	2
#13270	2
#13164	2
#12667	2
#12656	2

subgroup

index

Subgroups — #498643

Conjugacy classes of subgroups

Complements

Maximal subgroups