Group #146237 — subgroups

Subgroups — #146237

Order 512.

339 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 124 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	#13203	C₂	1	2	yes	maximal
2	#12905	C₂	1	2	yes	maximal
3	#9825	C₂	1	1	yes	maximal, characteristic
4	#8818	C₂	1	1	yes	maximal, characteristic
5	#7136	C₂	1	1	yes	maximal, characteristic
6	C₂×#494	C₂²	1	1	yes	characteristic
7	C₂×#366	—	2	4	no	—
8	C₂×#358	—	2	4	no	—
9	C₂×C₄×C₁₆	C₂²	1	2	yes	—
10	#1990	—	2	2	no	—
11	#1845	—	2	2	no	—
12	#1845	C₂²	1	1	yes	characteristic
13	#1845	C₂²	1	2	yes	—
14	C₂×C₈²	C₂²	1	1	yes	characteristic
15	C₂²×Q₁₆	—	2	2	no	—
16	C₂×C₄⋊Q₈	—	2	1	no	—
17	C₂²×C₁₆	D₄	1	4	yes	—
18	#494	—	2	1	no	—
19	#494	—	2	2	no	—
20	#494	Q₈	1	2	yes	—
21	C₂×#107	—	2	2	no	—
22	C₂×#107	—	2	2	no	—
23	C₂×#107	—	2	2	no	—
24	C₂×#107	—	4	4	no	—
25	C₂×C₄×C₈	D₄	1	2	yes	—
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	#384	—	2	2	no	—
28	#384	—	2	1	no	—
29	#366	—	4	8	no	—
30	#358	—	4	8	no	—
31	C₄×C₁₆	—	2	2	no	—
32	C₄×C₁₆	—	2	2	no	—
33	#340	—	4	4	no	—
34	C₈²	—	2	2	no	—
35	C₂²×Q₈	—	4	1	no	—
36	C₂×Q₁₆	—	2	4	no	—
37	C₂×Q₁₆	—	4	2	no	—
38	C₂×Q₁₆	—	4	4	no	—
39	C₂×Q₁₆	—	4	4	no	—
40	C₂²×C₈	D₈	1	4	yes	—
41	C₂²×C₈	C₂×D₄	1	2	yes	—
42	C₄⋊Q₈	—	2	2	no	—
43	C₄⋊Q₈	—	4	1	no	—
44	C₄⋊Q₈	—	4	2	no	—
45	C₂×C₄⋊C₄	—	4	2	no	—
46	C₂×C₄⋊C₄	—	4	2	no	—
47	C₂×C₄⋊C₄	—	4	1	no	—
48	C₂×C₄⋊C₄	—	4	2	no	—
49	C₂×C₄⋊C₄	—	4	4	no	—
50	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
51	C₂×C₁₆	—	2	4	no	—
52	C₂×C₁₆	—	2	4	no	—
53	C₂×C₁₆	D₈	1	8	yes	—
54	#107	—	4	4	no	—
55	#107	—	4	4	no	—
56	#107	—	4	4	no	—
57	#107	—	8	8	no	—
58	C₄×C₈	—	2	2	no	—
59	C₄×C₈	—	2	2	no	—
60	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
61	C₄×C₈	#41	1	1	yes	characteristic
62	C₄×C₈	#41	1	2	yes	—
63	C₂×Q₈	—	4	2	no	—
64	C₂×Q₈	—	8	1	no	—
65	C₂×Q₈	—	8	2	no	—
66	C₂×Q₈	—	8	2	no	—
67	C₂²×C₄	—	8	1	no	—
68	C₂²×C₄	—	8	1	no	—
69	C₂²×C₄	—	8	2	no	—
70	C₂²×C₄	#127	1	1	yes	characteristic
71	C₂²×C₄	C₂×D₈	1	2	yes	—
72	Q₁₆	—	4	8	no	—
73	Q₁₆	—	8	4	no	—
74	C₂×C₈	—	2	4	no	—
75	C₂×C₈	—	2	4	no	—
76	C₂×C₈	Q₃₂	1	8	yes	—
77	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₈	#123	1	2	yes	—
79	C₂×C₈	C₂×D₈	1	4	yes	—
80	C₂×C₈	#137	1	4	yes	—
81	C₄⋊C₄	—	4	4	no	—
82	C₄⋊C₄	—	8	2	no	—
83	C₄⋊C₄	—	8	2	no	—
84	C₄⋊C₄	—	8	2	no	—
85	C₄⋊C₄	—	8	2	no	—
86	C₄⋊C₄	—	8	8	no	—
87	C₄⋊C₄	—	8	4	no	—
88	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
89	C₄²	#123	1	1	yes	characteristic
90	C₄²	#137	1	2	yes	—
91	C₁₆	—	2	8	no	—
92	C₂³	#493	1	1	yes	characteristic
93	Q₈	—	8	4	no	—
94	Q₈	—	16	2	no	—
95	C₂×C₄	—	8	2	no	—
96	C₂×C₄	—	16	1	no	—
97	C₂×C₄	—	16	1	no	—
98	C₂×C₄	—	16	2	no	—
99	C₂×C₄	—	16	1	no	—
100	C₂×C₄	—	16	4	no	—
101	C₂×C₄	—	16	2	no	—
102	C₂×C₄	#399	1	1	yes	characteristic
103	C₂×C₄	#474	1	2	yes	—
104	C₂×C₄	#480	1	2	yes	—
105	C₂×C₄	#493	1	2	yes	—
106	C₂×C₄	#497	1	2	yes	—
107	C₂×C₄	#497	1	1	yes	characteristic
108	C₂×C₄	C₂×Q₃₂	1	4	yes	—
109	C₂×C₄	#509	1	4	yes	—
110	C₈	—	2	8	no	—
111	C₈	#474	1	4	yes	—
112	C₈	#480	1	4	yes	—
113	C₂²	#1712	1	1	yes	characteristic
114	C₂²	#1722	1	1	yes	characteristic
115	C₂²	#2093	1	1	yes	characteristic
116	C₂²	#2247	1	2	yes	—
117	C₂²	#2250	1	2	yes	—
118	C₄	—	16	2	no	—
119	C₄	—	32	2	no	—
120	C₄	—	32	1	no	—
121	C₄	#1722	1	2	yes	—
122	C₄	#2093	1	2	yes	—
123	C₄	#2227	1	4	yes	—
124	C₄	#2236	1	4	yes	—
125	C₂	#10673	1	1	yes	characteristic
126	C₂	#12516	1	1	yes	characteristic
127	C₂	#12519	1	1	yes	characteristic
128	C₂	#12534	1	2	yes	—
129	C₂	#13530	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13203

C₂

yes

maximal

#12905

C₂

yes

maximal

#9825

C₂

yes

maximal, characteristic

#8818

C₂

yes

maximal, characteristic

#7136

C₂

yes

maximal, characteristic

C₂×#494

C₂²

yes

characteristic

C₂×#366

—

C₂×#358

—

C₂×C₄×C₁₆

C₂²

yes

—

#1990

—

#1845

—

#1845

C₂²

yes

characteristic

#1845

C₂²

yes

—

C₂×C₈²

C₂²

yes

characteristic

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

D₄

yes

—

#494

—

#494

—

#494

Q₈

yes

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

—

#340

—

C₈²

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₈

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

C₂×D₈

yes

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Q₃₂

yes

—

C₂×C₈

C₂²⋊Q₈

yes

—

C₂×C₈

#123

yes

—

C₂×C₈

C₂×D₈

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊Q₈

yes

C₄²

yes

C₄²

yes

—

C₁₆

—

C₂³

#493

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

#399

yes

characteristic

103

C₂×C₄

#474

yes

—

104

C₂×C₄

#480

yes

—

105

C₂×C₄

#493

yes

—

106

C₂×C₄

#497

yes

—

107

C₂×C₄

#497

yes

characteristic

108

C₂×C₄

C₂×Q₃₂

yes

—

109

C₂×C₄

#509

yes

—

110

C₈

—

111

C₈

#474

yes

—

112

C₈

#480

yes

—

113

C₂²

#1712

yes

characteristic

114

C₂²

#1722

yes

characteristic

115

C₂²

#2093

yes

characteristic

116

C₂²

#2247

yes

—

117

C₂²

#2250

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#1722

yes

—

122

C₄

#2093

yes

—

123

C₄

#2227

yes

—

124

C₄

#2236

yes

—

125

C₂

#10673

yes

characteristic

126

C₂

#12516

yes

characteristic

127

C₂

#12519

yes

characteristic

128

C₂

#12534

yes

—

129

C₂

#13530

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#493

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ 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
#13203	2
#12905	2
#9825	2
#8818	2
#7136	2

subgroup

index

Subgroups — #146237

Conjugacy classes of subgroups

Maximal subgroups