Group #135787 — subgroups

Subgroups — #135787

Order 512.

310 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 162 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	#12906	C₂	1	2	yes	maximal
2	#12888	C₂	1	2	yes	maximal
3	C₂×#1312	C₂	1	1	yes	maximal, characteristic
4	#8818	C₂	1	1	yes	maximal, characteristic
5	#8814	C₂	1	1	yes	maximal, characteristic
6	C₂×#367	—	2	4	no	—
7	C₂×C₄×C₁₆	C₂²	1	2	yes	—
8	#1845	—	2	1	no	—
9	#1845	—	2	1	no	—
10	#1845	C₂²	1	1	yes	characteristic
11	#1845	C₂²	1	2	yes	—
12	#1845	C₂²	1	1	yes	characteristic
13	#1844	—	2	1	no	—
14	#1834	—	2	1	no	—
15	C₂×C₈²	C₂²	1	1	yes	characteristic
16	#1380	—	2	4	no	—
17	#1312	C₄	1	4	yes	complemented
18	C₂²×C₁₆	D₄	1	4	yes	—
19	C₂×#107	—	2	2	no	—
20	C₂×#107	—	2	2	no	—
21	C₂×#107	—	2	2	no	—
22	C₂×#107	—	2	2	no	—
23	C₂×#107	—	4	2	no	—
24	C₂×#107	—	4	2	no	—
25	C₂×#106	—	4	2	no	—
26	C₂×C₄×C₈	D₄	1	1	yes	characteristic
27	C₂×C₄×C₈	D₄	1	1	yes	characteristic
28	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
29	#384	—	2	1	no	—
30	#384	—	2	1	no	—
31	#384	—	2	2	no	—
32	#367	—	4	8	no	—
33	C₄×C₁₆	C₂×C₄	1	4	yes	—
34	C₄×C₁₆	Q₈	1	4	yes	—
35	#340	—	4	2	no	—
36	C₈²	C₂×C₄	1	2	yes	—
37	C₈²	Q₈	1	2	yes	—
38	C₂²×C₈	D₈	1	2	yes	—
39	C₂²×C₈	D₈	1	2	yes	—
40	C₂²×C₈	C₂×D₄	1	2	yes	—
41	C₂×C₄⋊C₄	—	4	1	no	—
42	C₂×C₄⋊C₄	—	4	2	no	—
43	C₂×C₄⋊C₄	—	4	1	no	—
44	C₂×C₄⋊C₄	—	4	1	no	—
45	C₂×C₄⋊C₄	—	4	1	no	—
46	C₂×C₄⋊C₄	—	4	1	no	—
47	C₂×C₄⋊C₄	—	4	1	no	—
48	C₂×C₄⋊C₄	—	4	2	no	—
49	C₂×C₄⋊C₄	—	4	2	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	C₄⋊C₄	1	8	yes	—
54	#107	—	4	4	no	—
55	#107	—	4	4	no	—
56	#107	—	4	4	no	—
57	#107	—	4	4	no	—
58	#107	—	8	4	no	—
59	#107	—	8	4	no	—
60	#106	—	8	4	no	—
61	C₄×C₈	C₄⋊C₄	1	2	yes	—
62	C₄×C₈	C₄⋊C₄	1	2	yes	—
63	C₄×C₈	Q₁₆	1	2	yes	—
64	C₄×C₈	Q₁₆	1	2	yes	—
65	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
66	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
67	C₄×C₈	C₂×Q₈	1	2	yes	—
68	C₂²×C₄	—	8	2	no	—
69	C₂²×C₄	—	8	1	no	—
70	C₂²×C₄	—	8	1	no	—
71	C₂²×C₄	#127	1	1	yes	characteristic
72	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
73	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	#107	1	4	yes	—
77	C₂×C₈	#107	1	4	yes	—
78	C₂×C₈	SD₃₂	1	4	yes	—
79	C₂×C₈	SD₃₂	1	4	yes	—
80	C₂×C₈	C₂×C₄⋊C₄	1	2	yes	—
81	C₂×C₈	C₄⋊Q₈	1	2	yes	—
82	C₂×C₈	AGL(1,ℤ/8)	1	4	yes	—
83	C₂×C₈	#137	1	4	yes	—
84	C₄⋊C₄	—	8	2	no	—
85	C₄⋊C₄	—	8	2	no	—
86	C₄⋊C₄	—	8	2	no	—
87	C₄⋊C₄	—	8	2	no	—
88	C₄⋊C₄	—	8	2	no	—
89	C₄⋊C₄	—	8	2	no	—
90	C₄⋊C₄	—	8	4	no	—
91	C₄⋊C₄	—	8	4	no	—
92	C₄⋊C₄	—	8	2	no	—
93	C₄⋊C₄	—	8	2	no	—
94	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₄²	C₄⋊Q₈	1	1	yes	characteristic
96	C₄²	C₂×Q₁₆	1	1	yes	characteristic
97	C₄²	C₂×Q₁₆	1	1	yes	characteristic
98	C₁₆	—	2	8	no	—
99	C₂³	#493	1	1	yes	characteristic
100	C₂×C₄	—	16	1	no	—
101	C₂×C₄	—	16	2	no	—
102	C₂×C₄	—	16	1	no	—
103	C₂×C₄	—	16	1	no	—
104	C₂×C₄	—	16	2	no	—
105	C₂×C₄	—	16	1	no	—
106	C₂×C₄	—	16	2	no	—
107	C₂×C₄	—	16	1	no	—
108	C₂×C₄	—	16	1	no	—
109	C₂×C₄	#384	1	1	yes	characteristic
110	C₂×C₄	C₂×#107	1	1	yes	characteristic
111	C₂×C₄	C₂×#107	1	1	yes	characteristic
112	C₂×C₄	#494	1	1	yes	characteristic
113	C₂×C₄	#496	1	2	yes	—
114	C₂×C₄	#497	1	2	yes	—
115	C₂×C₄	#500	1	1	yes	characteristic
116	C₂×C₄	#500	1	1	yes	characteristic
117	C₂×C₄	C₂×SD₃₂	1	2	yes	—
118	C₂×C₄	C₂×SD₃₂	1	2	yes	—
119	C₂×C₄	#509	1	2	yes	—
120	C₂×C₄	#510	1	2	yes	—
121	C₈	—	2	4	no	—
122	C₈	#367	1	8	yes	complemented
123	C₈	#428	1	4	yes	—
124	C₈	#501	1	4	yes	—
125	C₂²	#1721	1	1	yes	characteristic
126	C₂²	#1722	1	1	yes	characteristic
127	C₂²	#1845	1	1	yes	characteristic
128	C₂²	#2248	1	1	yes	characteristic
129	C₂²	#2248	1	1	yes	characteristic
130	C₂²	#2250	1	1	yes	characteristic
131	C₂²	#2251	1	1	yes	characteristic
132	C₄	—	32	1	no	—
133	C₄	—	32	2	no	—
134	C₄	—	32	1	no	—
135	C₄	#1723	1	2	yes	—
136	C₄	#1847	1	2	yes	—
137	C₄	C₂×#367	1	2	yes	—
138	C₄	#2159	1	2	yes	—
139	C₄	#2254	1	2	yes	—
140	C₄	#2255	1	2	yes	—
141	C₂	#8820	1	1	yes	characteristic
142	C₂	#12526	1	1	yes	characteristic
143	C₂	#12529	1	1	yes	characteristic
144	C₂	#12532	1	1	yes	characteristic
145	C₂	#12535	1	1	yes	characteristic
146	C₂	#12906	1	1	yes	characteristic
147	C₂	#12908	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12906

C₂

yes

maximal

#12888

C₂

yes

maximal

C₂×#1312

C₂

yes

maximal, characteristic

#8818

C₂

yes

maximal, characteristic

#8814

C₂

yes

maximal, characteristic

C₂×#367

—

C₂×C₄×C₁₆

C₂²

yes

—

#1845

—

#1845

—

#1845

C₂²

yes

characteristic

#1845

C₂²

yes

—

#1845

C₂²

yes

characteristic

#1844

—

#1834

—

C₂×C₈²

C₂²

yes

characteristic

#1380

—

#1312

C₄

yes

complemented

C₂²×C₁₆

D₄

yes

—

—

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

Q₈

yes

—

#340

—

C₈²

C₂×C₄

yes

—

C₈²

Q₈

yes

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

C₂×D₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

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

C₄⋊C₄

yes

—

—

—

—

—

—

—

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#107

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

SD₃₂

yes

—

C₂×C₈

SD₃₂

yes

—

C₂×C₈

C₂×C₄⋊C₄

yes

—

C₂×C₈

C₄⋊Q₈

yes

—

C₂×C₈

AGL(1,ℤ/8)

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₄⋊C₄

yes

characteristic

C₄²

C₄⋊Q₈

yes

characteristic

C₄²

C₂×Q₁₆

yes

characteristic

C₄²

C₂×Q₁₆

yes

characteristic

C₁₆

—

C₂³

#493

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#384

yes

characteristic

110

C₂×C₄

C₂×#107

yes

characteristic

111

C₂×C₄

C₂×#107

yes

characteristic

112

C₂×C₄

#494

yes

characteristic

113

C₂×C₄

#496

yes

—

114

C₂×C₄

#497

yes

—

115

C₂×C₄

#500

yes

characteristic

116

C₂×C₄

#500

yes

characteristic

117

C₂×C₄

C₂×SD₃₂

yes

—

118

C₂×C₄

C₂×SD₃₂

yes

—

119

C₂×C₄

#509

yes

—

120

C₂×C₄

#510

yes

—

121

C₈

—

122

C₈

#367

yes

complemented

123

C₈

#428

yes

—

124

C₈

#501

yes

—

125

C₂²

#1721

yes

characteristic

126

C₂²

#1722

yes

characteristic

127

C₂²

#1845

yes

characteristic

128

C₂²

#2248

yes

characteristic

129

C₂²

#2248

yes

characteristic

130

C₂²

#2250

yes

characteristic

131

C₂²

#2251

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#1723

yes

—

136

C₄

#1847

yes

—

137

C₄

C₂×#367

yes

—

138

C₄

#2159

yes

—

139

C₄

#2254

yes

—

140

C₄

#2255

yes

—

141

C₂

#8820

yes

characteristic

142

C₂

#12526

yes

characteristic

143

C₂

#12529

yes

characteristic

144

C₂

#12532

yes

characteristic

145

C₂

#12535

yes

characteristic

146

C₂

#12906

yes

characteristic

147

C₂

#12908

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#493
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ 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

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
17. #1312	C₄	semidirect
122. C₈	#367	semidirect

H ≅ G/N

type

17. #1312

C₄

semidirect

122. C₈

#367

semidirect

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

subgroup	index
#12906	2
#12888	2
C₂×#1312	2
#8818	2
#8814	2

subgroup

index

Subgroups — #135787

Conjugacy classes of subgroups

Complements

Maximal subgroups