Group #135683 — subgroups

Subgroups — #135683

Order 512.

226 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 114 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	#11410	C₂	1	1	yes	maximal, characteristic
2	#8823	C₂	1	1	yes	maximal, characteristic
3	#8814	C₂	1	1	yes	maximal, characteristic
4	#7217	C₂	1	4	yes	maximal
5	#2162	C₂²	1	2	yes	—
6	#1848	—	2	1	no	—
7	#1845	—	2	1	no	—
8	#1844	—	2	1	no	—
9	#1834	C₂²	1	2	yes	—
10	#1567	—	2	1	no	—
11	#1567	C₂²	1	2	yes	—
12	C₂×C₈²	C₂²	1	1	yes	characteristic
13	#1312	C₄	1	2	yes	complemented
14	C₈×C₁₆	C₄	1	2	yes	complemented
15	C₂×#108	—	4	1	no	—
16	C₂×#108	—	4	1	no	—
17	C₂×#107	—	4	2	no	—
18	C₂×#106	—	4	1	no	—
19	C₂×#106	—	4	1	no	—
20	#423	—	2	2	no	—
21	C₂×C₄×C₈	D₄	1	1	yes	characteristic
22	C₂×C₄×C₈	D₄	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
24	#384	—	2	2	no	—
25	C₄⋊C₁₆	—	2	2	no	—
26	C₂²⋊C₁₆	—	2	2	no	—
27	C₄×C₁₆	C₂×C₄	1	4	yes	—
28	#340	—	2	4	no	—
29	#335	—	2	4	no	—
30	#335	—	4	2	no	—
31	C₈²	C₂×C₄	1	1	yes	characteristic
32	C₈²	C₂×C₄	1	1	yes	characteristic
33	C₈²	Q₈	1	1	yes	characteristic
34	C₈²	Q₈	1	1	yes	characteristic
35	C₂×M₁₆	—	4	2	no	—
36	C₂²×C₈	D₈	1	1	yes	characteristic
37	C₂²×C₈	D₈	1	1	yes	characteristic
38	C₂²×C₈	SD₁₆	1	2	yes	—
39	C₂²×C₈	#41	1	2	yes	—
40	C₂×C₄⋊C₄	—	4	2	no	—
41	C₂×C₄⋊C₄	—	4	2	no	—
42	C₂×C₄⋊C₄	—	4	2	no	—
43	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
44	C₂×C₁₆	—	2	4	no	—
45	#108	—	8	2	no	—
46	#108	—	8	2	no	—
47	#107	—	8	4	no	—
48	#106	—	8	2	no	—
49	#106	—	8	2	no	—
50	C₄⋊C₈	—	4	2	no	—
51	C₄⋊C₈	—	4	2	no	—
52	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₈	SD₁₆	1	2	yes	—
57	C₄×C₈	Q₁₆	1	1	yes	characteristic
58	C₄×C₈	Q₁₆	1	1	yes	characteristic
59	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
61	C₄×C₈	#41	1	2	yes	—
62	C₂²×C₄	—	8	2	no	—
63	C₂²×C₄	#124	1	1	yes	characteristic
64	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
65	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
66	M₁₆	—	8	2	no	—
67	M₁₆	—	8	2	no	—
68	C₂×C₈	—	2	4	no	—
69	C₂×C₈	—	4	2	no	—
70	C₂×C₈	—	8	2	no	—
71	C₂×C₈	#106	1	2	yes	—
72	C₂×C₈	#106	1	2	yes	—
73	C₂×C₈	#107	1	1	yes	characteristic
74	C₂×C₈	#107	1	1	yes	characteristic
75	C₂×C₈	#107	1	1	yes	characteristic
76	C₂×C₈	#107	1	1	yes	characteristic
77	C₂×C₈	D₁₆	1	2	yes	—
78	C₂×C₈	SD₃₂	1	2	yes	—
79	C₂×C₈	SD₃₂	1	2	yes	—
80	C₂×C₈	Q₃₂	1	2	yes	—
81	C₂×C₈	#117	1	2	yes	—
82	C₂×C₈	#125	1	2	yes	—
83	C₄⋊C₄	—	8	4	no	—
84	C₄⋊C₄	—	8	4	no	—
85	C₄⋊C₄	—	8	2	no	—
86	C₄⋊C₄	—	8	2	no	—
87	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₄²	#124	1	1	yes	characteristic
89	C₄²	C₂×SD₁₆	1	1	yes	characteristic
90	C₄²	C₂×Q₁₆	1	1	yes	characteristic
91	C₁₆	—	4	4	no	—
92	C₂³	#486	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	16	2	no	—
96	C₂×C₄	—	16	2	no	—
97	C₂×C₄	—	16	2	no	—
98	C₂×C₄	#382	1	1	yes	characteristic
99	C₂×C₄	C₂×#106	1	1	yes	characteristic
100	C₂×C₄	C₂×#107	1	1	yes	characteristic
101	C₂×C₄	#486	1	1	yes	characteristic
102	C₂×C₄	#487	1	1	yes	characteristic
103	C₂×C₄	#487	1	1	yes	characteristic
104	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
105	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
106	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
107	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
108	C₈	—	2	2	no	—
109	C₈	—	2	2	no	—
110	C₈	—	2	2	no	—
111	C₈	—	16	2	no	—
112	C₈	#365	1	4	yes	—
113	C₈	#366	1	2	yes	—
114	C₈	#366	1	2	yes	—
115	C₈	#367	1	2	yes	—
116	C₈	#367	1	2	yes	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#1834	1	1	yes	characteristic
120	C₂²	#2240	1	1	yes	characteristic
121	C₂²	#2241	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	32	2	no	—
125	C₄	#1833	1	2	yes	—
126	C₄	C₂×#365	1	2	yes	—
127	C₄	C₂×#366	1	1	yes	characteristic
128	C₄	C₂×#367	1	1	yes	characteristic
129	C₄	#2240	1	1	yes	characteristic
130	C₄	#2241	1	1	yes	characteristic
131	C₂	—	2	1	no	—
132	C₂	—	2	1	no	—
133	C₂	#8790	1	1	yes	characteristic
134	C₂	#12886	1	1	yes	characteristic
135	C₂	#12888	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11410

C₂

yes

maximal, characteristic

#8823

C₂

yes

maximal, characteristic

#8814

C₂

yes

maximal, characteristic

#7217

C₂

yes

maximal

#2162

C₂²

yes

—

—

—

—

C₂²

yes

—

#1567

—

#1567

C₂²

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1312

C₄

yes

complemented

C₈×C₁₆

C₄

yes

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

#384

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

#340

—

#335

—

#335

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

SD₁₆

yes

—

C₂²×C₈

#41

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

—

C₂²×C₄

—

C₂²×C₄

#124

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#106

yes

—

C₂×C₈

#106

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

D₁₆

yes

—

C₂×C₈

SD₃₂

yes

—

C₂×C₈

SD₃₂

yes

—

C₂×C₈

Q₃₂

yes

—

C₂×C₈

#117

yes

—

C₂×C₈

#125

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₄⋊C₄

yes

characteristic

C₄²

#124

yes

characteristic

C₄²

C₂×SD₁₆

yes

characteristic

C₄²

C₂×Q₁₆

yes

characteristic

C₁₆

—

C₂³

#486

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

C₂×#107

yes

characteristic

101

C₂×C₄

#486

yes

characteristic

102

C₂×C₄

#487

yes

characteristic

103

C₂×C₄

#487

yes

characteristic

104

C₂×C₄

C₂×D₁₆

yes

characteristic

105

C₂×C₄

C₂×SD₃₂

yes

characteristic

106

C₂×C₄

C₂×SD₃₂

yes

characteristic

107

C₂×C₄

C₂×Q₃₂

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

#365

yes

—

113

C₈

#366

yes

—

114

C₈

#366

yes

—

115

C₈

#367

yes

—

116

C₈

#367

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

#1834

yes

characteristic

120

C₂²

#2240

yes

characteristic

121

C₂²

#2241

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#1833

yes

—

126

C₄

C₂×#365

yes

—

127

C₄

C₂×#366

yes

characteristic

128

C₄

C₂×#367

yes

characteristic

129

C₄

#2240

yes

characteristic

130

C₄

#2241

yes

characteristic

131

C₂

—

132

C₂

—

133

C₂

#8790

yes

characteristic

134

C₂

#12886

yes

characteristic

135

C₂

#12888

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1834
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#486
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1834

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#486

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
13. #1312	C₄	semidirect
14. C₈×C₁₆	C₄	semidirect

H ≅ G/N

type

13. #1312

C₄

semidirect

14. C₈×C₁₆

C₄

semidirect

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

subgroup	index
#11410	2
#8823	2
#8814	2
#7217	2

subgroup

index

Subgroups — #135683

Conjugacy classes of subgroups

Complements

Maximal subgroups