Group #135686 — subgroups

Subgroups — #135686

Order 512.

214 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 98 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	#11411	C₂	1	1	yes	maximal, characteristic
2	#8824	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	#1845	—	2	1	no	—
7	#1844	—	2	1	no	—
8	#1837	—	2	1	no	—
9	#1834	C₂²	1	2	yes	—
10	#1567	C₂²	1	2	yes	—
11	#1566	—	2	1	no	—
12	C₂×C₈²	C₂²	1	1	yes	characteristic
13	#1312	C₄	1	2	yes	complemented
14	#1311	C₄	1	2	yes	complemented
15	C₂×#107	—	4	2	no	—
16	C₂×#106	—	4	2	no	—
17	#423	—	2	2	no	—
18	C₂×C₄×C₈	D₄	1	1	yes	characteristic
19	C₂×C₄×C₈	D₄	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
21	#384	—	2	2	no	—
22	C₄⋊C₁₆	—	2	2	no	—
23	C₂²⋊C₁₆	—	2	2	no	—
24	C₄×C₁₆	C₂×C₄	1	4	yes	—
25	#341	—	4	2	no	—
26	#340	—	2	4	no	—
27	#335	—	2	4	no	—
28	#334	—	4	2	no	—
29	C₈²	C₂×C₄	1	1	yes	characteristic
30	C₈²	C₂×C₄	1	1	yes	characteristic
31	C₈²	Q₈	1	1	yes	characteristic
32	C₈²	Q₈	1	1	yes	characteristic
33	C₂×M₁₆	—	4	2	no	—
34	C₂²×C₈	D₈	1	2	yes	—
35	C₂²×C₈	SD₁₆	1	2	yes	—
36	C₂²×C₈	#41	1	2	yes	—
37	C₂×C₄⋊C₄	—	4	2	no	—
38	C₂×C₄⋊C₄	—	4	2	no	—
39	C₂×C₄⋊C₄	—	4	2	no	—
40	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
41	C₂×C₁₆	—	2	4	no	—
42	#107	—	8	4	no	—
43	#106	—	8	4	no	—
44	C₄⋊C₈	—	4	2	no	—
45	C₄⋊C₈	—	4	2	no	—
46	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
50	C₄×C₈	SD₁₆	1	2	yes	—
51	C₄×C₈	Q₁₆	1	2	yes	—
52	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
54	C₄×C₈	#41	1	2	yes	—
55	C₂²×C₄	—	8	2	no	—
56	C₂²×C₄	#124	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
58	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
59	M₁₆	—	8	2	no	—
60	M₁₆	—	8	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	4	no	—
64	C₂×C₈	—	4	2	no	—
65	C₂×C₈	—	8	2	no	—
66	C₂×C₈	#106	1	2	yes	—
67	C₂×C₈	#106	1	2	yes	—
68	C₂×C₈	#107	1	2	yes	—
69	C₂×C₈	#107	1	2	yes	—
70	C₂×C₈	#117	1	2	yes	—
71	C₂×C₈	#125	1	2	yes	—
72	C₄⋊C₄	—	8	4	no	—
73	C₄⋊C₄	—	8	4	no	—
74	C₄⋊C₄	—	8	4	no	—
75	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₄²	#124	1	1	yes	characteristic
77	C₄²	C₂×SD₁₆	1	1	yes	characteristic
78	C₄²	C₂×Q₁₆	1	1	yes	characteristic
79	C₁₆	—	4	4	no	—
80	C₂³	#486	1	1	yes	characteristic
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	16	2	no	—
84	C₂×C₄	—	16	2	no	—
85	C₂×C₄	—	16	2	no	—
86	C₂×C₄	#382	1	1	yes	characteristic
87	C₂×C₄	C₂×#106	1	1	yes	characteristic
88	C₂×C₄	C₂×#107	1	1	yes	characteristic
89	C₂×C₄	#486	1	1	yes	characteristic
90	C₂×C₄	#487	1	1	yes	characteristic
91	C₂×C₄	#487	1	1	yes	characteristic
92	C₂×C₄	#509	1	2	yes	—
93	C₂×C₄	#510	1	2	yes	—
94	C₈	—	2	2	no	—
95	C₈	—	2	2	no	—
96	C₈	—	2	2	no	—
97	C₈	—	2	2	no	—
98	C₈	—	2	2	no	—
99	C₈	—	16	2	no	—
100	C₈	#365	1	4	yes	—
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	#1834	1	1	yes	characteristic
104	C₂²	#2244	1	1	yes	characteristic
105	C₂²	#2245	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	2	1	no	—
108	C₄	—	32	2	no	—
109	C₄	#1833	1	2	yes	—
110	C₄	C₂×#365	1	2	yes	—
111	C₄	#2159	1	2	yes	—
112	C₄	#2243	1	2	yes	—
113	C₂	—	2	1	no	—
114	C₂	—	2	1	no	—
115	C₂	#8790	1	1	yes	characteristic
116	C₂	#12886	1	1	yes	characteristic
117	C₂	#12890	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11411

C₂

yes

maximal, characteristic

#8824

C₂

yes

maximal, characteristic

#8814

C₂

yes

maximal, characteristic

#7217

C₂

yes

maximal

#2162

C₂²

yes

—

—

—

—

C₂²

yes

—

#1567

C₂²

yes

—

#1566

—

C₂×C₈²

C₂²

yes

characteristic

#1312

C₄

yes

complemented

#1311

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

—

—

—

—

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

D₈

yes

—

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

—

#107

—

#106

—

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

—

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

—

C₂×C₈

—

C₂×C₈

#106

yes

—

C₂×C₈

#106

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#117

yes

—

C₂×C₈

#125

yes

—

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

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#510

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

100

C₈

#365

yes

—

101

C₂²

—

102

C₂²

—

103

C₂²

#1834

yes

characteristic

104

C₂²

#2244

yes

characteristic

105

C₂²

#2245

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

#1833

yes

—

110

C₄

C₂×#365

yes

—

111

C₄

#2159

yes

—

112

C₄

#2243

yes

—

113

C₂

—

114

C₂

—

115

C₂

#8790

yes

characteristic

116

C₂

#12886

yes

characteristic

117

C₂

#12890

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. #1311	C₄	semidirect

H ≅ G/N

type

13. #1312

C₄

semidirect

14. #1311

C₄

semidirect

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

subgroup	index
#11411	2
#8824	2
#8814	2
#7217	2

subgroup

index

Subgroups — #135686

Conjugacy classes of subgroups

Complements

Maximal subgroups