Group #146102 — subgroups

Subgroups — #146102

Order 512.

197 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 197 conjugacy classes of subgroups — too many to draw. Only its 52 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	#11963	C₂	1	2	yes	maximal
2	#9760	C₂	1	1	yes	maximal, characteristic
3	#8798	C₂	1	1	yes	maximal, characteristic, complemented
4	#7218	C₂	1	2	yes	maximal, complemented
5	#7198	C₂	1	1	yes	maximal, characteristic, complemented
6	#3147	C₂²	1	1	yes	characteristic
7	#2162	C₂²	1	2	yes	—
8	#1835	C₂²	1	1	yes	characteristic
9	#1568	C₂²	1	2	yes	—
10	#1517	—	2	2	no	—
11	#1452	C₂²	1	1	yes	characteristic
12	#1362	—	2	2	no	—
13	#1361	—	2	2	no	—
14	#1287	—	2	2	no	—
15	C₂×#135	—	2	2	no	—
16	#532	—	2	1	no	—
17	#495	—	2	2	no	—
18	#494	D₄	1	1	yes	characteristic
19	#493	D₄	1	1	yes	characteristic
20	#492	—	2	1	no	—
21	#432	—	2	2	no	—
22	#431	D₄	1	2	yes	—
23	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
24	#382	—	2	1	no	—
25	C₄⋊C₁₆	—	2	2	no	—
26	C₂²⋊C₁₆	—	2	2	no	—
27	C₄×C₁₆	—	2	2	no	—
28	#340	—	2	2	no	—
29	#335	—	2	2	no	—
30	#334	—	2	2	no	—
31	#327	—	4	4	no	—
32	#322	—	2	2	no	—
33	C₂×#41	—	4	1	no	—
34	#135	—	8	4	no	—
35	C₂×Q₁₆	—	2	2	no	—
36	C₂×SD₁₆	—	4	2	no	—
37	C₂×D₈	—	2	2	no	—
38	C₂²×C₈	#41	1	2	yes	—
39	C₄⋊Q₈	—	2	1	no	—
40	#127	—	2	1	no	—
41	#124	—	4	1	no	—
42	#121	—	4	2	no	—
43	C₄×D₄	—	4	2	no	—
44	C₂×C₄⋊C₄	—	4	1	no	—
45	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
46	C₂×C₁₆	—	4	2	no	—
47	C₄⋊C₈	—	2	2	no	—
48	C₄⋊C₈	—	4	2	no	—
49	C₂²⋊C₈	—	2	2	no	—
50	C₂²⋊C₈	—	4	2	no	—
51	#97	—	4	2	no	—
52	C₄×C₈	—	2	2	no	—
53	C₄×C₈	C₂×D₄	1	1	yes	characteristic
54	C₄×C₈	#41	1	1	yes	characteristic
55	C₄×C₈	#41	1	2	yes	—
56	#95	—	4	2	no	—
57	#41	—	8	4	no	—
58	C₂×Q₈	—	4	1	no	—
59	C₂×D₄	—	4	1	no	—
60	C₂×D₄	—	4	2	no	—
61	C₂×D₄	—	8	1	no	—
62	C₂²×C₄	—	8	1	no	—
63	C₂²×C₄	—	8	1	no	—
64	C₂²×C₄	#124	1	1	yes	characteristic
65	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
66	Q₁₆	—	8	2	no	—
67	SD₁₆	—	8	4	no	—
68	D₈	—	8	2	no	—
69	C₂×C₈	—	4	2	no	—
70	C₂×C₈	—	4	2	no	—
71	C₂×C₈	—	8	2	no	—
72	C₂×C₈	#123	1	2	yes	—
73	C₂×C₈	#126	1	2	yes	—
74	C₄⋊C₄	—	4	2	no	—
75	C₄⋊C₄	—	8	2	no	—
76	C₂²⋊C₄	—	8	2	no	—
77	C₄²	#121	1	1	yes	characteristic
78	C₄²	#123	1	1	yes	characteristic
79	C₄²	#135	1	2	yes	—
80	C₁₆	—	8	2	no	—
81	C₂³	—	8	1	no	—
82	C₂³	#490	1	1	yes	characteristic
83	Q₈	—	8	2	no	—
84	D₄	—	8	2	no	—
85	D₄	—	16	2	no	—
86	D₄	—	16	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	8	1	no	—
90	C₂×C₄	—	16	2	no	—
91	C₂×C₄	—	16	1	no	—
92	C₂×C₄	—	16	2	no	—
93	C₂×C₄	#397	1	1	yes	characteristic
94	C₂×C₄	#463	1	2	yes	—
95	C₂×C₄	#482	1	2	yes	—
96	C₂×C₄	#488	1	1	yes	characteristic
97	C₈	—	2	2	no	—
98	C₈	—	2	2	no	—
99	C₈	—	8	2	no	—
100	C₈	—	16	2	no	—
101	C₂²	—	2	1	no	—
102	C₂²	—	2	2	no	—
103	C₂²	—	16	1	no	—
104	C₂²	—	16	2	no	—
105	C₂²	#2073	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	2	2	no	—
108	C₄	—	2	2	no	—
109	C₄	—	16	1	no	—
110	C₄	—	32	1	no	—
111	C₄	#2081	1	2	yes	—
112	C₂	—	4	1	no	—
113	C₂	—	32	1	no	—
114	C₂	#10634	1	1	yes	characteristic
115	C₂	#13489	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11963

C₂

yes

maximal

#9760

C₂

yes

maximal, characteristic

#8798

C₂

yes

maximal, characteristic, complemented

#7218

C₂

yes

maximal, complemented

#7198

C₂

yes

maximal, characteristic, complemented

#3147

C₂²

yes

characteristic

#2162

C₂²

yes

—

#1835

C₂²

yes

characteristic

#1568

C₂²

yes

—

#1517

—

#1452

C₂²

yes

—

—

—

—

—

—

D₄

yes

characteristic

#493

D₄

yes

—

—

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

#382

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

—

—

—

—

—

C₂×#41

—

#135

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂²×C₈

#41

yes

—

C₄⋊Q₈

—

#127

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

C₄×C₈

yes

—

#95

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#124

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

—

Q₁₆

—

SD₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#123

yes

—

C₂×C₈

#126

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

yes

C₄²

yes

C₄²

yes

—

C₁₆

—

C₂³

—

C₂³

#490

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#397

yes

characteristic

C₂×C₄

#463

yes

—

C₂×C₄

#482

yes

—

C₂×C₄

#488

yes

characteristic

C₈

—

C₈

—

C₈

—

100

C₈

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

104

C₂²

—

105

C₂²

#2073

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#2081

yes

—

112

C₂

—

113

C₂

—

114

C₂

#10634

yes

characteristic

115

C₂

#13489

yes

—

	subgroup	G/H
center, socle	C₂²	#2073
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2073

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

#3147

C₂²

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
3. #8798	C₂	semidirect
4. #7218	C₂	semidirect
5. #7198	C₂	semidirect

H ≅ G/N

type

3. #8798

C₂

semidirect

4. #7218

C₂

semidirect

5. #7198

C₂

semidirect

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

subgroup	index
#11963	2
#9760	2
#8798	2
#7218	2
#7198	2

subgroup

index

Subgroups — #146102

Conjugacy classes of subgroups

Complements

Maximal subgroups