Group #129078 — subgroups

Subgroups — #129078

Order 512.

250 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 110 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	#12013	C₂	1	2	yes	maximal
2	#11415	C₂	1	2	yes	maximal
3	C₈×#97	C₂	1	1	yes	maximal, characteristic
4	#7149	C₂	1	2	yes	maximal
5	#2163	C₂²	1	4	yes	—
6	#1744	C₄	1	1	yes	characteristic
7	C₄×#97	C₂²	1	1	yes	characteristic
8	C₄²×C₈	C₄	1	1	yes	characteristic
9	C₂×#322	C₄	1	1	yes	characteristic
10	C₂×#322	C₂²	1	1	yes	characteristic
11	C₂×C₈²	C₄	1	1	yes	characteristic
12	C₂×C₈²	C₂²	1	1	yes	characteristic
13	#1370	C₄	1	4	yes	complemented
14	#1367	C₄	1	4	yes	complemented
15	#1313	—	2	4	no	—
16	C₂×#97	—	2	2	no	—
17	C₂×C₄×C₈	—	2	2	no	—
18	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
24	C₄³	C₂×C₄	1	1	yes	characteristic
25	C₄⋊C₁₆	C₂×C₄	1	8	yes	—
26	C₂²⋊C₁₆	—	2	4	no	—
27	#346	—	2	4	no	—
28	#336	—	2	2	no	—
29	#322	—	2	2	no	—
30	#322	D₄	1	2	yes	—
31	#322	Q₈	1	2	yes	—
32	C₈²	—	2	2	no	—
33	C₈²	C₂×C₄	1	2	yes	—
34	C₈²	C₂×C₄	1	2	yes	—
35	C₂²×C₈	—	2	1	no	—
36	C₂²×C₈	—	2	1	no	—
37	C₂²×C₈	—	2	1	no	—
38	C₂²×C₈	—	2	1	no	—
39	C₂²×C₈	M₁₆	1	2	yes	—
40	C₂²×C₈	#41	1	2	yes	—
41	C₂×C₄²	—	2	1	no	—
42	C₂×C₄²	—	2	1	no	—
43	C₂×C₄²	M₁₆	1	2	yes	—
44	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₁₆	—	2	8	no	—
46	#97	—	2	8	no	—
47	C₄×C₈	—	2	1	no	—
48	C₄×C₈	—	2	1	no	—
49	C₄×C₈	—	2	1	no	—
50	C₄×C₈	—	2	1	no	—
51	C₄×C₈	—	4	4	no	—
52	C₄×C₈	C₄²	1	2	yes	—
53	C₄×C₈	C₄²	1	1	yes	characteristic
54	C₄×C₈	C₄²	1	1	yes	characteristic
55	C₄×C₈	C₄⋊C₄	1	2	yes	—
56	C₄×C₈	C₄⋊C₄	1	2	yes	—
57	C₄×C₈	M₁₆	1	2	yes	—
58	C₄×C₈	M₁₆	1	2	yes	—
59	C₄×C₈	C₂²×C₄	1	2	yes	—
60	C₄×C₈	C₂×D₄	1	1	yes	characteristic
61	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	#117	1	1	yes	characteristic
64	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
65	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
66	C₂×C₈	—	2	4	no	—
67	C₂×C₈	—	2	4	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	2	4	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	4	2	no	—
74	C₂×C₈	—	4	2	no	—
75	C₂×C₈	—	4	2	no	—
76	C₂×C₈	#97	1	2	yes	—
77	C₂×C₈	#97	1	2	yes	—
78	C₂×C₈	C₄×D₄	1	2	yes	—
79	C₂×C₈	C₄×Q₈	1	2	yes	—
80	C₄²	—	2	4	no	—
81	C₄²	—	4	2	no	—
82	C₄²	—	4	2	no	—
83	C₄²	C₂×C₄²	1	1	yes	characteristic
84	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₄²	C₂×M₁₆	1	1	yes	characteristic
86	C₄²	C₂×M₁₆	1	1	yes	characteristic
87	C₁₆	—	4	8	no	—
88	C₂³	#432	1	1	yes	characteristic
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄	C₂×#97	1	1	yes	characteristic
97	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
98	C₂×C₄	#423	1	1	yes	characteristic
99	C₂×C₄	#423	1	1	yes	characteristic
100	C₂×C₄	#432	1	1	yes	characteristic
101	C₈	—	2	4	no	—
102	C₈	—	2	4	no	—
103	C₈	—	2	2	no	—
104	C₈	—	2	2	no	—
105	C₈	—	2	2	no	—
106	C₈	—	4	2	no	—
107	C₈	—	4	2	no	—
108	C₈	#443	1	4	yes	—
109	C₂²	—	2	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	#1767	1	1	yes	characteristic
112	C₂²	#2165	1	2	yes	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	2	1	no	—
116	C₄	—	2	1	no	—
117	C₄	—	4	4	no	—
118	C₄	#1772	1	2	yes	—
119	C₄	#2110	1	2	yes	—
120	C₂	—	2	2	no	—
121	C₂	#8267	1	1	yes	characteristic
122	C₂	#12657	1	1	yes	characteristic
123	C₂	#12659	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12013

C₂

yes

maximal

#11415

C₂

yes

maximal

C₈×#97

C₂

yes

maximal, characteristic

#7149

C₂

yes

maximal

#2163

C₂²

yes

—

#1744

C₄

yes

characteristic

C₄×#97

C₂²

yes

characteristic

C₄²×C₈

C₄

yes

characteristic

C₂×#322

C₄

yes

characteristic

C₂×#322

C₂²

yes

characteristic

C₂×C₈²

C₄

yes

characteristic

C₂×C₈²

C₂²

yes

characteristic

#1370

C₄

yes

complemented

#1367

C₄

yes

complemented

#1313

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

C₄⋊C₁₆

C₂×C₄

yes

—

C₂²⋊C₁₆

—

—

—

—

D₄

yes

—

#322

Q₈

yes

—

C₈²

—

C₈²

C₂×C₄

yes

—

C₈²

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

M₁₆

yes

—

C₂²×C₈

#41

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

M₁₆

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₁₆

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄²

yes

—

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

#117

yes

characteristic

C₂²×C₄

C₂×M₁₆

yes

characteristic

C₂²×C₄

C₂×M₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#97

yes

—

C₂×C₈

#97

yes

—

C₂×C₈

C₄×D₄

yes

—

C₂×C₈

C₄×Q₈

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×C₄²

yes

characteristic

C₄²

C₂×C₄⋊C₄

yes

characteristic

C₄²

C₂×M₁₆

yes

characteristic

C₄²

C₂×M₁₆

yes

characteristic

C₁₆

—

C₂³

#432

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

C₂×C₄

C₂×#97

yes

characteristic

C₂×C₄

C₄×M₁₆

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

#432

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

#443

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

#1767

yes

characteristic

112

C₂²

#2165

yes

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

#1772

yes

—

119

C₄

#2110

yes

—

120

C₂

—

121

C₂

#8267

yes

characteristic

122

C₂

#12657

yes

characteristic

123

C₂

#12659

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄×C₄⋊C₄
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1767
prime radical	C₂³	#432
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

C₄×C₄⋊C₄

commutator

C₄²

C₂×C₄²

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₄² ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₄ ◁ C₂×#322 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
13. #1370	C₄	semidirect
14. #1367	C₄	semidirect

H ≅ G/N

type

13. #1370

C₄

semidirect

14. #1367

C₄

semidirect

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

subgroup	index
#12013	2
#11415	2
C₈×#97	2
#7149	2

subgroup

index

#12013

#11415

C₈×#97

#7149

Subgroups — #129078

Conjugacy classes of subgroups

Complements

Maximal subgroups