Group #142125 — subgroups

Subgroups — #142125

Order 512.

270 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 94 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	#33257	C₂	1	1	yes	maximal, characteristic
2	#12900	C₂	1	1	yes	maximal, characteristic
3	#8267	C₂	1	1	yes	maximal, characteristic
4	#7279	C₂	1	4	yes	maximal
5	#2864	—	2	1	no	—
6	C₂×#364	—	2	2	no	—
7	#2150	C₂²	1	2	yes	—
8	#1942	—	2	2	no	—
9	#1941	C₄	1	4	yes	—
10	#1846	C₂²	1	2	yes	—
11	#1835	—	2	1	no	—
12	C₄²×C₈	C₂²	1	1	yes	characteristic
13	#1564	C₂²	1	2	yes	—
14	#1452	—	2	1	no	—
15	#1276	—	2	1	no	—
16	C₂×M₃₂	—	2	2	no	—
17	#431	—	2	2	no	—
18	C₂×#107	—	2	2	no	—
19	C₂×#106	C₂×C₄	1	4	yes	—
20	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₈	D₄	1	2	yes	—
22	C₂×C₄×C₈	D₄	1	1	yes	characteristic
23	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
24	#389	—	2	4	no	—
25	#387	—	4	4	no	—
26	#382	—	2	2	no	—
27	C₄³	D₄	1	1	yes	characteristic
28	#364	—	4	4	no	—
29	C₄⋊C₁₆	—	2	4	no	—
30	#340	—	2	4	no	—
31	#335	—	2	2	no	—
32	#334	Q₈	1	4	yes	—
33	#322	—	4	2	no	—
34	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄	—	2	4	no	—
38	C₂×C₄⋊C₄	—	4	4	no	—
39	C₂×C₄²	—	2	2	no	—
40	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
42	M₃₂	—	4	4	no	—
43	C₂×C₁₆	—	4	2	no	—
44	#107	—	4	2	no	—
45	#107	—	4	2	no	—
46	#106	—	4	4	no	—
47	C₄⋊C₈	—	2	4	no	—
48	C₂²⋊C₈	—	4	2	no	—
49	C₄×C₈	—	2	2	no	—
50	C₄×C₈	—	2	2	no	—
51	C₄×C₈	—	4	2	no	—
52	C₄×C₈	—	4	2	no	—
53	C₄×C₈	SD₁₆	1	2	yes	—
54	C₄×C₈	Q₁₆	1	2	yes	—
55	C₄×C₈	C₂×Q₈	1	2	yes	—
56	C₄×C₈	#41	1	2	yes	—
57	#95	—	4	4	no	—
58	#95	—	4	4	no	—
59	C₂²×C₄	—	2	2	no	—
60	C₂²×C₄	—	4	4	no	—
61	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	4	4	no	—
67	C₂×C₈	#103	1	4	yes	—
68	C₂×C₈	C₄×Q₈	1	2	yes	—
69	C₂×C₈	#124	1	1	yes	characteristic
70	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
72	C₂×C₈	#137	1	2	yes	—
73	C₄⋊C₄	—	4	4	no	—
74	C₄⋊C₄	—	4	4	no	—
75	C₄⋊C₄	—	8	8	no	—
76	C₄²	—	2	2	no	—
77	C₄²	—	2	2	no	—
78	C₄²	—	4	4	no	—
79	C₄²	C₂²⋊Q₈	1	2	yes	—
80	C₄²	C₂×SD₁₆	1	1	yes	characteristic
81	C₄²	C₂×Q₁₆	1	1	yes	characteristic
82	C₁₆	—	8	2	no	—
83	C₂³	#418	1	1	yes	characteristic
84	C₂×C₄	—	2	4	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	8	8	no	—
89	C₂×C₄	—	8	4	no	—
90	C₂×C₄	#391	1	1	yes	characteristic
91	C₂×C₄	#413	1	2	yes	—
92	C₂×C₄	C₂×#103	1	1	yes	characteristic
93	C₂×C₄	#417	1	2	yes	—
94	C₂×C₄	#476	1	2	yes	—
95	C₂×C₄	#477	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	2	2	no	—
98	C₈	—	8	4	no	—
99	C₈	#490	1	1	yes	characteristic
100	C₈	#491	1	1	yes	characteristic
101	C₈	#501	1	2	yes	—
102	C₂²	—	2	2	no	—
103	C₂²	#1557	1	2	yes	—
104	C₂²	#1987	1	1	yes	characteristic
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	4	4	no	—
108	C₄	—	16	4	no	—
109	C₄	#1550	1	2	yes	—
110	C₄	#1992	1	1	yes	characteristic
111	C₄	#1998	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#9902	1	1	yes	characteristic
114	C₂	#13271	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33257

C₂

yes

maximal, characteristic

#12900

C₂

yes

maximal, characteristic

#8267

C₂

yes

maximal, characteristic

#7279

C₂

yes

—

—

C₂²

yes

—

#1942

—

#1941

C₄

yes

—

#1846

C₂²

yes

—

#1835

—

C₄²×C₈

C₂²

yes

characteristic

#1564

C₂²

yes

—

#1452

—

#1276

—

C₂×M₃₂

—

#431

—

C₂×#107

—

C₂×#106

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

C₄³

D₄

yes

characteristic

#364

—

C₄⋊C₁₆

—

#340

—

#335

—

#334

Q₈

yes

—

#322

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

#107

—

#107

—

#106

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#103

yes

—

C₂×C₈

C₄×Q₈

yes

—

C₂×C₈

#124

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

AGL(1,ℤ/8)

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²⋊Q₈

yes

—

C₄²

C₂×SD₁₆

yes

characteristic

C₄²

C₂×Q₁₆

yes

characteristic

C₁₆

—

C₂³

#418

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#391

yes

characteristic

C₂×C₄

#413

yes

—

C₂×C₄

C₂×#103

yes

characteristic

C₂×C₄

#417

yes

—

C₂×C₄

#476

yes

—

C₂×C₄

#477

yes

—

C₈

—

C₈

—

C₈

—

C₈

#490

yes

characteristic

100

C₈

#491

yes

characteristic

101

C₈

#501

yes

—

102

C₂²

—

103

C₂²

#1557

yes

—

104

C₂²

#1987

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

#1550

yes

—

110

C₄

#1992

yes

characteristic

111

C₄

#1998

yes

characteristic

112

C₂

—

113

C₂

#9902

yes

characteristic

114

C₂

#13271

yes

—

	subgroup	G/H
center, socle	C₂²	#1987
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1987

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#418

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#33257	2
#12900	2
#8267	2
#7279	2

subgroup

index

Subgroups — #142125

Conjugacy classes of subgroups

Maximal subgroups