Group #99287 — subgroups

Subgroups — #99287

Order 512.

242 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 106 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	#8807	C₂	1	1	yes	maximal, characteristic
2	#8807	C₂	1	1	yes	maximal, characteristic
3	#8133	C₂	1	1	yes	maximal, characteristic
4	#7018	C₂	1	4	yes	maximal
5	#1831	C₂²	1	2	yes	—
6	#1831	C₂²	1	2	yes	—
7	#1744	C₂²	1	2	yes	—
8	C₄²×C₈	C₂²	1	1	yes	characteristic
9	C₂×#335	—	2	1	no	—
10	C₂×#335	—	2	1	no	—
11	C₂×#334	—	2	1	no	—
12	C₂×#334	—	2	1	no	—
13	#1273	C₄	1	4	yes	—
14	C₂×C₄⋊C₈	—	2	2	no	—
15	C₂×C₄⋊C₈	—	2	2	no	—
16	C₂×C₄×C₈	C₂×C₄	1	4	yes	—
17	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₈	D₄	1	1	yes	characteristic
19	C₂×C₄×C₈	D₄	1	1	yes	characteristic
20	C₂×C₄×C₈	Q₈	1	2	yes	—
21	C₄³	C₂³	1	1	yes	characteristic
22	#336	—	2	2	no	—
23	#336	—	2	2	no	—
24	#336	—	2	4	no	—
25	#336	—	2	4	no	—
26	#335	—	4	2	no	—
27	#335	—	4	2	no	—
28	#334	—	4	2	no	—
29	#334	—	4	2	no	—
30	C₂²×C₈	—	2	4	no	—
31	C₂²×C₈	—	4	2	no	—
32	C₂²×C₈	—	4	2	no	—
33	C₂²×C₈	C₄⋊C₄	1	2	yes	—
34	C₂²×C₈	C₄⋊C₄	1	2	yes	—
35	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄²	#41	1	2	yes	—
39	C₂×C₄²	#41	1	2	yes	—
40	C₄⋊C₈	—	4	2	no	—
41	C₄⋊C₈	—	4	2	no	—
42	C₄⋊C₈	—	4	2	no	—
43	C₄⋊C₈	—	4	2	no	—
44	C₄×C₈	—	2	4	no	—
45	C₄×C₈	—	2	8	no	—
46	C₄×C₈	—	4	2	no	—
47	C₄×C₈	D₈	1	2	yes	—
48	C₄×C₈	SD₁₆	1	2	yes	—
49	C₄×C₈	SD₁₆	1	2	yes	—
50	C₄×C₈	Q₁₆	1	2	yes	—
51	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄	#117	1	2	yes	—
53	C₂²×C₄	#124	1	1	yes	characteristic
54	C₂²×C₄	#124	1	1	yes	characteristic
55	C₂²×C₄	#125	1	2	yes	—
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	4	4	no	—
61	C₂×C₈	—	4	4	no	—
62	C₂×C₈	—	4	4	no	—
63	C₂×C₈	—	8	4	no	—
64	C₂×C₈	—	8	4	no	—
65	C₂×C₈	—	8	2	no	—
66	C₂×C₈	—	8	2	no	—
67	C₂×C₈	#106	1	4	yes	—
68	C₂×C₈	#107	1	4	yes	—
69	C₄²	—	2	2	no	—
70	C₄²	—	4	2	no	—
71	C₄²	—	4	2	no	—
72	C₄²	#131	1	4	yes	—
73	C₄²	C₂×D₈	1	1	yes	characteristic
74	C₄²	C₂×SD₁₆	1	1	yes	characteristic
75	C₄²	C₂×SD₁₆	1	1	yes	characteristic
76	C₄²	C₂×Q₁₆	1	1	yes	characteristic
77	C₂³	#382	1	1	yes	characteristic
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	#424	1	2	yes	—
86	C₂×C₄	C₂×#106	1	1	yes	characteristic
87	C₂×C₄	C₂×#107	1	1	yes	characteristic
88	C₂×C₄	#430	1	2	yes	—
89	C₂×C₄	#433	1	4	yes	—
90	C₂×C₄	#486	1	1	yes	characteristic
91	C₂×C₄	#486	1	1	yes	characteristic
92	C₂×C₄	#487	1	1	yes	characteristic
93	C₂×C₄	#487	1	1	yes	characteristic
94	C₈	—	2	4	no	—
95	C₈	—	2	4	no	—
96	C₈	—	8	4	no	—
97	C₈	—	16	2	no	—
98	C₈	—	16	2	no	—
99	C₂²	#1832	1	1	yes	characteristic
100	C₂²	#1833	1	2	yes	—
101	C₂²	#1833	1	2	yes	—
102	C₂²	#1834	1	1	yes	characteristic
103	C₂²	#1838	1	1	yes	characteristic
104	C₄	—	2	4	no	—
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	4	2	no	—
110	C₄	#1570	1	2	yes	—
111	C₄	#1571	1	2	yes	—
112	C₂	#8783	1	2	yes	—
113	C₂	#8787	1	1	yes	characteristic
114	C₂	#8790	1	1	yes	characteristic
115	C₂	#8793	1	1	yes	characteristic
116	C₂	#8799	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#8807

C₂

yes

maximal, characteristic

#8807

C₂

yes

maximal, characteristic

#8133

C₂

yes

maximal, characteristic

#7018

C₂

yes

maximal

#1831

C₂²

yes

—

#1831

C₂²

yes

—

#1744

C₂²

yes

—

C₄²×C₈

C₂²

yes

—

—

—

—

C₄

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

—

C₄³

C₂³

yes

—

—

—

—

—

—

—

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×Q₈

yes

characteristic

C₂×C₄²

#41

yes

—

C₂×C₄²

#41

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

#117

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#106

yes

—

C₂×C₈

#107

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#131

yes

—

C₄²

C₂×D₈

yes

characteristic

C₄²

C₂×SD₁₆

yes

characteristic

C₄²

C₂×SD₁₆

yes

characteristic

C₄²

C₂×Q₁₆

yes

characteristic

C₂³

#382

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#424

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#433

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

#1832

yes

characteristic

100

C₂²

#1833

yes

—

101

C₂²

#1833

yes

—

102

C₂²

#1834

yes

characteristic

103

C₂²

#1838

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#1570

yes

—

111

C₄

#1571

yes

—

112

C₂

#8783

yes

—

113

C₂

#8787

yes

characteristic

114

C₂

#8790

yes

characteristic

115

C₂

#8793

yes

characteristic

116

C₂

#8799

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³	#382
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#382

commutator

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₄³

C₂³

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

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

subgroup	index
#8807	2
#8133	2
#7018	2

subgroup

index

#8807

#8133

#7018

Subgroups — #99287

Conjugacy classes of subgroups

Maximal subgroups