Group #99298 — subgroups

Subgroups — #99298

Order 512.

372 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 206 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	#8806	C₂	1	2	yes	maximal
2	#8784	C₂	1	4	yes	maximal
3	C₄×C₈²	C₂	1	1	yes	maximal, characteristic
4	#1842	C₂²	1	4	yes	—
5	C₄²×C₈	C₂²	1	2	yes	—
6	C₄²×C₈	C₂²	1	1	yes	characteristic
7	C₂×#335	—	2	4	no	—
8	C₂×C₈²	C₄	1	2	yes	—
9	C₂×C₈²	C₄	1	1	yes	characteristic
10	C₂×C₈²	C₄	1	1	yes	characteristic
11	#1277	—	2	8	no	—
12	C₂×C₄⋊C₈	—	2	4	no	—
13	C₂×C₄⋊C₈	—	2	8	no	—
14	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₈	D₄	1	4	yes	—
19	C₂×C₄×C₈	D₄	1	2	yes	—
20	C₂×C₄×C₈	Q₈	1	4	yes	—
21	C₂×C₄×C₈	Q₈	1	2	yes	—
22	C₄³	C₂³	1	1	yes	characteristic
23	#335	—	4	8	no	—
24	C₈²	—	2	4	no	—
25	C₈²	—	2	2	no	—
26	C₈²	C₈	1	4	yes	complemented
27	C₂²×C₈	—	4	4	no	—
28	C₂²×C₈	C₄⋊C₄	1	4	yes	—
29	C₂²×C₈	C₄⋊C₄	1	2	yes	—
30	C₂²×C₈	C₄⋊C₄	1	4	yes	—
31	C₂²×C₈	C₄⋊C₄	1	2	yes	—
32	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄²	C₂×D₄	1	2	yes	—
35	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄²	C₂×Q₈	1	2	yes	—
37	C₄⋊C₈	—	4	8	no	—
38	C₄⋊C₈	—	4	16	no	—
39	C₄×C₈	—	2	8	no	—
40	C₄×C₈	—	2	4	no	—
41	C₄×C₈	—	2	2	no	—
42	C₄×C₈	—	2	2	no	—
43	C₄×C₈	—	2	2	no	—
44	C₄×C₈	—	2	8	no	—
45	C₄×C₈	C₂×C₈	1	4	yes	—
46	C₄×C₈	C₂×C₈	1	2	yes	—
47	C₄×C₈	M₁₆	1	4	yes	—
48	C₄×C₈	M₁₆	1	2	yes	—
49	C₄×C₈	D₈	1	4	yes	—
50	C₄×C₈	Q₁₆	1	4	yes	—
51	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄	C₂×C₄⋊C₄	1	2	yes	—
53	C₂²×C₄	#127	1	1	yes	characteristic
54	C₂²×C₄	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	2	4	no	—
58	C₂×C₈	—	2	4	no	—
59	C₂×C₈	—	2	4	no	—
60	C₂×C₈	—	2	4	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	8	4	no	—
63	C₂×C₈	—	8	8	no	—
64	C₂×C₈	C₄⋊C₈	1	8	yes	—
65	C₂×C₈	C₄⋊C₈	1	4	yes	—
66	C₂×C₈	#107	1	4	yes	—
67	C₂×C₈	#107	1	4	yes	—
68	C₂×C₈	#108	1	8	yes	—
69	C₂×C₈	#108	1	4	yes	—
70	C₄²	—	2	2	no	—
71	C₄²	—	2	4	no	—
72	C₄²	C₂²×C₈	1	1	yes	characteristic
73	C₄²	C₂×M₁₆	1	1	yes	characteristic
74	C₄²	C₂×M₁₆	1	2	yes	—
75	C₄²	C₂×D₈	1	2	yes	—
76	C₄²	C₂×Q₁₆	1	2	yes	—
77	C₄²	#135	1	8	yes	—
78	C₂³	#384	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₂×C₄	C₂×C₄⋊C₈	1	2	yes	—
84	C₂×C₄	#423	1	1	yes	characteristic
85	C₂×C₄	#423	1	2	yes	—
86	C₂×C₄	C₂×#107	1	2	yes	—
87	C₂×C₄	#427	1	4	yes	—
88	C₂×C₄	C₂×#108	1	2	yes	—
89	C₂×C₄	C₂×#108	1	4	yes	—
90	C₂×C₄	#493	1	1	yes	characteristic
91	C₂×C₄	#494	1	1	yes	characteristic
92	C₂×C₄	#495	1	4	yes	—
93	C₂×C₄	#499	1	4	yes	—
94	C₂×C₄	#500	1	2	yes	—
95	C₈	—	2	8	no	—
96	C₈	—	2	4	no	—
97	C₈	—	2	8	no	—
98	C₈	—	16	4	no	—
99	C₈	#335	1	8	yes	—
100	C₂²	#1842	1	1	yes	characteristic
101	C₂²	#1845	1	1	yes	characteristic
102	C₂²	#1846	1	2	yes	—
103	C₂²	#1848	1	1	yes	characteristic
104	C₂²	#1848	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	4	no	—
107	C₄	C₂×#335	1	2	yes	—
108	C₄	#1564	1	4	yes	—
109	C₄	#1567	1	2	yes	—
110	C₄	#1568	1	4	yes	—
111	C₄	#1709	1	2	yes	—
112	C₄	#1710	1	2	yes	—
113	C₂	#8806	1	1	yes	characteristic
114	C₂	#8807	1	2	yes	—
115	C₂	#8816	1	1	yes	characteristic
116	C₂	#8823	1	1	yes	characteristic
117	C₂	#8824	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#8806

C₂

yes

maximal

#8784

C₂

yes

maximal

C₄×C₈²

C₂

yes

maximal, characteristic

#1842

C₂²

yes

—

C₄²×C₈

C₂²

yes

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

—

C₂×C₈²

C₄

yes

—

C₂×C₈²

C₄

yes

characteristic

C₂×C₈²

C₄

yes

characteristic

#1277

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

Q₈

yes

—

C₂×C₄×C₈

Q₈

yes

—

C₄³

C₂³

yes

characteristic

#335

—

C₈²

—

C₈²

—

C₈²

C₈

yes

complemented

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₄⋊C₄

yes

—

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×D₄

yes

—

C₂×C₄²

C₂×Q₈

yes

characteristic

C₂×C₄²

C₂×Q₈

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

D₈

yes

—

C₄×C₈

Q₁₆

yes

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂×C₄⋊C₄

yes

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

C₄⋊Q₈

yes

—

C₂²×C₄

C₄⋊Q₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

#108

yes

—

C₂×C₈

#108

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×C₈

yes

characteristic

C₄²

C₂×M₁₆

yes

characteristic

C₄²

C₂×M₁₆

yes

—

C₄²

C₂×D₈

yes

—

C₄²

C₂×Q₁₆

yes

—

C₄²

#135

yes

—

C₂³

#384

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₈

yes

characteristic

C₂×C₄

C₂×C₄⋊C₈

yes

—

C₂×C₄

#423

yes

characteristic

C₂×C₄

#423

yes

—

C₂×C₄

C₂×#107

yes

—

C₂×C₄

#427

yes

—

C₂×C₄

C₂×#108

yes

—

C₂×C₄

C₂×#108

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#499

yes

—

C₂×C₄

#500

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#335

yes

—

100

C₂²

#1842

yes

characteristic

101

C₂²

#1845

yes

characteristic

102

C₂²

#1846

yes

—

103

C₂²

#1848

yes

characteristic

104

C₂²

#1848

yes

—

105

C₄

—

106

C₄

—

107

C₄

C₂×#335

yes

—

108

C₄

#1564

yes

—

109

C₄

#1567

yes

—

110

C₄

#1568

yes

—

111

C₄

#1709

yes

—

112

C₄

#1710

yes

—

113

C₂

#8806

yes

characteristic

114

C₂

#8807

yes

—

115

C₂

#8816

yes

characteristic

116

C₂

#8823

yes

characteristic

117

C₂

#8824

yes

—

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#384
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center

C₂²×C₄

#127

commutator

C₄²

C₂²×C₈

Frattini, cosocle

C₄³

C₂³

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
26. C₈²	C₈	semidirect

H ≅ G/N

type

26. C₈²

C₈

semidirect

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

subgroup	index
#8806	2
#8784	2
C₄×C₈²	2

subgroup

index

#8806

#8784

C₄×C₈²

Subgroups — #99298

Conjugacy classes of subgroups

Complements

Maximal subgroups