Group #94018 — subgroups

Subgroups — #94018

Order 512.

162 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 86 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	C₂×#1422	C₂	1	2	yes	maximal
2	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×M₆₄	—	2	2	no	—
4	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
5	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
6	C₂×C₈²	C₄	1	1	yes	characteristic
7	#1422	—	2	2	no	—
8	#1422	C₄	1	4	yes	—
9	C₈×C₁₆	—	2	2	no	—
10	C₂²×C₁₆	C₈	1	1	yes	characteristic
11	C₂²×C₁₆	C₈	1	1	yes	characteristic
12	C₂²×C₁₆	D₄	1	1	yes	characteristic
13	C₂²×C₁₆	Q₈	1	1	yes	characteristic
14	C₂×C₄×C₈	—	2	1	no	—
15	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
16	M₆₄	—	2	4	no	—
17	M₆₄	—	4	2	no	—
18	C₂×C₃₂	—	4	2	no	—
19	C₄×C₁₆	—	2	1	no	—
20	C₄×C₁₆	—	2	1	no	—
21	C₄×C₁₆	C₂×C₄	1	2	yes	—
22	C₄×C₁₆	D₄	1	2	yes	—
23	C₈²	—	2	1	no	—
24	C₈²	C₈	1	2	yes	—
25	C₂²×C₈	—	2	2	no	—
26	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄²	M₁₆	1	1	yes	characteristic
29	C₂×C₁₆	—	2	1	no	—
30	C₂×C₁₆	—	2	1	no	—
31	C₂×C₁₆	—	2	1	no	—
32	C₂×C₁₆	—	2	1	no	—
33	C₂×C₁₆	—	2	1	no	—
34	C₂×C₁₆	—	2	1	no	—
35	C₂×C₁₆	C₁₆	1	2	yes	—
36	C₂×C₁₆	C₁₆	1	2	yes	—
37	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
38	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
39	C₂×C₁₆	D₈	1	1	yes	characteristic
40	C₂×C₁₆	SD₁₆	1	2	yes	—
41	C₂×C₁₆	Q₁₆	1	1	yes	characteristic
42	C₄×C₈	—	2	2	no	—
43	C₄×C₈	—	2	2	no	—
44	C₄×C₈	C₄²	1	1	yes	characteristic
45	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
46	C₄×C₈	C₂×C₈	1	1	yes	characteristic
47	C₄×C₈	M₁₆	1	1	yes	characteristic
48	C₃₂	—	4	4	no	—
49	C₂²×C₄	—	2	1	no	—
50	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
51	C₂×C₈	—	2	4	no	—
52	C₂×C₈	—	2	8	no	—
53	C₂×C₈	#95	1	1	yes	characteristic
54	C₂×C₈	C₄×C₈	1	1	yes	characteristic
55	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
56	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₈	#106	1	1	yes	characteristic
58	C₂×C₈	#107	1	1	yes	characteristic
59	C₂×C₈	#108	1	1	yes	characteristic
60	C₂×C₈	#108	1	1	yes	characteristic
61	C₂×C₈	C₂×C₁₆	1	1	yes	characteristic
62	C₂×C₈	C₂×C₁₆	1	1	yes	characteristic
63	C₂×C₈	M₃₂	1	1	yes	characteristic
64	C₂×C₈	M₃₂	1	1	yes	characteristic
65	C₄²	—	2	1	no	—
66	C₄²	#97	1	1	yes	characteristic
67	C₄²	C₂²⋊C₈	1	1	yes	characteristic
68	C₁₆	—	2	2	no	—
69	C₁₆	—	2	2	no	—
70	C₁₆	—	2	2	no	—
71	C₁₆	#102	1	2	yes	—
72	C₁₆	#103	1	2	yes	—
73	C₂³	#364	1	1	yes	characteristic
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	#334	1	1	yes	characteristic
78	C₂×C₄	#335	1	1	yes	characteristic
79	C₂×C₄	#336	1	1	yes	characteristic
80	C₂×C₄	C₄⋊C₁₆	1	1	yes	characteristic
81	C₂×C₄	C₄⋊C₁₆	1	1	yes	characteristic
82	C₂×C₄	#364	1	1	yes	characteristic
83	C₈	—	2	8	no	—
84	C₈	#334	1	1	yes	characteristic
85	C₈	#335	1	1	yes	characteristic
86	C₈	#340	1	1	yes	characteristic
87	C₈	#341	1	1	yes	characteristic
88	C₈	C₄×C₁₆	1	1	yes	characteristic
89	C₈	#346	1	1	yes	characteristic
90	C₈	C₂²⋊C₁₆	1	1	yes	characteristic
91	C₈	C₂²⋊C₁₆	1	1	yes	characteristic
92	C₂²	—	2	2	no	—
93	C₂²	#1367	1	1	yes	characteristic
94	C₂²	#1370	1	1	yes	characteristic
95	C₂²	#1376	1	1	yes	characteristic
96	C₄	—	2	2	no	—
97	C₄	—	2	2	no	—
98	C₄	#1277	1	1	yes	characteristic
99	C₄	#1367	1	1	yes	characteristic
100	C₄	#1370	1	1	yes	characteristic
101	C₄	#1374	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	—	2	1	no	—
104	C₂	#7151	1	1	yes	characteristic
105	C₂	#7459	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1422

C₂

yes

maximal

C₂×C₈×C₁₆

C₂

yes

maximal, characteristic

C₂×M₆₄

—

C₂×C₄×C₁₆

C₄

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₈²

C₄

yes

characteristic

#1422

—

#1422

C₄

yes

—

C₈×C₁₆

—

C₂²×C₁₆

C₈

yes

characteristic

C₂²×C₁₆

C₈

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

Q₈

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

D₄

yes

—

C₈²

—

C₈²

C₈

yes

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₂×C₈

yes

characteristic

C₂×C₄²

M₁₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₁₆

yes

—

C₂×C₁₆

C₁₆

yes

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

D₈

yes

characteristic

C₂×C₁₆

SD₁₆

yes

—

C₂×C₁₆

Q₁₆

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₃₂

—

C₂²×C₄

—

C₂²×C₄

C₄⋊C₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#95

yes

characteristic

C₂×C₈

C₄×C₈

yes

characteristic

C₂×C₈

C₂²⋊C₈

yes

characteristic

C₂×C₈

C₄⋊C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂×C₁₆

yes

characteristic

C₂×C₈

C₂×C₁₆

yes

characteristic

C₂×C₈

M₃₂

yes

characteristic

C₂×C₈

M₃₂

yes

characteristic

C₄²

—

C₄²

#97

yes

characteristic

C₄²

C₂²⋊C₈

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

#102

yes

—

C₁₆

#103

yes

—

C₂³

#364

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄⋊C₁₆

yes

characteristic

C₂×C₄

C₄⋊C₁₆

yes

characteristic

C₂×C₄

#364

yes

characteristic

C₈

—

C₈

yes

C₈

yes

C₈

yes

C₈

yes

C₈

C₄×C₁₆

yes

characteristic

C₈

#346

yes

characteristic

C₈

C₂²⋊C₁₆

yes

characteristic

C₈

C₂²⋊C₁₆

yes

characteristic

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

C₄

yes

C₄

yes

100

C₄

#1370

yes

characteristic

101

C₄

#1374

yes

characteristic

102

C₂

—

103

C₂

—

104

C₂

#7151

yes

characteristic

105

C₂

#7459

yes

—

	subgroup	G/H
center	C₂×C₁₆	D₈
commutator	C₈	C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle	C₂²	#1370
prime radical	C₂³	#364
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₁₆

D₈

commutator

C₈

C₄×C₁₆

Frattini, cosocle

C₂×C₄×C₁₆

C₂²

C₂²

C₂³

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₂×#1422	2
C₂×C₈×C₁₆	2

subgroup

index

C₂×#1422

C₂×C₈×C₁₆

Subgroups — #94018

Conjugacy classes of subgroups

Maximal subgroups