Group #92897 — subgroups

Subgroups — #92897

Order 512.

210 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 126 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	#11410	C₂	1	1	yes	maximal, characteristic
2	C₈×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#2162	C₄	1	2	yes	—
5	C₄×M₃₂	C₄	1	2	yes	—
6	C₂×C₄×C₁₆	C₄	1	2	yes	—
7	C₂×C₈²	C₂²	1	1	yes	characteristic
8	#1312	—	2	1	no	—
9	#1311	—	2	1	no	—
10	C₈×C₁₆	—	2	1	no	—
11	C₈×C₁₆	—	2	1	no	—
12	C₈×C₁₆	—	2	1	no	—
13	C₈×C₁₆	—	2	1	no	—
14	C₂×M₃₂	C₈	1	4	yes	—
15	C₂²×C₁₆	—	2	2	no	—
16	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
19	C₄⋊C₁₆	C₈	1	4	yes	—
20	C₂²⋊C₁₆	—	2	2	no	—
21	#346	—	2	2	no	—
22	C₄×C₁₆	—	2	2	no	—
23	C₄×C₁₆	—	2	2	no	—
24	C₄×C₁₆	—	2	2	no	—
25	C₄×C₁₆	C₈	1	4	yes	—
26	C₈²	D₄	1	1	yes	characteristic
27	C₈²	D₄	1	1	yes	characteristic
28	C₈²	D₄	1	1	yes	characteristic
29	C₈²	Q₈	1	1	yes	characteristic
30	C₂²×C₈	C₂×C₈	1	2	yes	—
31	C₂²×C₈	M₁₆	1	2	yes	—
32	C₂²×C₈	M₁₆	1	2	yes	—
33	C₂×C₄²	C₄²	1	1	yes	characteristic
34	M₃₂	—	2	4	no	—
35	M₃₂	C₁₆	1	8	yes	complemented
36	C₂×C₁₆	—	2	8	no	—
37	C₂×C₁₆	—	2	4	no	—
38	C₂×C₁₆	—	2	4	no	—
39	C₂×C₁₆	—	2	4	no	—
40	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
42	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
43	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₈	C₂×C₈	1	2	yes	—
47	C₄×C₈	C₂×C₈	1	2	yes	—
48	C₄×C₈	M₁₆	1	2	yes	—
49	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
50	C₂²×C₄	#97	1	1	yes	characteristic
51	C₂²×C₄	#97	1	1	yes	characteristic
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	2	4	no	—
54	C₂×C₈	C₂²⋊C₈	1	2	yes	—
55	C₂×C₈	C₂²⋊C₈	1	2	yes	—
56	C₂×C₈	C₂²⋊C₈	1	2	yes	—
57	C₂×C₈	C₄⋊C₈	1	2	yes	—
58	C₂×C₈	C₄⋊C₈	1	2	yes	—
59	C₂×C₈	C₄⋊C₈	1	2	yes	—
60	C₂×C₈	C₂×C₁₆	1	4	yes	—
61	C₂×C₈	M₃₂	1	4	yes	—
62	C₄²	#95	1	1	yes	characteristic
63	C₄²	C₄×C₈	1	1	yes	characteristic
64	C₄²	C₄×C₈	1	1	yes	characteristic
65	C₄²	#97	1	1	yes	characteristic
66	C₁₆	—	2	8	no	—
67	C₁₆	—	2	8	no	—
68	C₁₆	—	4	4	no	—
69	C₂³	#322	1	1	yes	characteristic
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	C₈²	1	1	yes	characteristic
73	C₂×C₄	#322	1	1	yes	characteristic
74	C₂×C₄	#322	1	1	yes	characteristic
75	C₂×C₄	#336	1	1	yes	characteristic
76	C₂×C₄	#336	1	1	yes	characteristic
77	C₂×C₄	#336	1	1	yes	characteristic
78	C₂×C₄	C₄×C₁₆	1	2	yes	—
79	C₂×C₄	#346	1	2	yes	—
80	C₈	—	2	2	no	—
81	C₈	—	2	2	no	—
82	C₈	C₂²⋊C₁₆	1	4	yes	—
83	C₈	#350	1	4	yes	—
84	C₈	C₄⋊C₁₆	1	4	yes	—
85	C₈	#364	1	4	yes	—
86	C₂²	—	2	1	no	—
87	C₂²	—	2	1	no	—
88	C₂²	#1273	1	1	yes	characteristic
89	C₂²	#1312	1	2	yes	—
90	C₄	—	2	1	no	—
91	C₄	—	2	1	no	—
92	C₄	C₈×C₁₆	1	1	yes	characteristic
93	C₄	#1311	1	1	yes	characteristic
94	C₄	#1374	1	2	yes	—
95	C₄	#1376	1	2	yes	—
96	C₄	#1379	1	2	yes	—
97	C₂	—	2	2	no	—
98	C₂	#7136	1	1	yes	characteristic
99	C₂	#7138	1	1	yes	characteristic
100	C₂	#7141	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11410

C₂

yes

maximal, characteristic

C₈×M₃₂

C₂

yes

maximal, characteristic

C₂×C₈×C₁₆

C₂

yes

maximal, characteristic

#2162

C₄

yes

—

C₄×M₃₂

C₄

yes

—

C₂×C₄×C₁₆

C₄

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1312

—

#1311

—

C₈×C₁₆

—

C₈×C₁₆

—

C₈×C₁₆

—

C₈×C₁₆

—

C₂×M₃₂

C₈

yes

—

C₂²×C₁₆

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₄⋊C₁₆

C₈

yes

—

C₂²⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₈

yes

—

C₈²

D₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂²×C₈

C₂×C₈

yes

—

C₂²×C₈

M₁₆

yes

—

C₂²×C₈

M₁₆

yes

—

C₂×C₄²

C₄²

yes

characteristic

M₃₂

—

M₃₂

C₁₆

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₂²×C₄

C₄×C₈

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₂×C₁₆

yes

—

C₂×C₈

M₃₂

yes

—

C₄²

#95

yes

characteristic

C₄²

C₄×C₈

yes

characteristic

C₄²

C₄×C₈

yes

characteristic

C₄²

#97

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

#322

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈²

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄×C₁₆

yes

—

C₂×C₄

#346

yes

—

C₈

—

C₈

—

C₈

C₂²⋊C₁₆

yes

—

C₈

#350

yes

—

C₈

C₄⋊C₁₆

yes

—

C₈

#364

yes

—

C₂²

—

C₂²

—

C₂²

#1273

yes

characteristic

C₂²

#1312

yes

—

C₄

—

C₄

—

C₄

C₈×C₁₆

yes

C₄

yes

C₄

yes

—

C₄

#1376

yes

—

C₄

#1379

yes

—

C₂

—

C₂

yes

C₂

yes

100

C₂

#7141

yes

characteristic

	subgroup	G/H
center	C₈²	D₄
commutator	C₄	C₈×C₁₆
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1273
prime radical	C₂³	#322
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₈²

D₄

commutator

C₄

C₈×C₁₆

Frattini, cosocle

C₂×C₈²

C₂²

C₂²

C₂³

G ▷ C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₈² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
35. M₃₂	C₁₆	semidirect

H ≅ G/N

type

35. M₃₂

C₁₆

semidirect

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

subgroup	index
#11410	2
C₈×M₃₂	2
C₂×C₈×C₁₆	2

subgroup

index

#11410

C₈×M₃₂

C₂×C₈×C₁₆

Subgroups — #92897

Conjugacy classes of subgroups

Complements

Maximal subgroups