Group #93064 — subgroups

Subgroups — #93064

Order 512.

200 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 102 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×#1370	C₂	1	2	yes	maximal
2	#8818	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₁₆	—	2	2	no	—
4	#1845	—	2	2	no	—
5	#1845	C₄	1	2	yes	—
6	C₂×C₈²	C₂²	1	1	yes	characteristic
7	#1370	—	2	2	no	—
8	#1370	C₄	1	4	yes	complemented
9	C₂²×C₁₆	—	4	2	no	—
10	C₂×#107	—	4	2	no	—
11	C₂×#107	—	4	2	no	—
12	C₂×#107	—	4	2	no	—
13	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₈	D₄	1	2	yes	—
15	#384	—	2	2	no	—
16	C₄⋊C₁₆	—	2	4	no	—
17	C₄⋊C₁₆	—	4	2	no	—
18	C₈²	C₂×C₄	1	2	yes	—
19	C₈²	D₄	1	1	yes	characteristic
20	C₈²	Q₈	1	1	yes	characteristic
21	C₂²×C₈	—	2	1	no	—
22	C₂²×C₈	D₈	1	2	yes	—
23	C₂²×C₈	SD₁₆	1	2	yes	—
24	C₂×C₄⋊C₄	—	4	2	no	—
25	C₂×C₄⋊C₄	—	4	4	no	—
26	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₁₆	—	4	4	no	—
28	C₂×C₁₆	—	8	2	no	—
29	C₂×C₁₆	—	8	2	no	—
30	#107	—	8	4	no	—
31	#107	—	8	4	no	—
32	#107	—	8	2	no	—
33	#107	—	8	2	no	—
34	C₄×C₈	—	2	1	no	—
35	C₄×C₈	C₄²	1	1	yes	characteristic
36	C₄×C₈	C₂²⋊C₄	1	2	yes	—
37	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
38	C₄×C₈	C₄⋊C₄	1	2	yes	—
39	C₄×C₈	SD₁₆	1	2	yes	—
40	C₄×C₈	Q₁₆	1	2	yes	—
41	C₂²×C₄	—	8	2	no	—
42	C₂²×C₄	#100	1	1	yes	characteristic
43	C₂²×C₄	#102	1	2	yes	—
44	C₂×C₈	—	2	2	no	—
45	C₂×C₈	—	2	2	no	—
46	C₂×C₈	—	2	1	no	—
47	C₂×C₈	—	2	1	no	—
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	—	2	2	no	—
50	C₂×C₈	#102	1	2	yes	—
51	C₂×C₈	#103	1	2	yes	—
52	C₂×C₈	#106	1	2	yes	—
53	C₂×C₈	#107	1	2	yes	—
54	C₂×C₈	D₁₆	1	2	yes	—
55	C₂×C₈	SD₃₂	1	2	yes	—
56	C₂×C₈	SD₃₂	1	2	yes	—
57	C₂×C₈	Q₃₂	1	2	yes	—
58	C₄⋊C₄	—	8	4	no	—
59	C₄⋊C₄	—	8	4	no	—
60	C₄⋊C₄	—	8	4	no	—
61	C₄²	#95	1	1	yes	characteristic
62	C₄²	#101	1	1	yes	characteristic
63	C₄²	#103	1	2	yes	—
64	C₁₆	—	8	4	no	—
65	C₂³	#331	1	1	yes	characteristic
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	16	2	no	—
69	C₂×C₄	—	16	4	no	—
70	C₂×C₄	#332	1	2	yes	—
71	C₂×C₄	#333	1	1	yes	characteristic
72	C₂×C₄	#340	1	2	yes	—
73	C₂×C₄	#343	1	1	yes	characteristic
74	C₂×C₄	#357	1	2	yes	—
75	C₂×C₄	#358	1	2	yes	—
76	C₂×C₄	#361	1	2	yes	—
77	C₂×C₄	#362	1	2	yes	—
78	C₈	—	2	2	no	—
79	C₈	—	2	2	no	—
80	C₈	—	2	2	no	—
81	C₈	—	2	2	no	—
82	C₈	#357	1	2	yes	—
83	C₈	#358	1	2	yes	—
84	C₈	#366	1	2	yes	—
85	C₈	#367	1	2	yes	—
86	C₂²	#1296	1	1	yes	characteristic
87	C₂²	#1357	1	1	yes	characteristic
88	C₂²	#1358	1	1	yes	characteristic
89	C₂²	#1361	1	2	yes	—
90	C₂²	#1362	1	2	yes	—
91	C₄	—	2	1	no	—
92	C₄	—	2	1	no	—
93	C₄	—	32	2	no	—
94	C₄	#1364	1	2	yes	—
95	C₄	#1365	1	2	yes	—
96	C₄	#1380	1	2	yes	—
97	C₄	#1387	1	2	yes	—
98	C₂	#7096	1	1	yes	characteristic
99	C₂	#7097	1	2	yes	—
100	C₂	#7098	1	1	yes	characteristic
101	C₂	#7269	1	1	yes	characteristic
102	C₂	#7274	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1370

C₂

yes

maximal

#8818

C₂

yes

maximal, characteristic

C₂×C₄⋊C₁₆

—

#1845

—

#1845

C₄

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1370

—

#1370

C₄

yes

complemented

C₂²×C₁₆

—

C₂×#107

—

C₂×#107

—

C₂×#107

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

#384

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₈²

C₂×C₄

yes

—

C₈²

D₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

SD₁₆

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₄²

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₂²×C₄

—

C₂²×C₄

#100

yes

characteristic

C₂²×C₄

#102

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#102

yes

—

C₂×C₈

#103

yes

—

C₂×C₈

#106

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

D₁₆

yes

—

C₂×C₈

SD₃₂

yes

—

C₂×C₈

SD₃₂

yes

—

C₂×C₈

Q₃₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

yes

C₄²

yes

C₄²

yes

—

C₁₆

—

C₂³

#331

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#332

yes

—

C₂×C₄

#333

yes

characteristic

C₂×C₄

#340

yes

—

C₂×C₄

#343

yes

characteristic

C₂×C₄

#357

yes

—

C₂×C₄

#358

yes

—

C₂×C₄

#361

yes

—

C₂×C₄

#362

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#357

yes

—

C₈

#358

yes

—

C₈

#366

yes

—

C₈

#367

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#1362

yes

—

C₄

—

C₄

—

C₄

—

C₄

#1364

yes

—

C₄

#1365

yes

—

C₄

#1380

yes

—

C₄

#1387

yes

—

C₂

#7096

yes

characteristic

C₂

#7097

yes

—

100

C₂

#7098

yes

characteristic

101

C₂

#7269

yes

characteristic

102

C₂

#7274

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³	#331
commutator	C₄×C₈	C₄²
Frattini, cosocle	C₂×C₈²	C₂²
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#331

commutator

C₄×C₈

C₄²

Frattini, cosocle

C₂×C₈²

C₂²

LCS

G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
8. #1370	C₄	semidirect

H ≅ G/N

type

8. #1370

C₄

semidirect

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

subgroup	index
C₂×#1370	2
#8818	2

subgroup

index

C₂×#1370

#8818

Subgroups — #93064

Conjugacy classes of subgroups

Complements

Maximal subgroups