Group #93257 — subgroups

Subgroups — #93257

Order 512.

320 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 170 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	C₄×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	C₄×C₈²	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₁₆	C₄	1	4	yes	—
4	C₂×C₄×C₁₆	—	2	2	no	—
5	C₄²×C₈	C₄	1	2	yes	—
6	C₄²×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₈²	—	2	2	no	—
8	#1374	—	2	2	no	—
9	C₂²×C₁₆	—	2	4	no	—
10	C₂×C₄×C₈	—	2	2	no	—
11	C₂×C₄×C₈	—	2	2	no	—
12	C₂×C₄×C₈	C₈	1	4	yes	—
13	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₈	D₄	1	2	yes	—
15	C₂×C₄×C₈	D₄	1	1	yes	characteristic
16	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
17	C₄³	C₂×C₄	1	1	yes	characteristic
18	C₄⋊C₁₆	—	2	4	no	—
19	C₄⋊C₁₆	C₈	1	8	yes	complemented
20	C₄×C₁₆	—	4	2	no	—
21	C₄×C₁₆	—	4	2	no	—
22	C₈²	—	2	8	no	—
23	C₂²×C₈	—	2	4	no	—
24	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
25	C₂²×C₈	C₄⋊C₄	1	2	yes	—
26	C₂×C₄²	C₄²	1	1	yes	characteristic
27	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄²	C₂×C₈	1	2	yes	—
30	C₂×C₄²	M₁₆	1	2	yes	—
31	C₂×C₁₆	—	2	8	no	—
32	C₂×C₁₆	—	4	4	no	—
33	C₂×C₁₆	—	4	4	no	—
34	C₄×C₈	—	2	8	no	—
35	C₄×C₈	—	2	8	no	—
36	C₄×C₈	—	2	4	no	—
37	C₄×C₈	—	2	2	no	—
38	C₄×C₈	—	2	4	no	—
39	C₄×C₈	C₁₆	1	8	yes	complemented
40	C₄×C₈	C₂×C₈	1	4	yes	—
41	C₄×C₈	M₁₆	1	4	yes	—
42	C₄×C₈	D₈	1	1	yes	characteristic
43	C₄×C₈	SD₁₆	1	2	yes	—
44	C₄×C₈	Q₁₆	1	1	yes	characteristic
45	C₂²×C₄	#95	1	1	yes	characteristic
46	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
47	C₂²×C₄	#97	1	1	yes	characteristic
48	C₂²×C₄	C₂²⋊C₈	1	2	yes	—
49	C₂²×C₄	C₄⋊C₈	1	2	yes	—
50	C₂×C₈	—	2	8	no	—
51	C₂×C₈	—	2	8	no	—
52	C₂×C₈	—	2	2	no	—
53	C₂×C₈	—	2	2	no	—
54	C₂×C₈	—	2	8	no	—
55	C₂×C₈	C₂²⋊C₈	1	4	yes	—
56	C₂×C₈	#102	1	2	yes	—
57	C₂×C₈	#103	1	2	yes	—
58	C₂×C₈	#104	1	4	yes	—
59	C₂×C₈	C₄⋊C₈	1	4	yes	—
60	C₄²	—	2	2	no	—
61	C₄²	—	2	4	no	—
62	C₄²	C₄×C₈	1	2	yes	—
63	C₄²	#97	1	2	yes	—
64	C₄²	#106	1	1	yes	characteristic
65	C₄²	#107	1	1	yes	characteristic
66	C₄²	#108	1	2	yes	—
67	C₄²	C₂×C₁₆	1	4	yes	—
68	C₄²	M₃₂	1	4	yes	—
69	C₁₆	—	4	8	no	—
70	C₂³	#336	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	C₈²	1	1	yes	characteristic
75	C₂×C₄	#322	1	2	yes	—
76	C₂×C₄	#322	1	1	yes	characteristic
77	C₂×C₄	#334	1	2	yes	—
78	C₂×C₄	#335	1	2	yes	—
79	C₂×C₄	#336	1	2	yes	—
80	C₂×C₄	#339	1	2	yes	—
81	C₂×C₄	#340	1	1	yes	characteristic
82	C₂×C₄	#341	1	1	yes	characteristic
83	C₂×C₄	C₄×C₁₆	1	2	yes	—
84	C₂×C₄	#346	1	2	yes	—
85	C₂×C₄	C₂²⋊C₁₆	1	4	yes	—
86	C₂×C₄	C₄⋊C₁₆	1	4	yes	—
87	C₂×C₄	#364	1	4	yes	—
88	C₈	—	2	16	no	—
89	C₈	—	2	4	no	—
90	C₈	D₄⋊C₈	1	4	yes	—
91	C₈	Q₈⋊C₈	1	4	yes	—
92	C₂²	#1273	1	1	yes	characteristic
93	C₂²	#1276	1	1	yes	characteristic
94	C₂²	#1277	1	1	yes	characteristic
95	C₂²	#1374	1	2	yes	—
96	C₂²	#1376	1	2	yes	—
97	C₄	—	2	2	no	—
98	C₄	—	2	4	no	—
99	C₄	#1278	1	2	yes	—
100	C₄	#1279	1	2	yes	—
101	C₄	C₈×C₁₆	1	1	yes	characteristic
102	C₄	#1311	1	1	yes	characteristic
103	C₄	#1312	1	2	yes	—
104	C₄	#1367	1	4	yes	—
105	C₄	#1370	1	4	yes	—
106	C₂	#7015	1	1	yes	characteristic
107	C₂	#7136	1	1	yes	characteristic
108	C₂	#7138	1	1	yes	characteristic
109	C₂	#7148	1	2	yes	—
110	C₂	#7151	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₄⋊C₁₆

C₂

yes

maximal

C₄×C₈²

C₂

yes

maximal, characteristic

C₂×C₄⋊C₁₆

C₄

yes

—

C₂×C₄×C₁₆

—

C₄²×C₈

C₄

yes

—

C₄²×C₈

C₂²

yes

characteristic

C₂×C₈²

—

#1374

—

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₈

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₈

yes

complemented

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂×C₄²

C₄²

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₂×C₈

yes

—

C₂×C₄²

M₁₆

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₁₆

yes

complemented

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

D₈

yes

characteristic

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

characteristic

C₂²×C₄

#95

yes

characteristic

C₂²×C₄

C₄×C₈

yes

characteristic

C₂²×C₄

#97

yes

characteristic

C₂²×C₄

C₂²⋊C₈

yes

—

C₂²×C₄

C₄⋊C₈

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

#102

yes

—

C₂×C₈

#103

yes

—

C₂×C₈

#104

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₄²

—

C₄²

—

C₄²

C₄×C₈

yes

—

C₄²

#97

yes

—

C₄²

yes

C₄²

yes

C₄²

yes

—

C₄²

C₂×C₁₆

yes

—

C₄²

M₃₂

yes

—

C₁₆

—

C₂³

#336

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈²

yes

characteristic

C₂×C₄

#322

yes

—

C₂×C₄

#322

yes

characteristic

C₂×C₄

#334

yes

—

C₂×C₄

#335

yes

—

C₂×C₄

#336

yes

—

C₂×C₄

#339

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄×C₁₆

yes

—

C₂×C₄

#346

yes

—

C₂×C₄

C₂²⋊C₁₆

yes

—

C₂×C₄

C₄⋊C₁₆

yes

—

C₂×C₄

#364

yes

—

C₈

—

C₈

—

C₈

D₄⋊C₈

yes

—

C₈

Q₈⋊C₈

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#1376

yes

—

C₄

—

C₄

—

C₄

#1278

yes

—

100

C₄

#1279

yes

—

101

C₄

C₈×C₁₆

yes

characteristic

102

C₄

#1311

yes

characteristic

103

C₄

#1312

yes

—

104

C₄

#1367

yes

—

105

C₄

#1370

yes

—

106

C₂

#7015

yes

characteristic

107

C₂

#7136

yes

characteristic

108

C₂

#7138

yes

characteristic

109

C₂

#7148

yes

—

110

C₂

#7151

yes

—

	subgroup	G/H
center	C₂×C₄×C₈	D₄
commutator	C₄	C₈×C₁₆
Frattini, cosocle	C₄²×C₈	C₂²
socle, prime radical	C₂³	#336
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄×C₈

D₄

commutator

C₄

C₈×C₁₆

Frattini, cosocle

C₄²×C₈

C₂²

C₂³

G ▷ C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
19. C₄⋊C₁₆	C₈	semidirect
39. C₄×C₈	C₁₆	semidirect

H ≅ G/N

type

19. C₄⋊C₁₆

C₈

semidirect

39. C₄×C₈

C₁₆

semidirect

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

subgroup	index
C₄×C₄⋊C₁₆	2
C₄×C₈²	2

subgroup

index

C₄×C₄⋊C₁₆

C₄×C₈²

Subgroups — #93257

Conjugacy classes of subgroups

Complements

Maximal subgroups