Group #12661 — subgroups

Subgroups — C₁₆×C₄⋊C₄ #12661

Order 256.

199 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 143 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×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₆	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
5	#1374	C₂	1	2	yes	maximal
6	C₂²×C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₆	C₂²	1	2	yes	—
8	C₂²×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄	C₄	1	1	yes	characteristic
13	C₄⋊C₁₆	C₄	1	4	yes	complemented
14	C₄×C₁₆	—	2	4	no	—
15	C₄×C₁₆	C₄	1	4	yes	complemented
16	#336	C₄	1	2	yes	—
17	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈	C₂×C₄	1	2	yes	—
19	C₂²×C₈	C₂³	1	1	yes	characteristic
20	C₂×C₄⋊C₄	C₈	1	2	yes	—
21	C₂×C₄²	C₂×C₄	1	2	yes	—
22	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆	—	2	2	no	—
24	C₂×C₁₆	—	2	1	no	—
25	C₂×C₁₆	—	2	1	no	—
26	C₂×C₁₆	—	2	2	no	—
27	C₂×C₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₁₆	C₂×C₄	1	4	yes	—
29	C₂×C₁₆	C₂×C₄	1	2	yes	—
30	C₂×C₁₆	D₄	1	2	yes	—
31	C₂×C₁₆	Q₈	1	2	yes	—
32	C₄⋊C₈	C₈	1	4	yes	—
33	C₄×C₈	—	2	2	no	—
34	C₄×C₈	—	2	2	no	—
35	C₄×C₈	C₂×C₄	1	2	yes	—
36	C₄×C₈	C₂×C₄	1	2	yes	—
37	#95	C₈	1	2	yes	—
38	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄	C₂×C₈	1	4	yes	—
40	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈	—	2	2	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	2	2	no	—
46	C₂×C₈	C₄²	1	2	yes	—
47	C₂×C₈	C₂×C₈	1	4	yes	—
48	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈	#41	1	1	yes	characteristic
53	C₂×C₈	#41	1	1	yes	characteristic
54	C₄⋊C₄	C₁₆	1	8	yes	direct product factor
55	C₄²	—	2	2	no	—
56	C₄²	—	2	2	no	—
57	C₄²	C₄²	1	2	yes	—
58	C₄²	C₂×C₈	1	2	yes	—
59	C₁₆	—	2	2	no	—
60	C₁₆	—	2	4	no	—
61	C₁₆	C₄⋊C₄	1	4	yes	direct product factor
62	C₂³	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄	—	2	4	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	C₄×C₈	1	2	yes	—
67	C₂×C₄	C₄×C₈	1	2	yes	—
68	C₂×C₄	C₂×C₁₆	1	8	yes	—
69	C₂×C₄	C₂×C₁₆	1	4	yes	—
70	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
71	C₂×C₄	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄	#131	1	1	yes	characteristic
75	C₂×C₄	#131	1	1	yes	characteristic
76	C₈	—	2	2	no	—
77	C₈	—	2	4	no	—
78	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₈	#117	1	1	yes	characteristic
80	C₈	C₄×D₄	1	1	yes	characteristic
81	C₈	C₄×Q₈	1	1	yes	characteristic
82	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂²	C₈×D₄	1	1	yes	characteristic
84	C₂²	C₈×Q₈	1	1	yes	characteristic
85	C₂²	C₂²×C₁₆	1	2	yes	—
86	C₂²	#504	1	2	yes	—
87	C₄	—	2	2	no	—
88	C₄	—	2	8	no	—
89	C₄	C₄×C₁₆	1	4	yes	complemented
90	C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
91	C₄	#405	1	1	yes	characteristic
92	C₄	C₈×D₄	1	1	yes	characteristic
93	C₄	C₈×Q₈	1	1	yes	characteristic
94	C₂	C₈×C₄⋊C₄	1	1	yes	characteristic
95	C₂	C₂×C₄×C₁₆	1	1	yes	characteristic
96	C₂	#2108	1	1	yes	characteristic
97	C₂	D₄×C₁₆	1	2	yes	—
98	C₂	Q₈×C₁₆	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×C₄×C₁₆

C₂

yes

maximal

C₂×C₄×C₁₆

C₂

yes

maximal, characteristic

C₈×C₄⋊C₄

C₂

yes

maximal, characteristic

#1374

C₂

yes

maximal

C₂²×C₁₆

C₂²

yes

characteristic

C₂²×C₁₆

C₂²

yes

—

C₂²×C₁₆

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₄

yes

characteristic

C₄⋊C₁₆

C₄

yes

complemented

C₄×C₁₆

—

C₄×C₁₆

C₄

yes

complemented

#336

C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂×C₄⋊C₄

C₈

yes

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄

yes

—

C₂×C₁₆

C₂×C₄

yes

—

C₂×C₁₆

C₂×C₄

yes

—

C₂×C₁₆

D₄

yes

—

C₂×C₁₆

Q₈

yes

—

C₄⋊C₈

C₈

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

#95

C₈

yes

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²

yes

—

C₂×C₈

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

C₁₆

yes

direct product factor

C₄²

—

C₄²

—

C₄²

yes

—

C₄²

C₂×C₈

yes

—

C₁₆

—

C₁₆

—

C₁₆

C₄⋊C₄

yes

direct product factor

C₂³

C₂²×C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×C₈

yes

—

C₂×C₄

C₄×C₈

yes

—

C₂×C₄

C₂×C₁₆

yes

—

C₂×C₄

C₂×C₁₆

yes

—

C₂×C₄

C₂×C₄²

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×Q₈

yes

characteristic

C₂×C₄

C₂²×C₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

C₂×C₄⋊C₄

yes

characteristic

C₈

#117

yes

characteristic

C₈

C₄×D₄

yes

characteristic

C₈

C₄×Q₈

yes

characteristic

C₂²

C₂×C₄×C₈

yes

characteristic

C₂²

C₈×D₄

yes

characteristic

C₂²

C₈×Q₈

yes

characteristic

C₂²

C₂²×C₁₆

yes

—

C₂²

#504

yes

—

C₄

—

C₄

—

C₄

C₄×C₁₆

yes

complemented

C₄

C₄×C₄⋊C₄

yes

characteristic

C₄

#405

yes

characteristic

C₄

C₈×D₄

yes

characteristic

C₄

C₈×Q₈

yes

characteristic

C₂

C₈×C₄⋊C₄

yes

characteristic

C₂

C₂×C₄×C₁₆

yes

characteristic

C₂

#2108

yes

characteristic

C₂

D₄×C₁₆

yes

—

C₂

Q₈×C₁₆

yes

—

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

subgroup

G/H

center

C₂²×C₁₆

C₂²

commutator

C₂

C₂×C₄×C₁₆

Frattini, cosocle

C₂²×C₈

C₂³

socle, prime radical

C₂³

C₂²×C₈

LCS

G ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₆ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
13. C₄⋊C₁₆	C₄	semidirect
15. C₄×C₁₆	C₄	semidirect
54. C₄⋊C₄	C₁₆	direct (N × H)
61. C₁₆	C₄⋊C₄	direct (N × H)
89. C₄	C₄×C₁₆	semidirect

H ≅ G/N

type

13. C₄⋊C₁₆

C₄

semidirect

15. C₄×C₁₆

C₄

semidirect

54. C₄⋊C₄

C₁₆

direct (N × H)

61. C₁₆

C₄⋊C₄

direct (N × H)

89. C₄

C₄×C₁₆

semidirect

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

subgroup	index
C₂×C₄⋊C₁₆	2
C₂×C₄×C₁₆	2
C₈×C₄⋊C₄	2
#1374	2

subgroup

index

C₂×C₄⋊C₁₆

C₂×C₄×C₁₆

C₈×C₄⋊C₄

#1374

Subgroups — C₁₆×C₄⋊C₄ #12661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C16×C4⋊C4 #12661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×C₄⋊C₄ #12661