Group #12653 — subgroups

Subgroups — #12653

Order 256.

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

normal subgroups only

This group has 207 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	2	yes	maximal
2	C₂×#346	C₂	1	2	yes	maximal
3	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
4	#1374	C₂	1	2	yes	maximal
5	C₂²×C₁₆	C₂²	1	4	yes	—
6	C₂×C₄×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₄³	C₄	1	1	yes	characteristic
13	C₄⋊C₁₆	C₄	1	8	yes	complemented
14	#346	—	2	4	no	—
15	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈	C₂³	1	1	yes	characteristic
19	C₂×C₄²	C₈	1	2	yes	—
20	C₂×C₄²	C₈	1	2	yes	—
21	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆	—	2	4	no	—
25	C₂×C₁₆	—	2	4	no	—
26	C₂×C₁₆	C₂×C₄	1	8	yes	—
27	C₄×C₈	—	2	2	no	—
28	C₄×C₈	—	2	2	no	—
29	C₄×C₈	—	2	2	no	—
30	C₄×C₈	C₈	1	4	yes	—
31	C₄×C₈	C₂×C₄	1	4	yes	—
32	C₄×C₈	D₄	1	2	yes	—
33	C₄×C₈	Q₈	1	2	yes	—
34	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄	C₂×C₈	1	2	yes	—
36	C₂²×C₄	C₂×C₈	1	2	yes	—
37	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	C₄²	1	2	yes	—
44	C₂×C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₈	C₂×C₈	1	2	yes	—
46	C₂×C₈	C₂×C₈	1	2	yes	—
47	C₂×C₈	M₁₆	1	2	yes	—
48	C₂×C₈	C₂²×C₄	1	2	yes	—
49	C₂×C₈	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈	#41	1	2	yes	—
52	C₄²	—	2	4	no	—
53	C₄²	—	2	4	no	—
54	C₄²	—	2	1	no	—
55	C₄²	—	2	1	no	—
56	C₄²	C₄²	1	2	yes	—
57	C₄²	C₄⋊C₄	1	2	yes	—
58	C₄²	C₂×C₈	1	2	yes	—
59	C₄²	M₁₆	1	2	yes	—
60	C₁₆	—	2	8	no	—
61	C₂³	C₂²×C₈	1	1	yes	characteristic
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	C₄×C₈	1	2	yes	—
68	C₂×C₄	C₄×C₈	1	2	yes	—
69	C₂×C₄	C₄⋊C₈	1	4	yes	—
70	C₂×C₄	M₃₂	1	4	yes	—
71	C₂×C₄	M₃₂	1	4	yes	—
72	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄	#117	1	1	yes	characteristic
75	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
78	C₈	—	2	2	no	—
79	C₈	—	2	2	no	—
80	C₈	C₄⋊C₈	1	4	yes	—
81	C₈	C₄×D₄	1	2	yes	—
82	C₈	C₄×Q₈	1	2	yes	—
83	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
84	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
85	C₂²	#431	1	1	yes	characteristic
86	C₂²	C₂×M₃₂	1	2	yes	—
87	C₂²	C₂×M₃₂	1	2	yes	—
88	C₄	—	2	2	no	—
89	C₄	—	2	4	no	—
90	C₄	—	2	4	no	—
91	C₄	#346	1	4	yes	complemented
92	C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
93	C₄	C₄×M₁₆	1	1	yes	characteristic
94	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₄	#431	1	1	yes	characteristic
96	C₂	C₄×C₄⋊C₈	1	1	yes	characteristic
97	C₂	C₂×#346	1	1	yes	characteristic
98	C₂	C₄×M₃₂	1	1	yes	characteristic
99	C₂	#2150	1	2	yes	—
100	C₂	#2163	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₁₆

C₂

yes

maximal

C₂×#346

C₂

yes

maximal

C₄²×C₈

C₂

yes

maximal, characteristic

#1374

C₂

yes

maximal

C₂²×C₁₆

C₂²

yes

—

C₂×C₄×C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄³

C₄

yes

characteristic

C₄⋊C₁₆

C₄

yes

complemented

#346

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂×C₄²

C₈

yes

—

C₂×C₄²

C₈

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₈

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

Q₈

yes

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

—

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

C₄⋊C₄

yes

—

C₂×C₈

yes

—

C₂×C₈

yes

—

C₂×C₈

M₁₆

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

characteristic

C₂×C₈

#41

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

yes

—

C₄²

C₄⋊C₄

yes

—

C₄²

C₂×C₈

yes

—

C₄²

M₁₆

yes

—

C₁₆

—

C₂³

C₂²×C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×C₈

yes

—

C₂×C₄

C₄×C₈

yes

—

C₂×C₄

C₄⋊C₈

yes

—

C₂×C₄

M₃₂

yes

—

C₂×C₄

M₃₂

yes

—

C₂×C₄

C₂×C₄²

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

#117

yes

characteristic

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

characteristic

C₈

—

C₈

—

C₈

C₄⋊C₈

yes

—

C₈

C₄×D₄

yes

—

C₈

C₄×Q₈

yes

—

C₂²

C₂×C₄×C₈

yes

characteristic

C₂²

C₂×C₄⋊C₈

yes

characteristic

C₂²

#431

yes

characteristic

C₂²

C₂×M₃₂

yes

—

C₂²

C₂×M₃₂

yes

—

C₄

—

C₄

—

C₄

—

C₄

#346

yes

complemented

C₄

C₄×C₄⋊C₄

yes

characteristic

C₄

C₄×M₁₆

yes

characteristic

C₄

C₂×C₄⋊C₈

yes

characteristic

C₄

#431

yes

characteristic

C₂

C₄×C₄⋊C₈

yes

characteristic

C₂

C₂×#346

yes

characteristic

C₂

C₄×M₃₂

yes

characteristic

C₂

#2150

yes

—

100

C₂

#2163

yes

—

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

subgroup

G/H

center, Frattini, cosocle

C₂²×C₈

C₂³

commutator

C₂²

C₂×C₄×C₈

socle, prime radical

C₂³

C₂²×C₈

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
13. C₄⋊C₁₆	C₄	semidirect
91. C₄	#346	semidirect

H ≅ G/N

type

13. C₄⋊C₁₆

C₄

semidirect

91. C₄

#346

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×#346	2
C₄²×C₈	2
#1374	2

subgroup

index

C₂×C₄⋊C₁₆

C₂×#346

C₄²×C₈

#1374

Subgroups — #12653

Conjugacy classes of subgroups

Complements

Maximal subgroups