Group #12655 — subgroups

Subgroups — C₄×#364 #12655

Order 256.

199 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 127 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×#364	C₂	1	2	yes	maximal
2	C₄×M₃₂	C₂	1	2	yes	maximal
3	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
4	#1376	C₂	1	2	yes	maximal
5	C₂×M₃₂	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	#364	C₄	1	8	yes	direct product factor
14	#346	—	2	2	no	—
15	C₄×C₁₆	—	2	2	no	—
16	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈	C₂³	1	1	yes	characteristic
20	C₂×C₄²	—	2	2	no	—
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	M₃₂	—	2	4	no	—
25	M₃₂	C₂×C₄	1	8	yes	—
26	C₂×C₁₆	—	2	4	no	—
27	C₄×C₈	—	2	2	no	—
28	C₄×C₈	—	2	2	no	—
29	C₄×C₈	C₈	1	4	yes	—
30	C₄×C₈	C₈	1	4	yes	—
31	C₄×C₈	C₂×C₄	1	4	yes	—
32	C₄×C₈	D₄	1	1	yes	characteristic
33	C₄×C₈	D₄	1	1	yes	characteristic
34	C₄×C₈	Q₈	1	1	yes	characteristic
35	C₄×C₈	Q₈	1	1	yes	characteristic
36	C₂²×C₄	—	2	1	no	—
37	C₂²×C₄	—	2	1	no	—
38	C₂²×C₄	M₁₆	1	2	yes	—
39	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈	—	2	1	no	—
41	C₂×C₈	—	2	1	no	—
42	C₂×C₈	C₄²	1	2	yes	—
43	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈	C₂×C₈	1	2	yes	—
46	C₂×C₈	C₂×C₈	1	2	yes	—
47	C₂×C₈	C₂×C₈	1	2	yes	—
48	C₂×C₈	C₂×C₈	1	2	yes	—
49	C₂×C₈	M₁₆	1	2	yes	—
50	C₂×C₈	C₂²×C₄	1	2	yes	—
51	C₂×C₈	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈	#41	1	2	yes	—
54	C₄²	—	2	8	no	—
55	C₄²	—	2	1	no	—
56	C₄²	—	2	1	no	—
57	C₄²	C₄²	1	2	yes	—
58	C₄²	C₄⋊C₄	1	1	yes	characteristic
59	C₄²	C₄⋊C₄	1	1	yes	characteristic
60	C₄²	C₂×C₈	1	2	yes	—
61	C₄²	C₂×C₈	1	2	yes	—
62	C₁₆	—	2	8	no	—
63	C₂³	C₂×M₁₆	1	1	yes	characteristic
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	4	no	—
69	C₂×C₄	C₄⋊C₈	1	2	yes	—
70	C₂×C₄	C₄⋊C₈	1	2	yes	—
71	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄	#117	1	1	yes	characteristic
74	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
77	C₈	—	2	2	no	—
78	C₈	C₄×C₈	1	4	yes	—
79	C₈	C₄⋊C₈	1	2	yes	—
80	C₈	C₄⋊C₈	1	2	yes	—
81	C₈	C₄×D₄	1	2	yes	—
82	C₈	C₄×Q₈	1	2	yes	—
83	C₂²	—	2	2	no	—
84	C₂²	C₄×M₁₆	1	1	yes	characteristic
85	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
86	C₂²	#431	1	1	yes	characteristic
87	C₄	—	2	4	no	—
88	C₄	—	2	4	no	—
89	C₄	—	2	2	no	—
90	C₄	#364	1	4	yes	direct product factor
91	C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
92	C₄	C₂×C₄×C₈	1	1	yes	characteristic
93	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
94	C₄	#431	1	1	yes	characteristic
95	C₂	—	2	1	no	—
96	C₂	—	2	1	no	—
97	C₂	C₄×C₄⋊C₈	1	1	yes	characteristic
98	C₂	C₂×#364	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#364

C₂

yes

maximal

C₄×M₃₂

C₂

yes

maximal

C₄²×C₈

C₂

yes

maximal, characteristic

#1376

C₂

yes

maximal

C₂×M₃₂

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

#364

C₄

yes

direct product factor

#346

—

C₄×C₁₆

—

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

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

M₃₂

—

M₃₂

C₂×C₄

yes

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₈

yes

—

C₄×C₈

C₈

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

Q₈

yes

characteristic

C₄×C₈

Q₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

M₁₆

yes

—

C₂²×C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²

yes

—

C₂×C₈

C₄⋊C₄

yes

characteristic

C₂×C₈

C₄⋊C₄

yes

characteristic

C₂×C₈

yes

—

C₂×C₈

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

yes

—

C₄²

C₄⋊C₄

yes

characteristic

C₄²

C₄⋊C₄

yes

characteristic

C₄²

C₂×C₈

yes

—

C₄²

C₂×C₈

yes

—

C₁₆

—

C₂³

C₂×M₁₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊C₈

yes

—

C₂×C₄

C₄⋊C₈

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

C₂²×C₈

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

characteristic

C₈

—

C₈

C₄×C₈

yes

—

C₈

C₄⋊C₈

yes

—

C₈

C₄⋊C₈

yes

—

C₈

C₄×D₄

yes

—

C₈

C₄×Q₈

yes

—

C₂²

—

C₂²

C₄×M₁₆

yes

characteristic

C₂²

C₂×C₄⋊C₈

yes

characteristic

C₂²

#431

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#364

yes

direct product factor

C₄

C₄×C₄⋊C₄

yes

characteristic

C₄

C₂×C₄×C₈

yes

characteristic

C₄

C₂×C₄⋊C₈

yes

characteristic

C₄

#431

yes

characteristic

C₂

—

C₂

—

C₂

C₄×C₄⋊C₈

yes

characteristic

C₂

C₂×#364

yes

—

	subgroup	G/H
center	C₄×C₈	D₄
commutator	C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂²	C₂×C₄⋊C₈
prime radical	C₂³	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄×C₈

D₄

commutator

C₄

C₂×C₄×C₈

Frattini, cosocle

C₂²×C₈

C₂³

socle

C₂²

C₂×C₄⋊C₈

prime radical

C₂³

C₂×M₁₆

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
13. #364	C₄	direct (N × H)
90. C₄	#364	direct (N × H)

H ≅ G/N

type

13. #364

C₄

direct (N × H)

90. C₄

#364

direct (N × H)

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

subgroup	index
C₂×#364	2
C₄×M₃₂	2
C₄²×C₈	2
#1376	2

subgroup

index

C₂×#364

C₄×M₃₂

C₄²×C₈

#1376

Subgroups — C₄×#364 #12655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#364 #12655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#364 #12655