Group #1627 — subgroups

Subgroups — #1627

Order 128.

125 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 125 conjugacy classes of subgroups — too many to draw. Only its 38 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	D₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#486	C₂	1	1	yes	maximal, characteristic, complemented
3	#460	C₂	1	1	yes	maximal, characteristic, complemented
4	#459	C₂	1	1	yes	maximal, characteristic
5	#332	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈	—	2	1	no	—
9	C₂×SD₁₆	—	2	1	no	—
10	C₂×D₈	—	2	1	no	—
11	C₄⋊Q₈	—	2	1	no	—
12	C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	#127	C₂²	1	1	yes	characteristic
14	C₂²⋊Q₈	—	2	1	no	—
15	C₂²⋊Q₈	—	2	2	no	—
16	C₄×Q₈	C₂²	1	1	yes	characteristic
17	C₄×D₄	—	2	1	no	—
18	C₄×D₄	C₂²	1	1	yes	characteristic
19	C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊C₈	C₂²	1	1	yes	characteristic
21	#102	—	2	1	no	—
22	#102	—	2	1	no	—
23	#102	—	2	2	no	—
24	C₄×C₈	C₂²	1	1	yes	characteristic
25	C₂×Q₈	—	2	2	no	—
26	C₂×Q₈	—	2	1	no	—
27	C₂×Q₈	—	4	2	no	—
28	C₂×Q₈	—	4	2	no	—
29	C₂×Q₈	D₄	1	1	yes	characteristic
30	C₂×D₄	—	2	1	no	—
31	C₂×D₄	—	2	1	no	—
32	C₂×D₄	—	2	1	no	—
33	C₂×D₄	D₄	1	1	yes	characteristic
34	C₂²×C₄	—	2	2	no	—
35	C₂²×C₄	—	2	1	no	—
36	SD₁₆	—	4	2	no	—
37	D₈	—	4	2	no	—
38	C₂×C₈	—	2	1	no	—
39	C₂×C₈	—	2	1	no	—
40	C₂×C₈	—	2	1	no	—
41	C₄⋊C₄	—	2	2	no	—
42	C₄⋊C₄	—	2	1	no	—
43	C₄⋊C₄	—	2	1	no	—
44	C₄⋊C₄	D₄	1	1	yes	characteristic
45	C₄⋊C₄	D₄	1	1	yes	characteristic
46	C₄⋊C₄	D₄	1	2	yes	—
47	C₂²⋊C₄	—	2	2	no	—
48	C₂²⋊C₄	—	2	1	no	—
49	C₄²	—	2	1	no	—
50	C₄²	C₂³	1	1	yes	characteristic
51	C₂³	—	2	1	no	—
52	C₂³	—	4	1	no	—
53	Q₈	—	2	1	no	—
54	Q₈	—	4	4	no	—
55	Q₈	—	4	1	no	—
56	Q₈	—	4	1	no	—
57	Q₈	D₈	1	2	yes	complemented
58	D₄	—	2	1	no	—
59	D₄	—	4	2	no	—
60	D₄	—	4	1	no	—
61	D₄	—	4	1	no	—
62	D₄	—	4	1	no	—
63	D₄	—	4	1	no	—
64	D₄	SD₁₆	1	2	yes	complemented
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	4	4	no	—
70	C₂×C₄	—	4	1	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄	C₂×D₄	1	1	yes	characteristic
75	C₈	—	4	1	no	—
76	C₈	—	4	1	no	—
77	C₈	—	4	1	no	—
78	C₂²	—	2	2	no	—
79	C₂²	—	4	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	—	8	1	no	—
82	C₂²	—	8	1	no	—
83	C₂²	—	8	1	no	—
84	C₂²	#120	1	1	yes	characteristic
85	C₄	—	2	2	no	—
86	C₄	—	2	1	no	—
87	C₄	—	4	2	no	—
88	C₄	—	4	1	no	—
89	C₄	—	4	1	no	—
90	C₄	C₂×D₈	1	1	yes	characteristic
91	C₄	C₂×SD₁₆	1	1	yes	characteristic
92	C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
93	C₄	#137	1	1	yes	characteristic
94	C₂	—	4	2	no	—
95	C₂	—	16	1	no	—
96	C₂	#447	1	1	yes	characteristic
97	C₂	#448	1	1	yes	characteristic
98	C₂	#454	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×Q₈

C₂

yes

maximal, characteristic, complemented

#486

C₂

yes

maximal, characteristic, complemented

#460

C₂

yes

maximal, characteristic, complemented

#459

C₂

yes

maximal, characteristic

#332

C₂

yes

maximal, characteristic, complemented

Q₈⋊C₈

C₂

yes

maximal, characteristic, complemented

D₄⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²

yes

characteristic

#127

C₂²

yes

characteristic

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

C₂²

yes

characteristic

C₄×D₄

—

C₄×D₄

C₂²

yes

characteristic

C₄⋊C₈

C₂²

yes

characteristic

C₄⋊C₈

C₂²

yes

—

—

—

C₄×C₈

C₂²

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₄

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

SD₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄

yes

characteristic

C₄⋊C₄

D₄

yes

characteristic

C₄⋊C₄

D₄

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂³

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

D₈

yes

complemented

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

SD₁₆

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂×D₄

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#120

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

C₂×D₈

yes

characteristic

C₄

C₂×SD₁₆

yes

C₄

yes

C₄

yes

C₂

—

C₂

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center, socle	C₂²	#120
commutator, Frattini, cosocle	C₄²	C₂³
prime radical	#459	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#120

commutator, Frattini, cosocle

C₄²

C₂³

C₂

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂² ◁ C₄² ◁ G

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
1. D₄×Q₈	C₂	semidirect
2. #486	C₂	semidirect
3. #460	C₂	semidirect
5. #332	C₂	semidirect
6. Q₈⋊C₈	C₂	semidirect
7. D₄⋊C₈	C₂	semidirect
57. Q₈	D₈	semidirect
64. D₄	SD₁₆	semidirect

H ≅ G/N

type

1. D₄×Q₈

C₂

semidirect

2. #486

C₂

semidirect

3. #460

C₂

semidirect

5. #332

C₂

semidirect

6. Q₈⋊C₈

C₂

semidirect

7. D₄⋊C₈

C₂

semidirect

57. Q₈

D₈

semidirect

64. D₄

SD₁₆

semidirect

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

subgroup	index
D₄×Q₈	2
#486	2
#460	2
#459	2
#332	2
Q₈⋊C₈	2
D₄⋊C₈	2

subgroup

index

D₄×Q₈

Q₈⋊C₈

D₄⋊C₈

Subgroups — #1627

Conjugacy classes of subgroups

Complements

Maximal subgroups