Group #3387 — subgroups

Subgroups — #3387

Order 128.

163 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 163 conjugacy classes of subgroups — too many to draw. Only its 96 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	Q₈²	C₂	1	1	yes	maximal, characteristic
2	#557	C₂	1	1	yes	maximal, characteristic
3	#501	C₂	1	2	yes	maximal
4	#498	C₂	1	1	yes	maximal, characteristic
5	#479	C₂	1	2	yes	maximal
6	#477	C₂	1	1	yes	maximal, characteristic
7	#475	C₂	1	2	yes	maximal
8	#475	C₂	1	1	yes	maximal, characteristic
9	#446	C₂	1	1	yes	maximal, characteristic
10	#441	C₂	1	2	yes	maximal
11	C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₁₆	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈	—	2	1	no	—
14	C₄⋊Q₈	—	2	1	no	—
15	C₄⋊Q₈	—	2	2	no	—
16	C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈	C₂²	1	2	yes	—
18	C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	#125	—	2	2	no	—
20	#125	C₂²	1	2	yes	—
21	C₄×Q₈	—	2	1	no	—
22	C₄×Q₈	—	2	1	no	—
23	C₄×Q₈	C₂²	1	1	yes	characteristic
24	C₄×Q₈	C₂²	1	2	yes	—
25	C₄×Q₈	C₂²	1	1	yes	characteristic
26	C₄×Q₈	C₂²	1	2	yes	—
27	C₄×Q₈	C₂²	1	1	yes	characteristic
28	#107	C₂²	1	1	yes	characteristic
29	#107	C₂²	1	2	yes	—
30	#106	C₂²	1	2	yes	—
31	#106	C₂²	1	2	yes	—
32	#106	C₂²	1	2	yes	—
33	C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₄⋊C₈	C₂²	1	2	yes	—
35	#103	C₂²	1	1	yes	characteristic
36	#103	C₂²	1	1	yes	characteristic
37	#103	C₂²	1	2	yes	—
38	#103	C₂²	1	2	yes	—
39	#97	C₂²	1	2	yes	—
40	C₄×C₈	C₂²	1	1	yes	characteristic
41	C₂×Q₈	—	2	1	no	—
42	C₂×Q₈	—	2	2	no	—
43	C₂×Q₈	—	2	1	no	—
44	C₂×Q₈	D₄	1	1	yes	characteristic
45	C₂×Q₈	C₂³	1	1	yes	characteristic
46	C₂×Q₈	C₂³	1	1	yes	characteristic
47	Q₁₆	Q₈	1	4	yes	complemented
48	C₂×C₈	C₂³	1	1	yes	characteristic
49	C₂×C₈	C₂³	1	2	yes	—
50	C₂×C₈	C₂³	1	1	yes	characteristic
51	C₄⋊C₄	—	2	2	no	—
52	C₄⋊C₄	—	2	2	no	—
53	C₄⋊C₄	—	2	1	no	—
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	2	2	no	—
56	C₄⋊C₄	—	2	2	no	—
57	C₄⋊C₄	—	2	1	no	—
58	C₄⋊C₄	—	2	1	no	—
59	C₄⋊C₄	—	2	2	no	—
60	C₄⋊C₄	D₄	1	1	yes	characteristic
61	C₄⋊C₄	D₄	1	2	yes	—
62	C₄⋊C₄	C₂³	1	2	yes	—
63	C₄⋊C₄	C₂³	1	1	yes	characteristic
64	C₄⋊C₄	C₂³	1	2	yes	—
65	C₄⋊C₄	C₂³	1	1	yes	characteristic
66	C₄²	—	2	2	no	—
67	C₄²	—	2	1	no	—
68	C₄²	—	2	2	no	—
69	C₄²	—	2	1	no	—
70	C₄²	C₂³	1	1	yes	characteristic
71	C₄²	C₂³	1	2	yes	—
72	Q₈	—	2	2	no	—
73	Q₈	—	2	1	no	—
74	Q₈	—	2	1	no	—
75	Q₈	—	4	2	no	—
76	Q₈	—	8	2	no	—
77	Q₈	—	8	1	no	—
78	Q₈	C₂×Q₈	1	2	yes	—
79	Q₈	C₂×Q₈	1	2	yes	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	C₂×D₄	1	2	yes	—
87	C₂×C₄	C₂×D₄	1	2	yes	—
88	C₂×C₄	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄	C₂⁴	1	1	yes	characteristic
91	C₈	—	2	1	no	—
92	C₈	—	2	2	no	—
93	C₈	C₂×Q₈	1	2	yes	—
94	C₂²	C₂²×D₄	1	1	yes	characteristic
95	C₄	—	2	2	no	—
96	C₄	—	2	2	no	—
97	C₄	—	2	2	no	—
98	C₄	—	2	2	no	—
99	C₄	—	2	1	no	—
100	C₄	—	4	2	no	—
101	C₄	—	4	1	no	—
102	C₄	—	4	2	no	—
103	C₄	—	4	1	no	—
104	C₄	#137	1	2	yes	—
105	C₄	C₂²×Q₈	1	1	yes	characteristic
106	C₄	2₋¹⁺⁴	1	1	yes	characteristic
107	C₂	D₄×Q₈	1	1	yes	characteristic
108	C₂	C₂×#137	1	1	yes	characteristic
109	C₂	#577	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈²

C₂

yes

maximal, characteristic

#557

C₂

yes

maximal, characteristic

#501

C₂

yes

maximal

#498

C₂

yes

maximal, characteristic

#479

C₂

yes

maximal

#477

C₂

yes

maximal, characteristic

#475

C₂

yes

maximal

#475

C₂

yes

maximal, characteristic

#446

C₂

yes

maximal, characteristic

#441

C₂

yes

maximal

C₄×Q₁₆

C₂

yes

maximal, characteristic

C₂×Q₁₆

C₂²

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²

yes

characteristic

C₄⋊Q₈

C₂²

yes

—

C₄⋊Q₈

C₂²

yes

characteristic

#125

—

#125

C₂²

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂²

yes

characteristic

C₄×Q₈

C₂²

yes

—

C₄×Q₈

C₂²

yes

characteristic

C₄×Q₈

C₂²

yes

—

C₄×Q₈

C₂²

yes

characteristic

#107

C₂²

yes

characteristic

#107

C₂²

yes

—

#106

C₂²

yes

—

#106

C₂²

yes

—

#106

C₂²

yes

—

C₄⋊C₈

C₂²

yes

characteristic

C₄⋊C₈

C₂²

yes

—

#103

C₂²

yes

characteristic

#103

C₂²

yes

characteristic

#103

C₂²

yes

—

#103

C₂²

yes

—

#97

C₂²

yes

—

C₄×C₈

C₂²

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₄

yes

characteristic

C₂×Q₈

C₂³

yes

characteristic

C₂×Q₈

C₂³

yes

characteristic

Q₁₆

Q₈

yes

complemented

C₂×C₈

C₂³

yes

characteristic

C₂×C₈

C₂³

yes

—

C₂×C₈

C₂³

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄

yes

characteristic

C₄⋊C₄

D₄

yes

—

C₄⋊C₄

C₂³

yes

—

C₄⋊C₄

C₂³

yes

characteristic

C₄⋊C₄

C₂³

yes

—

C₄⋊C₄

C₂³

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂³

yes

characteristic

C₄²

C₂³

yes

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

C₂×Q₈

yes

—

Q₈

C₂×Q₈

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

C₂×D₄

yes

—

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂×D₄

yes

characteristic

C₂×C₄

C₂⁴

yes

characteristic

C₈

—

C₈

—

C₈

C₂×Q₈

yes

—

C₂²

C₂²×D₄

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

#137

yes

—

105

C₄

C₂²×Q₈

yes

characteristic

106

C₄

2₋¹⁺⁴

yes

characteristic

107

C₂

D₄×Q₈

yes

characteristic

108

C₂

C₂×#137

yes

characteristic

109

C₂

#577

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂²

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ G

LECS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
47. Q₁₆	Q₈	semidirect

H ≅ G/N

type

47. Q₁₆

Q₈

semidirect

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

subgroup	index
Q₈²	2
#557	2
#501	2
#498	2
#479	2
#477	2
#475	2
#446	2
#441	2
C₄×Q₁₆	2

subgroup

index

Q₈²

C₄×Q₁₆

Subgroups — #3387

Conjugacy classes of subgroups

Complements

Maximal subgroups