Group #13136731 — subgroups

Subgroups — #13136731

Order 1728.

195 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 195 conjugacy classes of subgroups — too many to draw. Only its 39 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	#11722448	C₂	1	2	yes	maximal
2	C₂×#88627	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#272	C₂²	1	1	yes	characteristic
4	C₂²⋊#1088	—	2	2	no	—
5	C₂×#6188	C₄	1	2	yes	complemented
6	C₄⋊#1088	—	2	2	no	—
7	C₄×#1088	C₄	1	4	yes	complemented
8	#88627	—	2	2	no	—
9	C₂²×#272	D₄	1	1	yes	characteristic
10	C₂×#1088	—	2	4	no	—
11	C₂×C₄×H(3,𝔽₃)	Q₈	1	1	yes	characteristic
12	C₂×#1084	C₂×C₄	1	1	yes	characteristic
13	C₄×#272	C₂×C₄	1	2	yes	—
14	C₄×#272	D₄	1	2	yes	—
15	#6188	—	2	2	no	—
16	C₃×H(3,ℤ/4)	—	9	1	no	maximal
17	C₂×C₃×C₄×D₃	—	12	1	no	—
18	C₂²×H(3,𝔽₃)	C₄⋊C₄	1	1	yes	characteristic
19	C₂×#272	—	4	1	no	—
20	C₂×#272	C₂²⋊C₄	1	2	yes	—
21	#1088	—	4	4	no	—
22	C₄×H(3,𝔽₃)	C₄⋊C₄	1	2	yes	—
23	#1084	C₄²	1	1	yes	characteristic
24	#1084	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₃×M₁₆	—	9	1	no	—
26	C₃×#117	—	9	2	no	—
27	C₂²×C₃×D₃	—	12	1	no	—
28	C₂×C₃²×C₄	—	4	1	no	—
29	C₂×C₃×Dic₃	—	12	1	no	—
30	C₃×C₄×D₃	—	12	2	no	—
31	C₃×C₄×D₃	—	12	2	no	—
32	H(3,ℤ/4)	—	9	1	no	Sylow
33	C₂×H(3,𝔽₃)	—	2	1	no	—
34	C₂×H(3,𝔽₃)	#95	1	1	yes	characteristic
35	#272	—	2	2	no	—
36	C₂²×C₃×C₄	—	9	1	no	—
37	C₂×C₄×D₃	—	12	1	no	—
38	C₃×M₁₆	—	18	2	no	—
39	C₂×C₃×C₈	—	9	2	no	—
40	C₃×C₄⋊C₄	—	18	2	no	—
41	C₃×C₂²⋊C₄	—	18	2	no	—
42	C₃×C₄²	—	9	4	no	—
43	C₂²×C₃²	—	4	1	no	—
44	C₂×C₃×D₃	—	12	2	no	—
45	C₂×C₃×D₃	—	12	4	no	—
46	C₃²×C₄	—	4	2	no	—
47	C₃×Dic₃	—	12	1	no	—
48	C₃×Dic₃	—	12	1	no	—
49	C₂×M₁₆	—	9	1	no	—
50	#117	—	9	2	no	—
51	H(3,𝔽₃)	H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
52	C₂³×C₃	—	9	1	no	—
53	C₂²×D₃	—	12	1	no	—
54	C₂×C₃×C₄	—	9	1	no	—
55	C₂×C₃×C₄	—	9	2	no	—
56	C₂×C₃×C₄	—	9	2	no	—
57	C₂×C₃×C₄	—	12	1	no	—
58	C₂×C₃×C₄	—	18	4	no	—
59	C₂×C₃×C₄	PSU(3,𝔽₂)	1	1	yes	characteristic
60	C₂×Dic₃	—	12	1	no	—
61	C₄×D₃	—	12	2	no	—
62	C₄×D₃	—	12	2	no	—
63	C₃×C₈	—	18	2	no	—
64	C₂×C₃²	—	4	2	no	—
65	C₂×C₃²	—	4	1	no	—
66	C₃×D₃	—	12	4	no	—
67	C₂²×C₄	—	9	1	no	—
68	M₁₆	—	18	2	no	—
69	C₂×C₈	—	9	2	no	—
70	C₄⋊C₄	—	18	2	no	—
71	C₂²⋊C₄	—	18	2	no	—
72	C₄²	—	9	4	no	—
73	C₂²×C₃	—	9	2	no	—
74	C₂²×C₃	—	12	1	no	—
75	C₂²×C₃	—	36	1	no	—
76	C₂²×C₃	#3777	1	1	yes	characteristic
77	C₂×D₃	—	12	2	no	—
78	C₂×D₃	—	12	4	no	—
79	C₃×C₄	—	9	1	no	—
80	C₃×C₄	—	9	1	no	—
81	C₃×C₄	—	12	2	no	—
82	C₃×C₄	—	36	4	no	—
83	C₃×C₄	#3777	1	2	yes	—
84	Dic₃	—	12	1	no	—
85	Dic₃	—	12	1	no	—
86	C₃²	—	4	1	no	—
87	C₂³	—	9	1	no	—
88	C₂×C₄	—	9	1	no	—
89	C₂×C₄	—	9	2	no	—
90	C₂×C₄	—	9	2	no	—
91	C₂×C₄	—	18	4	no	—
92	C₂×C₄	SU(3,𝔽₂)	1	1	yes	characteristic
93	C₈	—	18	2	no	—
94	C₂×C₃	—	2	1	no	—
95	C₂×C₃	—	12	2	no	—
96	C₂×C₃	—	12	1	no	—
97	C₂×C₃	—	18	2	no	—
98	C₂×C₃	#63747	1	1	yes	characteristic
99	D₃	—	12	4	no	—
100	C₂²	—	9	2	no	—
101	C₂²	—	36	1	no	—
102	C₂²	#88592	1	1	yes	characteristic
103	C₄	—	9	1	no	—
104	C₄	—	9	1	no	—
105	C₄	—	36	4	no	—
106	C₄	#88592	1	2	yes	—
107	C₃	—	12	1	no	—
108	C₃	#10590091	1	1	yes	characteristic
109	C₂	—	2	1	no	—
110	C₂	—	18	2	no	—
111	C₂	#11720711	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11722448

C₂

yes

maximal

C₂×#88627

C₂

yes

maximal, characteristic

C₂×C₄×#272

C₂²

yes

characteristic

C₂²⋊#1088

—

C₂×#6188

C₄

yes

complemented

C₄⋊#1088

—

C₄×#1088

C₄

yes

complemented

#88627

—

C₂²×#272

D₄

yes

characteristic

C₂×#1088

—

C₂×C₄×H(3,𝔽₃)

Q₈

yes

characteristic

C₂×#1084

C₂×C₄

yes

characteristic

C₄×#272

C₂×C₄

yes

—

C₄×#272

D₄

yes

—

#6188

—

C₃×H(3,ℤ/4)

—

maximal

C₂×C₃×C₄×D₃

—

C₂²×H(3,𝔽₃)

C₄⋊C₄

yes

characteristic

C₂×#272

—

C₂×#272

C₂²⋊C₄

yes

—

#1088

—

C₄×H(3,𝔽₃)

C₄⋊C₄

yes

—

#1084

C₄²

yes

characteristic

#1084

C₂²⋊C₄

yes

characteristic

C₂×C₃×M₁₆

—

C₃×#117

—

C₂²×C₃×D₃

—

C₂×C₃²×C₄

—

C₂×C₃×Dic₃

—

C₃×C₄×D₃

—

C₃×C₄×D₃

—

H(3,ℤ/4)

—

Sylow

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

#95

yes

characteristic

#272

—

C₂²×C₃×C₄

—

C₂×C₄×D₃

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₂²×C₃²

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃²×C₄

—

C₃×Dic₃

—

C₃×Dic₃

—

C₂×M₁₆

—

#117

—

H(3,𝔽₃)

H(3,ℤ/4)

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

PSU(3,𝔽₂)

yes

characteristic

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₃×D₃

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

#3777

yes

characteristic

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#3777

yes

—

Dic₃

—

Dic₃

—

C₃²

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

SU(3,𝔽₂)

yes

characteristic

C₈

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

#63747

yes

characteristic

D₃

—

100

C₂²

—

101

C₂²

—

102

C₂²

#88592

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

#88592

yes

—

107

C₃

—

108

C₃

#10590091

yes

characteristic

109

C₂

—

110

C₂

—

111

C₂

#11720711

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₃	#63747
commutator	#1084	C₄²
Frattini, hypercenter	C₂×C₃×C₄	PSU(3,𝔽₂)
Fitting	C₂×C₄×H(3,𝔽₃)	Q₈
prime radical	C₂²×#272	D₄
cosocle	C₂×C₄×#272	C₂²
hypocenter	H(3,𝔽₃)	H(3,ℤ/4)
LCS	G ▷ #1084 ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄

subgroup

G/H

C₂×C₃

C₄²

Frattini, hypercenter

C₂×C₃×C₄

PSU(3,𝔽₂)

Fitting

C₂×C₄×H(3,𝔽₃)

Q₈

prime radical

C₂²×#272

D₄

cosocle

C₂×C₄×#272

C₂²

G ▷ #1084 ▷ H(3,𝔽₃)

C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄

N	H ≅ G/N	type
5. C₂×#6188	C₄	semidirect
7. C₄×#1088	C₄	semidirect
51. H(3,𝔽₃)	H(3,ℤ/4)	semidirect

H ≅ G/N

type

5. C₂×#6188

C₄

semidirect

7. C₄×#1088

C₄

semidirect

51. H(3,𝔽₃)

H(3,ℤ/4)

semidirect

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

subgroup	index
#11722448	2
C₂×#88627	2
C₃×H(3,ℤ/4)	9

subgroup

index

#11722448

C₂×#88627

C₃×H(3,ℤ/4)

Subgroups — #13136731

Conjugacy classes of subgroups

Complements

Maximal subgroups