Group #13146093 — subgroups

Subgroups — #13146093

Order 1728.

896 conjugacy classes of subgroups; only the 451 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 451 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	C₃³×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	C₂	1	1	yes	maximal, characteristic
4	#10589413	—	3	13	no	maximal
5	C₂×C₃³×Q₈	—	2	1	no	—
6	C₃³×C₄⋊C₄	—	2	1	no	—
7	C₃³×C₄⋊C₄	C₄	1	2	yes	—
8	C₃³×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
10	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
11	C₃²×C₄⋊Q₈	D₃	1	13	yes	—
12	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	13	no	—
13	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	13	no	—
14	C₃³×Q₈	—	4	1	no	—
15	C₃³×Q₈	—	4	1	no	—
16	C₂×C₃³×C₄	—	2	2	no	—
17	C₂×C₃³×C₄	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
19	C₂×C₃³×C₄	D₄	1	1	yes	characteristic
20	C₃⋊(C₃⋊(C₃⋊C₈))	—	4	1	no	—
21	C₃⋊(C₃⋊(C₃⋊C₈))	—	4	1	no	—
22	#4547	—	9	13	no	—
23	C₂×C₃²×Q₈	—	2	13	no	—
24	C₃²×C₄⋊C₄	—	2	13	no	—
25	C₃²×C₄⋊C₄	Dic₃	1	26	yes	—
26	C₃²×C₄²	C₂×D₃	1	13	yes	—
27	C₂×C₃⋊(C₃⋊C₈)	—	6	13	no	—
28	C₂×C₃⋊(C₃⋊C₈)	—	6	13	no	—
29	C₂²×C₃³	C₂²⋊C₄	1	1	yes	characteristic
30	C₃³×C₄	—	2	1	no	—
31	C₃³×C₄	—	4	2	no	—
32	C₃³×C₄	D₈	1	1	yes	characteristic
33	C₃³×C₄	SD₁₆	1	1	yes	characteristic
34	C₃³×C₄	SD₁₆	1	1	yes	characteristic
35	C₃³×C₄	Q₁₆	1	1	yes	characteristic
36	C₃×C₄⋊Q₈	C₃⋊D₃	1	13	yes	—
37	C₄⋊(C₃⋊C₈)	—	9	13	no	—
38	C₄⋊(C₃⋊C₈)	—	9	13	no	—
39	C₃²×Q₈	—	4	13	no	—
40	C₃²×Q₈	—	4	13	no	—
41	C₂×C₃²×C₄	—	2	26	no	—
42	C₂×C₃²×C₄	C₂×Dic₃	1	13	yes	—
43	C₂×C₃²×C₄	#67	1	13	yes	—
44	C₂×C₃²×C₄	#67	1	13	yes	—
45	C₃⋊(C₃⋊C₈)	—	12	13	no	—
46	C₃⋊(C₃⋊C₈)	—	12	13	no	—
47	#332	—	27	1	no	Sylow
48	C₂×C₃³	#101	1	1	yes	characteristic
49	C₂×C₃³	#102	1	1	yes	characteristic
50	C₂×C₃³	#103	1	1	yes	characteristic
51	C₂×C₃×Q₈	—	2	13	no	—
52	C₃×C₄⋊C₄	—	2	13	no	—
53	C₃×C₄⋊C₄	C₃⋊Dic₃	1	26	yes	—
54	C₃×C₄²	C₂×C₃⋊D₃	1	13	yes	—
55	C₂×C₃⋊C₈	—	18	13	no	—
56	C₂×C₃⋊C₈	—	18	13	no	—
57	C₂²×C₃²	#217	1	13	yes	—
58	C₃²×C₄	—	2	13	no	—
59	C₃²×C₄	—	4	26	no	—
60	C₃²×C₄	#213	1	13	yes	—
61	C₃²×C₄	#214	1	13	yes	—
62	C₃²×C₄	#215	1	13	yes	—
63	C₃²×C₄	#216	1	13	yes	—
64	C₄⋊Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
65	C₄⋊C₈	—	27	1	no	—
66	C₄⋊C₈	—	27	1	no	—
67	C₃³	#332	1	1	yes	characteristic, complemented, Sylow
68	C₃×Q₈	—	4	13	no	—
69	C₃×Q₈	—	4	13	no	—
70	C₂×C₃×C₄	—	2	26	no	—
71	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	13	yes	—
72	C₂×C₃×C₄	#638	1	13	yes	—
73	C₂×C₃×C₄	#638	1	13	yes	—
74	C₃⋊C₈	—	36	13	no	—
75	C₃⋊C₈	—	36	13	no	—
76	C₂×C₃²	#832	1	13	yes	—
77	C₂×C₃²	#835	1	13	yes	—
78	C₂×C₃²	#836	1	13	yes	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×C₈	—	54	1	no	—
81	C₂×C₈	—	54	1	no	—
82	C₄⋊C₄	—	2	1	no	—
83	C₄⋊C₄	C₃⋊(C₃⋊Dic₃)	1	2	yes	—
84	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
85	C₂²×C₃	#3757	1	13	yes	—
86	C₃×C₄	—	2	13	no	—
87	C₃×C₄	—	4	26	no	—
88	C₃×C₄	#3753	1	13	yes	—
89	C₃×C₄	#3754	1	13	yes	—
90	C₃×C₄	#3755	1	13	yes	—
91	C₃×C₄	#3756	1	13	yes	—
92	C₃²	#4547	1	13	yes	complemented
93	Q₈	—	4	1	no	—
94	Q₈	—	4	1	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
97	C₂×C₄	#6312	1	1	yes	characteristic
98	C₂×C₄	#6312	1	1	yes	characteristic
99	C₈	—	108	1	no	—
100	C₈	—	108	1	no	—
101	C₂×C₃	#63659	1	13	yes	—
102	C₂×C₃	#63662	1	13	yes	—
103	C₂×C₃	#63663	1	13	yes	—
104	C₂²	#88864	1	1	yes	characteristic
105	C₄	—	2	1	no	—
106	C₄	—	4	2	no	—
107	C₄	#88860	1	1	yes	characteristic
108	C₄	#88861	1	1	yes	characteristic
109	C₄	#88862	1	1	yes	characteristic
110	C₄	#88863	1	1	yes	characteristic
111	C₃	#10589413	1	13	yes	complemented
112	C₂	#11722202	1	1	yes	characteristic
113	C₂	#11722205	1	1	yes	characteristic
114	C₂	#11722206	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))

C₂

yes

maximal, characteristic

C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))

C₂

yes

maximal, characteristic

#10589413

—

maximal

C₂×C₃³×Q₈

—

C₃³×C₄⋊C₄

—

C₃³×C₄⋊C₄

C₄

yes

—

C₃³×C₄²

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊(C₃⋊C₈))

—

C₂×C₃⋊(C₃⋊(C₃⋊C₈))

—

C₃²×C₄⋊Q₈

D₃

yes

—

C₄⋊(C₃⋊(C₃⋊C₈))

—

C₄⋊(C₃⋊(C₃⋊C₈))

—

C₃³×Q₈

—

C₃³×Q₈

—

C₂×C₃³×C₄

—

C₂×C₃³×C₄

C₂×C₄

yes

characteristic

C₂×C₃³×C₄

D₄

yes

characteristic

C₂×C₃³×C₄

D₄

yes

characteristic

C₃⋊(C₃⋊(C₃⋊C₈))

—

C₃⋊(C₃⋊(C₃⋊C₈))

—

#4547

—

C₂×C₃²×Q₈

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

Dic₃

yes

—

C₃²×C₄²

C₂×D₃

yes

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂×C₃⋊(C₃⋊C₈)

—

C₂²×C₃³

C₂²⋊C₄

yes

characteristic

C₃³×C₄

—

C₃³×C₄

—

C₃³×C₄

D₈

yes

characteristic

C₃³×C₄

SD₁₆

yes

characteristic

C₃³×C₄

SD₁₆

yes

characteristic

C₃³×C₄

Q₁₆

yes

characteristic

C₃×C₄⋊Q₈

C₃⋊D₃

yes

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₃²×Q₈

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃²×C₄

#67

yes

—

C₃⋊(C₃⋊C₈)

—

C₃⋊(C₃⋊C₈)

—

—

C₂×C₃³

yes

C₂×C₃³

yes

C₂×C₃³

yes

C₂×C₃×Q₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₃⋊Dic₃

yes

—

C₃×C₄²

C₂×C₃⋊D₃

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

#217

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

#213

yes

—

C₃²×C₄

#214

yes

—

C₃²×C₄

#215

yes

—

C₃²×C₄

#216

yes

—

C₄⋊Q₈

C₃⋊(C₃⋊D₃)

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₃³

#332

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃⋊Dic₃

yes

—

C₂×C₃×C₄

#638

yes

—

C₂×C₃×C₄

#638

yes

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

#832

yes

—

C₂×C₃²

#835

yes

—

C₂×C₃²

#836

yes

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

C₃⋊(C₃⋊Dic₃)

yes

—

C₄²

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

C₂²×C₃

#3757

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#3753

yes

—

C₃×C₄

#3754

yes

—

C₃×C₄

#3755

yes

—

C₃×C₄

#3756

yes

—

C₃²

#4547

yes

complemented

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

C₂×C₃⋊(C₃⋊Dic₃)

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

108

—

100

C₈

—

108

—

101

C₂×C₃

#63659

yes

—

102

C₂×C₃

#63662

yes

—

103

C₂×C₃

#63663

yes

—

104

C₂²

#88864

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

#88860

yes

characteristic

108

C₄

#88861

yes

characteristic

109

C₄

#88862

yes

characteristic

110

C₄

#88863

yes

characteristic

111

C₃

#10589413

yes

complemented

112

C₂

#11722202

yes

characteristic

113

C₂

#11722205

yes

characteristic

114

C₂

#11722206

yes

characteristic

	subgroup	G/H
center	C₂²	#88864
commutator	C₂×C₃³×C₄	C₂×C₄
Frattini	C₄²	C₂×C₃⋊(C₃⋊D₃)
Fitting	C₃³×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₃³	C₂²⋊C₄
cosocle	C₃³×C₄²	C₂²
hypercenter	C₄⋊Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#332
LCS	G ▷ C₂×C₃³×C₄ ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

#88864

commutator

C₂×C₃³×C₄

C₂×C₄

Frattini

C₄²

C₂×C₃⋊(C₃⋊D₃)

Fitting

C₃³×C₄⋊Q₈

C₂

socle, prime radical

C₂²×C₃³

C₂²⋊C₄

cosocle

C₃³×C₄²

C₂²

hypercenter

C₄⋊Q₈

C₃⋊(C₃⋊D₃)

hypocenter

C₃³

#332

LCS

G ▷ C₂×C₃³×C₄ ▷ C₂²×C₃³ ▷ C₃³

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
67. C₃³	#332	semidirect
92. C₃²	#4547	semidirect
111. C₃	#10589413	semidirect

H ≅ G/N

type

67. C₃³

#332

semidirect

92. C₃²

#4547

semidirect

111. C₃

#10589413

semidirect

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

subgroup	index
C₃³×C₄⋊Q₈	2
C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	2
#10589413	3

subgroup

index

C₃³×C₄⋊Q₈

C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))

#10589413

Subgroups — #13146093

Conjugacy classes of subgroups

Complements

Maximal subgroups