Group #11721846 — subgroups

Subgroups — #11721846

Order 864.

418 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 88 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×D₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×#6229	C₂	1	1	yes	maximal, characteristic, complemented
3	#88552	C₂	1	4	yes	maximal, complemented
4	C₄×#1084	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#936	—	3	4	no	maximal
6	C₂³×H(3,𝔽₃)	C₂²	1	2	yes	—
7	D₄×H(3,𝔽₃)	—	2	2	no	—
8	C₂×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
9	C₂×#1084	C₂²	1	2	yes	—
10	C₂×#1084	C₂²	1	2	yes	—
11	#6229	—	2	2	no	—
12	C₂×C₃²×D₄	D₃	1	4	yes	—
13	C₂×C₃×Dic₆	—	3	4	no	—
14	C₃×#217	—	3	16	no	—
15	C₃×C₄×Dic₃	—	3	4	no	—
16	C₂²×H(3,𝔽₃)	—	2	2	no	—
17	C₂²×H(3,𝔽₃)	—	2	4	no	—
18	C₂²×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
19	C₄×H(3,𝔽₃)	D₄	1	2	yes	—
20	#1084	—	2	2	no	—
21	#1084	—	2	2	no	—
22	C₃×#124	—	9	1	no	—
23	#936	—	3	4	no	—
24	C₂³×C₃²	C₂×D₃	1	8	yes	—
25	C₃²×D₄	—	2	8	no	—
26	C₂×C₃²×C₄	C₂×D₃	1	4	yes	—
27	C₂×C₃×Dic₃	—	3	8	no	—
28	C₂×C₃×Dic₃	—	3	8	no	—
29	C₃×Dic₆	—	6	8	no	—
30	C₂×H(3,𝔽₃)	—	4	2	no	—
31	C₂×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
32	C₂×H(3,𝔽₃)	#41	1	2	yes	—
33	C₂×C₃×Q₈	—	9	1	no	—
34	C₂×C₃×D₄	—	3	4	no	—
35	C₂×C₃×D₄	C₃⋊D₃	1	1	yes	characteristic
36	C₂×Dic₆	—	3	4	no	—
37	C₃×C₂²⋊C₄	—	9	4	no	—
38	C₃×C₄²	—	9	1	no	—
39	#217	—	3	16	no	—
40	C₄×Dic₃	—	3	4	no	—
41	C₂²×C₃²	—	2	16	no	—
42	C₂²×C₃²	—	2	8	no	—
43	C₂²×C₃²	C₂²×D₃	1	4	yes	—
44	C₃²×C₄	#67	1	8	yes	—
45	C₃×Dic₃	—	6	8	no	—
46	C₃×Dic₃	—	6	8	no	—
47	#124	—	9	1	no	Sylow
48	H(3,𝔽₃)	#124	1	1	yes	characteristic, complemented, Sylow
49	C₂³×C₃	—	3	8	no	—
50	C₂³×C₃	C₂×C₃⋊D₃	1	2	yes	—
51	C₃×Q₈	—	18	2	no	—
52	C₃×D₄	—	2	2	no	—
53	C₃×D₄	—	6	8	no	—
54	C₂×C₃×C₄	—	3	4	no	—
55	C₂×C₃×C₄	—	9	2	no	—
56	C₂×C₃×C₄	—	9	2	no	—
57	C₂×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
58	C₂×Dic₃	—	3	8	no	—
59	C₂×Dic₃	—	3	8	no	—
60	Dic₆	—	6	8	no	—
61	C₂×C₃²	—	4	8	no	—
62	C₂×C₃²	#237	1	8	yes	—
63	C₂×C₃²	C₂×#67	1	4	yes	—
64	C₂×Q₈	—	9	1	no	—
65	C₂×D₄	#272	1	1	yes	characteristic
66	C₂²⋊C₄	—	9	4	no	—
67	C₄²	—	9	1	no	—
68	C₂²×C₃	—	2	4	no	—
69	C₂²×C₃	—	2	2	no	—
70	C₂²×C₃	—	3	4	no	—
71	C₂²×C₃	—	6	8	no	—
72	C₂²×C₃	—	6	16	no	—
73	C₂²×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
74	C₃×C₄	—	3	8	no	—
75	C₃×C₄	—	18	2	no	—
76	C₃×C₄	—	18	2	no	—
77	C₃×C₄	#638	1	2	yes	—
78	Dic₃	—	6	8	no	—
79	Dic₃	—	6	8	no	—
80	C₃²	#936	1	4	yes	complemented
81	C₂³	C₂×#272	1	2	yes	—
82	Q₈	—	18	2	no	—
83	D₄	—	2	2	no	—
84	C₂×C₄	—	9	2	no	—
85	C₂×C₄	—	9	2	no	—
86	C₂×C₄	C₂×#272	1	1	yes	characteristic
87	C₂×C₃	—	3	8	no	—
88	C₂×C₃	—	3	4	no	—
89	C₂×C₃	—	4	2	no	—
90	C₂×C₃	—	12	8	no	—
91	C₂×C₃	#3830	1	2	yes	—
92	C₂×C₃	C₂×#638	1	1	yes	characteristic
93	C₂²	—	2	4	no	—
94	C₂²	—	2	2	no	—
95	C₂²	C₂²×#272	1	1	yes	characteristic
96	C₄	—	18	2	no	—
97	C₄	—	18	2	no	—
98	C₄	#6235	1	2	yes	—
99	C₃	—	3	4	no	—
100	C₃	#64145	1	1	yes	characteristic
101	C₂	—	4	2	no	—
102	C₂	#88744	1	2	yes	—
103	C₂	C₂×#6235	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×H(3,𝔽₃)

C₂

yes

maximal, characteristic

C₂×#6229

C₂

yes

maximal, characteristic, complemented

#88552

C₂

yes

maximal, complemented

C₄×#1084

C₂

yes

maximal, characteristic, complemented

C₃×#936

—

maximal

C₂³×H(3,𝔽₃)

C₂²

yes

—

D₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

C₂²

yes

characteristic

C₂×#1084

C₂²

yes

—

C₂×#1084

C₂²

yes

—

#6229

—

C₂×C₃²×D₄

D₃

yes

—

C₂×C₃×Dic₆

—

C₃×#217

—

C₃×C₄×Dic₃

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

C₂³

yes

characteristic

C₄×H(3,𝔽₃)

D₄

yes

—

—

—

—

—

C₂³×C₃²

C₂×D₃

yes

—

C₃²×D₄

—

C₂×C₃²×C₄

C₂×D₃

yes

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

C₂×D₄

yes

characteristic

C₂×H(3,𝔽₃)

#41

yes

—

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

C₂×C₃×D₄

C₃⋊D₃

yes

characteristic

C₂×Dic₆

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

#217

—

C₄×Dic₃

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂²×D₃

yes

—

C₃²×C₄

#67

yes

—

C₃×Dic₃

—

C₃×Dic₃

—

—

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂³×C₃

C₂×C₃⋊D₃

yes

—

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃⋊D₃

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

C₂×C₃²

—

C₂×C₃²

#237

yes

—

C₂×C₃²

C₂×#67

yes

—

C₂×Q₈

—

C₂×D₄

#272

yes

characteristic

C₂²⋊C₄

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂²×C₃⋊D₃

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#638

yes

—

Dic₃

—

Dic₃

—

C₃²

#936

yes

complemented

C₂³

C₂×#272

yes

—

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#272

yes

characteristic

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

#3830

yes

—

C₂×C₃

C₂×#638

yes

characteristic

C₂²

—

C₂²

—

C₂²

C₂²×#272

yes

characteristic

C₄

—

C₄

—

C₄

#6235

yes

—

C₃

—

100

C₃

#64145

yes

characteristic

101

C₂

—

102

C₂

#88744

yes

—

103

C₂

C₂×#6235

yes

characteristic

	subgroup	G/H
center, Frattini, socle	C₂²×C₃	C₂²×C₃⋊D₃
commutator, cosocle	C₂²×H(3,𝔽₃)	C₂³
Fitting, prime radical	C₂×D₄×H(3,𝔽₃)	C₂
hypercenter	C₂×C₃×D₄	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#124
LCS	G ▷ C₂²×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×D₄

subgroup

G/H

center, Frattini, socle

C₂²×C₃

C₂²×C₃⋊D₃

commutator, cosocle

C₂²×H(3,𝔽₃)

C₂³

Fitting, prime radical

C₂×D₄×H(3,𝔽₃)

C₂

hypercenter

C₂×C₃×D₄

C₃⋊D₃

G ▷ C₂²×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂²×C₃ ◁ C₂×C₃×D₄

N	H ≅ G/N	type
2. C₂×#6229	C₂	semidirect
3. #88552	C₂	semidirect
4. C₄×#1084	C₂	semidirect
48. H(3,𝔽₃)	#124	semidirect
80. C₃²	#936	semidirect

H ≅ G/N

type

2. C₂×#6229

C₂

semidirect

3. #88552

C₂

semidirect

4. C₄×#1084

C₂

semidirect

48. H(3,𝔽₃)

#124

semidirect

80. C₃²

#936

semidirect

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

subgroup	index
C₂×D₄×H(3,𝔽₃)	2
C₂×#6229	2
#88552	2
C₄×#1084	2
C₃×#936	3

subgroup

index

C₂×D₄×H(3,𝔽₃)

Subgroups — #11721846

Conjugacy classes of subgroups

Complements

Maximal subgroups