Group #13135925 — subgroups

Subgroups — #13135925

Order 1728.

352 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 115 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	H(3,𝔽₃)×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊#6180	C₂	1	2	yes	maximal
3	C₃×#4546	—	3	4	no	maximal
4	C₂×Q₈×H(3,𝔽₃)	C₄	1	2	yes	—
5	C₄⋊C₄×H(3,𝔽₃)	—	2	2	no	—
6	C₄²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₂×#6180	—	2	2	no	—
8	C₃²×C₄⋊Q₈	D₃	1	4	yes	—
9	C₃×C₄⋊(C₃⋊C₈)	—	3	8	no	—
10	Q₈×H(3,𝔽₃)	—	4	2	no	—
11	C₂×C₄×H(3,𝔽₃)	—	2	2	no	—
12	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×H(3,𝔽₃)	D₄	1	2	yes	—
14	#6180	—	4	2	no	—
15	C₃×#333	—	9	1	no	—
16	#4546	—	3	4	no	—
17	C₂×C₃²×Q₈	Dic₃	1	8	yes	—
18	C₃²×C₄⋊C₄	—	2	8	no	—
19	C₃²×C₄²	C₂×D₃	1	4	yes	—
20	C₂×C₃×C₃⋊C₈	—	6	8	no	—
21	C₂²×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×H(3,𝔽₃)	—	2	1	no	—
23	C₄×H(3,𝔽₃)	—	4	2	no	—
24	C₄×H(3,𝔽₃)	SD₁₆	1	2	yes	—
25	C₄×H(3,𝔽₃)	Q₁₆	1	2	yes	—
26	C₃×C₄⋊Q₈	—	3	4	no	—
27	C₃×C₄⋊Q₈	C₃⋊D₃	1	1	yes	characteristic
28	C₃×C₄⋊C₈	—	9	2	no	—
29	C₄⋊(C₃⋊C₈)	—	3	8	no	—
30	C₃²×Q₈	—	4	8	no	—
31	C₂×C₃²×C₄	—	2	8	no	—
32	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
33	C₂×C₃²×C₄	#67	1	8	yes	—
34	C₃×C₃⋊C₈	—	12	8	no	—
35	#333	—	9	1	no	Sylow
36	C₂×H(3,𝔽₃)	#100	1	1	yes	characteristic
37	C₂×H(3,𝔽₃)	#103	1	2	yes	—
38	C₂×C₃×Q₈	—	3	8	no	—
39	C₂×C₃×Q₈	C₃⋊Dic₃	1	2	yes	—
40	C₂×C₃×C₈	—	18	2	no	—
41	C₃×C₄⋊C₄	—	2	2	no	—
42	C₃×C₄⋊C₄	—	6	8	no	—
43	C₃×C₄²	—	3	4	no	—
44	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
45	C₂×C₃⋊C₈	—	6	8	no	—
46	C₂²×C₃²	#217	1	4	yes	—
47	C₃²×C₄	—	2	4	no	—
48	C₃²×C₄	—	4	8	no	—
49	C₃²×C₄	#215	1	8	yes	—
50	C₃²×C₄	#216	1	8	yes	—
51	C₄⋊Q₈	#272	1	1	yes	characteristic
52	C₄⋊C₈	—	9	2	no	—
53	H(3,𝔽₃)	#333	1	1	yes	characteristic, complemented, Sylow
54	C₃×Q₈	—	4	2	no	—
55	C₃×Q₈	—	12	8	no	—
56	C₂×C₃×C₄	—	2	2	no	—
57	C₂×C₃×C₄	—	3	4	no	—
58	C₂×C₃×C₄	—	3	8	no	—
59	C₂×C₃×C₄	—	6	8	no	—
60	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
61	C₂×C₃×C₄	#638	1	2	yes	—
62	C₃×C₈	—	36	2	no	—
63	C₃⋊C₈	—	12	8	no	—
64	C₂×C₃²	#833	1	4	yes	—
65	C₂×C₃²	#835	1	8	yes	—
66	C₂×Q₈	#1084	1	2	yes	—
67	C₂×C₈	—	18	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄²	C₂×#272	1	1	yes	characteristic
70	C₂²×C₃	—	3	4	no	—
71	C₂²×C₃	#3757	1	1	yes	characteristic
72	C₃×C₄	—	2	1	no	—
73	C₃×C₄	—	3	8	no	—
74	C₃×C₄	—	3	8	no	—
75	C₃×C₄	—	4	2	no	—
76	C₃×C₄	—	6	4	no	—
77	C₃×C₄	—	12	8	no	—
78	C₃×C₄	#3755	1	2	yes	—
79	C₃×C₄	#3756	1	2	yes	—
80	C₃²	#4546	1	4	yes	complemented
81	Q₈	—	4	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	C₂×#1084	1	1	yes	characteristic
84	C₂×C₄	#6235	1	2	yes	—
85	C₈	—	36	2	no	—
86	C₂×C₃	—	3	4	no	—
87	C₂×C₃	—	3	8	no	—
88	C₂×C₃	#63660	1	1	yes	characteristic
89	C₂×C₃	#63662	1	2	yes	—
90	C₂²	#88552	1	1	yes	characteristic
91	C₄	—	2	1	no	—
92	C₄	—	4	2	no	—
93	C₄	#88549	1	2	yes	—
94	C₄	#88550	1	2	yes	—
95	C₃	—	3	4	no	—
96	C₃	#10589412	1	1	yes	characteristic
97	C₂	#11720579	1	1	yes	characteristic
98	C₂	#11720583	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

H(3,𝔽₃)×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₄⋊#6180

C₂

yes

maximal

C₃×#4546

—

maximal

C₂×Q₈×H(3,𝔽₃)

C₄

yes

—

C₄⋊C₄×H(3,𝔽₃)

—

C₄²×H(3,𝔽₃)

C₂²

yes

characteristic

C₂×#6180

—

C₃²×C₄⋊Q₈

D₃

yes

—

C₃×C₄⋊(C₃⋊C₈)

—

Q₈×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

C₂×C₄

yes

characteristic

C₂×C₄×H(3,𝔽₃)

D₄

yes

—

#6180

—

C₃×#333

—

#4546

—

C₂×C₃²×Q₈

Dic₃

yes

—

C₃²×C₄⋊C₄

—

C₃²×C₄²

C₂×D₃

yes

—

C₂×C₃×C₃⋊C₈

—

C₂²×H(3,𝔽₃)

C₂²⋊C₄

yes

characteristic

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

SD₁₆

yes

—

C₄×H(3,𝔽₃)

Q₁₆

yes

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

C₃⋊D₃

yes

characteristic

C₃×C₄⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃²×C₄

#67

yes

—

C₃×C₃⋊C₈

—

—

C₂×H(3,𝔽₃)

yes

C₂×H(3,𝔽₃)

yes

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₃⋊Dic₃

yes

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₂×C₃⋊C₈

—

C₂²×C₃²

#217

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

#215

yes

—

C₃²×C₄

#216

yes

—

C₄⋊Q₈

#272

yes

characteristic

C₄⋊C₈

—

H(3,𝔽₃)

#333

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃⋊Dic₃

yes

characteristic

C₂×C₃×C₄

#638

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

#833

yes

—

C₂×C₃²

#835

yes

—

C₂×Q₈

#1084

yes

—

C₂×C₈

—

C₄⋊C₄

—

C₄²

C₂×#272

yes

characteristic

C₂²×C₃

—

C₂²×C₃

#3757

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#3755

yes

—

C₃×C₄

#3756

yes

—

C₃²

#4546

yes

complemented

Q₈

—

C₂×C₄

—

C₂×C₄

C₂×#1084

yes

characteristic

C₂×C₄

#6235

yes

—

C₈

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

#63660

yes

characteristic

C₂×C₃

#63662

yes

—

C₂²

#88552

yes

characteristic

C₄

—

C₄

—

C₄

#88549

yes

—

C₄

#88550

yes

—

C₃

—

C₃

yes

C₂

yes

C₂

yes

—

	subgroup	G/H
center, socle	C₂²×C₃	#3757
commutator	C₂×C₄×H(3,𝔽₃)	C₂×C₄
Frattini	C₃×C₄²	C₂×C₃⋊D₃
Fitting	H(3,𝔽₃)×C₄⋊Q₈	C₂
prime radical	C₂²×H(3,𝔽₃)	C₂²⋊C₄
cosocle	C₄²×H(3,𝔽₃)	C₂²
hypercenter	C₃×C₄⋊Q₈	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#333
LCS	G ▷ C₂×C₄×H(3,𝔽₃) ▷ C₂²×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄⋊Q₈

subgroup

G/H

center, socle

C₂²×C₃

#3757

commutator

C₂×C₄×H(3,𝔽₃)

C₂×C₄

Frattini

C₃×C₄²

C₂×C₃⋊D₃

Fitting

H(3,𝔽₃)×C₄⋊Q₈

C₂

prime radical

C₂²×H(3,𝔽₃)

C₂²⋊C₄

cosocle

C₄²×H(3,𝔽₃)

C₂²

hypercenter

C₃×C₄⋊Q₈

C₃⋊D₃

G ▷ C₂×C₄×H(3,𝔽₃) ▷ C₂²×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄⋊Q₈

N	H ≅ G/N	type
53. H(3,𝔽₃)	#333	semidirect
80. C₃²	#4546	semidirect

H ≅ G/N

type

53. H(3,𝔽₃)

#333

semidirect

80. C₃²

#4546

semidirect

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

subgroup	index
H(3,𝔽₃)×C₄⋊Q₈	2
C₄⋊#6180	2
C₃×#4546	3

subgroup

index

H(3,𝔽₃)×C₄⋊Q₈

C₄⋊#6180

C₃×#4546

Subgroups — #13135925

Conjugacy classes of subgroups

Complements

Maximal subgroups