Subgroups — C₂×C₃×#41⋊C₃ #64335

Order 288.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#41⋊C₃	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#41	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×#41⋊C₃	C₃	1	3	yes	maximal, direct product factor
5	C₂×C₃²×C₄	—	4	1	no	maximal
6	C₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
7	C₃×#41	—	3	2	no	—
8	C₃×#41	C₂×C₃	1	2	yes	complemented
9	C₂×C₃×Q₈	C₂×C₃	1	1	yes	characteristic
10	C₂×C₃×D₄	—	3	1	no	—
11	C₂²×C₃×C₄	—	3	1	no	—
12	#41⋊C₃	C₂×C₃	1	6	yes	direct product factor
13	C₂×SL(2,𝔽₃)	C₂×C₃	1	3	yes	complemented
14	C₂²×C₃²	—	4	1	no	—
15	C₃²×C₄	—	4	2	no	—
16	C₂×#41	C₃²	1	1	yes	characteristic, complemented, Sylow
17	C₂³×C₃	—	3	1	no	—
18	C₃×Q₈	—	3	1	no	—
19	C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
20	C₃×D₄	—	3	2	no	—
21	C₃×D₄	—	3	2	no	—
22	C₂×C₃×C₄	—	3	1	no	—
23	C₂×C₃×C₄	—	3	2	no	—
24	C₂×C₃×C₄	—	3	2	no	—
25	C₂×C₃×C₄	—	4	3	no	—
26	C₂×C₃×C₄	A₄	1	1	yes	characteristic
27	SL(2,𝔽₃)	C₂²×C₃	1	3	yes	complemented
28	C₂×C₃²	—	4	2	no	—
29	C₂×C₃²	—	4	1	no	—
30	#41	—	3	2	no	—
31	#41	C₂×C₃²	1	2	yes	complemented
32	C₂×Q₈	C₂×C₃²	1	1	yes	characteristic
33	C₂×D₄	—	3	1	no	—
34	C₂²×C₄	—	3	1	no	—
35	C₂²×C₃	—	3	2	no	—
36	C₂²×C₃	—	4	3	no	—
37	C₂²×C₃	—	6	2	no	—
38	C₂²×C₃	C₂×A₄	1	1	yes	characteristic
39	C₃×C₄	—	3	1	no	—
40	C₃×C₄	—	3	1	no	—
41	C₃×C₄	—	4	6	no	—
42	C₃×C₄	C₂×A₄	1	2	yes	—
43	C₃²	—	4	1	no	Sylow
44	C₂³	—	3	1	no	—
45	Q₈	—	3	1	no	—
46	Q₈	C₂²×C₃²	1	1	yes	characteristic
47	D₄	—	3	2	no	—
48	D₄	—	3	2	no	—
49	C₂×C₄	—	3	1	no	—
50	C₂×C₄	—	3	2	no	—
51	C₂×C₄	—	3	2	no	—
52	C₂×C₄	C₃×A₄	1	1	yes	characteristic
53	C₂×C₃	—	4	3	no	—
54	C₂×C₃	—	4	6	no	—
55	C₂×C₃	—	6	2	no	—
56	C₂×C₃	#41⋊C₃	1	2	yes	direct product factor
57	C₂×C₃	C₂²×A₄	1	1	yes	characteristic
58	C₂²	—	3	2	no	—
59	C₂²	—	6	2	no	—
60	C₂²	C₂×C₃×A₄	1	1	yes	characteristic
61	C₄	—	3	1	no	—
62	C₄	—	3	1	no	—
63	C₄	C₂×C₃×A₄	1	2	yes	—
64	C₃	—	4	3	no	—
65	C₃	C₂×#41⋊C₃	1	1	yes	characteristic, direct product factor
66	C₂	—	6	2	no	—
67	C₂	C₃×#41⋊C₃	1	2	yes	direct product factor
68	C₂	C₂²×C₃×A₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄	A₄
commutator, cosocle, hypocenter	Q₈	C₂²×C₃²
Frattini	C₂	C₂²×C₃×A₄
Fitting	C₂×C₃×#41	C₃
socle	C₂²×C₃	C₂×A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₃×C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃×#41⋊C₃	C₂	direct (N × H)
2. C₂×C₃×SL(2,𝔽₃)	C₂	semidirect
3. C₂×C₃×#41	C₃	semidirect
4. C₂×#41⋊C₃	C₃	direct (N × H)
6. C₃×SL(2,𝔽₃)	C₂²	semidirect
8. C₃×#41	C₂×C₃	semidirect
12. #41⋊C₃	C₂×C₃	direct (N × H)
13. C₂×SL(2,𝔽₃)	C₂×C₃	semidirect
16. C₂×#41	C₃²	semidirect
27. SL(2,𝔽₃)	C₂²×C₃	semidirect
31. #41	C₂×C₃²	semidirect
56. C₂×C₃	#41⋊C₃	direct (N × H)
65. C₃	C₂×#41⋊C₃	direct (N × H)
67. C₂	C₃×#41⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#41⋊C₃	2
C₂×C₃×SL(2,𝔽₃)	2
C₂×C₃×#41	3
C₂×#41⋊C₃	3
C₂×C₃²×C₄	4

Subgroups — C2×C3×#41⋊C3 #64335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#41⋊C₃ #64335