Subgroups — C₃×#1007 #64349

Order 288.

311 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#239	C₂	1	4	yes	maximal, complemented
3	C₃×D₃×Q₈	C₂	1	4	yes	maximal, complemented
4	C₃×#235	C₂	1	6	yes	maximal, complemented
5	C₃×2₋¹⁺⁴	—	3	1	no	maximal
6	#1007	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₃²×Q₈	C₂²	1	4	yes	—
8	C₂×C₃²×C₄	C₂²	1	3	yes	—
9	C₃×#67	C₂²	1	4	yes	—
10	C₃×D₁₂	C₂²	1	6	yes	—
11	C₃×C₄×D₃	C₂²	1	12	yes	—
12	C₃×Dic₆	C₂²	1	6	yes	—
13	C₃×#41	—	3	4	no	—
14	C₃×#41	—	3	6	no	—
15	C₂×C₃×Q₈	—	2	1	no	—
16	C₂×C₃×Q₈	—	3	4	no	—
17	C₂×C₃×Q₈	D₃	1	1	yes	characteristic, complemented
18	C₂×C₃×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
19	#239	C₂×C₃	1	4	yes	complemented
20	D₃×Q₈	C₂×C₃	1	4	yes	complemented
21	#235	C₂×C₃	1	6	yes	complemented
22	C₂²×C₃²	C₂³	1	1	yes	characteristic
23	C₂×C₃×D₃	C₂³	1	4	yes	—
24	C₃²×C₄	C₂³	1	6	yes	—
25	C₃×Dic₃	C₂³	1	4	yes	—
26	2₋¹⁺⁴	—	3	1	no	Sylow
27	C₃×Q₈	—	2	4	no	—
28	C₃×Q₈	—	3	6	no	—
29	C₃×Q₈	C₂×D₃	1	4	yes	—
30	C₃×Q₈	C₂²×C₃	1	4	yes	—
31	C₃×D₄	—	3	4	no	—
32	C₃×D₄	—	3	6	no	—
33	C₂×C₃×C₄	—	2	3	no	—
34	C₂×C₃×C₄	—	3	12	no	—
35	C₂×C₃×C₄	C₂×D₃	1	3	yes	—
36	C₂×C₃×C₄	C₂²×C₃	1	3	yes	—
37	#67	C₂²×C₃	1	4	yes	—
38	D₁₂	C₂²×C₃	1	6	yes	—
39	C₄×D₃	C₂²×C₃	1	12	yes	—
40	Dic₆	C₂²×C₃	1	6	yes	—
41	C₂×C₃²	—	2	1	no	—
42	C₂×C₃²	C₂⁴	1	1	yes	characteristic
43	C₃×D₃	—	2	4	no	—
44	#41	—	3	4	no	—
45	#41	—	3	6	no	—
46	C₂×Q₈	—	3	4	no	—
47	C₂×Q₈	C₃×D₃	1	1	yes	characteristic, complemented
48	C₂²×C₃	—	2	1	no	—
49	C₂²×C₃	—	3	4	no	—
50	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
51	C₂²×C₃	C₂³×C₃	1	1	yes	characteristic
52	C₂×D₃	C₂³×C₃	1	4	yes	—
53	C₃×C₄	—	2	6	no	—
54	C₃×C₄	—	3	4	no	—
55	C₃×C₄	C₂²×D₃	1	6	yes	—
56	C₃×C₄	C₂³×C₃	1	6	yes	—
57	Dic₃	C₂³×C₃	1	4	yes	—
58	C₃²	2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
59	Q₈	—	3	6	no	—
60	Q₈	C₂×C₃×D₃	1	4	yes	—
61	D₄	—	3	4	no	—
62	D₄	—	3	6	no	—
63	C₂×C₄	—	3	12	no	—
64	C₂×C₄	C₂×C₃×D₃	1	3	yes	—
65	C₂×C₃	—	2	2	no	—
66	C₂×C₃	—	2	1	no	—
67	C₂×C₃	—	2	1	no	—
68	C₂×C₃	—	2	1	no	—
69	C₂×C₃	—	6	4	no	—
70	C₂×C₃	C₂³×D₃	1	1	yes	characteristic
71	C₂×C₃	C₂⁴×C₃	1	1	yes	characteristic
72	D₃	—	2	4	no	—
73	C₂²	—	3	4	no	—
74	C₂²	C₂²×C₃×D₃	1	1	yes	characteristic
75	C₄	—	3	4	no	—
76	C₄	C₂²×C₃×D₃	1	6	yes	—
77	C₃	—	2	1	no	—
78	C₃	#1007	1	1	yes	characteristic, direct product factor
79	C₃	C₃×2₋¹⁺⁴	1	1	yes	characteristic, complemented
80	C₂	—	2	1	no	—
81	C₂	—	6	4	no	—
82	C₂	C₂³×C₃×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂³×D₃
commutator, cosocle	C₂×C₃	C₂⁴×C₃
Frattini	C₂	C₂³×C₃×D₃
Fitting	C₂×C₃²×Q₈	C₂
socle	C₂×C₃²	C₂⁴
hypercenter	C₂×C₃×Q₈	D₃
hypocenter	C₃	C₃×2₋¹⁺⁴
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×Q₈	C₂	semidirect
2. C₃×#239	C₂	semidirect
3. C₃×D₃×Q₈	C₂	semidirect
4. C₃×#235	C₂	semidirect
6. #1007	C₃	direct (N × H)
17. C₂×C₃×Q₈	D₃	semidirect
18. C₂×C₃×Q₈	C₂×C₃	semidirect
19. #239	C₂×C₃	semidirect
20. D₃×Q₈	C₂×C₃	semidirect
21. #235	C₂×C₃	semidirect
47. C₂×Q₈	C₃×D₃	semidirect
58. C₃²	2₋¹⁺⁴	semidirect
78. C₃	#1007	direct (N × H)
79. C₃	C₃×2₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×Q₈	2
C₃×#239	2
C₃×D₃×Q₈	2
C₃×#235	2
C₃×2₋¹⁺⁴	3
#1007	3

Subgroups — C3×#1007 #64349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#1007 #64349