Subgroups — C₂×C₃×#239 #64348

Order 288.

347 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	8	yes	maximal, direct product factor
3	C₂×C₃×D₁₂	C₂	1	3	yes	maximal, complemented
4	C₂×C₃×C₄×D₃	C₂	1	3	yes	maximal, complemented
5	C₂×C₃×#41	—	3	1	no	maximal
6	C₂×#239	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×D₃	C₂²	1	3	yes	—
8	C₃²×Q₈	C₂²	1	4	yes	complemented
9	C₂×C₃²×C₄	C₂²	1	3	yes	—
10	C₂×C₃×Dic₃	C₂²	1	1	yes	characteristic
11	C₃×D₁₂	C₂²	1	12	yes	complemented
12	C₃×C₄×D₃	C₂²	1	12	yes	complemented
13	C₃×#41	—	3	8	no	—
14	C₂×C₃×Q₈	—	2	1	no	—
15	C₂×C₃×Q₈	D₃	1	1	yes	characteristic, complemented
16	C₂×C₃×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
17	C₂×C₃×D₄	—	3	3	no	—
18	C₂²×C₃×C₄	—	3	3	no	—
19	#239	C₂×C₃	1	8	yes	direct product factor
20	C₂×D₁₂	C₂×C₃	1	3	yes	complemented
21	C₂×C₄×D₃	C₂×C₃	1	3	yes	complemented
22	C₂²×C₃²	C₂³	1	1	yes	characteristic
23	C₂×C₃×D₃	—	2	6	no	—
24	C₂×C₃×D₃	C₂³	1	6	yes	—
25	C₃²×C₄	C₂³	1	6	yes	—
26	C₃×Dic₃	C₂³	1	2	yes	—
27	C₂×#41	—	3	1	no	Sylow
28	C₂³×C₃	—	3	3	no	—
29	C₂²×D₃	C₂²×C₃	1	3	yes	—
30	C₃×Q₈	—	2	4	no	—
31	C₃×Q₈	C₂×D₃	1	4	yes	complemented
32	C₃×Q₈	C₂²×C₃	1	4	yes	complemented
33	C₃×D₄	—	3	12	no	—
34	C₂×C₃×C₄	—	2	3	no	—
35	C₂×C₃×C₄	—	3	1	no	—
36	C₂×C₃×C₄	—	3	12	no	—
37	C₂×C₃×C₄	C₂×D₃	1	3	yes	—
38	C₂×C₃×C₄	C₂²×C₃	1	3	yes	—
39	C₂×Dic₃	C₂²×C₃	1	1	yes	characteristic
40	D₁₂	C₂²×C₃	1	12	yes	complemented
41	C₄×D₃	C₂²×C₃	1	12	yes	complemented
42	C₂×C₃²	#41	1	2	yes	complemented
43	C₂×C₃²	C₂⁴	1	1	yes	characteristic
44	C₃×D₃	—	2	6	no	—
45	#41	—	3	8	no	—
46	C₂×Q₈	C₃×D₃	1	1	yes	characteristic, complemented
47	C₂×D₄	—	3	3	no	—
48	C₂²×C₄	—	3	3	no	—
49	C₂²×C₃	—	2	1	no	—
50	C₂²×C₃	—	3	6	no	—
51	C₂²×C₃	—	6	6	no	—
52	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
53	C₂²×C₃	C₂³×C₃	1	1	yes	characteristic
54	C₂×D₃	—	2	6	no	—
55	C₂×D₃	C₂³×C₃	1	6	yes	—
56	C₃×C₄	—	2	6	no	—
57	C₃×C₄	—	3	2	no	—
58	C₃×C₄	C₂²×D₃	1	6	yes	—
59	C₃×C₄	C₂³×C₃	1	6	yes	—
60	Dic₃	C₂³×C₃	1	2	yes	—
61	C₃²	C₂×#41	1	1	yes	characteristic, complemented, Sylow
62	C₂³	—	3	3	no	—
63	Q₈	C₂×C₃×D₃	1	4	yes	complemented
64	D₄	—	3	12	no	—
65	C₂×C₄	—	3	1	no	—
66	C₂×C₄	—	3	12	no	—
67	C₂×C₄	C₂×C₃×D₃	1	3	yes	—
68	C₂×C₃	—	2	1	no	—
69	C₂×C₃	—	2	2	no	—
70	C₂×C₃	—	6	6	no	—
71	C₂×C₃	#239	1	2	yes	direct product factor
72	C₂×C₃	C₃×#41	1	2	yes	complemented
73	C₂×C₃	C₂³×D₃	1	1	yes	characteristic
74	C₂×C₃	C₂⁴×C₃	1	1	yes	characteristic
75	D₃	—	2	6	no	—
76	C₂²	—	3	6	no	—
77	C₂²	—	6	6	no	—
78	C₂²	C₂²×C₃×D₃	1	1	yes	characteristic
79	C₄	—	3	2	no	—
80	C₄	C₂²×C₃×D₃	1	6	yes	—
81	C₃	—	2	1	no	—
82	C₃	C₂×#239	1	1	yes	characteristic, direct product factor
83	C₃	C₂×C₃×#41	1	1	yes	characteristic, complemented
84	C₂	—	6	6	no	—
85	C₂	C₃×#239	1	2	yes	direct product factor
86	C₂	C₂³×C₃×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂²×D₃
commutator, cosocle	C₂×C₃	C₂⁴×C₃
Frattini	C₂	C₂³×C₃×D₃
Fitting	C₂×C₃²×Q₈	C₂
socle	C₂²×C₃²	C₂³
hypercenter	C₂×C₃×Q₈	D₃
hypocenter	C₃	C₂×C₃×#41
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×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	direct (N × H)
3. C₂×C₃×D₁₂	C₂	semidirect
4. C₂×C₃×C₄×D₃	C₂	semidirect
6. C₂×#239	C₃	direct (N × H)
8. C₃²×Q₈	C₂²	semidirect
11. C₃×D₁₂	C₂²	semidirect
12. C₃×C₄×D₃	C₂²	semidirect
15. C₂×C₃×Q₈	D₃	semidirect
16. C₂×C₃×Q₈	C₂×C₃	semidirect
19. #239	C₂×C₃	direct (N × H)
20. C₂×D₁₂	C₂×C₃	semidirect
21. C₂×C₄×D₃	C₂×C₃	semidirect
31. C₃×Q₈	C₂×D₃	semidirect
32. C₃×Q₈	C₂²×C₃	semidirect
40. D₁₂	C₂²×C₃	semidirect
41. C₄×D₃	C₂²×C₃	semidirect
42. C₂×C₃²	#41	semidirect
46. C₂×Q₈	C₃×D₃	semidirect
61. C₃²	C₂×#41	semidirect
63. Q₈	C₂×C₃×D₃	semidirect
71. C₂×C₃	#239	direct (N × H)
72. C₂×C₃	C₃×#41	semidirect
82. C₃	C₂×#239	direct (N × H)
83. C₃	C₂×C₃×#41	semidirect
85. C₂	C₃×#239	direct (N × H)

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
C₂×C₃×D₁₂	2
C₂×C₃×C₄×D₃	2
C₂×C₃×#41	3
C₂×#239	3

Subgroups — C2×C3×#239 #64348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#239 #64348