Subgroups — C₂²×C₃×Dic₆ #64340

Order 288.

339 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂³×C₃	C₂×D₃
commutator, cosocle	C₂×C₃	C₂⁴×C₃
Frattini	C₂	C₂³×C₃×D₃
Fitting	C₂²×C₃²×C₄	C₂
socle, prime radical	C₂³×C₃²	C₂²
hypercenter	C₂²×C₃×C₄	D₃
hypocenter	C₃	C₂²×C₃×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×Dic₆	C₂	direct (N × H)
5. C₂²×Dic₆	C₃	direct (N × H)
9. C₃×Dic₆	C₂²	direct (N × H)
16. C₂×Dic₆	C₂×C₃	direct (N × H)
17. C₂²×C₃²	Q₈	semidirect
31. Dic₆	C₂²×C₃	direct (N × H)
32. C₂×C₃²	C₂×Q₈	semidirect
39. C₂²×C₃	Dic₆	direct (N × H)
40. C₂²×C₃	C₃×Q₈	semidirect
48. C₃²	C₂²×Q₈	semidirect
55. C₂×C₃	C₂×Dic₆	direct (N × H)
56. C₂×C₃	C₂×C₃×Q₈	semidirect
59. C₂²	C₃×Dic₆	direct (N × H)
64. C₃	C₂²×Dic₆	direct (N × H)
65. C₃	C₂²×C₃×Q₈	semidirect
66. C₂	C₂×C₃×Dic₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₄	2
C₂²×C₃×Dic₃	2
C₂×C₃×Dic₆	2
C₂²×C₃×Q₈	3
C₂²×Dic₆	3

Subgroups — C22×C3×Dic6 #64340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×Dic₆ #64340