Subgroups — C₂×C₃×Dic₁₂ #64028

Order 288.

131 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Dic₆	C₂	1	2	yes	maximal
2	C₂×C₃²×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×Dic₁₂	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×Q₁₆	—	3	1	no	maximal
5	C₂×Dic₁₂	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
7	C₂×C₃×Dic₃	—	2	2	no	—
8	C₃×Dic₆	—	2	2	no	—
9	C₃×Dic₆	C₂²	1	4	yes	—
10	C₃²×C₈	C₂²	1	2	yes	—
11	C₂×C₃×Q₈	—	3	2	no	—
12	C₂×Dic₆	C₂×C₃	1	2	yes	—
13	C₃×Q₁₆	—	3	4	no	—
14	C₂×C₃×C₈	—	2	1	no	—
15	C₂×C₃×C₈	D₃	1	1	yes	characteristic
16	C₂×C₃×C₈	C₂×C₃	1	1	yes	characteristic
17	Dic₁₂	C₂×C₃	1	4	yes	direct product factor
18	C₂²×C₃²	D₄	1	1	yes	characteristic
19	C₃²×C₄	D₄	1	1	yes	characteristic
20	C₃²×C₄	C₂³	1	1	yes	characteristic
21	C₃×Dic₃	—	2	4	no	—
22	C₂×Q₁₆	—	3	1	no	Sylow
23	C₃×Q₈	—	3	4	no	—
24	C₃×Q₈	—	6	2	no	—
25	C₂×C₃×C₄	—	2	1	no	—
26	C₂×C₃×C₄	—	6	2	no	—
27	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
28	C₂×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
29	C₂×Dic₃	—	2	2	no	—
30	Dic₆	—	2	2	no	—
31	Dic₆	C₂²×C₃	1	4	yes	—
32	C₃×C₈	—	2	2	no	—
33	C₃×C₈	C₂×D₃	1	2	yes	—
34	C₃×C₈	C₂²×C₃	1	2	yes	—
35	C₂×C₃²	Q₁₆	1	2	yes	complemented
36	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
37	C₂×Q₈	—	3	2	no	—
38	Q₁₆	—	3	4	no	—
39	C₂×C₈	C₃×D₃	1	1	yes	characteristic
40	C₂²×C₃	—	2	1	no	—
41	C₂²×C₃	D₁₂	1	1	yes	characteristic
42	C₂²×C₃	C₃×D₄	1	1	yes	characteristic
43	C₃×C₄	—	2	1	no	—
44	C₃×C₄	—	2	1	no	—
45	C₃×C₄	—	6	4	no	—
46	C₃×C₄	D₁₂	1	1	yes	characteristic
47	C₃×C₄	C₃×D₄	1	1	yes	characteristic
48	C₃×C₄	C₂²×D₃	1	1	yes	characteristic
49	C₃×C₄	C₂³×C₃	1	1	yes	characteristic
50	Dic₃	—	2	4	no	—
51	C₃²	C₂×Q₁₆	1	1	yes	characteristic, complemented, Sylow
52	Q₈	—	3	4	no	—
53	Q₈	—	6	2	no	—
54	C₂×C₄	—	6	2	no	—
55	C₂×C₄	C₂×C₃×D₃	1	1	yes	characteristic
56	C₈	C₂×C₃×D₃	1	2	yes	—
57	C₂×C₃	—	2	2	no	—
58	C₂×C₃	—	2	1	no	—
59	C₂×C₃	Dic₁₂	1	2	yes	direct product factor
60	C₂×C₃	C₃×Q₁₆	1	2	yes	complemented
61	C₂×C₃	C₂×D₁₂	1	1	yes	characteristic
62	C₂×C₃	C₂×C₃×D₄	1	1	yes	characteristic
63	C₂²	C₃×D₁₂	1	1	yes	characteristic
64	C₄	—	6	4	no	—
65	C₄	C₃×D₁₂	1	1	yes	characteristic
66	C₄	C₂²×C₃×D₃	1	1	yes	characteristic
67	C₃	—	2	1	no	—
68	C₃	C₂×Dic₁₂	1	1	yes	characteristic, direct product factor
69	C₃	C₂×C₃×Q₁₆	1	1	yes	characteristic, complemented
70	C₂	C₃×Dic₁₂	1	2	yes	direct product factor
71	C₂	C₂×C₃×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₁₂
commutator, cosocle	C₃×C₄	C₂³×C₃
Frattini	C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×C₈	C₂
socle, prime radical	C₂²×C₃²	D₄
hypercenter	C₂×C₃×C₈	D₃
hypocenter	C₃	C₂×C₃×Q₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈

Complements

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

N	H ≅ G/N	type
3. C₃×Dic₁₂	C₂	direct (N × H)
5. C₂×Dic₁₂	C₃	direct (N × H)
17. Dic₁₂	C₂×C₃	direct (N × H)
35. C₂×C₃²	Q₁₆	semidirect
51. C₃²	C₂×Q₁₆	semidirect
59. C₂×C₃	Dic₁₂	direct (N × H)
60. C₂×C₃	C₃×Q₁₆	semidirect
68. C₃	C₂×Dic₁₂	direct (N × H)
69. C₃	C₂×C₃×Q₁₆	semidirect
70. C₂	C₃×Dic₁₂	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×C₃×Dic₆	2
C₂×C₃²×C₈	2
C₃×Dic₁₂	2
C₂×C₃×Q₁₆	3
C₂×Dic₁₂	3

Subgroups — C2×C3×Dic12 #64028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×Dic₁₂ #64028