Subgroups — C₃×D₃×Q₁₆ #64040

Order 288.

129 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₃×Q₈	C₂	1	2	yes	maximal
2	C₃²×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#216	C₂	1	2	yes	maximal, complemented
4	C₃×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×D₃×C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃×Q₁₆	—	3	1	no	maximal
7	D₃×Q₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₃²×Q₈	C₂²	1	2	yes	—
9	C₃×C₄×D₃	—	2	2	no	—
10	C₃×C₄×D₃	C₂²	1	1	yes	characteristic
11	C₃×Dic₆	—	2	2	no	—
12	C₃×Dic₆	C₂²	1	2	yes	—
13	C₃²×C₈	C₂²	1	1	yes	characteristic
14	C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×Q₈	—	3	2	no	—
16	D₃×Q₈	C₂×C₃	1	2	yes	—
17	C₃×Q₁₆	—	2	1	no	—
18	C₃×Q₁₆	—	3	2	no	—
19	C₃×Q₁₆	—	3	1	no	—
20	C₃×Q₁₆	D₃	1	1	yes	characteristic, direct product factor
21	C₃×Q₁₆	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×C₃×C₈	—	3	1	no	—
23	#216	C₂×C₃	1	2	yes	complemented
24	Dic₁₂	C₂×C₃	1	1	yes	characteristic, complemented
25	D₃×C₈	C₂×C₃	1	1	yes	characteristic
26	C₂×C₃×D₃	D₄	1	1	yes	characteristic
27	C₃²×C₄	—	2	2	no	—
28	C₃²×C₄	C₂³	1	1	yes	characteristic
29	C₃×Dic₃	—	2	2	no	—
30	C₃×Dic₃	D₄	1	1	yes	characteristic
31	C₂×Q₁₆	—	3	1	no	Sylow
32	C₃×Q₈	—	2	2	no	—
33	C₃×Q₈	—	3	2	no	—
34	C₃×Q₈	—	6	2	no	—
35	C₃×Q₈	C₂×D₃	1	2	yes	—
36	C₃×Q₈	C₂²×C₃	1	2	yes	—
37	C₂×C₃×C₄	—	3	1	no	—
38	C₂×C₃×C₄	—	6	2	no	—
39	C₄×D₃	—	2	2	no	—
40	C₄×D₃	C₂²×C₃	1	1	yes	characteristic
41	Dic₆	—	2	2	no	—
42	Dic₆	C₂²×C₃	1	2	yes	—
43	C₃×C₈	—	2	1	no	—
44	C₃×C₈	—	3	1	no	—
45	C₃×C₈	C₂×D₃	1	1	yes	characteristic
46	C₃×C₈	C₂²×C₃	1	1	yes	characteristic
47	C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
49	C₃×D₃	Q₁₆	1	2	yes	direct product factor
50	C₂×Q₈	—	3	2	no	—
51	Q₁₆	—	3	2	no	—
52	Q₁₆	—	3	1	no	—
53	Q₁₆	C₃×D₃	1	1	yes	characteristic, direct product factor
54	C₂×C₈	—	3	1	no	—
55	C₂²×C₃	—	3	1	no	—
56	C₂×D₃	C₃×D₄	1	1	yes	characteristic
57	C₃×C₄	—	2	2	no	—
58	C₃×C₄	—	2	2	no	—
59	C₃×C₄	—	2	1	no	—
60	C₃×C₄	—	3	1	no	—
61	C₃×C₄	—	4	2	no	—
62	C₃×C₄	—	6	2	no	—
63	C₃×C₄	C₂²×D₃	1	1	yes	characteristic
64	C₃×C₄	C₂³×C₃	1	1	yes	characteristic
65	Dic₃	—	2	2	no	—
66	Dic₃	C₃×D₄	1	1	yes	characteristic
67	C₃²	C₂×Q₁₆	1	1	yes	characteristic, complemented, Sylow
68	Q₈	—	3	2	no	—
69	Q₈	—	6	2	no	—
70	Q₈	C₂×C₃×D₃	1	2	yes	—
71	C₂×C₄	—	3	1	no	—
72	C₂×C₄	—	6	2	no	—
73	C₈	—	3	1	no	—
74	C₈	C₂×C₃×D₃	1	1	yes	characteristic
75	C₂×C₃	—	2	1	no	—
76	C₂×C₃	—	3	2	no	—
77	C₂×C₃	D₃×D₄	1	1	yes	characteristic
78	C₂×C₃	C₂×C₃×D₄	1	1	yes	characteristic
79	D₃	C₃×Q₁₆	1	2	yes	direct product factor
80	C₂²	—	3	1	no	—
81	C₄	—	2	2	no	—
82	C₄	—	3	1	no	—
83	C₄	—	6	2	no	—
84	C₄	C₂²×C₃×D₃	1	1	yes	characteristic
85	C₃	—	2	1	no	—
86	C₃	D₃×Q₁₆	1	1	yes	characteristic, direct product factor
87	C₃	C₂×C₃×Q₁₆	1	1	yes	characteristic, complemented
88	C₂	—	3	2	no	—
89	C₂	C₃×D₃×D₄	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃²×Q₁₆	C₂	semidirect
3. C₃×#216	C₂	semidirect
4. C₃×Dic₁₂	C₂	semidirect
7. D₃×Q₁₆	C₃	direct (N × H)
20. C₃×Q₁₆	D₃	direct (N × H)
21. C₃×Q₁₆	C₂×C₃	semidirect
23. #216	C₂×C₃	semidirect
24. Dic₁₂	C₂×C₃	semidirect
49. C₃×D₃	Q₁₆	direct (N × H)
53. Q₁₆	C₃×D₃	direct (N × H)
67. C₃²	C₂×Q₁₆	semidirect
79. D₃	C₃×Q₁₆	direct (N × H)
86. C₃	D₃×Q₁₆	direct (N × H)
87. C₃	C₂×C₃×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₃×Q₈	2
C₃²×Q₁₆	2
C₃×#216	2
C₃×Dic₁₂	2
C₃×D₃×C₈	2
C₂×C₃×Q₁₆	3
D₃×Q₁₆	3

Subgroups — C3×D3×Q16 #64040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₃×Q₁₆ #64040