Subgroups — C₃×Dic₇₂ #11720345

Order 864.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Dic₃₆	C₂	1	2	yes	maximal
2	C₃×C₉×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×Dic₂₄	—	3	1	no	maximal
4	Dic₇₂	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₃×Dic₁₈	—	2	2	no	—
6	C₃×C₈×C₉	C₂²	1	1	yes	characteristic
7	C₃×Dic₁₂	—	3	2	no	—
8	C₃²×C₁₆	D₃	1	1	yes	characteristic
9	Dic₃₆	C₂×C₃	1	2	yes	—
10	C₉×C₁₆	—	2	1	no	—
11	C₉×C₁₆	C₂×C₃	1	1	yes	characteristic
12	C₃×C₄×C₉	D₄	1	1	yes	characteristic
13	C₃×Dic₉	—	4	2	no	—
14	C₃×Q₃₂	—	9	1	no	—
15	Dic₂₄	—	3	1	no	—
16	C₃×Dic₆	—	6	2	no	—
17	C₃²×C₈	C₂×D₃	1	1	yes	characteristic
18	Dic₁₈	—	2	2	no	—
19	C₈×C₉	—	2	1	no	—
20	C₈×C₉	C₂²×C₃	1	1	yes	characteristic
21	C₂×C₃×C₉	D₈	1	1	yes	characteristic
22	C₃×Q₁₆	—	9	2	no	—
23	Dic₁₂	—	3	2	no	—
24	C₃×C₁₆	—	2	1	no	—
25	C₃×C₁₆	D₉	1	1	yes	characteristic
26	C₃×C₁₆	C₃×D₃	1	1	yes	characteristic
27	C₃²×C₄	D₁₂	1	1	yes	characteristic
28	C₃×Dic₃	—	12	2	no	—
29	C₄×C₉	—	2	1	no	—
30	C₄×C₉	C₃×D₄	1	1	yes	characteristic
31	Dic₉	—	4	2	no	—
32	Q₃₂	—	9	1	no	Sylow
33	C₃×C₉	Q₃₂	1	1	yes	characteristic, complemented, Sylow
34	C₃×Q₈	—	18	2	no	—
35	Dic₆	—	6	2	no	—
36	C₃×C₈	—	2	1	no	—
37	C₃×C₈	C₂×D₉	1	1	yes	characteristic
38	C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
39	C₂×C₃²	D₂₄	1	1	yes	characteristic
40	C₂×C₉	—	2	1	no	—
41	C₂×C₉	C₃×D₈	1	1	yes	characteristic
42	Q₁₆	—	9	2	no	—
43	C₁₆	C₃×D₉	1	1	yes	characteristic
44	C₃×C₄	—	2	1	no	—
45	C₃×C₄	—	36	2	no	—
46	C₃×C₄	D₃₆	1	1	yes	characteristic
47	C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
48	Dic₃	—	12	2	no	—
49	C₃²	Dic₂₄	1	1	yes	characteristic
50	C₉	—	2	1	no	—
51	C₉	C₃×Q₃₂	1	1	yes	characteristic, complemented
52	Q₈	—	18	2	no	—
53	C₈	C₂×C₃×D₉	1	1	yes	characteristic
54	C₂×C₃	—	2	1	no	—
55	C₂×C₃	D₇₂	1	1	yes	characteristic
56	C₂×C₃	C₃×D₂₄	1	1	yes	characteristic
57	C₄	—	36	2	no	—
58	C₄	C₃×D₃₆	1	1	yes	characteristic
59	C₃	—	2	1	no	—
60	C₃	Dic₇₂	1	1	yes	characteristic, direct product factor
61	C₃	C₃×Dic₂₄	1	1	yes	characteristic
62	C₂	C₃×D₇₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	D₇₂
commutator, cosocle	C₈×C₉	C₂²×C₃
Frattini	C₃×C₈	C₂×C₃×D₃
Fitting	C₃×C₉×C₁₆	C₂
socle, prime radical	C₂×C₃²	D₂₄
hypercenter	C₃×C₁₆	D₉
hypocenter	C₉	C₃×Q₃₂
LCS	G ▷ C₈×C₉ ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×C₈ ◁ C₃×C₁₆

Complements

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

N	H ≅ G/N	type
4. Dic₇₂	C₃	direct (N × H)
33. C₃×C₉	Q₃₂	semidirect
51. C₉	C₃×Q₃₂	semidirect
60. C₃	Dic₇₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×Dic₃₆	2
C₃×C₉×C₁₆	2
C₃×Dic₂₄	3
Dic₇₂	3

Subgroups — C3×Dic72 #11720345

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Dic₇₂ #11720345