Subgroups — C₃×C₉×Q₆₄ #13136120

Order 1728.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×Q₃₂	C₂	1	2	yes	maximal
2	C₃×C₉×C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₃²×Q₆₄	C₃	1	1	yes	maximal, characteristic
4	C₉×Q₆₄	C₃	1	3	yes	maximal, direct product factor
5	C₃×C₉×Q₁₆	—	2	2	no	—
6	C₃×C₉×C₁₆	C₂²	1	1	yes	characteristic
7	C₃²×Q₃₂	C₂×C₃	1	2	yes	—
8	C₃²×C₃₂	C₂×C₃	1	1	yes	characteristic
9	C₉×Q₃₂	C₂×C₃	1	6	yes	—
10	C₉×C₃₂	C₂×C₃	1	3	yes	—
11	C₃×Q₈×C₉	—	4	2	no	—
12	C₃×C₈×C₉	D₄	1	1	yes	characteristic
13	C₃×Q₆₄	C₉	1	3	yes	direct product factor
14	C₃×Q₆₄	C₃²	1	1	yes	characteristic
15	C₃²×Q₁₆	—	2	2	no	—
16	C₃²×C₁₆	C₂²×C₃	1	1	yes	characteristic
17	C₉×Q₁₆	—	2	6	no	—
18	C₉×C₁₆	C₂²×C₃	1	3	yes	—
19	C₃×C₄×C₉	—	8	2	no	—
20	C₃×C₄×C₉	D₈	1	1	yes	characteristic
21	C₃×Q₃₂	C₂×C₉	1	6	yes	—
22	C₃×Q₃₂	C₂×C₃²	1	2	yes	—
23	C₃×C₃₂	C₂×C₉	1	3	yes	—
24	C₃×C₃₂	C₂×C₃²	1	1	yes	characteristic
25	C₃²×Q₈	—	4	2	no	—
26	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
27	Q₈×C₉	—	4	6	no	—
28	C₈×C₉	C₃×D₄	1	3	yes	—
29	Q₆₄	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
30	C₂×C₃×C₉	D₁₆	1	1	yes	characteristic
31	C₃×Q₁₆	—	2	6	no	—
32	C₃×Q₁₆	—	2	2	no	—
33	C₃×C₁₆	C₂²×C₉	1	3	yes	—
34	C₃×C₁₆	C₂²×C₃²	1	1	yes	characteristic
35	C₃²×C₄	—	8	2	no	—
36	C₃²×C₄	C₃×D₈	1	1	yes	characteristic
37	C₄×C₉	—	8	6	no	—
38	C₄×C₉	C₃×D₈	1	3	yes	—
39	Q₃₂	C₂×C₃×C₉	1	2	yes	—
40	C₃₂	C₂×C₃×C₉	1	1	yes	characteristic
41	C₃×C₉	Q₆₄	1	1	yes	characteristic, direct product factor, Sylow
42	C₃×Q₈	—	4	6	no	—
43	C₃×Q₈	—	4	2	no	—
44	C₃×C₈	D₄×C₉	1	3	yes	—
45	C₃×C₈	C₃²×D₄	1	1	yes	characteristic
46	C₂×C₃²	C₃×D₁₆	1	1	yes	characteristic
47	C₂×C₉	C₃×D₁₆	1	3	yes	—
48	Q₁₆	—	2	2	no	—
49	C₁₆	C₂²×C₃×C₉	1	1	yes	characteristic
50	C₃×C₄	—	8	6	no	—
51	C₃×C₄	—	8	2	no	—
52	C₃×C₄	C₉×D₈	1	3	yes	—
53	C₃×C₄	C₃²×D₈	1	1	yes	characteristic
54	C₃²	C₃×Q₆₄	1	1	yes	characteristic
55	C₉	C₃×Q₆₄	1	3	yes	direct product factor
56	Q₈	—	4	2	no	—
57	C₈	C₃×D₄×C₉	1	1	yes	characteristic
58	C₂×C₃	C₉×D₁₆	1	3	yes	—
59	C₂×C₃	C₃²×D₁₆	1	1	yes	characteristic
60	C₄	—	8	2	no	—
61	C₄	C₃×C₉×D₈	1	1	yes	characteristic
62	C₃	C₉×Q₆₄	1	3	yes	direct product factor
63	C₃	C₃²×Q₆₄	1	1	yes	characteristic
64	C₂	C₃×C₉×D₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₉	D₁₆
commutator	C₁₆	C₂²×C₃×C₉
Frattini, cosocle	C₃×C₁₆	C₂²×C₃²
socle, prime radical	C₂×C₃²	C₃×D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₉ ◁ C₃×C₄×C₉ ◁ C₃×C₈×C₉ ◁ C₃×C₉×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₉×Q₆₄	C₃	direct (N × H)
13. C₃×Q₆₄	C₉	direct (N × H)
29. Q₆₄	C₃×C₉	direct (N × H)
41. C₃×C₉	Q₆₄	direct (N × H)
55. C₉	C₃×Q₆₄	direct (N × H)
62. C₃	C₉×Q₆₄	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×C₉×Q₃₂	2
C₃×C₉×C₃₂	2
C₃²×Q₆₄	3
C₉×Q₆₄	3

Subgroups — C3×C9×Q64 #13136120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×Q₆₄ #13136120