Subgroups — C₃²×C₉²×C₁₆ #72190096

Order 11664.

5330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₈×C₉²	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₉×C₁₆	C₃	1	4	yes	maximal
3	C₃×C₉²×C₁₆	C₃	1	36	yes	maximal, direct product factor
4	C₃²×C₄×C₉²	C₄	1	1	yes	characteristic
5	C₃³×C₈×C₉	C₂×C₃	1	4	yes	—
6	C₃×C₈×C₉²	C₂×C₃	1	36	yes	—
7	C₂×C₃²×C₉²	C₈	1	1	yes	characteristic
8	C₃⁴×C₁₆	C₃²	1	1	yes	characteristic
9	C₃²×C₉×C₁₆	C₉	1	108	yes	direct product factor
10	C₃²×C₉×C₁₆	C₃²	1	48	yes	—
11	C₉²×C₁₆	C₃²	1	81	yes	direct product factor
12	C₃³×C₄×C₉	C₃×C₄	1	4	yes	—
13	C₃×C₄×C₉²	C₃×C₄	1	36	yes	—
14	C₃²×C₉²	C₁₆	1	1	yes	characteristic, direct product factor, Sylow
15	C₃⁴×C₈	C₂×C₃²	1	1	yes	characteristic
16	C₃²×C₈×C₉	C₂×C₉	1	108	yes	—
17	C₃²×C₈×C₉	C₂×C₃²	1	48	yes	—
18	C₈×C₉²	C₂×C₃²	1	81	yes	—
19	C₂×C₃³×C₉	C₃×C₈	1	4	yes	—
20	C₂×C₃×C₉²	C₃×C₈	1	36	yes	—
21	C₃³×C₁₆	C₃×C₉	1	36	yes	—
22	C₃³×C₁₆	C₃³	1	4	yes	—
23	C₃×C₉×C₁₆	C₃×C₉	1	432	yes	direct product factor
24	C₃×C₉×C₁₆	C₃³	1	36	yes	—
25	C₃⁴×C₄	C₃²×C₄	1	1	yes	characteristic
26	C₃²×C₄×C₉	C₄×C₉	1	108	yes	—
27	C₃²×C₄×C₉	C₃²×C₄	1	48	yes	—
28	C₄×C₉²	C₃²×C₄	1	81	yes	—
29	C₃³×C₉	C₃×C₁₆	1	4	yes	—
30	C₃×C₉²	C₃×C₁₆	1	36	yes	direct product factor
31	C₃³×C₈	C₂×C₃×C₉	1	36	yes	—
32	C₃³×C₈	C₂×C₃³	1	4	yes	—
33	C₃×C₈×C₉	C₂×C₃×C₉	1	432	yes	—
34	C₃×C₈×C₉	C₂×C₃³	1	36	yes	—
35	C₂×C₃⁴	C₃²×C₈	1	1	yes	characteristic
36	C₂×C₃²×C₉	C₈×C₉	1	108	yes	—
37	C₂×C₃²×C₉	C₃²×C₈	1	48	yes	—
38	C₂×C₉²	C₃²×C₈	1	81	yes	—
39	C₃²×C₁₆	C₉²	1	81	yes	direct product factor
40	C₃²×C₁₆	C₃²×C₉	1	48	yes	—
41	C₃²×C₁₆	C₃⁴	1	1	yes	characteristic
42	C₉×C₁₆	C₃²×C₉	1	108	yes	direct product factor
43	C₃³×C₄	C₃×C₄×C₉	1	36	yes	—
44	C₃³×C₄	C₃³×C₄	1	4	yes	—
45	C₃×C₄×C₉	C₃×C₄×C₉	1	432	yes	—
46	C₃×C₄×C₉	C₃³×C₄	1	36	yes	—
47	C₃⁴	C₃²×C₁₆	1	1	yes	characteristic
48	C₃²×C₉	C₉×C₁₆	1	108	yes	direct product factor
49	C₃²×C₉	C₃²×C₁₆	1	48	yes	—
50	C₉²	C₃²×C₁₆	1	81	yes	direct product factor
51	C₃²×C₈	C₂×C₉²	1	81	yes	—
52	C₃²×C₈	C₂×C₃²×C₉	1	48	yes	—
53	C₃²×C₈	C₂×C₃⁴	1	1	yes	characteristic
54	C₈×C₉	C₂×C₃²×C₉	1	108	yes	—
55	C₂×C₃³	C₃×C₈×C₉	1	36	yes	—
56	C₂×C₃³	C₃³×C₈	1	4	yes	—
57	C₂×C₃×C₉	C₃×C₈×C₉	1	432	yes	—
58	C₂×C₃×C₉	C₃³×C₈	1	36	yes	—
59	C₃×C₁₆	C₃×C₉²	1	36	yes	direct product factor
60	C₃×C₁₆	C₃³×C₉	1	4	yes	—
61	C₃²×C₄	C₄×C₉²	1	81	yes	—
62	C₃²×C₄	C₃²×C₄×C₉	1	48	yes	—
63	C₃²×C₄	C₃⁴×C₄	1	1	yes	characteristic
64	C₄×C₉	C₃²×C₄×C₉	1	108	yes	—
65	C₃³	C₃×C₉×C₁₆	1	36	yes	—
66	C₃³	C₃³×C₁₆	1	4	yes	—
67	C₃×C₉	C₃×C₉×C₁₆	1	432	yes	direct product factor
68	C₃×C₉	C₃³×C₁₆	1	36	yes	—
69	C₃×C₈	C₂×C₃×C₉²	1	36	yes	—
70	C₃×C₈	C₂×C₃³×C₉	1	4	yes	—
71	C₂×C₃²	C₈×C₉²	1	81	yes	—
72	C₂×C₃²	C₃²×C₈×C₉	1	48	yes	—
73	C₂×C₃²	C₃⁴×C₈	1	1	yes	characteristic
74	C₂×C₉	C₃²×C₈×C₉	1	108	yes	—
75	C₁₆	C₃²×C₉²	1	1	yes	characteristic, direct product factor, Sylow
76	C₃×C₄	C₃×C₄×C₉²	1	36	yes	—
77	C₃×C₄	C₃³×C₄×C₉	1	4	yes	—
78	C₃²	C₉²×C₁₆	1	81	yes	direct product factor
79	C₃²	C₃²×C₉×C₁₆	1	48	yes	—
80	C₃²	C₃⁴×C₁₆	1	1	yes	characteristic
81	C₉	C₃²×C₉×C₁₆	1	108	yes	direct product factor
82	C₈	C₂×C₃²×C₉²	1	1	yes	characteristic
83	C₂×C₃	C₃×C₈×C₉²	1	36	yes	—
84	C₂×C₃	C₃³×C₈×C₉	1	4	yes	—
85	C₄	C₃²×C₄×C₉²	1	1	yes	characteristic
86	C₃	C₃×C₉²×C₁₆	1	36	yes	direct product factor
87	C₃	C₃³×C₉×C₁₆	1	4	yes	—
88	C₂	C₃²×C₈×C₉²	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃²×C₈	C₂×C₃⁴
socle, prime radical	C₂×C₃⁴	C₃²×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₉²×C₁₆	C₃	direct (N × H)
9. C₃²×C₉×C₁₆	C₉	direct (N × H)
11. C₉²×C₁₆	C₃²	direct (N × H)
14. C₃²×C₉²	C₁₆	direct (N × H)
23. C₃×C₉×C₁₆	C₃×C₉	direct (N × H)
30. C₃×C₉²	C₃×C₁₆	direct (N × H)
39. C₃²×C₁₆	C₉²	direct (N × H)
42. C₉×C₁₆	C₃²×C₉	direct (N × H)
48. C₃²×C₉	C₉×C₁₆	direct (N × H)
50. C₉²	C₃²×C₁₆	direct (N × H)
59. C₃×C₁₆	C₃×C₉²	direct (N × H)
67. C₃×C₉	C₃×C₉×C₁₆	direct (N × H)
75. C₁₆	C₃²×C₉²	direct (N × H)
78. C₃²	C₉²×C₁₆	direct (N × H)
81. C₉	C₃²×C₉×C₁₆	direct (N × H)
86. C₃	C₃×C₉²×C₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₈×C₉²	2
C₃³×C₉×C₁₆	3
C₃×C₉²×C₁₆	3

Subgroups — C32×C92×C16 #72190096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₉²×C₁₆ #72190096