Subgroups — C₃²×D₇₂ #13047793

Order 1296.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×D₃₆	C₂	1	2	yes	maximal, complemented
2	C₃²×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×D₂₄	—	3	1	no	maximal
4	C₃×D₇₂	C₃	1	4	yes	maximal, direct product factor
5	C₂×C₃²×D₉	—	2	2	no	—
6	C₃²×C₄×C₉	C₂²	1	1	yes	characteristic
7	C₃²×D₁₂	—	3	2	no	—
8	C₃³×C₈	D₃	1	1	yes	characteristic
9	C₃×D₃₆	C₂×C₃	1	8	yes	complemented
10	C₃×C₈×C₉	—	2	4	no	—
11	C₃×C₈×C₉	C₂×C₃	1	4	yes	complemented
12	C₂×C₃²×C₉	D₄	1	1	yes	characteristic
13	C₃²×D₉	—	4	2	no	—
14	C₃²×D₈	—	9	1	no	—
15	C₃×D₂₄	—	3	4	no	—
16	D₇₂	C₃²	1	1	yes	characteristic, direct product factor
17	C₂×C₃²×D₃	—	6	2	no	—
18	C₃³×C₄	C₂×D₃	1	1	yes	characteristic
19	C₂×C₃×D₉	—	2	8	no	—
20	C₃×C₄×C₉	—	2	4	no	—
21	C₃×C₄×C₉	C₂²×C₃	1	4	yes	—
22	C₃²×C₉	D₈	1	1	yes	characteristic, complemented, Sylow
23	C₃²×D₄	—	9	2	no	—
24	C₃×D₁₂	—	3	8	no	—
25	C₃²×C₈	—	2	4	no	—
26	C₃²×C₈	D₉	1	1	yes	characteristic, complemented
27	C₃²×C₈	C₃×D₃	1	4	yes	—
28	D₃₆	C₂×C₃²	1	2	yes	complemented
29	C₈×C₉	—	2	4	no	—
30	C₈×C₉	C₂×C₃²	1	1	yes	characteristic, complemented
31	C₂×C₃³	D₁₂	1	1	yes	characteristic
32	C₃²×D₃	—	12	2	no	—
33	C₂×C₃×C₉	—	2	4	no	—
34	C₂×C₃×C₉	C₃×D₄	1	4	yes	—
35	C₃×D₉	—	4	8	no	—
36	C₃×D₈	—	9	4	no	—
37	D₂₄	—	3	1	no	—
38	C₂²×C₃²	—	18	2	no	—
39	C₂×C₃×D₃	—	6	8	no	—
40	C₃²×C₄	—	2	4	no	—
41	C₃²×C₄	C₂×D₉	1	1	yes	characteristic
42	C₃²×C₄	C₂×C₃×D₃	1	4	yes	—
43	C₂×D₉	—	2	2	no	—
44	C₄×C₉	—	2	4	no	—
45	C₄×C₉	C₂²×C₃²	1	1	yes	characteristic
46	C₃³	D₂₄	1	1	yes	characteristic
47	C₃×C₉	—	2	4	no	—
48	C₃×C₉	C₃×D₈	1	4	yes	complemented
49	C₃×D₄	—	9	8	no	—
50	D₁₂	—	3	2	no	—
51	C₃×C₈	—	2	4	no	—
52	C₃×C₈	C₃×D₉	1	4	yes	complemented
53	C₃×C₈	C₃²×D₃	1	1	yes	characteristic
54	C₂×C₃²	—	2	4	no	—
55	C₂×C₃²	—	36	2	no	—
56	C₂×C₃²	D₃₆	1	1	yes	characteristic
57	C₂×C₃²	C₃×D₁₂	1	4	yes	—
58	C₃×D₃	—	12	8	no	—
59	C₂×C₉	—	2	4	no	—
60	C₂×C₉	C₃²×D₄	1	1	yes	characteristic
61	D₉	—	4	2	no	—
62	D₈	—	9	1	no	Sylow
63	C₂²×C₃	—	18	8	no	—
64	C₂×D₃	—	6	2	no	—
65	C₃×C₄	—	2	4	no	—
66	C₃×C₄	C₂×C₃×D₉	1	4	yes	—
67	C₃×C₄	C₂×C₃²×D₃	1	1	yes	characteristic
68	C₃²	—	2	4	no	—
69	C₃²	D₇₂	1	1	yes	characteristic, direct product factor
70	C₃²	C₃×D₂₄	1	4	yes	—
71	C₉	—	2	4	no	—
72	C₉	C₃²×D₈	1	1	yes	characteristic, complemented
73	D₄	—	9	2	no	—
74	C₈	C₃²×D₉	1	1	yes	characteristic, complemented
75	C₂×C₃	—	2	4	no	—
76	C₂×C₃	—	36	8	no	—
77	C₂×C₃	C₃×D₃₆	1	4	yes	—
78	C₂×C₃	C₃²×D₁₂	1	1	yes	characteristic
79	D₃	—	12	2	no	—
80	C₂²	—	18	2	no	—
81	C₄	C₂×C₃²×D₉	1	1	yes	characteristic
82	C₃	—	2	4	no	—
83	C₃	C₃×D₇₂	1	4	yes	direct product factor
84	C₃	C₃²×D₂₄	1	1	yes	characteristic
85	C₂	—	36	2	no	—
86	C₂	C₃²×D₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²	D₃₆
commutator, cosocle	C₄×C₉	C₂²×C₃²
Frattini	C₃×C₄	C₂×C₃²×D₃
Fitting	C₃²×C₈×C₉	C₂
socle	C₂×C₃³	D₁₂
hypercenter	C₃²×C₈	D₉
hypocenter	C₉	C₃²×D₈
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×C₄ ◁ C₃²×C₈

Complements

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

N	H ≅ G/N	type
1. C₃²×D₃₆	C₂	semidirect
2. C₃²×C₈×C₉	C₂	semidirect
4. C₃×D₇₂	C₃	direct (N × H)
9. C₃×D₃₆	C₂×C₃	semidirect
11. C₃×C₈×C₉	C₂×C₃	semidirect
16. D₇₂	C₃²	direct (N × H)
22. C₃²×C₉	D₈	semidirect
26. C₃²×C₈	D₉	semidirect
28. D₃₆	C₂×C₃²	semidirect
30. C₈×C₉	C₂×C₃²	semidirect
48. C₃×C₉	C₃×D₈	semidirect
52. C₃×C₈	C₃×D₉	semidirect
69. C₃²	D₇₂	direct (N × H)
72. C₉	C₃²×D₈	semidirect
74. C₈	C₃²×D₉	semidirect
83. C₃	C₃×D₇₂	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×D₃₆	2
C₃²×C₈×C₉	2
C₃²×D₂₄	3
C₃×D₇₂	3

Subgroups — C32×D72 #13047793

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×D₇₂ #13047793