Subgroups — #11757775

Order 972.

225 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₂²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₂²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₉×A₄	C₃	1	3	yes	maximal, complemented
4	C₂²×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃²⋊(C₂²⋊C₉)	C₃	1	1	yes	maximal, characteristic, complemented
6	(C₃×A₄)⋊C₉	C₃	1	6	yes	maximal, complemented
7	C₉×H(3,𝔽₃)	—	4	1	no	maximal, Sylow
8	C₂×C₃²⋊C₉	—	3	1	no	—
9	C₂²×C₃³	C₃²	1	1	yes	characteristic
10	C₃²×A₄	C₃²	1	3	yes	—
11	C₂²×C₃×C₉	C₃²	1	3	yes	—
12	C₂²⋊H(3,𝔽₃)	C₉	1	9	yes	complemented
13	C₃×C₂²⋊C₉	—	3	3	no	—
14	C₃×C₂²⋊C₉	C₃²	1	2	yes	—
15	C₃×C₂²⋊C₉	C₃²	1	3	yes	—
16	C₃×C₂²⋊C₉	C₃²	1	1	yes	characteristic
17	C₉×A₄	—	3	9	no	—
18	C₃×H(3,𝔽₃)	—	4	1	no	—
19	C₃²×C₉	—	4	1	no	—
20	C₃²×C₉	—	4	3	no	—
21	C₃²⋊C₉	—	4	1	no	—
22	C₃²⋊C₉	—	4	6	no	—
23	C₃²⋊C₉	A₄	1	1	yes	characteristic, complemented
24	C₂×C₃³	—	3	1	no	—
25	C₂×C₃×C₉	—	3	3	no	—
26	C₂²×C₃²	—	3	1	no	—
27	C₂²×C₃²	—	3	2	no	—
28	C₂²×C₃²	C₃×C₉	1	3	yes	complemented
29	C₂²×C₃²	C₃³	1	1	yes	characteristic
30	C₃×A₄	—	3	3	no	—
31	C₃×A₄	—	3	6	no	—
32	C₃×A₄	C₃×C₉	1	9	yes	complemented
33	C₂²×C₉	—	3	3	no	—
34	C₂²⋊C₉	—	3	3	no	—
35	C₂²⋊C₉	—	3	2	no	—
36	C₂²⋊C₉	H(3,𝔽₃)	1	3	yes	complemented
37	C₃³	—	4	3	no	—
38	C₃³	C₃×A₄	1	1	yes	characteristic
39	H(3,𝔽₃)	—	4	9	no	—
40	C₃×C₉	—	4	1	no	—
41	C₃×C₉	—	4	2	no	—
42	C₃×C₉	—	4	3	no	—
43	C₃×C₉	—	12	9	no	—
44	C₃×C₉	—	12	3	no	—
45	C₃×C₉	C₃×A₄	1	3	yes	—
46	C₂×C₃²	—	3	1	no	—
47	C₂×C₃²	—	3	3	no	—
48	C₂×C₃²	—	9	1	no	—
49	C₂×C₃²	—	9	2	no	—
50	C₂×C₉	—	9	3	no	—
51	C₂²×C₃	—	3	3	no	—
52	C₂²×C₃	C₃²×C₉	1	1	yes	characteristic
53	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
54	C₂²×C₃	#737	1	2	yes	—
55	A₄	—	3	9	no	—
56	C₃²	—	3	1	no	—
57	C₃²	—	3	2	no	—
58	C₃²	—	4	9	no	—
59	C₃²	—	12	3	no	—
60	C₃²	—	12	6	no	—
61	C₃²	C₉×A₄	1	3	yes	complemented
62	C₃²	C₃²×A₄	1	1	yes	characteristic
63	C₉	—	3	3	no	—
64	C₉	—	4	3	no	—
65	C₉	—	12	3	no	—
66	C₉	—	12	2	no	—
67	C₂×C₃	—	3	1	no	—
68	C₂×C₃	—	3	1	no	—
69	C₂×C₃	—	3	2	no	—
70	C₂×C₃	—	9	3	no	—
71	C₂²	C₉×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
72	C₃	—	3	3	no	—
73	C₃	—	12	9	no	—
74	C₃	C₃×C₉×A₄	1	1	yes	characteristic
75	C₃	A₄×H(3,𝔽₃)	1	1	yes	characteristic
76	C₃	#66440	1	2	yes	—
77	C₂	—	3	1	no	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×A₄
commutator	C₂²×C₃	C₃²×C₉
Fitting	C₂²×C₃²⋊C₉	C₃
socle, cosocle	C₂²×C₃²	C₃³
prime radical	C₃×C₂²⋊H(3,𝔽₃)	C₃
hypercenter	C₃²⋊C₉	A₄
hypocenter	C₂²	C₉×H(3,𝔽₃)
LCS	G ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
2. C₃²×C₂²⋊C₉	C₃	semidirect
3. C₃×C₉×A₄	C₃	semidirect
4. C₂²×C₃²⋊C₉	C₃	semidirect
5. C₃²⋊(C₂²⋊C₉)	C₃	semidirect
6. (C₃×A₄)⋊C₉	C₃	semidirect
12. C₂²⋊H(3,𝔽₃)	C₉	semidirect
23. C₃²⋊C₉	A₄	semidirect
28. C₂²×C₃²	C₃×C₉	semidirect
32. C₃×A₄	C₃×C₉	semidirect
36. C₂²⋊C₉	H(3,𝔽₃)	semidirect
61. C₃²	C₉×A₄	semidirect
71. C₂²	C₉×H(3,𝔽₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₂²⋊H(3,𝔽₃)	3
C₃²×C₂²⋊C₉	3
C₃×C₉×A₄	3
C₂²×C₃²⋊C₉	3
C₃²⋊(C₂²⋊C₉)	3
(C₃×A₄)⋊C₉	3
C₉×H(3,𝔽₃)	4