Subgroups — #11757416

Order 972.

203 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×A₄	C₃	1	1	yes	maximal, characteristic
2	C₂²×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²⋊(C₂²⋊C₉)	C₃	1	2	yes	maximal, complemented
4	#6866	—	4	1	no	maximal, Sylow
5	C₂×C₃²⋊C₉	—	3	1	no	—
6	C₂²×C₃³	C₃²	1	1	yes	characteristic
7	C₃²×A₄	—	3	1	no	—
8	C₃²×A₄	—	3	2	no	—
9	C₃²×A₄	—	3	3	no	—
10	C₃²×A₄	—	3	6	no	—
11	C₃²×A₄	C₉	1	3	yes	complemented
12	C₂²×C₃×C₉	—	3	1	no	—
13	C₃×C₂²⋊C₉	—	3	2	no	—
14	C₃⁴	—	4	1	no	—
15	C₃²⋊C₉	—	4	2	no	—
16	C₃²⋊C₉	A₄	1	1	yes	characteristic, complemented
17	C₂×C₃³	—	3	1	no	—
18	C₂×C₃×C₉	—	3	3	no	—
19	C₂²×C₃²	—	3	2	no	—
20	C₂²×C₃²	—	3	1	no	—
21	C₂²×C₃²	C₃×C₉	1	1	yes	characteristic
22	C₂²×C₃²	H(3,𝔽₃)	1	1	yes	characteristic
23	C₂²×C₃²	3₋¹⁺²	1	2	yes	—
24	C₃×A₄	—	3	18	no	—
25	C₃×A₄	—	3	3	no	—
26	C₃×A₄	—	3	3	no	—
27	C₃×A₄	—	3	6	no	—
28	C₃×A₄	—	3	9	no	—
29	C₂²×C₉	—	9	1	no	—
30	C₂²⋊C₉	—	9	2	no	—
31	C₃³	—	4	3	no	—
32	C₃³	—	12	3	no	—
33	C₃³	—	12	6	no	—
34	C₃³	—	12	2	no	—
35	C₃³	—	12	1	no	—
36	C₃³	C₃×A₄	1	1	yes	characteristic
37	C₃×C₉	—	3	1	no	—
38	C₃×C₉	—	12	2	no	—
39	C₂×C₃²	—	3	1	no	—
40	C₂×C₃²	—	3	1	no	—
41	C₂×C₃²	—	3	2	no	—
42	C₂×C₃²	—	9	1	no	—
43	C₂×C₃²	—	9	2	no	—
44	C₂×C₉	—	9	3	no	—
45	C₂²×C₃	—	3	1	no	—
46	C₂²×C₃	—	3	2	no	—
47	C₂²×C₃	C₃²⋊C₉	1	1	yes	characteristic
48	C₂²×C₃	AΣL(1,𝔽₂₇)	1	2	yes	—
49	C₂²×C₃	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
50	A₄	—	3	9	no	—
51	C₃²	—	3	1	no	—
52	C₃²	—	3	2	no	—
53	C₃²	—	12	9	no	—
54	C₃²	—	12	18	no	—
55	C₃²	—	12	3	no	—
56	C₃²	—	12	3	no	—
57	C₃²	—	12	6	no	—
58	C₃²	C₉×A₄	1	1	yes	characteristic
59	C₃²	C₉⋊A₄	1	2	yes	—
60	C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
61	C₉	—	9	1	no	—
62	C₉	—	36	2	no	—
63	C₂×C₃	—	3	1	no	—
64	C₂×C₃	—	3	1	no	—
65	C₂×C₃	—	3	2	no	—
66	C₂×C₃	—	9	1	no	—
67	C₂×C₃	—	9	2	no	—
68	C₂²	#6866	1	1	yes	characteristic, complemented, Sylow
69	C₃	—	3	1	no	—
70	C₃	—	3	2	no	—
71	C₃	—	12	9	no	—
72	C₃	(C₃×A₄)⋊C₉	1	1	yes	characteristic
73	C₃	#66363	1	1	yes	characteristic
74	C₃	#66366	1	2	yes	—
75	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃²	C₂²⋊H(3,𝔽₃)
commutator	C₂²×C₃²	C₃×C₉
Frattini	C₃³	C₃×A₄
Fitting	C₂²×C₃²⋊C₉	C₃
socle	C₂²×C₃²	H(3,𝔽₃)
prime radical	C₃³×A₄	C₃
cosocle	C₂²×C₃³	C₃²
hypercenter	C₃²⋊C₉	A₄
hypocenter	C₂²	#6866
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²⋊C₉	C₃	semidirect
3. C₃²⋊(C₂²⋊C₉)	C₃	semidirect
11. C₃²×A₄	C₉	semidirect
16. C₃²⋊C₉	A₄	semidirect
68. C₂²	#6866	semidirect

Maximal subgroups

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

subgroup	index
C₃³×A₄	3
C₂²×C₃²⋊C₉	3
C₃²⋊(C₂²⋊C₉)	3
#6866	4