Subgroups — C₃²×#994 #11724639

Order 864.

452 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#41⋊C₃	C₂	1	3	yes	maximal, complemented
2	C₃²×H(5,𝔽₂)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#994	C₃	1	12	yes	maximal, direct product factor
4	C₃³×Q₈	—	4	1	no	maximal
5	C₃²×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
6	C₃²×#41	—	3	1	no	—
7	C₃²×#41	C₂×C₃	1	3	yes	—
8	C₂×C₃²×D₄	—	3	3	no	—
9	C₃×#41⋊C₃	C₂×C₃	1	36	yes	complemented
10	C₃³×C₄	—	4	3	no	—
11	C₃×H(5,𝔽₂)	C₃²	1	4	yes	complemented
12	#994	C₃²	1	9	yes	direct product factor
13	C₂³×C₃²	—	3	2	no	—
14	C₃²×Q₈	—	4	12	no	—
15	C₃²×Q₈	A₄	1	1	yes	characteristic
16	C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
17	C₃²×D₄	—	3	3	no	—
18	C₃²×D₄	—	3	3	no	—
19	C₂×C₃²×C₄	—	3	3	no	—
20	C₃×SL(2,𝔽₃)	C₂²×C₃	1	12	yes	complemented
21	C₂×C₃³	—	4	1	no	—
22	C₃×#41	—	3	4	no	—
23	C₃×#41	C₂×C₃²	1	12	yes	—
24	C₂×C₃×D₄	—	3	12	no	—
25	#41⋊C₃	C₂×C₃²	1	27	yes	complemented
26	C₂²×C₃²	—	3	3	no	—
27	C₂²×C₃²	—	12	2	no	—
28	C₃²×C₄	—	3	1	no	—
29	C₃²×C₄	—	4	36	no	—
30	C₃²×C₄	C₂×A₄	1	3	yes	—
31	H(5,𝔽₂)	C₃³	1	1	yes	characteristic, complemented, Sylow
32	C₃³	—	4	1	no	Sylow
33	C₂³×C₃	—	3	8	no	—
34	C₃×Q₈	—	4	9	no	—
35	C₃×Q₈	C₃×A₄	1	4	yes	—
36	C₃×Q₈	C₂²×C₃²	1	4	yes	—
37	C₃×D₄	—	3	12	no	—
38	C₃×D₄	—	3	12	no	—
39	C₂×C₃×C₄	—	3	12	no	—
40	SL(2,𝔽₃)	C₂²×C₃²	1	9	yes	complemented
41	C₂×C₃²	—	4	12	no	—
42	C₂×C₃²	—	6	3	no	—
43	C₂×C₃²	C₂²×A₄	1	1	yes	characteristic
44	#41	—	3	1	no	—
45	#41	C₂×C₃³	1	3	yes	—
46	C₂×D₄	—	3	3	no	—
47	C₂²×C₃	—	3	12	no	—
48	C₂²×C₃	—	12	8	no	—
49	C₃×C₄	—	3	4	no	—
50	C₃×C₄	—	4	27	no	—
51	C₃×C₄	C₂×C₃×A₄	1	12	yes	—
52	C₃²	—	4	12	no	—
53	C₃²	#994	1	1	yes	characteristic, direct product factor
54	C₂³	—	3	2	no	—
55	Q₈	C₃²×A₄	1	1	yes	characteristic
56	Q₈	C₂²×C₃³	1	1	yes	characteristic
57	D₄	—	3	3	no	—
58	D₄	—	3	3	no	—
59	C₂×C₄	—	3	3	no	—
60	C₂×C₃	—	4	9	no	—
61	C₂×C₃	—	6	12	no	—
62	C₂×C₃	C₂²×C₃×A₄	1	4	yes	—
63	C₂²	—	3	3	no	—
64	C₂²	—	12	2	no	—
65	C₄	—	3	1	no	—
66	C₄	C₂×C₃²×A₄	1	3	yes	—
67	C₃	—	4	9	no	—
68	C₃	C₃×#994	1	4	yes	direct product factor
69	C₂	—	6	3	no	—
70	C₂	C₂²×C₃²×A₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃²	C₂²×A₄
commutator, cosocle, hypocenter	Q₈	C₂²×C₃³
Frattini	C₂	C₂²×C₃²×A₄
Fitting	C₃²×H(5,𝔽₂)	C₃
hypercenter	C₃²×Q₈	A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₃² ◁ C₃²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₃²×#41⋊C₃	C₂	semidirect
2. C₃²×H(5,𝔽₂)	C₃	semidirect
3. C₃×#994	C₃	direct (N × H)
5. C₃²×SL(2,𝔽₃)	C₂²	semidirect
9. C₃×#41⋊C₃	C₂×C₃	semidirect
11. C₃×H(5,𝔽₂)	C₃²	semidirect
12. #994	C₃²	direct (N × H)
20. C₃×SL(2,𝔽₃)	C₂²×C₃	semidirect
25. #41⋊C₃	C₂×C₃²	semidirect
31. H(5,𝔽₂)	C₃³	semidirect
40. SL(2,𝔽₃)	C₂²×C₃²	semidirect
53. C₃²	#994	direct (N × H)
68. C₃	C₃×#994	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×#41⋊C₃	2
C₃²×H(5,𝔽₂)	3
C₃×#994	3
C₃³×Q₈	4

Subgroups — C32×#994 #11724639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#994 #11724639