Subgroups — #14517273

Order 1944.

779 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×Q₈	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×Q₈⋊H(3,𝔽₃)	C₃	1	12	yes	maximal, complemented
3	C₂×C₃²⋊H(3,𝔽₃)	—	4	1	no	maximal
4	C₃⁴×C₄	—	3	1	no	—
5	C₃²⋊H(3,𝔽₃)	—	4	1	no	Sylow
6	C₃³×Q₈	—	3	12	no	—
7	C₃³×Q₈	C₃²	1	4	yes	—
8	C₃²×SL(2,𝔽₃)	C₃²	1	9	yes	complemented
9	Q₈⋊H(3,𝔽₃)	—	3	36	no	—
10	C₂×C₃⁴	A₄	1	1	yes	characteristic
11	C₂×C₃×H(3,𝔽₃)	—	4	12	no	—
12	C₃³×C₄	—	3	36	no	—
13	C₃³×C₄	—	3	4	no	—
14	C₃⁴	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
15	C₃×H(3,𝔽₃)	—	4	12	no	—
16	C₃²×Q₈	—	3	27	no	—
17	C₃²×Q₈	—	3	12	no	—
18	C₃²×Q₈	H(3,𝔽₃)	1	12	yes	complemented
19	C₃²×Q₈	C₃³	1	1	yes	characteristic
20	C₃×SL(2,𝔽₃)	—	3	36	no	—
21	C₂×C₃³	—	3	12	no	—
22	C₂×C₃³	—	4	9	no	—
23	C₂×C₃³	C₃×A₄	1	4	yes	—
24	C₂×H(3,𝔽₃)	—	12	36	no	—
25	C₃²×C₄	—	3	81	no	—
26	C₃²×C₄	—	3	36	no	—
27	C₃²×C₄	—	3	12	no	—
28	C₃²×C₄	—	3	1	no	—
29	C₃³	—	3	12	no	—
30	C₃³	—	4	9	no	—
31	C₃³	C₃×SL(2,𝔽₃)	1	4	yes	—
32	H(3,𝔽₃)	—	12	36	no	—
33	C₃×Q₈	—	3	12	no	—
34	C₃×Q₈	C₃×H(3,𝔽₃)	1	4	yes	—
35	SL(2,𝔽₃)	—	9	9	no	—
36	C₂×C₃²	—	3	27	no	—
37	C₂×C₃²	—	3	12	no	—
38	C₂×C₃²	—	12	36	no	—
39	C₂×C₃²	C₂²⋊H(3,𝔽₃)	1	12	yes	—
40	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
41	C₃×C₄	—	3	36	no	—
42	C₃×C₄	—	3	4	no	—
43	C₃²	—	3	27	no	—
44	C₃²	—	3	12	no	—
45	C₃²	—	12	36	no	—
46	C₃²	Q₈⋊H(3,𝔽₃)	1	12	yes	complemented
47	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
48	Q₈	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₃	—	3	12	no	—
50	C₂×C₃	—	36	9	no	—
51	C₂×C₃	C₃×C₂²⋊H(3,𝔽₃)	1	4	yes	—
52	C₄	—	3	1	no	—
53	C₃	—	3	12	no	—
54	C₃	—	36	9	no	—
55	C₃	C₃×Q₈⋊H(3,𝔽₃)	1	4	yes	—
56	C₂	#11757866	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃²	C₃²×A₄
commutator, cosocle	C₃²×Q₈	C₃³
Fitting	C₃⁴×Q₈	C₃
hypercenter	C₂×C₃⁴	A₄
hypocenter	Q₈	C₃²⋊H(3,𝔽₃)
LCS	G ▷ C₃²×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃⁴

Complements

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

N	H ≅ G/N	type
1. C₃⁴×Q₈	C₃	semidirect
2. C₃×Q₈⋊H(3,𝔽₃)	C₃	semidirect
8. C₃²×SL(2,𝔽₃)	C₃²	semidirect
14. C₃⁴	SL(2,𝔽₃)	semidirect
18. C₃²×Q₈	H(3,𝔽₃)	semidirect
46. C₃²	Q₈⋊H(3,𝔽₃)	semidirect
48. Q₈	C₃²⋊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
C₃⁴×Q₈	3
C₃×Q₈⋊H(3,𝔽₃)	3
C₂×C₃²⋊H(3,𝔽₃)	4