Subgroups — #14515862

Order 1944.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10619912	C₃	1	1	yes	maximal, characteristic
2	C₃×Q₈×C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
3	#10619570	C₃	1	2	yes	maximal, complemented
4	C₂×#6873	—	4	1	no	maximal
5	C₃×C₄×C₂₇	—	3	1	no	—
6	#6873	—	4	1	no	Sylow
7	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
8	Q₈⋊H(3,𝔽₃)	C₉	1	1	yes	characteristic
9	#6204	C₉	1	2	yes	—
10	C₉⋊SL(2,𝔽₃)	—	3	2	no	—
11	C₉×SL(2,𝔽₃)	—	3	1	no	—
12	Q₈×C₂₇	—	3	1	no	—
13	Q₈⋊C₂₇	—	3	2	no	—
14	C₂×#737	—	4	1	no	—
15	C₂×C₂₇⋊C₃	—	4	2	no	—
16	C₂×C₃×C₂₇	A₄	1	1	yes	characteristic
17	C₃×C₄×C₉	—	3	1	no	—
18	C₄×C₂₇	—	3	3	no	—
19	#737	—	4	1	no	—
20	C₂₇⋊C₃	—	4	2	no	—
21	C₃×C₂₇	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
22	C₃²×Q₈	C₃×C₉	1	1	yes	characteristic
23	C₃×SL(2,𝔽₃)	—	3	1	no	—
24	Q₈×C₉	H(3,𝔽₃)	1	1	yes	characteristic
25	Q₈×C₉	3₋¹⁺²	1	2	yes	—
26	Q₈⋊C₉	—	3	2	no	—
27	C₂×3₋¹⁺²	—	4	2	no	—
28	C₂×3₋¹⁺²	—	12	2	no	—
29	C₂×H(3,𝔽₃)	—	4	1	no	—
30	C₂×C₃×C₉	—	12	1	no	—
31	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
32	C₂×C₂₇	—	3	1	no	—
33	C₂×C₂₇	—	12	2	no	—
34	C₃²×C₄	—	3	1	no	—
35	C₄×C₉	—	3	2	no	—
36	C₄×C₉	—	3	1	no	—
37	3₋¹⁺²	—	4	2	no	—
38	3₋¹⁺²	—	12	2	no	—
39	H(3,𝔽₃)	—	4	1	no	—
40	C₃×C₉	—	12	1	no	—
41	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
42	C₂₇	—	3	1	no	—
43	C₂₇	—	12	2	no	—
44	C₃×Q₈	—	3	1	no	—
45	C₃×Q₈	C₃²⋊C₉	1	1	yes	characteristic
46	SL(2,𝔽₃)	—	9	1	no	—
47	C₂×C₃²	—	12	1	no	—
48	C₂×C₃²	C₉×A₄	1	1	yes	characteristic
49	C₂×C₉	—	12	2	no	—
50	C₂×C₉	C₉⋊A₄	1	2	yes	—
51	C₂×C₉	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
52	C₃×C₄	—	3	3	no	—
53	C₃×C₄	—	3	1	no	—
54	C₃²	—	12	1	no	—
55	C₃²	C₉×SL(2,𝔽₃)	1	1	yes	characteristic
56	C₉	—	12	2	no	—
57	C₉	C₉⋊SL(2,𝔽₃)	1	2	yes	—
58	C₉	Q₈⋊H(3,𝔽₃)	1	1	yes	characteristic
59	Q₈	#6873	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₃	—	3	1	no	—
61	C₂×C₃	—	36	1	no	—
62	C₂×C₃	(C₃×A₄)⋊C₉	1	1	yes	characteristic
63	C₄	—	3	1	no	—
64	C₃	—	3	1	no	—
65	C₃	—	36	1	no	—
66	C₃	(C₃×SL(2,𝔽₃))⋊C₉	1	1	yes	characteristic
67	C₂	#11757427	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	C₂²⋊H(3,𝔽₃)
commutator	C₃²×Q₈	C₃×C₉
Frattini	C₂×C₃×C₉	C₃×A₄
Fitting	C₃×Q₈×C₂₇	C₃
socle	C₂×C₃	(C₃×A₄)⋊C₉
prime radical	Q₈⋊H(3,𝔽₃)	C₉
cosocle	C₃×Q₈×C₉	C₃²
hypercenter	C₂×C₃×C₂₇	A₄
hypocenter	Q₈	#6873
LCS	G ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₃×C₉ ◁ C₂×C₃×C₂₇

Complements

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

N	H ≅ G/N	type
2. C₃×Q₈×C₂₇	C₃	semidirect
3. #10619570	C₃	semidirect
21. C₃×C₂₇	SL(2,𝔽₃)	semidirect
59. Q₈	#6873	semidirect

Maximal subgroups

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

subgroup	index
#10619912	3
C₃×Q₈×C₂₇	3
#10619570	3
C₂×#6873	4