Subgroups — #14517267

Order 1944.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#737	C₃	1	1	yes	maximal, characteristic
2	#10619570	C₃	1	8	yes	maximal, complemented
3	C₃×Q₈⋊C₂₇	C₃	1	4	yes	maximal, complemented
4	C₂×#6903	—	4	1	no	maximal
5	C₄×#737	—	3	1	no	—
6	#6903	—	4	1	no	Sylow
7	Q₈×3₋¹⁺²	C₉	1	8	yes	—
8	Q₈×H(3,𝔽₃)	C₉	1	1	yes	characteristic
9	C₃×Q₈×C₉	C₃²	1	4	yes	—
10	Q₈⋊C₂₇	C₃²	1	1	yes	characteristic
11	Q₈⋊C₂₇	C₃²	1	8	yes	—
12	C₂×#737	A₄	1	1	yes	characteristic
13	C₂×C₂₇⋊C₃	—	4	8	no	—
14	C₂×C₃×C₂₇	—	4	4	no	—
15	C₄×3₋¹⁺²	—	3	8	no	—
16	C₄×H(3,𝔽₃)	—	3	1	no	—
17	C₃×C₄×C₉	—	3	4	no	—
18	#737	SL(2,𝔽₃)	1	1	yes	characteristic
19	C₂₇⋊C₃	—	4	8	no	—
20	C₃×C₂₇	—	4	4	no	—
21	C₃²×Q₈	C₃×C₉	1	4	yes	—
22	Q₈×C₉	C₃×C₉	1	8	yes	—
23	Q₈×C₉	C₃³	1	1	yes	characteristic
24	C₂×3₋¹⁺²	C₂²⋊C₉	1	8	yes	—
25	C₂×H(3,𝔽₃)	C₂²⋊C₉	1	1	yes	characteristic
26	C₂×C₃×C₉	C₃×A₄	1	4	yes	—
27	C₂×C₂₇	—	4	1	no	—
28	C₂×C₂₇	—	4	8	no	—
29	C₃²×C₄	—	3	4	no	—
30	C₄×C₉	—	3	8	no	—
31	C₄×C₉	—	3	1	no	—
32	3₋¹⁺²	Q₈⋊C₉	1	8	yes	—
33	H(3,𝔽₃)	Q₈⋊C₉	1	1	yes	characteristic
34	C₃×C₉	C₃×SL(2,𝔽₃)	1	4	yes	—
35	C₂₇	—	4	1	no	—
36	C₂₇	—	4	8	no	—
37	C₃×Q₈	—	3	4	no	—
38	C₃×Q₈	C₃²×C₉	1	1	yes	characteristic
39	C₂×C₃²	C₃×C₂²⋊C₉	1	4	yes	—
40	C₂×C₉	C₃×C₂²⋊C₉	1	8	yes	—
41	C₂×C₉	C₃²×A₄	1	1	yes	characteristic
42	C₃×C₄	—	3	1	no	—
43	C₃×C₄	—	9	4	no	—
44	C₃²	C₃×Q₈⋊C₉	1	4	yes	—
45	C₉	C₃×Q₈⋊C₉	1	8	yes	—
46	C₉	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
47	Q₈	#6903	1	1	yes	characteristic, complemented, Sylow
48	C₂×C₃	—	3	4	no	—
49	C₂×C₃	C₃²×C₂²⋊C₉	1	1	yes	characteristic
50	C₄	—	3	1	no	—
51	C₃	—	3	4	no	—
52	C₃	C₃²×Q₈⋊C₉	1	1	yes	characteristic
53	C₂	#11757860	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₉	C₃²×A₄
commutator	C₃×Q₈	C₃²×C₉
Fitting	Q₈×#737	C₃
socle	C₂×C₃	C₃²×C₂²⋊C₉
prime radical	C₂×H(3,𝔽₃)	C₂²⋊C₉
cosocle	Q₈×C₉	C₃³
hypercenter	C₂×#737	A₄
hypocenter	Q₈	#6903
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₉ ◁ C₂×#737

Complements

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

N	H ≅ G/N	type
2. #10619570	C₃	semidirect
3. C₃×Q₈⋊C₂₇	C₃	semidirect
47. Q₈	#6903	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#737	3
#10619570	3
C₃×Q₈⋊C₂₇	3
C₂×#6903	4