Subgroups — #14515882

Order 1944.

143 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×Q₈×C₉	C₃	1	1	yes	maximal, characteristic
2	#10619570	C₃	1	3	yes	maximal, complemented
3	C₂×#6869	—	4	1	no	maximal
4	C₃²×C₄×C₉	—	3	1	no	—
5	#6869	—	4	1	no	Sylow
6	C₃³×Q₈	C₉	1	1	yes	characteristic
7	C₃×Q₈×C₉	—	3	1	no	—
8	C₃×Q₈×C₉	—	3	2	no	—
9	C₃×Q₈×C₉	C₉	1	2	yes	—
10	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
11	Q₈⋊C₂₇	—	3	3	no	—
12	C₂×C₃²×C₉	A₄	1	1	yes	characteristic
13	C₂×C₂₇⋊C₃	—	4	3	no	—
14	C₃³×C₄	—	3	1	no	—
15	C₃×C₄×C₉	—	3	6	no	—
16	C₃×C₄×C₉	—	3	3	no	—
17	C₃×C₄×C₉	—	3	2	no	—
18	C₃×C₄×C₉	—	3	1	no	—
19	C₃²×C₉	SL(2,𝔽₃)	1	1	yes	characteristic
20	C₂₇⋊C₃	—	4	3	no	—
21	C₃²×Q₈	—	3	1	no	—
22	C₃²×Q₈	—	3	3	no	—
23	C₃²×Q₈	C₃×C₉	1	1	yes	characteristic
24	Q₈×C₉	—	3	2	no	—
25	Q₈×C₉	H(3,𝔽₃)	1	1	yes	characteristic
26	Q₈×C₉	3₋¹⁺²	1	2	yes	—
27	C₂×C₃³	C₂²⋊C₉	1	1	yes	characteristic
28	C₂×C₃×C₉	—	3	2	no	—
29	C₂×C₃×C₉	—	3	1	no	—
30	C₂×C₃×C₉	C₂²⋊C₉	1	2	yes	—
31	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
32	C₂×C₂₇	—	12	3	no	—
33	C₃²×C₄	—	3	3	no	—
34	C₃²×C₄	—	3	9	no	—
35	C₃²×C₄	—	3	1	no	—
36	C₄×C₉	—	3	6	no	—
37	C₄×C₉	—	3	2	no	—
38	C₄×C₉	—	3	1	no	—
39	C₃³	Q₈⋊C₉	1	1	yes	characteristic
40	C₃×C₉	—	3	2	no	—
41	C₃×C₉	—	3	1	no	—
42	C₃×C₉	Q₈⋊C₉	1	2	yes	—
43	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
44	C₂₇	—	12	3	no	—
45	C₃×Q₈	—	3	3	no	—
46	C₃×Q₈	—	3	1	no	—
47	C₃×Q₈	C₃²⋊C₉	1	1	yes	characteristic
48	C₂×C₃²	—	3	1	no	—
49	C₂×C₃²	—	3	3	no	—
50	C₂×C₃²	C₃×C₂²⋊C₉	1	1	yes	characteristic
51	C₂×C₉	—	3	2	no	—
52	C₂×C₉	#1094	1	2	yes	—
53	C₂×C₉	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
54	C₃×C₄	—	3	9	no	—
55	C₃×C₄	—	3	3	no	—
56	C₃×C₄	—	3	1	no	—
57	C₃²	—	3	3	no	—
58	C₃²	—	3	1	no	—
59	C₃²	C₃×Q₈⋊C₉	1	1	yes	characteristic
60	C₉	—	3	2	no	—
61	C₉	#6204	1	2	yes	—
62	C₉	Q₈⋊H(3,𝔽₃)	1	1	yes	characteristic
63	Q₈	#6869	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₃	—	3	3	no	—
65	C₂×C₃	—	3	1	no	—
66	C₂×C₃	C₃²⋊(C₂²⋊C₉)	1	1	yes	characteristic
67	C₄	—	3	1	no	—
68	C₃	—	3	3	no	—
69	C₃	—	3	1	no	—
70	C₃	#10619584	1	1	yes	characteristic
71	C₂	#11757445	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	C₃²×Q₈×C₉	C₃
socle	C₂×C₃	C₃²⋊(C₂²⋊C₉)
prime radical	C₂×C₃³	C₂²⋊C₉
cosocle	C₃×Q₈×C₉	C₃²
hypercenter	C₂×C₃²×C₉	A₄
hypocenter	Q₈	#6869
LCS	G ▷ C₃²×Q₈ ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₃×C₉ ◁ C₂×C₃²×C₉

Complements

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

N	H ≅ G/N	type
2. #10619570	C₃	semidirect
63. Q₈	#6869	semidirect

Maximal subgroups

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

subgroup	index
C₃²×Q₈×C₉	3
#10619570	3
C₂×#6869	4