Subgroups — #14517228

Order 1944.

257 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	#10619584	C₃	1	3	yes	maximal
3	C₉⋊(Q₈⋊C₉)	C₃	1	6	yes	maximal, complemented
4	C₉×Q₈⋊C₉	C₃	1	3	yes	maximal, complemented
5	C₂×#6887	—	4	1	no	maximal
6	C₃²×C₄×C₉	—	3	1	no	—
7	#6887	—	4	1	no	Sylow
8	C₃³×Q₈	C₃²	1	1	yes	characteristic
9	C₃×Q₈×C₉	C₉	1	9	yes	complemented
10	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
11	C₃×Q₈×C₉	C₃²	1	2	yes	—
12	C₃×Q₈⋊C₉	C₃²	1	9	yes	—
13	C₂×C₃²×C₉	A₄	1	1	yes	characteristic
14	C₂×C₉⋊C₉	—	4	6	no	—
15	C₂×C₃²⋊C₉	—	4	3	no	—
16	C₂×C₉²	—	4	3	no	—
17	C₃³×C₄	—	3	1	no	—
18	C₃×C₄×C₉	—	3	9	no	—
19	C₃×C₄×C₉	—	3	2	no	—
20	C₃×C₄×C₉	—	3	1	no	—
21	C₃²×C₉	SL(2,𝔽₃)	1	1	yes	characteristic
22	C₉⋊C₉	—	4	6	no	—
23	C₃²⋊C₉	—	4	3	no	—
24	C₉²	—	4	3	no	—
25	C₃²×Q₈	—	3	3	no	—
26	C₃²×Q₈	C₃×C₉	1	3	yes	—
27	C₃²×Q₈	C₃³	1	1	yes	characteristic
28	Q₈×C₉	C₃×C₉	1	3	yes	—
29	Q₈×C₉	C₃×C₉	1	6	yes	—
30	Q₈⋊C₉	—	3	9	no	—
31	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
32	C₂×C₃×C₉	—	4	9	no	—
33	C₂×C₃×C₉	C₂²⋊C₉	1	9	yes	—
34	C₂×C₃×C₉	C₃×A₄	1	2	yes	—
35	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
36	C₃²×C₄	—	3	9	no	—
37	C₃²×C₄	—	3	3	no	—
38	C₃²×C₄	—	3	1	no	—
39	C₄×C₉	—	3	6	no	—
40	C₄×C₉	—	3	3	no	—
41	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
42	C₃×C₉	—	4	9	no	—
43	C₃×C₉	Q₈⋊C₉	1	9	yes	complemented
44	C₃×C₉	C₃×SL(2,𝔽₃)	1	2	yes	—
45	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
46	C₃×Q₈	—	3	3	no	—
47	C₃×Q₈	C₃²×C₉	1	1	yes	characteristic
48	C₃×Q₈	#737	1	3	yes	—
49	C₂×C₃²	—	3	3	no	—
50	C₂×C₃²	C₃×C₂²⋊C₉	1	3	yes	—
51	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
52	C₂×C₉	—	12	9	no	—
53	C₂×C₉	C₃×C₂²⋊C₉	1	6	yes	—
54	C₂×C₉	C₃×C₂²⋊C₉	1	3	yes	—
55	C₃×C₄	—	3	9	no	—
56	C₃×C₄	—	3	3	no	—
57	C₃×C₄	—	3	1	no	—
58	C₃²	—	3	3	no	—
59	C₃²	C₃×Q₈⋊C₉	1	3	yes	—
60	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
61	C₉	—	12	9	no	—
62	C₉	C₃×Q₈⋊C₉	1	6	yes	—
63	C₉	C₃×Q₈⋊C₉	1	3	yes	—
64	Q₈	#6887	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₃	—	3	3	no	—
66	C₂×C₃	#66436	1	3	yes	—
67	C₂×C₃	C₃²×C₂²⋊C₉	1	1	yes	characteristic
68	C₄	—	3	1	no	—
69	C₃	—	3	3	no	—
70	C₃	#10619912	1	3	yes	—
71	C₃	C₃²×Q₈⋊C₉	1	1	yes	characteristic
72	C₂	#11757822	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₉	C₃×A₄
commutator	C₃×Q₈	C₃²×C₉
Frattini, socle	C₂×C₃²	C₃²×A₄
Fitting	C₃²×Q₈×C₉	C₃
prime radical	C₂×C₃³	C₃×A₄
cosocle	C₃²×Q₈	C₃³
hypercenter	C₂×C₃²×C₉	A₄
hypocenter	Q₈	#6887
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃×C₉ ◁ C₂×C₃²×C₉

Complements

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

N	H ≅ G/N	type
3. C₉⋊(Q₈⋊C₉)	C₃	semidirect
4. C₉×Q₈⋊C₉	C₃	semidirect
9. C₃×Q₈×C₉	C₉	semidirect
43. C₃×C₉	Q₈⋊C₉	semidirect
64. Q₈	#6887	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
#10619584	3
C₉⋊(Q₈⋊C₉)	3
C₉×Q₈⋊C₉	3
C₂×#6887	4