Subgroups — #14517254

Order 1944.

203 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	#10619584	C₃	1	3	yes	maximal
3	C₉⋊(Q₈⋊C₉)	C₃	1	2	yes	maximal, complemented
4	(Q₈⋊C₉)⋊C₉	C₃	1	6	yes	maximal, complemented
5	C₉×Q₈⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂×#6899	—	4	1	no	maximal
7	C₃×C₄×3₋¹⁺²	—	3	1	no	—
8	#6899	—	4	1	no	Sylow
9	C₃³×Q₈	C₃²	1	1	yes	characteristic
10	Q₈×3₋¹⁺²	—	3	3	no	—
11	C₃×Q₈×C₉	C₃²	1	2	yes	—
12	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
13	C₃×Q₈⋊C₉	C₃²	1	3	yes	—
14	C₃×Q₈⋊C₉	C₃²	1	6	yes	—
15	C₂×C₃×3₋¹⁺²	A₄	1	1	yes	characteristic
16	C₂×C₉⋊C₉	—	4	6	no	—
17	C₂×C₉⋊C₉	—	4	2	no	—
18	C₂×C₃²⋊C₉	—	4	3	no	—
19	C₂×C₉²	—	4	1	no	—
20	C₃³×C₄	—	3	1	no	—
21	C₄×3₋¹⁺²	—	3	9	no	—
22	C₃×C₄×C₉	—	3	2	no	—
23	C₃×C₄×C₉	—	3	1	no	—
24	C₃×3₋¹⁺²	SL(2,𝔽₃)	1	1	yes	characteristic
25	C₉⋊C₉	—	4	6	no	—
26	C₉⋊C₉	—	4	2	no	—
27	C₃²⋊C₉	—	4	3	no	—
28	C₉²	—	4	1	no	—
29	C₃²×Q₈	—	3	1	no	—
30	C₃²×Q₈	—	3	3	no	—
31	C₃²×Q₈	C₃³	1	1	yes	characteristic
32	Q₈×C₉	—	3	2	no	—
33	Q₈×C₉	3₋¹⁺²	1	3	yes	—
34	Q₈⋊C₉	—	3	3	no	—
35	Q₈⋊C₉	—	3	6	no	—
36	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
37	C₂×3₋¹⁺²	—	3	3	no	—
38	C₂×C₃×C₉	—	4	3	no	—
39	C₂×C₃×C₉	—	4	6	no	—
40	C₂×C₃×C₉	C₃×A₄	1	2	yes	—
41	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
42	C₃²×C₄	—	3	1	no	—
43	C₃²×C₄	—	3	3	no	—
44	C₃²×C₄	—	9	3	no	—
45	C₄×C₉	—	3	6	no	—
46	C₄×C₉	—	3	3	no	—
47	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
48	3₋¹⁺²	—	3	3	no	—
49	C₃×C₉	—	4	3	no	—
50	C₃×C₉	—	4	6	no	—
51	C₃×C₉	C₃×SL(2,𝔽₃)	1	2	yes	—
52	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
53	C₃×Q₈	—	9	1	no	—
54	C₃×Q₈	C₃×3₋¹⁺²	1	1	yes	characteristic
55	C₃×Q₈	#737	1	3	yes	—
56	C₂×C₃²	—	3	3	no	—
57	C₂×C₃²	—	3	1	no	—
58	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
59	C₂×C₉	—	3	2	no	—
60	C₂×C₉	—	12	3	no	—
61	C₂×C₉	—	12	6	no	—
62	C₂×C₉	#1094	1	3	yes	—
63	C₃×C₄	—	3	3	no	—
64	C₃×C₄	—	3	1	no	—
65	C₃×C₄	—	9	3	no	—
66	C₃²	—	3	3	no	—
67	C₃²	—	3	1	no	—
68	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
69	C₉	—	3	2	no	—
70	C₉	—	12	3	no	—
71	C₉	—	12	6	no	—
72	C₉	#6204	1	3	yes	—
73	Q₈	#6899	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₃	—	9	1	no	—
75	C₂×C₃	#66440	1	3	yes	—
76	C₂×C₃	C₃×#1094	1	1	yes	characteristic
77	C₄	—	3	1	no	—
78	C₃	—	9	1	no	—
79	C₃	#10619916	1	3	yes	—
80	C₃	C₃×#6204	1	1	yes	characteristic
81	C₂	#11757847	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃²	C₃²×A₄
commutator, cosocle	C₃²×Q₈	C₃³
Fitting	C₃×Q₈×3₋¹⁺²	C₃
prime radical	C₂×C₃³	C₃×A₄
hypercenter	C₂×C₃×3₋¹⁺²	A₄
hypocenter	Q₈	#6899
LCS	G ▷ C₃²×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
3. C₉⋊(Q₈⋊C₉)	C₃	semidirect
4. (Q₈⋊C₉)⋊C₉	C₃	semidirect
5. C₉×Q₈⋊C₉	C₃	semidirect
73. Q₈	#6899	semidirect

Maximal subgroups

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

subgroup	index
C₃×Q₈×3₋¹⁺²	3
#10619584	3
C₉⋊(Q₈⋊C₉)	3
(Q₈⋊C₉)⋊C₉	3
C₉×Q₈⋊C₉	3
C₂×#6899	4