Subgroups — #14517227

Order 1944.

236 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	C₃×#6204	C₃	1	1	yes	maximal, characteristic
3	#10619584	C₃	1	2	yes	maximal
4	C₉⋊(Q₈⋊C₉)	C₃	1	3	yes	maximal, complemented
5	(Q₈⋊C₉)⋊C₉	C₃	1	6	yes	maximal, complemented
6	C₂×#6894	—	4	1	no	maximal
7	C₃²×C₄×C₉	—	3	1	no	—
8	#6894	—	4	1	no	Sylow
9	C₃³×Q₈	C₃²	1	1	yes	characteristic
10	C₃×Q₈×C₉	—	3	3	no	—
11	C₃×Q₈×C₉	C₃²	1	2	yes	—
12	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
13	#6204	—	3	3	no	—
14	C₃×Q₈⋊C₉	C₃²	1	3	yes	—
15	C₃×Q₈⋊C₉	C₃²	1	6	yes	—
16	C₂×C₃×3₋¹⁺²	—	4	1	no	—
17	C₂×C₃²×C₉	A₄	1	1	yes	characteristic
18	C₂×C₉⋊C₉	—	4	6	no	—
19	C₂×C₉⋊C₉	—	4	3	no	—
20	C₂×C₃²⋊C₉	—	4	2	no	—
21	C₃³×C₄	—	3	1	no	—
22	C₃×C₄×C₉	—	3	9	no	—
23	C₃×C₄×C₉	—	3	2	no	—
24	C₃×C₄×C₉	—	3	1	no	—
25	C₃×3₋¹⁺²	—	4	1	no	—
26	C₃²×C₉	SL(2,𝔽₃)	1	1	yes	characteristic
27	C₉⋊C₉	—	4	6	no	—
28	C₉⋊C₉	—	4	3	no	—
29	C₃²⋊C₉	—	4	2	no	—
30	C₃²×Q₈	—	3	1	no	—
31	C₃²×Q₈	—	3	2	no	—
32	C₃²×Q₈	3₋¹⁺²	1	3	yes	—
33	C₃²×Q₈	C₃³	1	1	yes	characteristic
34	Q₈×C₉	—	3	2	no	—
35	Q₈×C₉	3₋¹⁺²	1	3	yes	complemented
36	Q₈⋊C₉	—	3	3	no	—
37	Q₈⋊C₉	—	3	6	no	—
38	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
39	C₂×3₋¹⁺²	—	12	3	no	—
40	C₂×C₃×C₉	—	3	3	no	—
41	C₂×C₃×C₉	—	4	6	no	—
42	C₂×C₃×C₉	—	4	3	no	—
43	C₂×C₃×C₉	C₃×A₄	1	2	yes	—
44	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
45	C₃²×C₄	—	3	3	no	—
46	C₃²×C₄	—	3	6	no	—
47	C₃²×C₄	—	3	3	no	—
48	C₃²×C₄	—	3	1	no	—
49	C₄×C₉	—	3	6	no	—
50	C₄×C₉	—	3	3	no	—
51	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
52	3₋¹⁺²	—	12	3	no	—
53	C₃×C₉	—	3	3	no	—
54	C₃×C₉	—	4	6	no	—
55	C₃×C₉	—	4	3	no	—
56	C₃×C₉	C₃×SL(2,𝔽₃)	1	2	yes	—
57	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
58	C₃×Q₈	—	3	3	no	—
59	C₃×Q₈	C₃×3₋¹⁺²	1	1	yes	characteristic
60	C₃×Q₈	C₃×3₋¹⁺²	1	1	yes	characteristic
61	C₃×Q₈	#737	1	2	yes	—
62	C₂×C₃²	—	3	2	no	—
63	C₂×C₃²	—	3	1	no	—
64	C₂×C₃²	C₉⋊A₄	1	3	yes	—
65	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
66	C₂×C₉	—	3	2	no	—
67	C₂×C₉	—	12	3	no	—
68	C₂×C₉	—	12	6	no	—
69	C₂×C₉	#1094	1	3	yes	—
70	C₃×C₄	—	3	9	no	—
71	C₃×C₄	—	3	1	no	—
72	C₃×C₄	—	3	2	no	—
73	C₃×C₄	—	3	1	no	—
74	C₃²	—	3	2	no	—
75	C₃²	—	3	1	no	—
76	C₃²	C₉⋊SL(2,𝔽₃)	1	3	yes	—
77	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
78	C₉	—	3	2	no	—
79	C₉	—	12	3	no	—
80	C₉	—	12	6	no	—
81	C₉	#6204	1	3	yes	complemented
82	Q₈	#6894	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₃	—	3	3	no	—
84	C₂×C₃	C₃×C₉⋊A₄	1	1	yes	characteristic
85	C₂×C₃	#66436	1	2	yes	—
86	C₂×C₃	C₃×#1094	1	1	yes	characteristic
87	C₄	—	3	1	no	—
88	C₃	—	3	3	no	—
89	C₃	C₃×C₉⋊SL(2,𝔽₃)	1	1	yes	characteristic
90	C₃	#10619912	1	2	yes	—
91	C₃	C₃×#6204	1	1	yes	characteristic
92	C₂	#11757821	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃²	C₃²×A₄
commutator, cosocle	C₃²×Q₈	C₃³
Fitting	C₃²×Q₈×C₉	C₃
prime radical	C₂×C₃³	C₃×A₄
hypercenter	C₂×C₃²×C₉	A₄
hypocenter	Q₈	#6894
LCS	G ▷ C₃²×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₉

Complements

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

N	H ≅ G/N	type
4. C₉⋊(Q₈⋊C₉)	C₃	semidirect
5. (Q₈⋊C₉)⋊C₉	C₃	semidirect
35. Q₈×C₉	3₋¹⁺²	semidirect
81. C₉	#6204	semidirect
82. Q₈	#6894	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
C₃×#6204	3
#10619584	3
C₉⋊(Q₈⋊C₉)	3
(Q₈⋊C₉)⋊C₉	3
C₂×#6894	4