Subgroups — #14517236

Order 1944.

284 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, complemented
2	C₃×Q₈⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×#6204	C₃	1	2	yes	maximal, complemented
4	(C₃×SL(2,𝔽₃))⋊C₉	C₃	1	9	yes	maximal, complemented
5	C₂×#6893	—	4	1	no	maximal
6	C₃²×C₄×C₉	—	3	1	no	—
7	#6893	—	4	1	no	Sylow
8	C₃³×Q₈	C₃²	1	1	yes	characteristic
9	C₃²×SL(2,𝔽₃)	C₃²	1	3	yes	—
10	C₃×Q₈×C₉	—	3	3	no	—
11	C₃×Q₈×C₉	C₃²	1	3	yes	—
12	Q₈⋊H(3,𝔽₃)	—	3	3	no	—
13	#6204	—	3	6	no	—
14	C₃×Q₈⋊C₉	C₃²	1	6	yes	—
15	C₂×C₃×3₋¹⁺²	—	4	2	no	—
16	C₂×C₃×H(3,𝔽₃)	—	4	1	no	—
17	C₂×C₃²×C₉	A₄	1	1	yes	characteristic
18	C₂×C₃²⋊C₉	—	4	9	no	—
19	C₃³×C₄	—	3	1	no	—
20	C₃×C₄×C₉	—	3	9	no	—
21	C₃×C₄×C₉	—	3	3	no	—
22	C₃×3₋¹⁺²	—	4	2	no	—
23	C₃×H(3,𝔽₃)	—	4	1	no	—
24	C₃²×C₉	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
25	C₃²⋊C₉	—	4	9	no	—
26	C₃²×Q₈	—	3	3	no	—
27	C₃²×Q₈	H(3,𝔽₃)	1	3	yes	—
28	C₃²×Q₈	C₃³	1	1	yes	characteristic
29	C₃×SL(2,𝔽₃)	—	3	3	no	—
30	C₃×SL(2,𝔽₃)	—	3	9	no	—
31	Q₈×C₉	—	3	3	no	—
32	Q₈⋊C₉	—	3	6	no	—
33	C₂×C₃³	—	4	3	no	—
34	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
35	C₂×3₋¹⁺²	—	12	6	no	—
36	C₂×H(3,𝔽₃)	—	12	3	no	—
37	C₂×C₃×C₉	—	3	3	no	—
38	C₂×C₃×C₉	—	4	6	no	—
39	C₂×C₃×C₉	C₃×A₄	1	3	yes	—
40	C₃²×C₄	—	3	9	no	—
41	C₃²×C₄	—	3	3	no	—
42	C₃²×C₄	—	3	1	no	—
43	C₄×C₉	—	3	9	no	—
44	C₃³	—	4	3	no	—
45	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
46	3₋¹⁺²	—	12	6	no	—
47	H(3,𝔽₃)	—	12	3	no	—
48	C₃×C₉	—	3	3	no	—
49	C₃×C₉	—	4	6	no	—
50	C₃×C₉	C₃×SL(2,𝔽₃)	1	3	yes	—
51	C₃×Q₈	—	3	3	no	—
52	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
53	C₃×Q₈	#737	1	3	yes	—
54	SL(2,𝔽₃)	—	9	3	no	—
55	C₂×C₃²	—	3	3	no	—
56	C₂×C₃²	—	12	9	no	—
57	C₂×C₃²	—	12	3	no	—
58	C₂×C₃²	C₂²⋊H(3,𝔽₃)	1	3	yes	—
59	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
60	C₂×C₉	—	3	3	no	—
61	C₂×C₉	—	12	6	no	—
62	C₃×C₄	—	3	9	no	—
63	C₃×C₄	—	3	3	no	—
64	C₃×C₄	—	3	1	no	—
65	C₃²	—	3	3	no	—
66	C₃²	—	12	9	no	—
67	C₃²	—	12	3	no	—
68	C₃²	Q₈⋊H(3,𝔽₃)	1	3	yes	—
69	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
70	C₉	—	3	3	no	—
71	C₉	—	12	6	no	—
72	Q₈	#6893	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₃	—	3	3	no	—
74	C₂×C₃	—	36	3	no	—
75	C₂×C₃	#66436	1	3	yes	—
76	C₂×C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
77	C₄	—	3	1	no	—
78	C₃	—	3	3	no	—
79	C₃	—	36	3	no	—
80	C₃	#10619912	1	3	yes	—
81	C₃	C₃×Q₈⋊H(3,𝔽₃)	1	1	yes	characteristic
82	C₂	#11757829	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	C₃×Q₈⋊H(3,𝔽₃)	C₃
hypercenter	C₂×C₃²×C₉	A₄
hypocenter	Q₈	#6893
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
1. C₃²×Q₈×C₉	C₃	semidirect
3. C₃×#6204	C₃	semidirect
4. (C₃×SL(2,𝔽₃))⋊C₉	C₃	semidirect
24. C₃²×C₉	SL(2,𝔽₃)	semidirect
72. Q₈	#6893	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×Q₈⋊H(3,𝔽₃)	3
C₃×#6204	3
(C₃×SL(2,𝔽₃))⋊C₉	3
C₂×#6893	4