Subgroups — #14517169

Order 1944.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉⋊SL(2,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	Q₈×C₉²	C₃	1	1	yes	maximal, characteristic, complemented
3	(C₃×SL(2,𝔽₃))⋊C₉	C₃	1	3	yes	maximal
4	C₉⋊(Q₈⋊C₉)	C₃	1	2	yes	maximal, complemented
5	(Q₈⋊C₉)⋊C₉	C₃	1	6	yes	maximal, complemented
6	C₂×#6899	—	4	1	no	maximal
7	C₄×C₉²	—	3	1	no	—
8	#6899	—	4	1	no	Sylow
9	C₃²×SL(2,𝔽₃)	C₃²	1	1	yes	characteristic
10	C₃×Q₈×C₉	C₃²	1	3	yes	—
11	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
12	C₃×Q₈⋊C₉	C₃²	1	2	yes	—
13	C₃×Q₈⋊C₉	C₃²	1	6	yes	—
14	C₉⋊SL(2,𝔽₃)	—	3	3	no	—
15	C₂×C₃×3₋¹⁺²	—	4	1	no	—
16	C₂×C₉⋊C₉	—	4	6	no	—
17	C₂×C₉⋊C₉	—	4	2	no	—
18	C₂×C₃²⋊C₉	—	4	3	no	—
19	C₂×C₉²	A₄	1	1	yes	characteristic
20	C₃×C₄×C₉	—	3	3	no	—
21	C₃×C₄×C₉	—	3	1	no	—
22	C₃×3₋¹⁺²	—	4	1	no	—
23	C₉⋊C₉	—	4	2	no	—
24	C₉⋊C₉	—	4	6	no	—
25	C₃²⋊C₉	—	4	3	no	—
26	C₉²	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
27	C₃²×Q₈	C₃³	1	1	yes	characteristic
28	C₃×SL(2,𝔽₃)	—	3	1	no	—
29	C₃×SL(2,𝔽₃)	—	3	3	no	—
30	Q₈×C₉	—	3	3	no	—
31	Q₈×C₉	3₋¹⁺²	1	3	yes	—
32	Q₈⋊C₉	—	3	2	no	—
33	Q₈⋊C₉	—	3	6	no	—
34	C₂×C₃³	—	4	1	no	—
35	C₂×3₋¹⁺²	—	12	3	no	—
36	C₂×C₃×C₉	—	4	6	no	—
37	C₂×C₃×C₉	—	4	2	no	—
38	C₂×C₃×C₉	C₃×A₄	1	3	yes	—
39	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
40	C₃²×C₄	—	3	1	no	—
41	C₄×C₉	—	3	9	no	—
42	C₄×C₉	—	3	3	no	—
43	C₃³	—	4	1	no	—
44	3₋¹⁺²	—	12	3	no	—
45	C₃×C₉	—	4	6	no	—
46	C₃×C₉	—	4	2	no	—
47	C₃×C₉	C₃×SL(2,𝔽₃)	1	3	yes	—
48	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
49	C₃×Q₈	C₃×3₋¹⁺²	1	1	yes	characteristic
50	C₃×Q₈	#737	1	3	yes	—
51	SL(2,𝔽₃)	—	9	1	no	—
52	C₂×C₃²	—	12	1	no	—
53	C₂×C₃²	—	12	3	no	—
54	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
55	C₂×C₉	—	3	3	no	—
56	C₂×C₉	—	12	2	no	—
57	C₂×C₉	—	12	6	no	—
58	C₂×C₉	C₉⋊A₄	1	3	yes	—
59	C₃×C₄	—	3	1	no	—
60	C₃×C₄	—	3	3	no	—
61	C₃²	—	12	1	no	—
62	C₃²	—	12	3	no	—
63	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
64	C₉	—	3	3	no	—
65	C₉	—	12	2	no	—
66	C₉	—	12	6	no	—
67	C₉	C₉⋊SL(2,𝔽₃)	1	3	yes	—
68	Q₈	#6899	1	1	yes	characteristic, complemented, Sylow
69	C₂×C₃	—	36	1	no	—
70	C₂×C₃	C₃×C₉⋊A₄	1	1	yes	characteristic
71	C₂×C₃	#66436	1	3	yes	—
72	C₄	—	3	1	no	—
73	C₃	—	36	1	no	—
74	C₃	C₃×C₉⋊SL(2,𝔽₃)	1	1	yes	characteristic
75	C₃	#10619912	1	3	yes	—
76	C₂	#11757762	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃²	C₃²×A₄
commutator, cosocle	C₃²×Q₈	C₃³
Fitting	Q₈×C₉²	C₃
prime radical	C₃²×SL(2,𝔽₃)	C₃²
hypercenter	C₂×C₉²	A₄
hypocenter	Q₈	#6899
LCS	G ▷ C₃²×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₉²

Complements

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

N	H ≅ G/N	type
2. Q₈×C₉²	C₃	semidirect
4. C₉⋊(Q₈⋊C₉)	C₃	semidirect
5. (Q₈⋊C₉)⋊C₉	C₃	semidirect
26. C₉²	SL(2,𝔽₃)	semidirect
68. 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×C₉⋊SL(2,𝔽₃)	3
Q₈×C₉²	3
(C₃×SL(2,𝔽₃))⋊C₉	3
C₉⋊(Q₈⋊C₉)	3
(Q₈⋊C₉)⋊C₉	3
C₂×#6899	4