Subgroups — #14517197

Order 1944.

182 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×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	(C₃×SL(2,𝔽₃))⋊C₉	C₃	1	3	yes	maximal
4	C₉⋊(Q₈⋊C₉)	C₃	1	3	yes	maximal, complemented
5	C₉⋊(Q₈⋊C₉)	C₃	1	1	yes	maximal, characteristic, complemented
6	(Q₈⋊C₉)⋊C₉	C₃	1	3	yes	maximal, complemented
7	C₉×Q₈⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
8	C₂×#6899	—	4	1	no	maximal
9	C₄×C₉⋊C₉	—	3	1	no	—
10	#6899	—	4	1	no	Sylow
11	C₃²×SL(2,𝔽₃)	C₃²	1	1	yes	characteristic
12	C₃×Q₈×C₉	C₃²	1	3	yes	—
13	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
14	C₃×Q₈⋊C₉	C₃²	1	1	yes	characteristic
15	C₃×Q₈⋊C₉	C₃²	1	1	yes	characteristic
16	C₃×Q₈⋊C₉	C₃²	1	3	yes	—
17	C₃×Q₈⋊C₉	C₃²	1	3	yes	—
18	C₉⋊SL(2,𝔽₃)	—	3	3	no	—
19	C₂×C₃×3₋¹⁺²	—	4	1	no	—
20	C₂×C₉⋊C₉	—	4	3	no	—
21	C₂×C₉⋊C₉	—	4	3	no	—
22	C₂×C₉⋊C₉	—	4	1	no	—
23	C₂×C₉⋊C₉	A₄	1	1	yes	characteristic
24	C₂×C₃²⋊C₉	—	4	3	no	—
25	C₂×C₉²	—	4	1	no	—
26	C₃×C₄×C₉	—	3	1	no	—
27	C₃×C₄×C₉	—	3	3	no	—
28	C₃×3₋¹⁺²	—	4	1	no	—
29	C₉⋊C₉	—	4	3	no	—
30	C₉⋊C₉	—	4	3	no	—
31	C₉⋊C₉	—	4	1	no	—
32	C₉⋊C₉	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
33	C₃²⋊C₉	—	4	3	no	—
34	C₉²	—	4	1	no	—
35	C₃²×Q₈	C₃³	1	1	yes	characteristic
36	C₃×SL(2,𝔽₃)	—	3	1	no	—
37	C₃×SL(2,𝔽₃)	—	3	3	no	—
38	Q₈×C₉	—	3	3	no	—
39	Q₈×C₉	3₋¹⁺²	1	3	yes	—
40	Q₈⋊C₉	—	3	1	no	—
41	Q₈⋊C₉	—	3	1	no	—
42	Q₈⋊C₉	—	3	3	no	—
43	Q₈⋊C₉	—	3	3	no	—
44	C₂×C₃³	—	4	1	no	—
45	C₂×3₋¹⁺²	—	12	3	no	—
46	C₂×C₃×C₉	—	4	3	no	—
47	C₂×C₃×C₉	—	4	3	no	—
48	C₂×C₃×C₉	—	4	1	no	—
49	C₂×C₃×C₉	—	4	1	no	—
50	C₂×C₃×C₉	C₃×A₄	1	3	yes	—
51	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
52	C₃²×C₄	—	3	1	no	—
53	C₄×C₉	—	3	3	no	—
54	C₄×C₉	—	9	3	no	—
55	C₃³	—	4	1	no	—
56	3₋¹⁺²	—	12	3	no	—
57	C₃×C₉	—	4	3	no	—
58	C₃×C₉	—	4	3	no	—
59	C₃×C₉	—	4	1	no	—
60	C₃×C₉	—	4	1	no	—
61	C₃×C₉	C₃×SL(2,𝔽₃)	1	3	yes	—
62	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
63	C₃×Q₈	C₃×3₋¹⁺²	1	1	yes	characteristic
64	C₃×Q₈	#737	1	3	yes	—
65	SL(2,𝔽₃)	—	9	1	no	—
66	C₂×C₃²	—	12	1	no	—
67	C₂×C₃²	—	12	3	no	—
68	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
69	C₂×C₉	—	3	3	no	—
70	C₂×C₉	—	12	1	no	—
71	C₂×C₉	—	12	1	no	—
72	C₂×C₉	—	12	3	no	—
73	C₂×C₉	—	12	3	no	—
74	C₂×C₉	C₉⋊A₄	1	3	yes	—
75	C₃×C₄	—	3	1	no	—
76	C₃×C₄	—	3	3	no	—
77	C₃²	—	12	1	no	—
78	C₃²	—	12	3	no	—
79	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
80	C₉	—	3	3	no	—
81	C₉	—	12	1	no	—
82	C₉	—	12	1	no	—
83	C₉	—	12	3	no	—
84	C₉	—	12	3	no	—
85	C₉	C₉⋊SL(2,𝔽₃)	1	3	yes	—
86	Q₈	#6899	1	1	yes	characteristic, complemented, Sylow
87	C₂×C₃	—	36	1	no	—
88	C₂×C₃	C₃×C₉⋊A₄	1	1	yes	characteristic
89	C₂×C₃	#66440	1	3	yes	—
90	C₄	—	3	1	no	—
91	C₃	—	36	1	no	—
92	C₃	C₃×C₉⋊SL(2,𝔽₃)	1	1	yes	characteristic
93	C₃	#10619916	1	3	yes	—
94	C₂	#11757790	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃²	C₃²×A₄
commutator, cosocle	C₃²×Q₈	C₃³
Fitting	Q₈×C₉⋊C₉	C₃
prime radical	C₃²×SL(2,𝔽₃)	C₃²
hypercenter	C₂×C₉⋊C₉	A₄
hypocenter	Q₈	#6899
LCS	G ▷ C₃²×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₉⋊C₉

Complements

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

N	H ≅ G/N	type
2. Q₈×C₉⋊C₉	C₃	semidirect
4. C₉⋊(Q₈⋊C₉)	C₃	semidirect
5. C₉⋊(Q₈⋊C₉)	C₃	semidirect
6. (Q₈⋊C₉)⋊C₉	C₃	semidirect
7. C₉×Q₈⋊C₉	C₃	semidirect
32. C₉⋊C₉	SL(2,𝔽₃)	semidirect
86. 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×C₉⋊C₉	3
(C₃×SL(2,𝔽₃))⋊C₉	3
C₉⋊(Q₈⋊C₉)	3
(Q₈⋊C₉)⋊C₉	3
C₉×Q₈⋊C₉	3
C₂×#6899	4