Subgroups — #14517252

Order 1944.

248 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	C₃×#6204	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×#6900	—	4	1	no	maximal
7	C₃×C₄×3₋¹⁺²	—	3	1	no	—
8	#6900	—	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	#6204	—	3	9	no	—
14	C₃×Q₈⋊C₉	C₃²	1	6	yes	—
15	C₃×Q₈⋊C₉	C₃²	1	3	yes	—
16	C₂×C₃×3₋¹⁺²	—	4	3	no	—
17	C₂×C₃×3₋¹⁺²	A₄	1	1	yes	characteristic
18	C₂×C₉⋊C₉	—	4	6	no	—
19	C₂×C₉⋊C₉	—	4	2	no	—
20	C₂×C₉²	—	4	1	no	—
21	C₃³×C₄	—	3	1	no	—
22	C₄×3₋¹⁺²	—	3	9	no	—
23	C₃×C₄×C₉	—	3	2	no	—
24	C₃×C₄×C₉	—	3	1	no	—
25	C₃×3₋¹⁺²	—	4	3	no	—
26	C₃×3₋¹⁺²	SL(2,𝔽₃)	1	1	yes	characteristic
27	C₉⋊C₉	—	4	6	no	—
28	C₉⋊C₉	—	4	2	no	—
29	C₉²	—	4	1	no	—
30	C₃²×Q₈	—	3	1	no	—
31	C₃²×Q₈	—	3	3	no	—
32	C₃²×Q₈	C₃³	1	1	yes	characteristic
33	Q₈×C₉	—	3	2	no	—
34	Q₈×C₉	3₋¹⁺²	1	3	yes	complemented
35	Q₈⋊C₉	—	3	6	no	—
36	Q₈⋊C₉	3₋¹⁺²	1	9	yes	complemented
37	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
38	C₂×3₋¹⁺²	—	3	3	no	—
39	C₂×3₋¹⁺²	—	12	9	no	—
40	C₂×C₃×C₉	—	4	3	no	—
41	C₂×C₃×C₉	—	4	6	no	—
42	C₂×C₃×C₉	C₃×A₄	1	2	yes	—
43	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
44	C₃²×C₄	—	3	1	no	—
45	C₃²×C₄	—	3	3	no	—
46	C₃²×C₄	—	9	3	no	—
47	C₄×C₉	—	3	6	no	—
48	C₄×C₉	—	3	3	no	—
49	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
50	3₋¹⁺²	—	3	3	no	—
51	3₋¹⁺²	—	12	9	no	—
52	C₃×C₉	—	4	3	no	—
53	C₃×C₉	—	4	6	no	—
54	C₃×C₉	C₃×SL(2,𝔽₃)	1	2	yes	—
55	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
56	C₃×Q₈	—	9	1	no	—
57	C₃×Q₈	C₃×3₋¹⁺²	1	3	yes	—
58	C₃×Q₈	C₃×3₋¹⁺²	1	1	yes	characteristic
59	C₂×C₃²	—	3	3	no	—
60	C₂×C₃²	—	3	1	no	—
61	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
62	C₂×C₉	—	3	2	no	—
63	C₂×C₉	—	4	9	no	—
64	C₂×C₉	—	12	6	no	—
65	C₂×C₉	#1094	1	3	yes	—
66	C₃×C₄	—	3	3	no	—
67	C₃×C₄	—	3	1	no	—
68	C₃×C₄	—	9	3	no	—
69	C₃²	—	3	3	no	—
70	C₃²	—	3	1	no	—
71	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
72	C₉	—	3	2	no	—
73	C₉	—	4	9	no	—
74	C₉	—	12	6	no	—
75	C₉	#6204	1	3	yes	complemented
76	Q₈	#6900	1	1	yes	characteristic, complemented, Sylow
77	C₂×C₃	—	9	1	no	—
78	C₂×C₃	A₄×3₋¹⁺²	1	3	yes	—
79	C₂×C₃	C₃×#1094	1	1	yes	characteristic
80	C₄	—	3	1	no	—
81	C₃	—	9	1	no	—
82	C₃	SL(2,𝔽₃)×3₋¹⁺²	1	3	yes	—
83	C₃	C₃×#6204	1	1	yes	characteristic
84	C₂	#11757845	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	#6900
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
34. Q₈×C₉	3₋¹⁺²	semidirect
36. Q₈⋊C₉	3₋¹⁺²	semidirect
75. C₉	#6204	semidirect
76. Q₈	#6900	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
C₃×#6204	3
C₉⋊(Q₈⋊C₉)	3
(Q₈⋊C₉)⋊C₉	3
C₉×Q₈⋊C₉	3
C₂×#6900	4