Subgroups — #14517271

Order 1944.

590 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×Q₈	C₃	1	1	yes	maximal, characteristic
2	C₃×#6204	C₃	1	3	yes	maximal, complemented
3	#10619584	C₃	1	9	yes	maximal, complemented
4	C₂×#6891	—	4	1	no	maximal
5	C₃⁴×C₄	—	3	1	no	—
6	#6891	—	4	1	no	Sylow
7	C₃³×Q₈	—	3	3	no	—
8	C₃³×Q₈	—	3	9	no	—
9	C₃³×Q₈	C₃²	1	1	yes	characteristic
10	C₃³×Q₈	C₃²	1	3	yes	—
11	#6204	—	3	9	no	—
12	C₃×Q₈⋊C₉	C₃²	1	9	yes	complemented
13	C₂×C₃⁴	A₄	1	1	yes	characteristic
14	C₂×C₃×3₋¹⁺²	—	4	3	no	—
15	C₂×C₃²⋊C₉	—	4	9	no	—
16	C₃³×C₄	—	3	27	no	—
17	C₃³×C₄	—	3	9	no	—
18	C₃³×C₄	—	3	3	no	—
19	C₃³×C₄	—	3	1	no	—
20	C₃⁴	SL(2,𝔽₃)	1	1	yes	characteristic
21	C₃×3₋¹⁺²	—	4	3	no	—
22	C₃²⋊C₉	—	4	9	no	—
23	C₃²×Q₈	—	3	27	no	—
24	C₃²×Q₈	—	3	3	no	—
25	C₃²×Q₈	—	3	3	no	—
26	C₃²×Q₈	—	3	6	no	—
27	C₃²×Q₈	H(3,𝔽₃)	1	3	yes	—
28	C₃²×Q₈	3₋¹⁺²	1	9	yes	complemented
29	C₃²×Q₈	C₃³	1	1	yes	characteristic
30	Q₈⋊C₉	—	3	9	no	—
31	C₂×C₃³	—	3	9	no	—
32	C₂×C₃³	—	3	3	no	—
33	C₂×C₃³	C₃×A₄	1	3	yes	—
34	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
35	C₂×3₋¹⁺²	—	12	9	no	—
36	C₂×C₃×C₉	—	4	9	no	—
37	C₃²×C₄	—	3	81	no	—
38	C₃²×C₄	—	3	18	no	—
39	C₃²×C₄	—	3	9	no	—
40	C₃²×C₄	—	3	9	no	—
41	C₃²×C₄	—	3	9	no	—
42	C₃²×C₄	—	3	3	no	—
43	C₃²×C₄	—	3	1	no	—
44	C₃³	—	3	9	no	—
45	C₃³	—	3	3	no	—
46	C₃³	C₃×SL(2,𝔽₃)	1	3	yes	—
47	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
48	3₋¹⁺²	—	12	9	no	—
49	C₃×C₉	—	4	9	no	—
50	C₃×Q₈	—	3	3	no	—
51	C₃×Q₈	—	3	9	no	—
52	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
53	C₃×Q₈	C₃×3₋¹⁺²	1	3	yes	—
54	C₂×C₃²	—	3	27	no	—
55	C₂×C₃²	—	3	6	no	—
56	C₂×C₃²	—	3	3	no	—
57	C₂×C₃²	—	3	3	no	—
58	C₂×C₃²	#1094	1	9	yes	—
59	C₂×C₃²	C₂²⋊H(3,𝔽₃)	1	3	yes	—
60	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
61	C₂×C₉	—	12	9	no	—
62	C₃×C₄	—	3	27	no	—
63	C₃×C₄	—	3	9	no	—
64	C₃×C₄	—	3	3	no	—
65	C₃×C₄	—	3	1	no	—
66	C₃²	—	3	27	no	—
67	C₃²	—	3	6	no	—
68	C₃²	—	3	3	no	—
69	C₃²	—	3	3	no	—
70	C₃²	#6204	1	9	yes	complemented
71	C₃²	Q₈⋊H(3,𝔽₃)	1	3	yes	—
72	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
73	C₉	—	12	9	no	—
74	Q₈	#6891	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₃	—	3	9	no	—
76	C₂×C₃	—	3	3	no	—
77	C₂×C₃	C₃×#1094	1	3	yes	—
78	C₂×C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
79	C₄	—	3	1	no	—
80	C₃	—	3	9	no	—
81	C₃	—	3	3	no	—
82	C₃	C₃×#6204	1	3	yes	—
83	C₃	C₃×Q₈⋊H(3,𝔽₃)	1	1	yes	characteristic
84	C₂	#11757864	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃²	C₃²×A₄
commutator, cosocle	C₃²×Q₈	C₃³
Fitting	C₃⁴×Q₈	C₃
prime radical, hypercenter	C₂×C₃⁴	A₄
hypocenter	Q₈	#6891
LCS	G ▷ C₃²×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃⁴

Complements

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

N	H ≅ G/N	type
2. C₃×#6204	C₃	semidirect
3. #10619584	C₃	semidirect
12. C₃×Q₈⋊C₉	C₃²	semidirect
28. C₃²×Q₈	3₋¹⁺²	semidirect
70. C₃²	#6204	semidirect
74. Q₈	#6891	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃⁴×Q₈	3
C₃×#6204	3
#10619584	3
C₂×#6891	4