Subgroups — #14517521

Order 1944.

774 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃²⋊H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	#11757605	C₂	1	2	yes	maximal
3	#10620175	C₃	1	1	yes	maximal, characteristic, complemented
4	#10619956	—	3	4	no	maximal
5	C₂×C₃²⋊H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₃⁴×C₄	C₂×C₃	1	1	yes	characteristic
7	C₃⋊(C₃⋊(C₃⋊Dic₃))	C₂×C₃	1	2	yes	—
8	C₃×C₄×H(3,𝔽₃)	—	2	4	no	—
9	C₃×C₄×H(3,𝔽₃)	D₃	1	4	yes	—
10	C₃⋊((C₃⋊Dic₃)⋊C₃)	—	3	8	no	—
11	C₃²⋊H(3,𝔽₃)	Q₈	1	1	yes	characteristic, complemented, Sylow
12	#6308	—	3	4	no	—
13	#6308	—	9	12	no	—
14	C₃×#634	—	9	1	no	—
15	#634⋊C₃	—	9	12	no	—
16	C₂×C₃⁴	C₂²×C₃	1	1	yes	characteristic
17	C₂×C₃×H(3,𝔽₃)	—	2	4	no	—
18	C₂×C₃×H(3,𝔽₃)	C₂×D₃	1	4	yes	—
19	C₃³×C₄	—	2	4	no	—
20	C₃³×C₄	—	3	12	no	—
21	C₃³×C₄	C₃×D₃	1	4	yes	—
22	C₃³×C₄	C₃⋊D₃	1	1	yes	characteristic
23	C₃⋊(C₃⋊Dic₃)	—	3	8	no	—
24	C₃⋊(C₃⋊Dic₃)	—	9	24	no	—
25	C₃×C₃⋊Dic₃	—	9	2	no	—
26	C₄×H(3,𝔽₃)	—	3	12	no	—
27	C₄×H(3,𝔽₃)	—	6	12	no	—
28	(C₃⋊Dic₃)⋊C₃	—	9	24	no	—
29	C₃⁴	C₃×Q₈	1	1	yes	characteristic, complemented
30	C₃×H(3,𝔽₃)	—	2	4	no	—
31	C₃×H(3,𝔽₃)	Dic₆	1	4	yes	complemented
32	#634	—	9	1	no	—
33	#634	—	9	12	no	—
34	#634	—	27	12	no	—
35	#634	—	27	27	no	—
36	C₃×Dic₆	—	27	4	no	—
37	C₂×C₃³	—	2	4	no	—
38	C₂×C₃³	—	3	12	no	—
39	C₂×C₃³	C₂×C₃×D₃	1	4	yes	—
40	C₂×C₃³	C₂×C₃⋊D₃	1	1	yes	characteristic
41	C₂×H(3,𝔽₃)	—	3	12	no	—
42	C₂×H(3,𝔽₃)	—	6	12	no	—
43	C₃²×C₄	—	3	27	no	—
44	C₃²×C₄	—	3	12	no	—
45	C₃²×C₄	—	3	4	no	—
46	C₃²×C₄	—	6	4	no	—
47	C₃²×C₄	—	6	12	no	—
48	C₃²×C₄	(C₃⋊D₃)⋊C₃	1	12	yes	—
49	C₃²×C₄	C₃×C₃⋊D₃	1	1	yes	characteristic
50	C₃⋊Dic₃	—	9	2	no	—
51	C₃⋊Dic₃	—	9	24	no	—
52	C₃⋊Dic₃	—	27	24	no	—
53	C₃⋊Dic₃	—	27	54	no	—
54	C₃×Dic₃	—	27	8	no	—
55	C₃³	—	2	4	no	—
56	C₃³	—	3	12	no	—
57	C₃³	C₃×Dic₆	1	4	yes	—
58	C₃³	#634	1	1	yes	characteristic, complemented
59	H(3,𝔽₃)	—	3	12	no	—
60	H(3,𝔽₃)	—	6	12	no	—
61	C₃×Q₈	—	81	1	no	—
62	Dic₆	—	27	4	no	—
63	Dic₆	—	81	12	no	—
64	C₂×C₃²	—	3	27	no	—
65	C₂×C₃²	—	3	12	no	—
66	C₂×C₃²	—	3	4	no	—
67	C₂×C₃²	—	6	4	no	—
68	C₂×C₃²	—	6	12	no	—
69	C₂×C₃²	C₂×(C₃⋊D₃)⋊C₃	1	12	yes	—
70	C₂×C₃²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
71	C₃×C₄	—	3	12	no	—
72	C₃×C₄	—	9	1	no	—
73	C₃×C₄	—	18	4	no	—
74	C₃×C₄	—	81	2	no	—
75	C₃×C₄	C₃⋊((C₃⋊D₃)⋊C₃)	1	4	yes	—
76	Dic₃	—	27	8	no	—
77	Dic₃	—	81	24	no	—
78	C₃²	—	3	27	no	—
79	C₃²	—	3	12	no	—
80	C₃²	—	3	4	no	—
81	C₃²	—	6	4	no	—
82	C₃²	—	6	12	no	—
83	C₃²	#634⋊C₃	1	12	yes	complemented
84	C₃²	C₃×#634	1	1	yes	characteristic
85	Q₈	—	81	1	no	Sylow
86	C₂×C₃	—	3	12	no	—
87	C₂×C₃	—	9	1	no	—
88	C₂×C₃	—	18	4	no	—
89	C₂×C₃	C₂×C₃⋊((C₃⋊D₃)⋊C₃)	1	4	yes	—
90	C₄	—	81	2	no	—
91	C₄	#92301	1	1	yes	characteristic
92	C₃	—	3	12	no	—
93	C₃	—	9	1	no	—
94	C₃	—	18	4	no	—
95	C₃	#10619956	1	4	yes	—
96	C₂	C₂×#92301	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#92301
commutator, cosocle	C₂×C₃⁴	C₂²×C₃
Frattini, socle	C₂×C₃²	C₂×C₃×C₃⋊D₃
Fitting	C₄×C₃²⋊H(3,𝔽₃)	C₂
prime radical	C₂×C₃²⋊H(3,𝔽₃)	C₂²
hypercenter	C₄	#92301
hypocenter	C₃⁴	C₃×Q₈
LCS	G ▷ C₂×C₃⁴ ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
3. #10620175	C₃	semidirect
11. C₃²⋊H(3,𝔽₃)	Q₈	semidirect
29. C₃⁴	C₃×Q₈	semidirect
31. C₃×H(3,𝔽₃)	Dic₆	semidirect
58. C₃³	#634	semidirect
83. C₃²	#634⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₃²⋊H(3,𝔽₃)	2
#11757605	2
#10620175	3
#10619956	3