Subgroups — Q₁₆×C₃⋊(C₃⋊D₃) #11723958

Order 864.

840 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₃⋊(C₃⋊D₃)	C₂	1	2	yes	maximal
2	C₃³×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#88863	C₂	1	2	yes	maximal, complemented
4	#88853	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic
6	Q₁₆×C₃⋊D₃	—	3	13	no	maximal
7	C₃³×Q₈	C₂²	1	2	yes	—
8	C₄×C₃⋊(C₃⋊D₃)	—	2	2	no	—
9	C₄×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
10	#6308	—	2	2	no	—
11	#6308	C₂²	1	2	yes	—
12	C₃³×C₈	C₂²	1	1	yes	characteristic
13	C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
14	Q₈×C₃⋊D₃	—	3	26	no	—
15	C₃²×Q₁₆	D₃	1	13	yes	complemented
16	#3756	—	3	26	no	—
17	#3746	—	3	13	no	—
18	C₈×C₃⋊D₃	—	3	13	no	—
19	C₂×C₃⋊(C₃⋊D₃)	D₄	1	1	yes	characteristic
20	C₃³×C₄	—	2	2	no	—
21	C₃³×C₄	C₂³	1	1	yes	characteristic
22	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
23	C₃⋊(C₃⋊Dic₃)	D₄	1	1	yes	characteristic
24	D₃×Q₁₆	—	9	13	no	—
25	C₃²×Q₈	C₂×D₃	1	26	yes	—
26	C₄×C₃⋊D₃	—	3	13	no	—
27	C₄×C₃⋊D₃	—	6	26	no	—
28	#634	—	3	26	no	—
29	#634	—	6	26	no	—
30	C₃²×C₈	C₂×D₃	1	13	yes	—
31	C₃⋊(C₃⋊C₈)	—	3	13	no	—
32	C₂×C₃³	C₂×D₄	1	1	yes	characteristic
33	C₃⋊(C₃⋊D₃)	Q₁₆	1	2	yes	direct product factor
34	D₃×Q₈	—	9	26	no	—
35	C₃×Q₁₆	C₃⋊D₃	1	13	yes	complemented
36	#216	—	9	26	no	—
37	Dic₁₂	—	9	13	no	—
38	D₃×C₈	—	9	13	no	—
39	C₂×C₃⋊D₃	—	3	13	no	—
40	C₃²×C₄	—	2	26	no	—
41	C₃²×C₄	C₂²×D₃	1	13	yes	—
42	C₃⋊Dic₃	—	3	13	no	—
43	C₃⋊Dic₃	—	6	26	no	—
44	C₂×Q₁₆	—	27	1	no	Sylow
45	C₃³	C₂×Q₁₆	1	1	yes	characteristic, complemented, Sylow
46	C₃×Q₈	C₂×C₃⋊D₃	1	26	yes	—
47	C₄×D₃	—	9	13	no	—
48	C₄×D₃	—	18	26	no	—
49	Dic₆	—	9	26	no	—
50	Dic₆	—	18	26	no	—
51	C₃×C₈	C₂×C₃⋊D₃	1	13	yes	—
52	C₃⋊C₈	—	9	13	no	—
53	C₂×C₃²	D₃×D₄	1	13	yes	—
54	C₃⋊D₃	—	3	26	no	—
55	C₂×Q₈	—	27	2	no	—
56	Q₁₆	—	27	2	no	—
57	Q₁₆	—	27	1	no	—
58	Q₁₆	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, direct product factor
59	C₂×C₈	—	27	1	no	—
60	C₂×D₃	—	9	13	no	—
61	C₃×C₄	—	2	26	no	—
62	C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
63	Dic₃	—	9	13	no	—
64	Dic₃	—	18	26	no	—
65	C₃²	D₃×Q₁₆	1	13	yes	complemented
66	Q₈	—	27	2	no	—
67	Q₈	—	54	2	no	—
68	Q₈	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
69	C₂×C₄	—	27	1	no	—
70	C₂×C₄	—	54	2	no	—
71	C₈	—	27	1	no	—
72	C₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
73	C₂×C₃	D₄×C₃⋊D₃	1	13	yes	—
74	D₃	—	9	26	no	—
75	C₂²	—	27	1	no	—
76	C₄	—	2	2	no	—
77	C₄	—	27	1	no	—
78	C₄	—	54	2	no	—
79	C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
80	C₃	Q₁₆×C₃⋊D₃	1	13	yes	complemented
81	C₂	—	27	2	no	—
82	C₂	D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×C₃⋊(C₃⋊D₃)
commutator, cosocle	C₃³×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₃³×Q₁₆	C₂
socle	C₂×C₃³	C₂×D₄
prime radical	C₂×C₃⋊(C₃⋊D₃)	D₄
hypercenter	Q₁₆	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₂×Q₁₆
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₃³×Q₁₆	C₂	semidirect
3. #88863	C₂	semidirect
4. #88853	C₂	semidirect
15. C₃²×Q₁₆	D₃	semidirect
33. C₃⋊(C₃⋊D₃)	Q₁₆	direct (N × H)
35. C₃×Q₁₆	C₃⋊D₃	semidirect
45. C₃³	C₂×Q₁₆	semidirect
58. Q₁₆	C₃⋊(C₃⋊D₃)	direct (N × H)
65. C₃²	D₃×Q₁₆	semidirect
80. C₃	Q₁₆×C₃⋊D₃	semidirect

Maximal subgroups

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

subgroup	index
Q₈×C₃⋊(C₃⋊D₃)	2
C₃³×Q₁₆	2
#88863	2
#88853	2
C₈×C₃⋊(C₃⋊D₃)	2
Q₁₆×C₃⋊D₃	3

Subgroups — Q16×C3⋊(C3⋊D3) #11723958

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×C₃⋊(C₃⋊D₃) #11723958