Subgroups — #3417⋊C₃ #73282

Order 384.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#578⋊C₃	C₂	1	2	yes	maximal, complemented
2	#504⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#3417	C₃	1	1	yes	maximal, characteristic, complemented, Sylow
4	2₋¹⁺⁴⋊C₃	—	2	2	no	—
5	#131⋊C₃	C₂²	1	1	yes	characteristic
6	C₃×D₁₆	—	4	1	no	maximal
7	#578	C₂×C₃	1	2	yes	complemented
8	#510	—	3	2	no	—
9	#508	—	3	1	no	—
10	C₂×Q₃₂	—	3	1	no	—
11	#504	C₂×C₃	1	1	yes	characteristic, complemented
12	#41⋊C₃	D₄	1	1	yes	characteristic
13	C₂×SL(2,𝔽₃)	—	4	2	no	—
14	C₃×D₈	—	4	2	no	—
15	C₃×C₁₆	—	4	1	no	—
16	2₋¹⁺⁴	—	2	2	no	—
17	#137	—	6	2	no	—
18	#135	—	3	2	no	—
19	C₂×Q₁₆	—	3	2	no	—
20	#131	C₂²×C₃	1	1	yes	characteristic
21	Q₃₂	—	3	2	no	—
22	Q₃₂	—	3	1	no	—
23	SD₃₂	—	3	2	no	—
24	D₁₆	A₄	1	1	yes	characteristic
25	M₃₂	—	3	1	no	—
26	C₂×C₁₆	—	3	1	no	—
27	C₃×D₄	—	8	2	no	—
28	SL(2,𝔽₃)	D₈	1	1	yes	characteristic
29	C₃×C₈	—	4	1	no	—
30	#41	—	6	2	no	—
31	#41	—	12	2	no	—
32	#41	C₃×D₄	1	1	yes	characteristic
33	C₂×Q₈	—	4	2	no	—
34	C₂×Q₈	—	6	2	no	—
35	Q₁₆	—	3	2	no	—
36	Q₁₆	—	6	2	no	—
37	SD₁₆	—	6	2	no	—
38	D₈	C₂×A₄	1	2	yes	—
39	M₁₆	—	3	1	no	—
40	C₂×C₈	—	3	1	no	—
41	C₁₆	—	3	1	no	—
42	C₁₆	C₂×A₄	1	1	yes	characteristic
43	C₂²×C₃	—	16	2	no	—
44	C₃×C₄	—	4	1	no	—
45	Q₈	—	6	2	no	—
46	Q₈	—	12	2	no	—
47	Q₈	C₃×D₈	1	1	yes	characteristic
48	D₄	—	2	2	no	—
49	D₄	—	3	1	no	—
50	D₄	—	12	2	no	—
51	C₂×C₄	—	3	1	no	—
52	C₂×C₄	—	12	2	no	—
53	C₂×C₄	—	12	2	no	—
54	C₈	—	3	1	no	—
55	C₈	C₂²×A₄	1	1	yes	characteristic
56	C₂×C₃	—	4	1	no	—
57	C₂×C₃	—	32	2	no	—
58	C₂²	—	3	1	no	—
59	C₂²	—	4	2	no	—
60	C₄	—	3	1	no	—
61	C₄	—	12	2	no	—
62	C₄	D₄×A₄	1	1	yes	characteristic
63	C₃	—	4	1	no	Sylow
64	C₂	—	6	1	no	—
65	C₂	—	8	2	no	—
66	C₂	A₄×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	A₄×D₈
commutator, cosocle	#131	C₂²×C₃
Frattini	C₈	C₂²×A₄
Fitting	#3417	C₃
hypercenter	D₁₆	A₄
hypocenter	Q₈	C₃×D₈
LCS	G ▷ #131 ▷ #41 ▷ Q₈
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₁₆

Complements

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

N	H ≅ G/N	type
1. #578⋊C₃	C₂	semidirect
2. #504⋊C₃	C₂	semidirect
3. #3417	C₃	semidirect
7. #578	C₂×C₃	semidirect
11. #504	C₂×C₃	semidirect

Maximal subgroups

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

subgroup	index
#578⋊C₃	2
#504⋊C₃	2
#3417	3
C₃×D₁₆	4

Subgroups — #3417⋊C3 #73282

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3417⋊C₃ #73282