Subgroups — #67567

Order 384.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#364	C₂	1	1	yes	maximal, characteristic
2	#4463	C₂	1	1	yes	maximal, characteristic
3	#4461	C₂	1	1	yes	maximal, characteristic
4	#1369	—	3	1	no	maximal, Sylow
5	C₃×M₃₂	—	2	1	no	—
6	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
7	#812	—	2	1	no	—
8	C₂×C₃⋊C₁₆	C₄	1	1	yes	characteristic
9	C₂×C₃⋊C₁₆	C₄	1	1	yes	characteristic
10	#364	—	3	1	no	—
11	#364	D₃	1	1	yes	characteristic
12	#346	—	3	1	no	—
13	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃×C₈	D₄	1	1	yes	characteristic
15	C₃×C₄²	Q₈	1	1	yes	characteristic
16	C₃×C₁₆	—	4	1	no	—
17	C₃⋊C₁₆	—	2	1	no	—
18	C₃⋊C₁₆	—	4	1	no	—
19	C₃⋊C₁₆	C₈	1	2	yes	—
20	M₃₂	—	2	1	no	—
21	M₃₂	—	6	1	no	—
22	C₂×C₁₆	—	3	1	no	—
23	C₂×C₁₆	—	3	1	no	—
24	C₄×C₈	C₂×D₃	1	1	yes	characteristic
25	C₂×C₃×C₄	—	2	1	no	—
26	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
27	C₃×C₈	C₂×C₈	1	1	yes	characteristic
28	C₃×C₈	M₁₆	1	1	yes	characteristic
29	C₃×C₈	SD₁₆	1	1	yes	characteristic
30	C₃×C₈	SD₁₆	1	1	yes	characteristic
31	C₂×C₈	C₄×D₃	1	1	yes	characteristic
32	C₂×C₈	#67	1	1	yes	characteristic
33	C₄²	Dic₆	1	1	yes	characteristic
34	C₁₆	—	3	2	no	—
35	C₁₆	—	4	1	no	—
36	C₁₆	—	6	1	no	—
37	C₁₆	—	12	1	no	—
38	C₂²×C₃	#108	1	1	yes	characteristic
39	C₃×C₄	—	4	1	no	—
40	C₃×C₄	C₄⋊C₈	1	1	yes	characteristic
41	C₃×C₄	#106	1	1	yes	characteristic
42	C₂×C₄	—	2	1	no	—
43	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
44	C₈	D₃×C₈	1	1	yes	characteristic
45	C₈	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
46	C₈	#214	1	1	yes	characteristic
47	C₈	#215	1	1	yes	characteristic
48	C₂×C₃	—	2	1	no	—
49	C₂×C₃	#334	1	1	yes	characteristic
50	C₂²	#822	1	1	yes	characteristic
51	C₄	—	4	1	no	—
52	C₄	#808	1	1	yes	characteristic
53	C₄	Dic₃⋊C₈	1	1	yes	characteristic
54	C₃	#1369	1	1	yes	characteristic, complemented, Sylow
55	C₂	—	2	1	no	—
56	C₂	#4480	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#808
commutator	C₃×C₈	C₂×C₈
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×#364	C₂
socle	C₂×C₃	#334
prime radical	C₂²×C₃	#108
cosocle	C₃×C₄×C₈	C₂²
hypercenter	#364	D₃
hypocenter	C₃	#1369
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
54. C₃	#1369	semidirect

Maximal subgroups

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

subgroup	index
C₃×#364	2
#4463	2
#4461	2
#1369	3