Subgroups — #67513

Order 384.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#346	C₂	1	1	yes	maximal, characteristic
2	#4457	C₂	1	2	yes	maximal
3	#1368	—	3	1	no	maximal, Sylow
4	C₂×C₃×C₁₆	C₄	1	1	yes	characteristic
5	C₂×C₃×C₁₆	C₄	1	1	yes	characteristic
6	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₄⋊(C₃⋊C₈)	—	2	2	no	—
8	#346	D₃	1	1	yes	characteristic
9	#334	—	3	2	no	—
10	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
11	C₂×C₃×C₈	Q₈	1	1	yes	characteristic
12	C₃×C₄²	D₄	1	1	yes	characteristic
13	C₂×C₃⋊C₈	—	4	2	no	—
14	C₃×C₁₆	—	2	1	no	—
15	C₃×C₁₆	C₈	1	2	yes	complemented
16	C₂×C₁₆	Dic₃	1	1	yes	characteristic
17	C₂×C₁₆	Dic₃	1	1	yes	characteristic
18	C₄⋊C₈	—	6	2	no	—
19	C₄×C₈	C₂×D₃	1	1	yes	characteristic
20	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₃×C₄	SD₁₆	1	2	yes	—
22	C₃×C₈	—	2	1	no	—
23	C₃×C₈	C₂×C₈	1	1	yes	characteristic
24	C₃×C₈	M₁₆	1	1	yes	characteristic
25	C₃⋊C₈	—	8	2	no	—
26	C₂×C₈	—	12	2	no	—
27	C₂×C₈	Dic₆	1	1	yes	characteristic
28	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
29	C₄²	D₁₂	1	1	yes	characteristic
30	C₁₆	—	2	1	no	—
31	C₁₆	C₃⋊C₈	1	2	yes	complemented
32	C₂²×C₃	#106	1	1	yes	characteristic
33	C₃×C₄	—	2	2	no	—
34	C₃×C₄	C₄⋊C₈	1	1	yes	characteristic
35	C₃×C₄	#108	1	1	yes	characteristic
36	C₂×C₄	#204	1	2	yes	—
37	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
38	C₈	—	2	1	no	—
39	C₈	—	24	2	no	—
40	C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
41	C₈	#208	1	1	yes	characteristic
42	C₂×C₃	#334	1	1	yes	characteristic
43	C₂×C₃	#365	1	2	yes	—
44	C₂²	#817	1	1	yes	characteristic
45	C₄	—	2	2	no	—
46	C₄	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
47	C₄	#819	1	1	yes	characteristic
48	C₃	#1368	1	1	yes	characteristic, complemented, Sylow
49	C₂	#4457	1	1	yes	characteristic
50	C₂	#4513	1	2	yes	—

	subgroup	G/H
center	C₂²	#817
commutator	C₃×C₈	C₂×C₈
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×#346	C₂
socle, prime radical	C₂²×C₃	#106
cosocle	C₃×C₄×C₈	C₂²
hypercenter	#346	D₃
hypocenter	C₃	#1368
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
15. C₃×C₁₆	C₈	semidirect
31. C₁₆	C₃⋊C₈	semidirect
48. C₃	#1368	semidirect

Maximal subgroups

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

subgroup	index
C₃×#346	2
#4457	2
#1368	3