Subgroups — #67600

Order 384.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×M₆₄	C₂	1	1	yes	maximal, characteristic
2	#4502	C₂	1	1	yes	maximal, characteristic
3	#4499	C₂	1	1	yes	maximal, characteristic
4	#1398	—	3	1	no	maximal, Sylow
5	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
6	C₈×Dic₃	C₄	1	1	yes	characteristic
7	C₂×C₃⋊C₁₆	C₄	1	1	yes	characteristic
8	C₃×C₃₂	C₄	1	2	yes	complemented
9	C₃⋊C₃₂	C₄	1	2	yes	complemented
10	M₆₄	—	3	1	no	—
11	M₆₄	D₃	1	1	yes	characteristic
12	#346	—	3	1	no	—
13	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
14	C₄×Dic₃	C₈	1	1	yes	characteristic
15	C₂×C₃⋊C₈	C₈	1	1	yes	characteristic
16	C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
17	C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₃⋊C₁₆	C₈	1	2	yes	—
19	C₂×C₁₆	—	3	1	no	—
20	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
21	C₄×C₈	—	3	1	no	—
22	C₃₂	—	3	2	no	—
23	C₃₂	Dic₃	1	2	yes	complemented
24	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
25	C₂×Dic₃	—	2	1	no	—
26	C₃×C₈	C₄²	1	1	yes	characteristic
27	C₃×C₈	C₂×C₈	1	1	yes	characteristic
28	C₃⋊C₈	—	2	1	no	—
29	C₂×C₈	—	3	1	no	—
30	C₂×C₈	C₄×D₃	1	1	yes	characteristic
31	C₄²	—	3	1	no	—
32	C₁₆	—	3	2	no	—
33	C₁₆	C₄×D₃	1	1	yes	characteristic
34	C₁₆	C₂×Dic₃	1	1	yes	characteristic
35	C₂²×C₃	M₃₂	1	1	yes	characteristic
36	C₃×C₄	C₄×C₈	1	1	yes	characteristic
37	C₃×C₄	M₃₂	1	1	yes	characteristic
38	Dic₃	—	4	1	no	—
39	C₂×C₄	—	6	1	no	—
40	C₂×C₄	D₃×C₈	1	1	yes	characteristic
41	C₈	—	6	1	no	—
42	C₈	D₃×C₈	1	1	yes	characteristic
43	C₈	C₄×Dic₃	1	1	yes	characteristic
44	C₂×C₃	—	2	1	no	—
45	C₂×C₃	#346	1	1	yes	characteristic
46	C₂²	#798	1	1	yes	characteristic
47	C₄	—	12	1	no	—
48	C₄	#798	1	1	yes	characteristic
49	C₄	C₈×Dic₃	1	1	yes	characteristic
50	C₃	#1398	1	1	yes	characteristic, complemented, Sylow
51	C₂	—	2	1	no	—
52	C₂	#4502	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄×Dic₃
commutator	C₃×C₄	C₄×C₈
Frattini	C₂×C₁₆	C₂×D₃
Fitting	C₃×M₆₄	C₂
socle	C₂×C₃	#346
prime radical	C₂²×C₃	M₃₂
cosocle	C₂×C₃×C₁₆	C₂²
hypercenter	M₆₄	D₃
hypocenter	C₃	#1398
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₈ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
8. C₃×C₃₂	C₄	semidirect
9. C₃⋊C₃₂	C₄	semidirect
23. C₃₂	Dic₃	semidirect
50. C₃	#1398	semidirect

Maximal subgroups

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

subgroup	index
C₃×M₆₄	2
#4502	2
#4499	2
#1398	3