Subgroups — #67601

Order 384.

52 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	#4506	C₂	1	1	yes	maximal, characteristic
3	#4505	C₂	1	1	yes	maximal, characteristic
4	#1426	—	3	1	no	maximal, Sylow
5	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
6	#819	—	2	1	no	—
7	#818	—	2	1	no	—
8	C₃×C₃₂	C₄	1	2	yes	complemented
9	M₆₄	D₃	1	1	yes	characteristic
10	#368	—	3	1	no	—
11	#367	—	3	1	no	—
12	C₂×C₃×C₈	D₄	1	1	yes	characteristic
13	C₄⋊Dic₃	—	4	1	no	—
14	#208	—	4	1	no	—
15	C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
16	C₃×C₁₆	Q₈	1	1	yes	characteristic
17	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
18	#108	—	6	1	no	—
19	#107	—	6	1	no	—
20	C₃₂	Dic₃	1	2	yes	complemented
21	C₂×C₃×C₄	D₈	1	1	yes	characteristic
22	C₂×Dic₃	—	8	1	no	—
23	C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
24	C₃×C₈	Q₁₆	1	1	yes	characteristic
25	C₃⋊C₈	—	8	1	no	—
26	M₁₆	—	12	1	no	—
27	C₂×C₈	D₁₂	1	1	yes	characteristic
28	C₄⋊C₄	—	12	1	no	—
29	C₁₆	Dic₆	1	1	yes	characteristic
30	C₁₆	C₂×Dic₃	1	1	yes	characteristic
31	C₂²×C₃	SD₃₂	1	1	yes	characteristic
32	C₃×C₄	#107	1	1	yes	characteristic
33	C₃×C₄	SD₃₂	1	1	yes	characteristic
34	Dic₃	—	16	1	no	—
35	C₂×C₄	—	24	1	no	—
36	C₂×C₄	D₂₄	1	1	yes	characteristic
37	C₈	—	24	1	no	—
38	C₈	Dic₁₂	1	1	yes	characteristic
39	C₈	C₄⋊Dic₃	1	1	yes	characteristic
40	C₂×C₃	—	2	1	no	—
41	C₂×C₃	#367	1	1	yes	characteristic
42	C₂²	#800	1	1	yes	characteristic
43	C₄	—	48	1	no	—
44	C₄	#800	1	1	yes	characteristic
45	C₄	#818	1	1	yes	characteristic
46	C₃	#1426	1	1	yes	characteristic, complemented, Sylow
47	C₂	—	2	1	no	—
48	C₂	#4505	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4505
commutator	C₃×C₁₆	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₃
Fitting	C₃×M₆₄	C₂
socle	C₂×C₃	#367
prime radical	C₂²×C₃	SD₃₂
cosocle	C₂×C₃×C₁₆	C₂²
hypercenter	M₆₄	D₃
hypocenter	C₃	#1426
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
8. C₃×C₃₂	C₄	semidirect
20. C₃₂	Dic₃	semidirect
46. C₃	#1426	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
#4506	2
#4505	2
#1426	3