Subgroups — #67501

Order 384.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#4458	C₂	1	2	yes	maximal
3	#1372	—	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	C₄×C₁₆	D₃	1	1	yes	characteristic
9	#335	—	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	D₈	1	1	yes	characteristic
22	C₂×C₃×C₄	Q₁₆	1	1	yes	characteristic
23	C₃×C₈	—	2	1	no	—
24	C₃×C₈	C₂×C₈	1	1	yes	characteristic
25	C₃×C₈	M₁₆	1	1	yes	characteristic
26	C₃⋊C₈	—	8	2	no	—
27	C₂×C₈	—	12	2	no	—
28	C₂×C₈	Dic₆	1	1	yes	characteristic
29	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
30	C₄²	D₁₂	1	1	yes	characteristic
31	C₁₆	—	2	1	no	—
32	C₁₆	C₃⋊C₈	1	2	yes	complemented
33	C₂²×C₃	#107	1	1	yes	characteristic
34	C₃×C₄	—	2	1	no	—
35	C₃×C₄	C₄⋊C₈	1	1	yes	characteristic
36	C₃×C₄	#108	1	1	yes	characteristic
37	C₃×C₄	SD₃₂	1	2	yes	—
38	C₂×C₄	D₂₄	1	1	yes	characteristic
39	C₂×C₄	Dic₁₂	1	1	yes	characteristic
40	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
41	C₈	—	2	1	no	—
42	C₈	—	24	2	no	—
43	C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
44	C₈	#208	1	1	yes	characteristic
45	C₂×C₃	#335	1	1	yes	characteristic
46	C₂×C₃	#367	1	1	yes	characteristic
47	C₂×C₃	#368	1	1	yes	characteristic
48	C₂²	#818	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	#800	1	2	yes	—
51	C₄	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
52	C₄	#819	1	1	yes	characteristic
53	C₃	#1372	1	1	yes	characteristic, complemented, Sylow
54	C₂	#4458	1	1	yes	characteristic
55	C₂	#4505	1	1	yes	characteristic
56	C₂	#4506	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₂₄
commutator	C₃×C₈	C₂×C₈
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×C₄×C₁₆	C₂
socle, prime radical	C₂²×C₃	#107
cosocle	C₃×C₄×C₈	C₂²
hypercenter	C₄×C₁₆	D₃
hypocenter	C₃	#1372
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
15. C₃×C₁₆	C₈	semidirect
32. C₁₆	C₃⋊C₈	semidirect
53. C₃	#1372	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₁₆	2
#4458	2
#1372	3