Subgroups — #67502

Order 384.

62 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	#4463	C₂	1	2	yes	maximal
3	#1373	—	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	#812	—	2	2	no	—
8	#364	—	3	2	no	—
9	C₄×C₁₆	D₃	1	1	yes	characteristic
10	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
11	C₂×C₃×C₈	D₄	1	1	yes	characteristic
12	C₃×C₄²	Q₈	1	1	yes	characteristic
13	C₃×C₁₆	—	2	1	no	—
14	C₃×C₁₆	C₈	1	2	yes	—
15	C₃⋊C₁₆	—	4	2	no	—
16	M₃₂	—	6	2	no	—
17	C₂×C₁₆	Dic₃	1	1	yes	characteristic
18	C₂×C₁₆	Dic₃	1	1	yes	characteristic
19	C₄×C₈	C₂×D₃	1	1	yes	characteristic
20	C₂×C₃×C₄	—	2	1	no	—
21	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
22	C₃×C₈	C₂×C₈	1	1	yes	characteristic
23	C₃×C₈	M₁₆	1	1	yes	characteristic
24	C₃×C₈	D₈	1	1	yes	characteristic
25	C₃×C₈	Q₁₆	1	1	yes	characteristic
26	C₂×C₈	D₁₂	1	1	yes	characteristic
27	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
28	C₄²	Dic₆	1	1	yes	characteristic
29	C₁₆	—	2	1	no	—
30	C₁₆	—	12	2	no	—
31	C₁₆	C₃⋊C₈	1	2	yes	—
32	C₂²×C₃	#108	1	1	yes	characteristic
33	C₃×C₄	—	2	2	no	—
34	C₃×C₄	C₄⋊C₈	1	1	yes	characteristic
35	C₃×C₄	#107	1	1	yes	characteristic
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	C₄⋊Dic₃	1	1	yes	characteristic
38	C₈	D₂₄	1	1	yes	characteristic
39	C₈	Dic₁₂	1	1	yes	characteristic
40	C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
41	C₈	#208	1	1	yes	characteristic
42	C₂×C₃	—	2	1	no	—
43	C₂×C₃	#335	1	1	yes	characteristic
44	C₂²	#819	1	1	yes	characteristic
45	C₄	—	2	2	no	—
46	C₄	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
47	C₄	#818	1	1	yes	characteristic
48	C₃	#1373	1	1	yes	characteristic, complemented, Sylow
49	C₂	—	2	1	no	—
50	C₂	#4458	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₂₄
commutator	C₃×C₈	C₂×C₈
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×C₄×C₁₆	C₂
socle	C₂×C₃	#335
prime radical	C₂²×C₃	#108
cosocle	C₃×C₄×C₈	C₂²
hypercenter	C₄×C₁₆	D₃
hypocenter	C₃	#1373
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
48. C₃	#1373	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
#4463	2
#1373	3