Subgroups — #68064

Order 384.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4682	C₂	1	2	yes	maximal
2	#4681	C₂	1	4	yes	maximal
3	C₃×C₈²	C₂	1	1	yes	maximal, characteristic
4	#1710	—	3	1	no	maximal, Sylow
5	#869	C₂²	1	4	yes	—
6	C₃×C₄×C₈	C₂²	1	2	yes	—
7	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
8	#818	—	2	4	no	—
9	#816	—	2	8	no	—
10	#500	—	3	2	no	—
11	#487	—	3	4	no	—
12	C₈²	D₃	1	1	yes	characteristic
13	C₂×Dic₆	—	2	4	no	—
14	C₂×C₃×C₈	D₄	1	4	yes	—
15	C₂×C₃×C₈	D₄	1	2	yes	—
16	C₃×C₄²	C₂³	1	1	yes	characteristic
17	C₄⋊Dic₃	—	2	4	no	—
18	C₄⋊Dic₃	—	2	4	no	—
19	C₄⋊Q₈	—	3	4	no	—
20	#107	—	6	4	no	—
21	#103	—	6	8	no	—
22	C₄×C₈	C₂×D₃	1	2	yes	—
23	C₄×C₈	C₂×D₃	1	1	yes	characteristic
24	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
25	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
26	C₂×Dic₃	—	4	4	no	—
27	Dic₆	—	4	8	no	—
28	C₃×C₈	—	2	4	no	—
29	C₃×C₈	Q₁₆	1	4	yes	—
30	C₂×Q₈	—	6	4	no	—
31	C₂×C₈	D₁₂	1	4	yes	—
32	C₂×C₈	D₁₂	1	2	yes	—
33	C₄⋊C₄	—	6	4	no	—
34	C₄⋊C₄	—	6	4	no	—
35	C₄²	C₂²×D₃	1	1	yes	characteristic
36	C₂²×C₃	#127	1	1	yes	characteristic
37	C₃×C₄	C₂×Q₁₆	1	2	yes	—
38	C₃×C₄	#135	1	4	yes	—
39	Dic₃	—	8	4	no	—
40	Q₈	—	12	8	no	—
41	C₂×C₄	—	12	4	no	—
42	C₂×C₄	C₂×D₁₂	1	2	yes	—
43	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
44	C₈	—	2	4	no	—
45	C₈	Dic₁₂	1	4	yes	—
46	C₂×C₃	#494	1	1	yes	characteristic
47	C₂×C₃	#495	1	2	yes	—
48	C₂²	#874	1	1	yes	characteristic
49	C₄	—	24	4	no	—
50	C₄	#904	1	4	yes	—
51	C₄	C₂×Dic₁₂	1	2	yes	—
52	C₃	#1710	1	1	yes	characteristic, complemented, Sylow
53	C₂	#4696	1	2	yes	—
54	C₂	#4699	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#874
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×C₈²	C₂
socle, prime radical	C₂²×C₃	#127
hypercenter	C₈²	D₃
hypocenter	C₃	#1710
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
52. C₃	#1710	semidirect

Maximal subgroups

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

subgroup	index
#4682	2
#4681	2
C₃×C₈²	2
#1710	3