Subgroups — #68717

Order 384.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5122	C₂	1	3	yes	maximal
2	C₃×#338	C₂	1	1	yes	maximal, characteristic
3	#4557	C₂	1	3	yes	maximal
4	#1747	—	3	1	no	maximal, Sylow
5	C₂×C₃×M₁₆	C₂²	1	3	yes	—
6	#926	C₂²	1	3	yes	—
7	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
8	#815	C₄	1	6	yes	complemented
9	C₈×Dic₃	C₄	1	6	yes	complemented
10	#405	—	3	3	no	—
11	#338	D₃	1	1	yes	characteristic
12	H(3,ℤ/4)	—	3	3	no	—
13	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
14	C₃×M₁₆	—	2	6	no	—
15	C₂×C₃×C₈	C₂×C₄	1	6	yes	—
16	#217	—	2	3	no	—
17	C₄⋊Dic₃	—	2	3	no	—
18	C₄×Dic₃	C₂×C₄	1	6	yes	—
19	C₂×C₃⋊C₈	C₂×C₄	1	3	yes	—
20	C₂×C₃⋊C₈	C₂×C₄	1	3	yes	—
21	C₂×M₁₆	C₂×D₃	1	3	yes	—
22	C₂²×C₈	—	3	1	no	—
23	#117	—	3	3	no	—
24	#97	—	3	6	no	—
25	C₄×C₈	—	3	6	no	—
26	C₂³×C₃	#41	1	1	yes	characteristic
27	C₂×C₃×C₄	C₂²×C₄	1	3	yes	—
28	C₂×C₃×C₄	#41	1	3	yes	—
29	C₂×Dic₃	—	2	6	no	—
30	C₃×C₈	—	2	6	no	—
31	C₃⋊C₈	C₄²	1	1	yes	characteristic
32	C₃⋊C₈	C₄²	1	3	yes	—
33	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
34	M₁₆	—	2	6	no	—
35	C₂×C₈	—	3	3	no	—
36	C₂×C₈	—	3	3	no	—
37	C₂×C₈	C₄×D₃	1	6	yes	—
38	C₄⋊C₄	—	6	3	no	—
39	C₂²⋊C₄	—	6	3	no	—
40	C₄²	—	3	6	no	—
41	C₂²×C₃	—	4	1	no	—
42	C₂²×C₃	#117	1	3	yes	—
43	C₃×C₄	C₂×C₄²	1	1	yes	characteristic
44	C₃×C₄	#117	1	3	yes	—
45	Dic₃	—	4	6	no	—
46	C₂³	#239	1	1	yes	characteristic
47	C₂×C₄	—	6	6	no	—
48	C₂×C₄	C₂×C₄×D₃	1	3	yes	—
49	C₂×C₄	#237	1	3	yes	—
50	C₈	—	2	6	no	—
51	C₈	—	3	1	no	—
52	C₈	—	3	3	no	—
53	C₂×C₃	—	2	3	no	—
54	C₂×C₃	#376	1	1	yes	characteristic
55	C₂²	—	4	1	no	—
56	C₂²	#892	1	3	yes	—
57	C₄	—	12	6	no	—
58	C₄	C₄²×D₃	1	1	yes	characteristic
59	C₄	#877	1	3	yes	—
60	C₃	#1747	1	1	yes	characteristic, complemented, Sylow
61	C₂	—	2	3	no	—
62	C₂	#4648	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄²×D₃
commutator	C₃×C₄	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#338	C₂
socle	C₂×C₃	#376
prime radical	C₂³×C₃	#41
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#338	D₃
hypocenter	C₃	#1747
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
8. #815	C₄	semidirect
9. C₈×Dic₃	C₄	semidirect
60. C₃	#1747	semidirect

Maximal subgroups

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

subgroup	index
#5122	2
C₃×#338	2
#4557	2
#1747	3