Subgroups — C₂×#3757 #64161

Order 288.

396 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃²	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃⋊Dic₃	C₂	1	2	yes	maximal, complemented
3	#3757	C₂	1	4	yes	maximal, direct product factor
4	C₂×#217	—	3	4	no	maximal
5	C₂³×C₃²	—	2	4	no	—
6	C₂³×C₃²	C₄	1	4	yes	complemented
7	C₂³×C₃²	C₂²	1	2	yes	—
8	C₂³×C₃²	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊Dic₃	—	2	4	no	—
10	C₂×C₃⋊Dic₃	C₂²	1	4	yes	complemented
11	C₂⁴×C₃	D₃	1	4	yes	—
12	C₂²×Dic₃	—	3	8	no	—
13	#217	—	3	16	no	—
14	C₂²×C₃²	—	2	8	no	—
15	C₂²×C₃²	—	2	4	no	—
16	C₂²×C₃²	C₂×C₄	1	2	yes	—
17	C₂²×C₃²	C₂×C₄	1	4	yes	complemented
18	C₂²×C₃²	D₄	1	4	yes	—
19	C₂²×C₃²	C₂³	1	1	yes	characteristic
20	C₃⋊Dic₃	—	2	4	no	—
21	C₂×C₂²⋊C₄	—	9	1	no	Sylow
22	C₂³×C₃	—	2	16	no	—
23	C₂³×C₃	Dic₃	1	16	yes	complemented
24	C₂³×C₃	C₂×D₃	1	8	yes	—
25	C₂³×C₃	C₂×D₃	1	4	yes	—
26	C₂×Dic₃	—	3	16	no	—
27	C₂×Dic₃	—	6	16	no	—
28	C₂×C₃²	—	2	4	no	—
29	C₂×C₃²	C₂²⋊C₄	1	4	yes	complemented
30	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₃²	C₂×D₄	1	2	yes	—
32	C₂⁴	C₃⋊D₃	1	1	yes	characteristic
33	C₂²×C₄	—	9	2	no	—
34	C₂²⋊C₄	—	9	4	no	—
35	C₂²×C₃	—	2	32	no	—
36	C₂²×C₃	—	2	16	no	—
37	C₂²×C₃	C₂×Dic₃	1	8	yes	—
38	C₂²×C₃	C₂×Dic₃	1	16	yes	complemented
39	C₂²×C₃	#67	1	16	yes	—
40	C₂²×C₃	C₂²×D₃	1	4	yes	—
41	Dic₃	—	6	16	no	—
42	C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
43	C₂³	—	2	4	no	—
44	C₂³	C₃⋊Dic₃	1	4	yes	complemented
45	C₂³	C₂×C₃⋊D₃	1	2	yes	—
46	C₂³	C₂×C₃⋊D₃	1	1	yes	characteristic
47	C₂×C₄	—	9	4	no	—
48	C₂×C₄	—	18	4	no	—
49	C₂×C₃	—	2	16	no	—
50	C₂×C₃	#217	1	16	yes	complemented
51	C₂×C₃	C₂²×Dic₃	1	4	yes	—
52	C₂×C₃	C₂×#67	1	8	yes	—
53	C₂²	—	2	8	no	—
54	C₂²	—	2	4	no	—
55	C₂²	C₂×C₃⋊Dic₃	1	2	yes	—
56	C₂²	C₂×C₃⋊Dic₃	1	4	yes	complemented
57	C₂²	#638	1	4	yes	—
58	C₂²	C₂²×C₃⋊D₃	1	1	yes	characteristic
59	C₄	—	18	4	no	—
60	C₃	C₂×#217	1	4	yes	complemented
61	C₂	—	2	4	no	—
62	C₂	#3757	1	4	yes	direct product factor
63	C₂	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
64	C₂	C₂×#638	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂²×C₄
Frattini	C₂²	C₂²×C₃⋊D₃
Fitting, prime radical	C₂⁴×C₃²	C₂
socle	C₂³×C₃²	C₂²
cosocle	C₂²×C₃²	C₂³
hypercenter	C₂⁴	C₃⋊D₃
hypocenter	C₃²	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃⋊Dic₃	C₂	semidirect
3. #3757	C₂	direct (N × H)
6. C₂³×C₃²	C₄	semidirect
10. C₂×C₃⋊Dic₃	C₂²	semidirect
17. C₂²×C₃²	C₂×C₄	semidirect
23. C₂³×C₃	Dic₃	semidirect
29. C₂×C₃²	C₂²⋊C₄	semidirect
38. C₂²×C₃	C₂×Dic₃	semidirect
42. C₃²	C₂×C₂²⋊C₄	semidirect
44. C₂³	C₃⋊Dic₃	semidirect
50. C₂×C₃	#217	semidirect
56. C₂²	C₂×C₃⋊Dic₃	semidirect
60. C₃	C₂×#217	semidirect
62. C₂	#3757	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃²	2
C₂²×C₃⋊Dic₃	2
#3757	2
C₂×#217	3

Subgroups — C2×#3757 #64161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3757 #64161