Subgroups — C₂×#3748 #64130

Order 288.

204 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₄	C₂	1	1	yes	maximal, characteristic
2	#3748	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃⋊(C₃⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₂×#208	—	3	4	no	maximal
5	C₂³×C₃²	C₄	1	1	yes	characteristic
6	C₂×C₃²×C₄	C₄	1	2	yes	—
7	C₂×C₃²×C₄	C₄	1	1	yes	characteristic
8	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃²×C₄	C₂²	1	2	yes	—
10	C₃⋊(C₃⋊C₈)	C₂²	1	4	yes	complemented
11	C₂²×C₃×C₄	D₃	1	4	yes	—
12	#208	—	3	16	no	—
13	C₂×C₃⋊C₈	—	3	8	no	—
14	C₂²×C₃²	—	2	2	no	—
15	C₂²×C₃²	C₂×C₄	1	2	yes	—
16	C₂²×C₃²	C₂×C₄	1	1	yes	characteristic
17	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
18	C₃²×C₄	C₂×C₄	1	2	yes	—
19	C₃²×C₄	C₂³	1	1	yes	characteristic
20	C₂×M₁₆	—	9	1	no	Sylow
21	C₂³×C₃	Dic₃	1	4	yes	—
22	C₂×C₃×C₄	Dic₃	1	8	yes	—
23	C₂×C₃×C₄	Dic₃	1	4	yes	—
24	C₂×C₃×C₄	C₂×D₃	1	8	yes	—
25	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
26	C₃⋊C₈	—	3	16	no	—
27	C₂×C₃²	—	2	2	no	—
28	C₂×C₃²	M₁₆	1	2	yes	complemented
29	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄	C₃⋊D₃	1	1	yes	characteristic
31	M₁₆	—	9	4	no	—
32	C₂×C₈	—	9	2	no	—
33	C₂²×C₃	—	2	8	no	—
34	C₂²×C₃	C₂×Dic₃	1	8	yes	—
35	C₂²×C₃	C₂×Dic₃	1	4	yes	—
36	C₃×C₄	C₂×Dic₃	1	8	yes	—
37	C₃×C₄	C₂×Dic₃	1	4	yes	—
38	C₃×C₄	C₂²×D₃	1	4	yes	—
39	C₃²	C₂×M₁₆	1	1	yes	characteristic, complemented, Sylow
40	C₂³	C₃⋊Dic₃	1	1	yes	characteristic
41	C₂×C₄	C₃⋊Dic₃	1	1	yes	characteristic
42	C₂×C₄	C₃⋊Dic₃	1	2	yes	—
43	C₂×C₄	C₂×C₃⋊D₃	1	2	yes	—
44	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
45	C₈	—	9	4	no	—
46	C₂×C₃	—	2	8	no	—
47	C₂×C₃	#208	1	8	yes	complemented
48	C₂×C₃	C₂²×Dic₃	1	4	yes	—
49	C₂²	—	2	2	no	—
50	C₂²	C₂×C₃⋊Dic₃	1	2	yes	—
51	C₂²	C₂×C₃⋊Dic₃	1	1	yes	characteristic
52	C₄	C₂×C₃⋊Dic₃	1	2	yes	—
53	C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
54	C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
55	C₃	C₂×#208	1	4	yes	complemented
56	C₂	—	2	2	no	—
57	C₂	#3748	1	2	yes	direct product factor
58	C₂	C₂²×C₃⋊Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂²×C₄
Frattini	C₄	C₂²×C₃⋊D₃
Fitting	C₂²×C₃²×C₄	C₂
socle	C₂²×C₃²	C₂×C₄
prime radical	C₂³×C₃²	C₄
cosocle	C₃²×C₄	C₂³
hypercenter	C₂²×C₄	C₃⋊D₃
hypocenter	C₃²	C₂×M₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. #3748	C₂	direct (N × H)
3. C₂×C₃⋊(C₃⋊C₈)	C₂	semidirect
10. C₃⋊(C₃⋊C₈)	C₂²	semidirect
28. C₂×C₃²	M₁₆	semidirect
39. C₃²	C₂×M₁₆	semidirect
47. C₂×C₃	#208	semidirect
55. C₃	C₂×#208	semidirect
57. C₂	#3748	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃²×C₄	2
#3748	2
C₂×C₃⋊(C₃⋊C₈)	2
C₂×#208	3

Subgroups — C2×#3748 #64130

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3748 #64130