Subgroups — C₂×#3745 #64112

Order 288.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₃⋊D₃)	C₂	1	2	yes	maximal, complemented
2	C₂×C₃²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#3745	C₂	1	4	yes	maximal, direct product factor
4	C₂×D₂₄	—	3	4	no	maximal
5	C₂²×C₃⋊D₃	—	2	2	no	—
6	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
7	C₄⋊(C₃⋊D₃)	—	2	2	no	—
8	C₄⋊(C₃⋊D₃)	C₂²	1	4	yes	complemented
9	C₃²×C₈	C₂²	1	2	yes	complemented
10	C₂×D₁₂	—	3	8	no	—
11	C₂×C₃×C₈	D₃	1	4	yes	complemented
12	D₂₄	—	3	16	no	—
13	C₂²×C₃²	D₄	1	1	yes	characteristic
14	C₂×C₃⋊D₃	—	2	4	no	—
15	C₂×C₃⋊D₃	—	4	4	no	—
16	C₃²×C₄	D₄	1	1	yes	characteristic
17	C₃²×C₄	C₂³	1	1	yes	characteristic
18	C₂×D₈	—	9	1	no	Sylow
19	C₂²×D₃	—	6	8	no	—
20	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
21	D₁₂	—	3	16	no	—
22	D₁₂	—	6	8	no	—
23	C₃×C₈	C₂×D₃	1	8	yes	complemented
24	C₂×C₃²	D₈	1	2	yes	complemented
25	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
26	C₃⋊D₃	—	4	4	no	—
27	C₂×D₄	—	9	2	no	—
28	D₈	—	9	4	no	—
29	C₂×C₈	C₃⋊D₃	1	1	yes	characteristic, complemented
30	C₂²×C₃	D₁₂	1	4	yes	—
31	C₂×D₃	—	6	16	no	—
32	C₂×D₃	—	12	16	no	—
33	C₃×C₄	D₁₂	1	4	yes	—
34	C₃×C₄	C₂²×D₃	1	4	yes	—
35	C₃²	C₂×D₈	1	1	yes	characteristic, complemented, Sylow
36	C₂³	—	18	2	no	—
37	D₄	—	9	4	no	—
38	D₄	—	18	2	no	—
39	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
40	C₈	C₂×C₃⋊D₃	1	2	yes	complemented
41	C₂×C₃	D₂₄	1	8	yes	complemented
42	C₂×C₃	C₂×D₁₂	1	4	yes	—
43	D₃	—	12	16	no	—
44	C₂²	—	18	4	no	—
45	C₂²	—	36	4	no	—
46	C₂²	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
47	C₄	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
48	C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
49	C₃	C₂×D₂₄	1	4	yes	complemented
50	C₂	—	36	4	no	—
51	C₂	#3745	1	2	yes	direct product factor
52	C₂	C₂×C₄⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊(C₃⋊D₃)
commutator, cosocle	C₃²×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×C₈	C₂
socle	C₂²×C₃²	D₄
hypercenter	C₂×C₈	C₃⋊D₃
hypocenter	C₃²	C₂×D₈
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊(C₃⋊D₃)	C₂	semidirect
2. C₂×C₃²×C₈	C₂	semidirect
3. #3745	C₂	direct (N × H)
8. C₄⋊(C₃⋊D₃)	C₂²	semidirect
9. C₃²×C₈	C₂²	semidirect
11. C₂×C₃×C₈	D₃	semidirect
23. C₃×C₈	C₂×D₃	semidirect
24. C₂×C₃²	D₈	semidirect
29. C₂×C₈	C₃⋊D₃	semidirect
35. C₃²	C₂×D₈	semidirect
40. C₈	C₂×C₃⋊D₃	semidirect
41. C₂×C₃	D₂₄	semidirect
49. C₃	C₂×D₂₄	semidirect
51. C₂	#3745	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₃⋊D₃)	2
C₂×C₃²×C₈	2
#3745	2
C₂×D₂₄	3

Subgroups — C2×#3745 #64112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3745 #64112