Subgroups — C₂²×C₃⋊(C₃⋊C₈) #64129

Order 288.

228 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	C₂×C₃⋊(C₃⋊C₈)	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃⋊C₈	—	3	4	no	maximal
4	C₂³×C₃²	C₄	1	1	yes	characteristic
5	C₂×C₃²×C₄	C₄	1	3	yes	—
6	C₂×C₃²×C₄	C₂²	1	3	yes	—
7	C₃⋊(C₃⋊C₈)	C₂²	1	4	yes	direct product factor
8	C₂²×C₃×C₄	D₃	1	4	yes	—
9	C₂×C₃⋊C₈	—	3	24	no	—
10	C₂²×C₃²	C₈	1	4	yes	complemented
11	C₂²×C₃²	C₂×C₄	1	3	yes	—
12	C₃²×C₄	C₂×C₄	1	3	yes	—
13	C₃²×C₄	C₂³	1	1	yes	characteristic
14	C₂²×C₈	—	9	1	no	Sylow
15	C₂³×C₃	Dic₃	1	4	yes	—
16	C₂×C₃×C₄	Dic₃	1	12	yes	—
17	C₂×C₃×C₄	C₂×D₃	1	12	yes	—
18	C₃⋊C₈	—	3	16	no	—
19	C₂×C₃²	C₂×C₈	1	6	yes	complemented
20	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
21	C₂²×C₄	C₃⋊D₃	1	1	yes	characteristic
22	C₂×C₈	—	9	6	no	—
23	C₂²×C₃	C₃⋊C₈	1	16	yes	complemented
24	C₂²×C₃	C₂×Dic₃	1	12	yes	—
25	C₃×C₄	C₂×Dic₃	1	12	yes	—
26	C₃×C₄	C₂²×D₃	1	4	yes	—
27	C₃²	C₂²×C₈	1	1	yes	characteristic, complemented, Sylow
28	C₂³	C₃⋊Dic₃	1	1	yes	characteristic
29	C₂×C₄	C₃⋊Dic₃	1	3	yes	—
30	C₂×C₄	C₂×C₃⋊D₃	1	3	yes	—
31	C₈	—	9	4	no	—
32	C₂×C₃	C₂×C₃⋊C₈	1	24	yes	complemented
33	C₂×C₃	C₂²×Dic₃	1	4	yes	—
34	C₂²	C₃⋊(C₃⋊C₈)	1	4	yes	direct product factor
35	C₂²	C₂×C₃⋊Dic₃	1	3	yes	—
36	C₄	C₂×C₃⋊Dic₃	1	3	yes	—
37	C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
38	C₃	C₂²×C₃⋊C₈	1	4	yes	complemented
39	C₂	C₂×C₃⋊(C₃⋊C₈)	1	6	yes	direct product factor
40	C₂	C₂²×C₃⋊Dic₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃⋊(C₃⋊C₈)	C₂	direct (N × H)
7. C₃⋊(C₃⋊C₈)	C₂²	direct (N × H)
10. C₂²×C₃²	C₈	semidirect
19. C₂×C₃²	C₂×C₈	semidirect
23. C₂²×C₃	C₃⋊C₈	semidirect
27. C₃²	C₂²×C₈	semidirect
32. C₂×C₃	C₂×C₃⋊C₈	semidirect
34. C₂²	C₃⋊(C₃⋊C₈)	direct (N × H)
38. C₃	C₂²×C₃⋊C₈	semidirect
39. C₂	C₂×C₃⋊(C₃⋊C₈)	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
C₂×C₃⋊(C₃⋊C₈)	2
C₂²×C₃⋊C₈	3

Subgroups — C22×C3⋊(C3⋊C8) #64129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃⋊(C₃⋊C₈) #64129