Subgroups — #63641

Order 288.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	Dic₃⋊C₈	—	3	4	no	maximal
5	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
6	C₂×C₃⋊Dic₃	C₄	1	1	yes	characteristic
7	C₂×C₃⋊Dic₃	C₄	1	1	yes	characteristic
8	C₃²×C₈	—	2	1	no	—
9	C₃⋊(C₃⋊C₈)	—	2	1	no	—
10	C₂×C₃×C₈	D₃	1	4	yes	—
11	C₄×Dic₃	—	3	4	no	—
12	C₂×C₃⋊C₈	—	3	4	no	—
13	C₂²×C₃²	C₂×C₄	1	1	yes	characteristic
14	C₃²×C₄	D₄	1	1	yes	characteristic
15	C₃²×C₄	Q₈	1	1	yes	characteristic
16	C₃⋊Dic₃	—	2	1	no	—
17	C₃⋊Dic₃	C₈	1	2	yes	complemented
18	C₄⋊C₈	—	9	1	no	Sylow
19	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
20	C₂×Dic₃	—	3	4	no	—
21	C₂×Dic₃	—	3	4	no	—
22	C₃×C₈	—	2	4	no	—
23	C₃⋊C₈	—	6	4	no	—
24	C₂×C₃²	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₃²	C₂×C₈	1	1	yes	characteristic
26	C₂×C₃²	M₁₆	1	1	yes	characteristic
27	C₂×C₈	—	9	1	no	—
28	C₂×C₈	C₃⋊D₃	1	1	yes	characteristic
29	C₄²	—	9	1	no	—
30	C₂²×C₃	C₄×D₃	1	4	yes	—
31	C₃×C₄	Dic₆	1	4	yes	—
32	C₃×C₄	#67	1	4	yes	—
33	Dic₃	—	3	8	no	—
34	Dic₃	—	6	4	no	—
35	C₃²	C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₄	—	9	1	no	—
37	C₂×C₄	—	9	1	no	—
38	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
39	C₈	—	2	1	no	—
40	C₈	—	18	1	no	—
41	C₂×C₃	D₃×C₈	1	4	yes	—
42	C₂×C₃	ΓL(1,𝔽₂₅)	1	4	yes	—
43	C₂×C₃	Dic₃⋊C₄	1	4	yes	—
44	C₂²	C₄×C₃⋊D₃	1	1	yes	characteristic
45	C₄	—	9	2	no	—
46	C₄	—	18	1	no	—
47	C₄	#634	1	1	yes	characteristic
48	C₄	#638	1	1	yes	characteristic
49	C₃	Dic₃⋊C₈	1	4	yes	complemented
50	C₂	C₈×C₃⋊D₃	1	1	yes	characteristic
51	C₂	#3743	1	1	yes	characteristic
52	C₂	#3750	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
17. C₃⋊Dic₃	C₈	semidirect
35. C₃²	C₄⋊C₈	semidirect
49. C₃	Dic₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₈	2
C₄×C₃⋊Dic₃	2
C₂×C₃⋊(C₃⋊C₈)	2
Dic₃⋊C₈	3