Subgroups — #63643

Order 288.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#634	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×C₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
4	#816	—	3	4	no	maximal
5	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
6	C₂×C₃⋊Dic₃	—	2	1	no	—
7	C₂×C₃⋊Dic₃	—	2	1	no	—
8	#634	—	2	1	no	—
9	#634	C₄	1	2	yes	complemented
10	C₃²×C₈	—	2	1	no	—
11	C₂×Dic₆	—	3	4	no	—
12	C₂×C₃×C₈	D₃	1	4	yes	—
13	C₄⋊Dic₃	—	3	4	no	—
14	C₂²×C₃²	D₄	1	1	yes	characteristic
15	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
16	C₃²×C₄	D₄	1	1	yes	characteristic
17	C₃⋊Dic₃	—	2	2	no	—
18	C₃⋊Dic₃	—	4	1	no	—
19	#103	—	9	1	no	Sylow
20	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
21	C₂×Dic₃	—	6	4	no	—
22	C₂×Dic₃	—	6	4	no	—
23	Dic₆	—	3	8	no	—
24	Dic₆	—	6	4	no	—
25	C₃×C₈	—	2	4	no	—
26	C₂×C₃²	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₃²	SD₁₆	1	1	yes	characteristic
28	C₂×C₃²	Q₁₆	1	1	yes	characteristic
29	C₂×Q₈	—	9	1	no	—
30	C₂×C₈	C₃⋊D₃	1	1	yes	characteristic
31	C₄⋊C₄	—	9	1	no	—
32	C₂²×C₃	D₁₂	1	4	yes	—
33	C₃×C₄	C₄×D₃	1	4	yes	—
34	C₃×C₄	#67	1	4	yes	—
35	Dic₃	—	6	8	no	—
36	Dic₃	—	12	4	no	—
37	C₃²	#103	1	1	yes	characteristic, complemented, Sylow
38	Q₈	—	9	2	no	—
39	Q₈	—	18	1	no	—
40	C₂×C₄	—	18	1	no	—
41	C₂×C₄	—	18	1	no	—
42	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
43	C₈	—	2	1	no	—
44	C₂×C₃	#204	1	4	yes	—
45	C₂×C₃	Dic₁₂	1	4	yes	—
46	C₂×C₃	(C₂×D₃)⋊C₄	1	4	yes	—
47	C₂²	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
48	C₄	—	18	2	no	—
49	C₄	—	36	1	no	—
50	C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
51	C₄	#638	1	1	yes	characteristic
52	C₃	#816	1	4	yes	complemented
53	C₂	#3744	1	1	yes	characteristic
54	C₂	#3746	1	1	yes	characteristic
55	C₂	#3752	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊(C₃⋊D₃)
commutator	C₃²×C₄	C₂×C₄
Frattini	C₂×C₄	C₂×C₃⋊D₃
Fitting	C₂×C₃²×C₈	C₂
socle, prime radical	C₂²×C₃²	D₄
cosocle	C₂×C₃²×C₄	C₂²
hypercenter	C₂×C₈	C₃⋊D₃
hypocenter	C₃²	#103
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
9. #634	C₄	semidirect
37. C₃²	#103	semidirect
52. C₃	#816	semidirect

Maximal subgroups

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

subgroup	index
C₂×#634	2
C₂×C₃²×C₈	2
C₄⋊(C₃⋊Dic₃)	2
#816	3