Subgroups — #63376

Order 288.

63 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₉	C₂	1	1	yes	maximal, characteristic
3	C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
4	#816	—	3	1	no	maximal
5	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
6	C₂×Dic₉	—	2	1	no	—
7	C₂×Dic₉	—	2	1	no	—
8	Dic₁₈	—	2	1	no	—
9	Dic₁₈	C₄	1	2	yes	complemented
10	C₈×C₉	—	2	1	no	—
11	C₂×Dic₆	—	3	1	no	—
12	C₂×C₃×C₈	D₃	1	1	yes	characteristic
13	C₄⋊Dic₃	—	3	1	no	—
14	C₂²×C₉	D₄	1	1	yes	characteristic
15	C₄×C₉	C₂×C₄	1	1	yes	characteristic
16	C₄×C₉	D₄	1	1	yes	characteristic
17	Dic₉	—	2	2	no	—
18	Dic₉	—	4	1	no	—
19	#103	—	9	1	no	Sylow
20	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
21	C₂×Dic₃	—	6	1	no	—
22	C₂×Dic₃	—	6	1	no	—
23	Dic₆	—	3	2	no	—
24	Dic₆	—	6	1	no	—
25	C₃×C₈	—	2	1	no	—
26	C₂×C₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₉	SD₁₆	1	1	yes	characteristic
28	C₂×C₉	Q₁₆	1	1	yes	characteristic
29	C₂×Q₈	—	9	1	no	—
30	C₂×C₈	D₉	1	1	yes	characteristic
31	C₄⋊C₄	—	9	1	no	—
32	C₂²×C₃	D₁₂	1	1	yes	characteristic
33	C₃×C₄	C₄×D₃	1	1	yes	characteristic
34	C₃×C₄	#67	1	1	yes	characteristic
35	Dic₃	—	6	2	no	—
36	Dic₃	—	12	1	no	—
37	C₉	#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×D₉	1	1	yes	characteristic
43	C₈	—	2	1	no	—
44	C₂×C₃	#204	1	1	yes	characteristic
45	C₂×C₃	Dic₁₂	1	1	yes	characteristic
46	C₂×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
47	C₂²	D₃₆	1	1	yes	characteristic
48	C₄	—	18	2	no	—
49	C₄	—	36	1	no	—
50	C₄	C₄×D₉	1	1	yes	characteristic
51	C₄	#611	1	1	yes	characteristic
52	C₃	#816	1	1	yes	characteristic
53	C₂	Dic₃₆	1	1	yes	characteristic
54	C₂	#3664	1	1	yes	characteristic
55	C₂	(C₂×D₉)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃₆
commutator	C₄×C₉	C₂×C₄
Frattini	C₂×C₃×C₄	C₂×D₃
Fitting	C₂×C₈×C₉	C₂
socle, prime radical	C₂²×C₃	D₁₂
cosocle	C₂×C₄×C₉	C₂²
hypercenter	C₂×C₈	D₉
hypocenter	C₉	#103
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
9. Dic₁₈	C₄	semidirect
37. C₉	#103	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₈	2
C₂×C₈×C₉	2
C₄⋊Dic₉	2
#816	3