Subgroups — #63471

Order 288.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3697	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#3664	C₂	1	2	yes	maximal, complemented
5	Dic₃₆	C₂	1	2	yes	maximal, complemented
6	#909	—	3	1	no	maximal
7	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
8	#611	—	2	1	no	—
9	C₂×Dic₉	—	2	1	no	—
10	D₃₆	C₂²	1	1	yes	characteristic
11	C₄×D₉	—	2	1	no	—
12	Dic₁₈	—	2	1	no	—
13	Dic₁₈	C₂²	1	1	yes	characteristic
14	Dic₁₈	C₂²	1	2	yes	—
15	C₈×C₉	C₂²	1	2	yes	—
16	#235	—	3	1	no	—
17	C₂×Dic₆	—	3	1	no	—
18	C₃×M₁₆	D₃	1	1	yes	characteristic
19	Dic₁₂	—	3	2	no	—
20	#204	—	3	2	no	—
21	C₂²×C₉	D₄	1	1	yes	characteristic
22	C₂×D₉	—	2	1	no	—
23	C₄×C₉	D₄	1	1	yes	characteristic
24	C₄×C₉	C₂³	1	1	yes	characteristic
25	Dic₉	—	2	2	no	—
26	Dic₉	—	2	1	no	—
27	#137	—	9	1	no	Sylow
28	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
29	#67	—	6	1	no	—
30	C₂×Dic₃	—	6	1	no	—
31	D₁₂	—	3	1	no	—
32	C₄×D₃	—	6	1	no	—
33	Dic₆	—	3	2	no	—
34	Dic₆	—	3	1	no	—
35	Dic₆	—	6	1	no	—
36	C₃×C₈	C₂×D₃	1	2	yes	—
37	C₂×C₉	—	2	1	no	—
38	C₂×C₉	C₂×D₄	1	1	yes	characteristic
39	D₉	—	4	1	no	—
40	#41	—	9	1	no	—
41	C₂×Q₈	—	9	1	no	—
42	Q₁₆	—	9	2	no	—
43	SD₁₆	—	9	2	no	—
44	M₁₆	D₉	1	1	yes	characteristic, complemented
45	C₂²×C₃	D₁₂	1	1	yes	characteristic
46	C₂×D₃	—	6	1	no	—
47	C₃×C₄	D₁₂	1	1	yes	characteristic
48	C₃×C₄	C₂²×D₃	1	1	yes	characteristic
49	Dic₃	—	6	2	no	—
50	Dic₃	—	6	1	no	—
51	C₉	#137	1	1	yes	characteristic, complemented, Sylow
52	Q₈	—	9	2	no	—
53	Q₈	—	9	1	no	—
54	Q₈	—	18	1	no	—
55	D₄	—	9	1	no	—
56	D₄	—	18	1	no	—
57	C₂×C₄	—	18	1	no	—
58	C₂×C₄	—	18	1	no	—
59	C₂×C₄	C₂×D₉	1	1	yes	characteristic
60	C₈	C₂×D₉	1	2	yes	—
61	C₂×C₃	—	2	1	no	—
62	C₂×C₃	C₂×D₁₂	1	1	yes	characteristic
63	D₃	—	12	1	no	—
64	C₂²	—	18	1	no	—
65	C₂²	D₃₆	1	1	yes	characteristic
66	C₄	—	18	1	no	—
67	C₄	—	18	2	no	—
68	C₄	D₃₆	1	1	yes	characteristic
69	C₄	C₂²×D₉	1	1	yes	characteristic
70	C₃	#909	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	—	36	1	no	—
73	C₂	C₂×D₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₃₆
commutator, cosocle	C₄×C₉	C₂³
Frattini	C₃×C₄	C₂²×D₃
Fitting	C₉×M₁₆	C₂
socle	C₂×C₃	C₂×D₁₂
prime radical	#3697	C₂
hypercenter	M₁₆	D₉
hypocenter	C₉	#137
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₈	C₂	semidirect
3. C₉×M₁₆	C₂	semidirect
4. #3664	C₂	semidirect
5. Dic₃₆	C₂	semidirect
44. M₁₆	D₉	semidirect
51. C₉	#137	semidirect

Maximal subgroups

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

subgroup	index
#3697	2
C₂×Dic₁₈	2
C₉×M₁₆	2
#3664	2
Dic₃₆	2
#909	3