Subgroups — #63713

Order 288.

219 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#3701	C₂	1	4	yes	maximal, complemented
3	Q₈×D₉	C₂	1	4	yes	maximal, complemented
4	#3697	C₂	1	6	yes	maximal, complemented
5	#1007	—	3	1	no	maximal
6	Q₈×C₉	C₂²	1	4	yes	—
7	C₂×C₄×C₉	C₂²	1	3	yes	—
8	#611	C₂²	1	4	yes	—
9	D₃₆	C₂²	1	6	yes	—
10	C₄×D₉	C₂²	1	12	yes	—
11	Dic₁₈	C₂²	1	6	yes	—
12	C₂×C₃×Q₈	D₃	1	1	yes	characteristic
13	#239	—	3	4	no	—
14	D₃×Q₈	—	3	4	no	—
15	#235	—	3	6	no	—
16	C₂²×C₉	C₂³	1	1	yes	characteristic
17	C₂×D₉	C₂³	1	4	yes	—
18	C₄×C₉	C₂³	1	6	yes	—
19	Dic₉	C₂³	1	4	yes	—
20	2₋¹⁺⁴	—	9	1	no	Sylow
21	C₃×Q₈	C₂×D₃	1	4	yes	—
22	C₂×C₃×C₄	C₂×D₃	1	3	yes	—
23	#67	—	3	4	no	—
24	D₁₂	—	3	6	no	—
25	C₄×D₃	—	3	12	no	—
26	Dic₆	—	3	6	no	—
27	C₂×C₉	—	2	1	no	—
28	C₂×C₉	C₂⁴	1	1	yes	characteristic
29	D₉	—	2	4	no	—
30	#41	—	9	4	no	—
31	#41	—	9	6	no	—
32	C₂×Q₈	—	9	4	no	—
33	C₂×Q₈	D₉	1	1	yes	characteristic, complemented
34	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
35	C₂×D₃	—	3	4	no	—
36	C₃×C₄	C₂²×D₃	1	6	yes	—
37	Dic₃	—	3	4	no	—
38	C₉	2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
39	Q₈	—	9	6	no	—
40	Q₈	C₂×D₉	1	4	yes	—
41	D₄	—	9	4	no	—
42	D₄	—	9	6	no	—
43	C₂×C₄	—	9	12	no	—
44	C₂×C₄	C₂×D₉	1	3	yes	—
45	C₂×C₃	—	2	1	no	—
46	C₂×C₃	C₂³×D₃	1	1	yes	characteristic
47	D₃	—	6	4	no	—
48	C₂²	—	9	4	no	—
49	C₂²	C₂²×D₉	1	1	yes	characteristic
50	C₄	—	9	4	no	—
51	C₄	C₂²×D₉	1	6	yes	—
52	C₃	#1007	1	1	yes	characteristic
53	C₂	—	2	1	no	—
54	C₂	—	18	4	no	—
55	C₂	C₂³×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂³×D₉
commutator, cosocle	C₂×C₉	C₂⁴
Frattini, socle	C₂×C₃	C₂³×D₃
Fitting	C₂×Q₈×C₉	C₂
hypercenter	C₂×Q₈	D₉
hypocenter	C₉	2₋¹⁺⁴
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×C₉	C₂	semidirect
2. #3701	C₂	semidirect
3. Q₈×D₉	C₂	semidirect
4. #3697	C₂	semidirect
33. C₂×Q₈	D₉	semidirect
38. C₉	2₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₉	2
#3701	2
Q₈×D₉	2
#3697	2
#1007	3