Subgroups — #63488

Order 288.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	#3676	C₂	1	2	yes	maximal
3	C₄⋊Dic₉	C₂	1	2	yes	maximal, complemented
4	C₄×Dic₉	C₂	1	2	yes	maximal, complemented
5	#926	—	3	1	no	maximal
6	C₂³×C₉	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₉	C₄	1	4	yes	complemented
8	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂×Dic₉	C₂²	1	4	yes	—
11	C₂²×C₃×C₄	D₃	1	1	yes	characteristic
12	#217	—	3	2	no	—
13	C₄⋊Dic₃	—	3	2	no	—
14	C₄×Dic₃	—	3	2	no	—
15	C₂²×C₉	—	2	2	no	—
16	C₂²×C₉	C₂×C₄	1	2	yes	—
17	C₂²×C₉	C₂³	1	1	yes	characteristic
18	C₄×C₉	C₂×C₄	1	2	yes	—
19	C₄×C₉	C₂×C₄	1	2	yes	—
20	Dic₉	—	2	4	no	—
21	#117	—	9	1	no	Sylow
22	C₂³×C₃	C₂×D₃	1	1	yes	characteristic
23	C₂×C₃×C₄	Dic₃	1	4	yes	—
24	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
25	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
26	C₂×Dic₃	—	3	4	no	—
27	C₂×C₉	—	2	2	no	—
28	C₂×C₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₉	#41	1	2	yes	—
30	C₂²×C₄	D₉	1	1	yes	characteristic
31	C₄⋊C₄	—	9	2	no	—
32	C₂²⋊C₄	—	9	2	no	—
33	C₄²	—	9	2	no	—
34	C₂²×C₃	—	2	2	no	—
35	C₂²×C₃	C₂×Dic₃	1	2	yes	—
36	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
37	C₃×C₄	C₂×Dic₃	1	2	yes	—
38	C₃×C₄	C₂×Dic₃	1	2	yes	—
39	Dic₃	—	6	4	no	—
40	C₉	#117	1	1	yes	characteristic, complemented, Sylow
41	C₂³	C₂×D₉	1	1	yes	characteristic
42	C₂×C₄	—	9	4	no	—
43	C₂×C₄	Dic₉	1	4	yes	complemented
44	C₂×C₄	C₂×D₉	1	1	yes	characteristic
45	C₂×C₄	C₂×D₉	1	1	yes	characteristic
46	C₂×C₃	—	2	2	no	—
47	C₂×C₃	#235	1	2	yes	—
48	C₂×C₃	C₂²×Dic₃	1	1	yes	characteristic
49	C₂²	—	2	2	no	—
50	C₂²	C₂×Dic₉	1	2	yes	—
51	C₂²	C₂²×D₉	1	1	yes	characteristic
52	C₄	—	18	4	no	—
53	C₄	C₂×Dic₉	1	2	yes	—
54	C₄	C₂×Dic₉	1	2	yes	—
55	C₃	#926	1	1	yes	characteristic
56	C₂	—	2	2	no	—
57	C₂	#3697	1	2	yes	—
58	C₂	C₂²×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₉
commutator	C₂×C₉	C₂²×C₄
Frattini, socle	C₂²×C₃	C₂²×D₃
Fitting	C₂²×C₄×C₉	C₂
prime radical	C₂³×C₃	C₂×D₃
cosocle	C₂²×C₉	C₂³
hypercenter	C₂²×C₄	D₉
hypocenter	C₉	#117
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
3. C₄⋊Dic₉	C₂	semidirect
4. C₄×Dic₉	C₂	semidirect
7. C₂×C₄×C₉	C₄	semidirect
40. C₉	#117	semidirect
43. C₂×C₄	Dic₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₉	2
#3676	2
C₄⋊Dic₉	2
C₄×Dic₉	2
#926	3