Subgroups — #63505

Order 288.

90 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	#3697	C₂	1	1	yes	maximal, characteristic
3	#3675	C₂	1	2	yes	maximal, complemented
4	#3674	C₂	1	2	yes	maximal, complemented
5	#3667	C₂	1	1	yes	maximal, characteristic, complemented
6	#942	—	3	1	no	maximal
7	Q₈×C₉	—	2	1	no	—
8	Q₈×C₉	C₂²	1	2	yes	—
9	C₂×C₄×C₉	—	2	1	no	—
10	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
11	#611	—	2	1	no	—
12	D₃₆	C₂²	1	1	yes	characteristic
13	C₄×D₉	—	2	1	no	—
14	Dic₁₈	C₂²	1	1	yes	characteristic
15	C₉⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃×Q₈	D₃	1	1	yes	characteristic
17	#235	—	3	1	no	—
18	#216	—	3	2	no	—
19	#215	—	3	2	no	—
20	#208	—	3	1	no	—
21	C₂²×C₉	D₄	1	1	yes	characteristic
22	C₂×D₉	—	2	1	no	—
23	C₄×C₉	—	2	2	no	—
24	C₄×C₉	D₄	1	1	yes	characteristic
25	C₄×C₉	C₂³	1	1	yes	characteristic
26	Dic₉	—	2	1	no	—
27	#137	—	9	1	no	Sylow
28	C₃×Q₈	—	2	1	no	—
29	C₃×Q₈	C₂×D₃	1	2	yes	—
30	C₂×C₃×C₄	—	2	1	no	—
31	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
32	#67	—	6	1	no	—
33	D₁₂	—	3	1	no	—
34	C₄×D₃	—	6	1	no	—
35	Dic₆	—	3	1	no	—
36	C₃⋊C₈	—	3	2	no	—
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	D₉	1	1	yes	characteristic, complemented
42	Q₁₆	—	9	2	no	—
43	SD₁₆	—	9	2	no	—
44	M₁₆	—	9	1	no	—
45	C₂²×C₃	#67	1	1	yes	characteristic
46	C₂×D₃	—	6	1	no	—
47	C₃×C₄	—	2	2	no	—
48	C₃×C₄	#67	1	1	yes	characteristic
49	C₃×C₄	C₂²×D₃	1	1	yes	characteristic
50	Dic₃	—	6	1	no	—
51	C₉	#137	1	1	yes	characteristic, complemented, Sylow
52	Q₈	—	2	1	no	—
53	Q₈	—	9	1	no	—
54	Q₈	C₂×D₉	1	2	yes	—
55	D₄	—	9	1	no	—
56	D₄	—	18	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	18	1	no	—
59	C₂×C₄	C₂×D₉	1	1	yes	characteristic
60	C₈	—	9	2	no	—
61	C₂×C₃	—	2	1	no	—
62	C₂×C₃	C₂×#67	1	1	yes	characteristic
63	D₃	—	12	1	no	—
64	C₂²	—	18	1	no	—
65	C₂²	#611	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	—	18	1	no	—
68	C₄	#611	1	1	yes	characteristic
69	C₄	C₂²×D₉	1	1	yes	characteristic
70	C₃	#942	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	—	36	1	no	—
73	C₂	C₂×#611	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#611
commutator, cosocle	C₄×C₉	C₂³
Frattini	C₃×C₄	C₂²×D₃
Fitting	C₂×Q₈×C₉	C₂
socle	C₂×C₃	C₂×#67
prime radical	#3697	C₂
hypercenter	C₂×Q₈	D₉
hypocenter	C₉	#137
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×C₉	C₂	semidirect
3. #3675	C₂	semidirect
4. #3674	C₂	semidirect
5. #3667	C₂	semidirect
41. C₂×Q₈	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
C₂×Q₈×C₉	2
#3697	2
#3675	2
#3674	2
#3667	2
#942	3