Subgroups — #63451

Order 288.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic
3	Dic₉⋊C₄	C₂	1	1	yes	maximal, characteristic
4	Dic₉⋊C₄	C₂	1	1	yes	maximal, characteristic
5	Dic₉⋊C₄	C₂	1	1	yes	maximal, characteristic
6	Dic₉⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₄×Dic₉	C₂	1	1	yes	maximal, characteristic
8	#889	—	3	1	no	maximal
9	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂×Dic₉	C₂²	1	1	yes	characteristic
13	C₂×Dic₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₉	C₂²	1	1	yes	characteristic
16	C₃×C₄⋊C₄	D₃	1	1	yes	characteristic
17	C₄⋊Dic₃	—	3	1	no	—
18	Dic₃⋊C₄	—	3	1	no	—
19	Dic₃⋊C₄	—	3	1	no	—
20	Dic₃⋊C₄	—	3	1	no	—
21	Dic₃⋊C₄	—	3	1	no	—
22	C₄×Dic₃	—	3	1	no	—
23	C₂²×C₉	C₂³	1	1	yes	characteristic
24	C₄×C₉	—	2	1	no	—
25	C₄×C₉	—	2	1	no	—
26	C₄×C₉	—	2	1	no	—
27	Dic₉	—	2	1	no	—
28	Dic₉	—	2	1	no	—
29	Dic₉	—	2	1	no	—
30	Dic₉	Q₈	1	2	yes	—
31	#125	—	9	1	no	Sylow
32	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
33	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
34	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
35	C₂×Dic₃	—	3	1	no	—
36	C₂×Dic₃	—	3	1	no	—
37	C₂×Dic₃	—	3	1	no	—
38	C₂×Dic₃	—	3	1	no	—
39	C₂×C₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₉	#41	1	1	yes	characteristic
41	C₂×C₉	#41	1	1	yes	characteristic
42	C₄⋊C₄	—	9	1	no	—
43	C₄⋊C₄	—	9	1	no	—
44	C₄⋊C₄	—	9	1	no	—
45	C₄⋊C₄	—	9	1	no	—
46	C₄⋊C₄	—	9	1	no	—
47	C₄⋊C₄	D₉	1	1	yes	characteristic
48	C₄²	—	9	1	no	—
49	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
50	C₃×C₄	—	2	1	no	—
51	C₃×C₄	—	2	1	no	—
52	C₃×C₄	—	2	1	no	—
53	Dic₃	—	3	2	no	—
54	Dic₃	—	6	1	no	—
55	Dic₃	—	6	1	no	—
56	Dic₃	—	6	1	no	—
57	C₉	#125	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₄	—	9	1	no	—
59	C₂×C₄	—	9	1	no	—
60	C₂×C₄	—	9	1	no	—
61	C₂×C₄	—	9	1	no	—
62	C₂×C₄	C₂×D₉	1	1	yes	characteristic
63	C₂×C₄	C₂×D₉	1	1	yes	characteristic
64	C₂×C₄	C₂×D₉	1	1	yes	characteristic
65	C₂×C₃	#235	1	1	yes	characteristic
66	C₂×C₃	#237	1	1	yes	characteristic
67	C₂×C₃	D₃×Q₈	1	1	yes	characteristic
68	C₂²	C₂²×D₉	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	2	1	no	—
71	C₄	—	2	1	no	—
72	C₄	—	9	2	no	—
73	C₄	—	18	1	no	—
74	C₄	—	18	1	no	—
75	C₄	—	18	1	no	—
76	C₃	#889	1	1	yes	characteristic
77	C₂	#3697	1	1	yes	characteristic
78	C₂	#3699	1	1	yes	characteristic
79	C₂	Q₈×D₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₉
commutator, cosocle	C₂²×C₉	C₂³
Frattini, socle, prime radical	C₂²×C₃	C₂²×D₃
Fitting	C₉×C₄⋊C₄	C₂
hypercenter	C₄⋊C₄	D₉
hypocenter	C₉	#125
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
57. C₉	#125	semidirect

Maximal subgroups

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

subgroup	index
C₉×C₄⋊C₄	2
C₄⋊Dic₉	2
Dic₉⋊C₄	2
C₄×Dic₉	2
#889	3