Subgroups — #10588864

Order 576.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₉	C₂	1	2	yes	maximal
3	#63395	C₂	1	2	yes	maximal
4	#63377	C₂	1	1	yes	maximal, characteristic
5	#63375	C₂	1	1	yes	maximal, characteristic
6	#5184	—	3	1	no	maximal
7	C₂×Q₈×C₉	C₂²	1	2	yes	—
8	C₉×Q₁₆	C₄	1	4	yes	complemented
9	C₂×C₈×C₉	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₉	—	2	2	no	—
11	C₄⋊Dic₉	C₂²	1	2	yes	—
12	C₄×Dic₉	—	2	2	no	—
13	C₄×Dic₉	C₂²	1	1	yes	characteristic
14	C₂×C₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×Q₁₆	D₃	1	1	yes	characteristic
16	Q₈×Dic₃	—	3	2	no	—
17	#835	—	3	2	no	—
18	#817	—	3	1	no	—
19	#815	—	3	1	no	—
20	Q₈×C₉	—	2	2	no	—
21	Q₈×C₉	C₂×C₄	1	4	yes	—
22	C₂×C₄×C₉	—	2	2	no	—
23	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
24	C₂×Dic₉	—	2	2	no	—
25	C₂×Dic₉	D₄	1	2	yes	—
26	C₈×C₉	C₂×C₄	1	2	yes	—
27	C₉⋊C₈	—	2	1	no	—
28	#441	—	9	1	no	Sylow
29	C₂×C₃×Q₈	C₂×D₃	1	2	yes	—
30	C₃×Q₁₆	Dic₃	1	4	yes	—
31	C₂×C₃×C₈	C₂×D₃	1	1	yes	characteristic
32	C₄⋊Dic₃	—	3	2	no	—
33	C₄⋊Dic₃	—	6	2	no	—
34	C₄×Dic₃	—	3	1	no	—
35	C₄×Dic₃	—	6	2	no	—
36	C₂×C₃⋊C₈	—	3	1	no	—
37	C₂²×C₉	C₂×D₄	1	1	yes	characteristic
38	C₄×C₉	—	2	4	no	—
39	C₄×C₉	C₂²×C₄	1	1	yes	characteristic
40	C₄×C₉	#41	1	1	yes	characteristic
41	Dic₉	—	2	2	no	—
42	Dic₉	—	4	2	no	—
43	C₂×Q₁₆	D₉	1	1	yes	characteristic
44	C₄×Q₈	—	9	2	no	—
45	#106	—	9	1	no	—
46	#103	—	9	2	no	—
47	#97	—	9	1	no	—
48	C₃×Q₈	—	2	2	no	—
49	C₃×Q₈	C₂×Dic₃	1	4	yes	—
50	C₂×C₃×C₄	—	2	2	no	—
51	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
52	C₂×Dic₃	—	3	2	no	—
53	C₂×Dic₃	—	6	2	no	—
54	C₃×C₈	C₂×Dic₃	1	2	yes	—
55	C₃⋊C₈	—	6	1	no	—
56	C₂×C₉	C₄×D₄	1	1	yes	characteristic
57	C₂×C₉	#137	1	2	yes	—
58	C₂×Q₈	C₂×D₉	1	2	yes	—
59	Q₁₆	Dic₉	1	4	yes	complemented
60	C₂×C₈	—	9	1	no	—
61	C₂×C₈	C₂×D₉	1	1	yes	characteristic
62	C₄⋊C₄	—	9	2	no	—
63	C₄⋊C₄	—	18	2	no	—
64	C₄²	—	9	1	no	—
65	C₄²	—	18	2	no	—
66	C₂²×C₃	D₃×D₄	1	1	yes	characteristic
67	C₃×C₄	—	2	4	no	—
68	C₃×C₄	#237	1	1	yes	characteristic
69	C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
70	Dic₃	—	6	2	no	—
71	Dic₃	—	12	2	no	—
72	C₉	#441	1	1	yes	characteristic, complemented, Sylow
73	Q₈	—	2	2	no	—
74	Q₈	C₂×Dic₉	1	4	yes	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	9	2	no	—
77	C₂×C₄	—	18	2	no	—
78	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
79	C₈	—	18	1	no	—
80	C₈	C₂×Dic₉	1	2	yes	—
81	C₂×C₃	#918	1	2	yes	—
82	C₂×C₃	D₄×Dic₃	1	1	yes	characteristic
83	C₂²	D₄×D₉	1	1	yes	characteristic
84	C₄	—	2	4	no	—
85	C₄	—	18	2	no	—
86	C₄	—	36	2	no	—
87	C₄	#3699	1	1	yes	characteristic
88	C₄	C₂²×Dic₉	1	1	yes	characteristic
89	C₃	#5184	1	1	yes	characteristic
90	C₂	#63480	1	2	yes	—
91	C₂	D₄×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₉
commutator	C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×Q₁₆	C₂
socle, prime radical	C₂²×C₃	D₃×D₄
cosocle	C₂×C₄×C₉	C₂³
hypercenter	C₂×Q₁₆	D₉
hypocenter	C₉	#441
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
8. C₉×Q₁₆	C₄	semidirect
59. Q₁₆	Dic₉	semidirect
72. C₉	#441	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×Q₁₆	2
Q₈×Dic₉	2
#63395	2
#63377	2
#63375	2
#5184	3