Subgroups — #10588758

Order 576.

147 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#63431	C₂	1	1	yes	maximal, characteristic
3	#63366	C₂	1	4	yes	maximal
4	C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#5078	—	3	1	no	maximal
6	C₂×Q₈×C₉	—	2	1	no	—
7	C₂×Dic₁₈	—	2	1	no	—
8	C₉×C₄⋊C₄	—	2	1	no	—
9	C₉×C₄⋊C₄	C₂²	1	2	yes	—
10	C₄²×C₉	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₉	—	2	1	no	—
12	C₄⋊Dic₉	C₂²	1	2	yes	—
13	C₂×C₉⋊C₈	C₂²	1	2	yes	—
14	C₃×C₄⋊Q₈	D₃	1	1	yes	characteristic
15	#869	—	3	1	no	—
16	#807	—	3	4	no	—
17	C₄×C₃⋊C₈	—	3	1	no	—
18	Q₈×C₉	—	4	2	no	—
19	C₂×C₄×C₉	—	2	2	no	—
20	C₂×C₄×C₉	D₄	1	1	yes	characteristic
21	C₂×C₄×C₉	D₄	1	1	yes	characteristic
22	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
23	C₂×Dic₉	—	2	2	no	—
24	Dic₁₈	—	4	2	no	—
25	C₉⋊C₈	Q₈	1	4	yes	complemented
26	#500	—	9	1	no	Sylow
27	C₂×C₃×Q₈	—	2	1	no	—
28	C₂×Dic₆	—	6	1	no	—
29	C₃×C₄⋊C₄	—	2	1	no	—
30	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
31	C₃×C₄²	C₂×D₃	1	1	yes	characteristic
32	C₄⋊Dic₃	—	3	2	no	—
33	C₄⋊Dic₃	—	6	1	no	—
34	C₂×C₃⋊C₈	—	3	2	no	—
35	C₂²×C₉	C₂×D₄	1	1	yes	characteristic
36	C₄×C₉	—	4	2	no	—
37	C₄×C₉	D₈	1	2	yes	—
38	C₄×C₉	Q₁₆	1	2	yes	—
39	C₄×C₉	C₂×Q₈	1	2	yes	—
40	Dic₉	—	4	2	no	—
41	C₄⋊Q₈	—	9	1	no	—
42	C₄⋊Q₈	D₉	1	1	yes	characteristic
43	#107	—	9	4	no	—
44	C₄×C₈	—	9	1	no	—
45	C₃×Q₈	—	4	2	no	—
46	C₂×C₃×C₄	—	2	2	no	—
47	C₂×C₃×C₄	#67	1	1	yes	characteristic
48	C₂×C₃×C₄	#67	1	1	yes	characteristic
49	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
50	C₂×Dic₃	—	6	2	no	—
51	Dic₆	—	12	2	no	—
52	C₃⋊C₈	—	3	4	no	—
53	C₂×C₉	C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₉	C₂×D₈	1	1	yes	characteristic
55	C₂×C₉	C₂×Q₁₆	1	1	yes	characteristic
56	C₂×Q₈	—	2	1	no	—
57	C₂×Q₈	—	18	1	no	—
58	C₂×C₈	—	9	2	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	9	2	no	—
61	C₄⋊C₄	—	18	1	no	—
62	C₄⋊C₄	C₂×D₉	1	2	yes	—
63	C₄²	C₂×D₉	1	1	yes	characteristic
64	C₂²×C₃	C₂×#67	1	1	yes	characteristic
65	C₃×C₄	—	4	2	no	—
66	C₃×C₄	#213	1	2	yes	—
67	C₃×C₄	#216	1	2	yes	—
68	C₃×C₄	D₃×Q₈	1	2	yes	—
69	Dic₃	—	12	2	no	—
70	C₉	#500	1	1	yes	characteristic, complemented, Sylow
71	Q₈	—	4	2	no	—
72	Q₈	—	36	2	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	18	2	no	—
75	C₂×C₄	#611	1	1	yes	characteristic
76	C₂×C₄	#611	1	1	yes	characteristic
77	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
78	C₈	—	9	4	no	—
79	C₂×C₃	C₂×#213	1	1	yes	characteristic
80	C₂×C₃	C₂×#216	1	1	yes	characteristic
81	C₂×C₃	#944	1	1	yes	characteristic
82	C₂²	C₂×#611	1	1	yes	characteristic
83	C₄	—	4	2	no	—
84	C₄	—	36	2	no	—
85	C₄	#3673	1	2	yes	—
86	C₄	#3674	1	2	yes	—
87	C₄	Q₈×D₉	1	2	yes	—
88	C₃	#5078	1	1	yes	characteristic
89	C₂	C₂×#3673	1	1	yes	characteristic
90	C₂	C₂×#3674	1	1	yes	characteristic
91	C₂	#63506	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#611
commutator, cosocle	C₂×C₄×C₉	C₂³
Frattini	C₂×C₃×C₄	C₂²×D₃
Fitting	C₉×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₃	C₂×#67
hypercenter	C₄⋊Q₈	D₉
hypocenter	C₉	#500
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
25. C₉⋊C₈	Q₈	semidirect
70. C₉	#500	semidirect

Maximal subgroups

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

subgroup	index
C₉×C₄⋊Q₈	2
#63431	2
#63366	2
C₄×C₉⋊C₈	2
#5078	3