Subgroups — #10589672

Order 576.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₉	C₂	1	2	yes	maximal
3	#63452	C₂	1	6	yes	maximal
4	#63431	C₂	1	3	yes	maximal
5	C₄×Dic₁₈	C₂	1	3	yes	maximal
6	#5570	—	3	1	no	maximal
7	C₂×Q₈×C₉	C₂²	1	1	yes	characteristic
8	C₂×Dic₁₈	C₂²	1	3	yes	—
9	C₉×C₄⋊C₄	C₂²	1	3	yes	—
10	C₄²×C₉	C₂²	1	3	yes	—
11	C₄⋊Dic₉	C₂²	1	6	yes	—
12	C₄⋊Dic₉	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₉	C₂²	1	6	yes	—
14	Dic₉⋊C₄	C₂²	1	6	yes	—
15	C₄×Dic₉	C₂²	1	6	yes	—
16	C₃×C₄×Q₈	D₃	1	1	yes	characteristic
17	Q₈×Dic₃	—	3	2	no	—
18	#890	—	3	6	no	—
19	#869	—	3	3	no	—
20	C₄×Dic₆	—	3	3	no	—
21	Q₈×C₉	Q₈	1	4	yes	complemented
22	C₂×C₄×C₉	C₂³	1	3	yes	—
23	C₂×C₄×C₉	C₂³	1	3	yes	—
24	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
25	C₂×Dic₉	C₂³	1	2	yes	—
26	C₂×Dic₉	C₂³	1	6	yes	—
27	Dic₁₈	—	2	6	no	—
28	#557	—	9	1	no	Sylow
29	C₂×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
30	C₂×Dic₆	—	3	3	no	—
31	C₃×C₄⋊C₄	C₂×D₃	1	3	yes	—
32	C₃×C₄²	C₂×D₃	1	3	yes	—
33	C₄⋊Dic₃	—	3	1	no	—
34	C₄⋊Dic₃	—	3	6	no	—
35	C₄⋊Dic₃	—	3	6	no	—
36	Dic₃⋊C₄	—	3	6	no	—
37	C₄×Dic₃	—	3	6	no	—
38	C₂²×C₉	C₂⁴	1	1	yes	characteristic
39	C₄×C₉	—	2	3	no	—
40	C₄×C₉	C₂×Q₈	1	6	yes	—
41	C₄×C₉	#41	1	2	yes	—
42	Dic₉	—	2	6	no	—
43	Dic₉	—	2	2	no	—
44	C₄⋊Q₈	—	9	3	no	—
45	#125	—	9	6	no	—
46	C₄×Q₈	—	9	2	no	—
47	C₄×Q₈	—	9	3	no	—
48	C₄×Q₈	D₉	1	1	yes	characteristic
49	C₃×Q₈	Dic₆	1	4	yes	—
50	C₂×C₃×C₄	C₂²×D₃	1	3	yes	—
51	C₂×C₃×C₄	C₂²×D₃	1	3	yes	—
52	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
53	C₂×Dic₃	—	3	2	no	—
54	C₂×Dic₃	—	3	6	no	—
55	Dic₆	—	6	6	no	—
56	C₂×C₉	C₂²×Q₈	1	1	yes	characteristic
57	C₂×C₉	C₂×#41	1	1	yes	characteristic
58	C₂×C₉	2₋¹⁺⁴	1	1	yes	characteristic
59	C₂×Q₈	—	9	3	no	—
60	C₂×Q₈	C₂×D₉	1	1	yes	characteristic
61	C₄⋊C₄	—	9	1	no	—
62	C₄⋊C₄	—	9	6	no	—
63	C₄⋊C₄	—	9	6	no	—
64	C₄⋊C₄	—	9	6	no	—
65	C₄⋊C₄	C₂×D₉	1	3	yes	—
66	C₄²	—	9	6	no	—
67	C₄²	C₂×D₉	1	3	yes	—
68	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
69	C₃×C₄	—	2	3	no	—
70	C₃×C₄	C₂×Dic₆	1	6	yes	—
71	C₃×C₄	#239	1	2	yes	—
72	Dic₃	—	6	6	no	—
73	Dic₃	—	6	2	no	—
74	C₉	#557	1	1	yes	characteristic, complemented, Sylow
75	Q₈	—	18	6	no	—
76	Q₈	Dic₁₈	1	4	yes	complemented
77	C₂×C₄	—	9	2	no	—
78	C₂×C₄	—	9	6	no	—
79	C₂×C₄	C₂²×D₉	1	3	yes	—
80	C₂×C₄	C₂²×D₉	1	3	yes	—
81	C₂×C₄	C₂²×D₉	1	1	yes	characteristic
82	C₂×C₃	C₂²×Dic₆	1	1	yes	characteristic
83	C₂×C₃	C₂×#239	1	1	yes	characteristic
84	C₂×C₃	#1010	1	1	yes	characteristic
85	C₂²	C₂³×D₉	1	1	yes	characteristic
86	C₄	—	2	3	no	—
87	C₄	—	18	6	no	—
88	C₄	—	18	2	no	—
89	C₄	C₂×Dic₁₈	1	6	yes	—
90	C₄	#3701	1	2	yes	—
91	C₃	#5570	1	1	yes	characteristic
92	C₂	C₂²×Dic₁₈	1	1	yes	characteristic
93	C₂	C₂×#3701	1	1	yes	characteristic
94	C₂	#63716	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₉
commutator, cosocle	C₂²×C₉	C₂⁴
Frattini, socle, prime radical	C₂²×C₃	C₂³×D₃
Fitting	C₄×Q₈×C₉	C₂
hypercenter	C₄×Q₈	D₉
hypocenter	C₉	#557
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
21. Q₈×C₉	Q₈	semidirect
74. C₉	#557	semidirect
76. Q₈	Dic₁₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₈×C₉	2
Q₈×Dic₉	2
#63452	2
#63431	2
C₄×Dic₁₈	2
#5570	3