Subgroups — C₉×C₃⋊#564 #13153148

Order 1728.

572 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×#125	C₂	1	1	yes	maximal, characteristic
2	C₉×#890	C₂	1	4	yes	maximal
3	C₉×#889	C₂	1	4	yes	maximal
4	C₉×#889	C₂	1	4	yes	maximal
5	C₉×#870	C₂	1	2	yes	maximal
6	C₃×C₃⋊#564	C₃	1	1	yes	maximal, characteristic
7	C₉×#564	—	3	1	no	maximal
8	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
9	C₃×C₉×C₄⋊C₄	C₂²	1	4	yes	—
10	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
11	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₉×C₄⋊Dic₃	C₂²	1	4	yes	—
14	C₉×Dic₃⋊C₄	C₂²	1	4	yes	—
15	C₉×Dic₃⋊C₄	C₂²	1	4	yes	—
16	C₉×Dic₃⋊C₄	C₂²	1	4	yes	—
17	C₉×Dic₃⋊C₄	C₂²	1	4	yes	—
18	C₄×C₉×Dic₃	C₂²	1	4	yes	—
19	C₃²×#125	C₂×C₃	1	1	yes	characteristic
20	C₃×#890	C₂×C₃	1	4	yes	—
21	C₃×#889	C₂×C₃	1	4	yes	—
22	C₃×#889	C₂×C₃	1	4	yes	—
23	C₃×#870	C₂×C₃	1	2	yes	—
24	C₉×#125	—	2	1	no	—
25	C₉×#125	—	3	4	no	—
26	C₉×#125	—	3	4	no	—
27	C₉×#125	—	3	4	no	—
28	C₉×#125	—	3	2	no	—
29	C₉×#125	D₃	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
31	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
32	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
33	C₂×C₉×Dic₃	C₂³	1	4	yes	—
34	C₂×C₉×Dic₃	C₂³	1	4	yes	—
35	C₃×#564	—	3	1	no	—
36	C₃⋊#564	C₉	1	1	yes	characteristic, direct product factor
37	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
38	C₃²×C₄⋊C₄	C₂²×C₃	1	4	yes	—
39	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
40	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
41	C₃×C₄⋊Dic₃	C₂²×C₃	1	4	yes	—
42	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
43	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
44	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
45	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
46	C₃×Dic₃⋊C₄	C₂²×C₃	1	4	yes	—
47	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
48	C₉×C₄⋊C₄	—	2	2	no	—
49	C₉×C₄⋊C₄	—	2	4	no	—
50	C₉×C₄⋊C₄	—	3	4	no	—
51	C₉×C₄⋊C₄	—	3	4	no	—
52	C₉×C₄⋊C₄	—	3	4	no	—
53	C₉×C₄⋊C₄	—	3	4	no	—
54	C₉×C₄⋊C₄	—	3	4	no	—
55	C₉×C₄⋊C₄	—	3	2	no	—
56	C₉×C₄⋊C₄	—	3	2	no	—
57	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
58	C₉×C₄⋊C₄	C₂×D₃	1	4	yes	—
59	C₄²×C₉	—	2	1	no	—
60	C₄²×C₉	—	3	4	no	—
61	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
62	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
63	C₃×C₄×C₉	—	2	1	no	—
64	C₃×C₄×C₉	—	2	4	no	—
65	C₃×C₄×C₉	—	2	2	no	—
66	C₉×Dic₃	—	2	4	no	—
67	C₉×Dic₃	—	2	4	no	—
68	C₃×#125	—	2	1	no	—
69	C₃×#125	—	3	4	no	—
70	C₃×#125	—	3	4	no	—
71	C₃×#125	—	3	4	no	—
72	C₃×#125	—	3	2	no	—
73	C₃×#125	C₂×C₉	1	1	yes	characteristic
74	C₃×#125	C₃×D₃	1	1	yes	characteristic
75	#890	C₂×C₉	1	4	yes	—
76	#889	C₂×C₉	1	4	yes	—
77	#889	C₂×C₉	1	4	yes	—
78	#870	C₂×C₉	1	2	yes	—
79	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
80	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
82	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
83	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
84	C₂×C₄×C₉	—	2	1	no	—
85	C₂×C₄×C₉	—	2	2	no	—
86	C₂×C₄×C₉	—	2	4	no	—
87	C₂×C₄×C₉	—	3	4	no	—
88	C₂×C₄×C₉	—	3	4	no	—
89	C₂×C₄×C₉	C₂²×D₃	1	4	yes	—
90	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
91	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
92	#564	—	3	1	no	Sylow
93	C₂×C₃×C₉	2₋¹⁺⁴	1	1	yes	characteristic
94	C₂×C₃×C₉	2₋¹⁺⁴	1	2	yes	—
95	C₃×C₄⋊C₄	—	2	2	no	—
96	C₃×C₄⋊C₄	—	2	4	no	—
97	C₃×C₄⋊C₄	—	3	4	no	—
98	C₃×C₄⋊C₄	—	3	4	no	—
99	C₃×C₄⋊C₄	—	3	4	no	—
100	C₃×C₄⋊C₄	—	3	4	no	—
101	C₃×C₄⋊C₄	—	3	4	no	—
102	C₃×C₄⋊C₄	—	3	2	no	—
103	C₃×C₄⋊C₄	—	3	2	no	—
104	C₃×C₄⋊C₄	C₂²×C₉	1	2	yes	—
105	C₃×C₄⋊C₄	C₂²×C₉	1	4	yes	—
106	C₃×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
107	C₃×C₄⋊C₄	C₂×C₃×D₃	1	4	yes	—
108	C₃×C₄²	—	2	1	no	—
109	C₃×C₄²	—	3	4	no	—
110	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
111	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
112	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
113	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
114	C₄⋊Dic₃	C₂²×C₉	1	4	yes	—
115	Dic₃⋊C₄	C₂²×C₉	1	4	yes	—
116	Dic₃⋊C₄	C₂²×C₉	1	4	yes	—
117	Dic₃⋊C₄	C₂²×C₉	1	4	yes	—
118	Dic₃⋊C₄	C₂²×C₉	1	4	yes	—
119	C₄×Dic₃	C₂²×C₉	1	4	yes	—
120	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
121	C₃²×C₄	—	2	1	no	—
122	C₃²×C₄	—	2	4	no	—
123	C₃²×C₄	—	2	2	no	—
124	C₃×Dic₃	—	2	4	no	—
125	C₃×Dic₃	—	2	4	no	—
126	C₂²×C₉	—	2	1	no	—
127	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
128	C₄×C₉	—	2	2	no	—
129	C₄×C₉	—	2	1	no	—
130	C₄×C₉	—	2	4	no	—
131	C₄×C₉	—	2	2	no	—
132	C₄×C₉	—	4	2	no	—
133	C₄×C₉	—	4	4	no	—
134	C₄×C₉	—	6	4	no	—
135	C₄×C₉	—	6	4	no	—
136	#125	—	3	4	no	—
137	#125	—	3	4	no	—
138	#125	—	3	4	no	—
139	#125	—	3	2	no	—
140	#125	D₃×C₉	1	1	yes	characteristic
141	C₃×C₉	#564	1	1	yes	characteristic, complemented, Sylow
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	2	2	no	—
144	C₂×C₃×C₄	—	2	4	no	—
145	C₂×C₃×C₄	—	3	4	no	—
146	C₂×C₃×C₄	—	3	4	no	—
147	C₂×C₃×C₄	C₂³×C₉	1	4	yes	—
148	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
149	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
150	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
151	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
152	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
153	C₂×Dic₃	C₂³×C₉	1	4	yes	—
154	C₂×Dic₃	C₂³×C₉	1	4	yes	—
155	C₂×C₃²	C₃×2₋¹⁺⁴	1	1	yes	characteristic
156	C₂×C₃²	C₃×2₋¹⁺⁴	1	2	yes	—
157	C₂×C₉	—	2	1	no	—
158	C₂×C₉	—	2	2	no	—
159	C₂×C₉	#1007	1	1	yes	characteristic
160	C₂×C₉	#1010	1	2	yes	—
161	C₄⋊C₄	—	3	4	no	—
162	C₄⋊C₄	—	3	4	no	—
163	C₄⋊C₄	—	3	4	no	—
164	C₄⋊C₄	—	3	4	no	—
165	C₄⋊C₄	—	3	4	no	—
166	C₄⋊C₄	—	3	2	no	—
167	C₄⋊C₄	—	3	2	no	—
168	C₄⋊C₄	C₂×D₃×C₉	1	2	yes	—
169	C₄⋊C₄	C₂×D₃×C₉	1	4	yes	—
170	C₄²	—	3	4	no	—
171	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
172	C₂²×C₃	—	2	1	no	—
173	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
174	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	—	2	4	no	—
178	C₃×C₄	—	2	2	no	—
179	C₃×C₄	—	2	1	no	—
180	C₃×C₄	—	2	4	no	—
181	C₃×C₄	—	2	2	no	—
182	C₃×C₄	—	4	2	no	—
183	C₃×C₄	—	4	4	no	—
184	C₃×C₄	—	6	4	no	—
185	C₃×C₄	—	6	4	no	—
186	Dic₃	—	2	4	no	—
187	Dic₃	—	2	4	no	—
188	C₃²	C₃×#564	1	1	yes	characteristic
189	C₉	—	2	1	no	—
190	C₉	C₃⋊#564	1	1	yes	characteristic, direct product factor
191	C₂×C₄	—	3	4	no	—
192	C₂×C₄	—	3	4	no	—
193	C₂×C₄	C₂²×D₃×C₉	1	4	yes	—
194	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
195	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
196	C₂×C₃	—	2	1	no	—
197	C₂×C₃	—	2	2	no	—
198	C₂×C₃	C₉×2₋¹⁺⁴	1	1	yes	characteristic
199	C₂×C₃	C₉×2₋¹⁺⁴	1	2	yes	—
200	C₂×C₃	C₃×#1007	1	1	yes	characteristic
201	C₂×C₃	C₃×#1010	1	2	yes	—
202	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	4	no	—
205	C₄	—	2	2	no	—
206	C₄	—	6	4	no	—
207	C₄	—	6	4	no	—
208	C₃	—	2	1	no	—
209	C₃	C₉×#564	1	1	yes	characteristic, complemented
210	C₃	C₃×C₃⋊#564	1	1	yes	characteristic
211	C₂	C₉×#1007	1	1	yes	characteristic
212	C₂	C₉×#1010	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂³×D₃
commutator	C₂²×C₃	C₂⁴×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₃×C₉×#125	C₂
socle, prime radical, cosocle	C₂²×C₃²	C₂⁴×C₃
hypercenter	C₉×#125	D₃
hypocenter	C₃	C₉×#564
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×#125

Complements

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

N	H ≅ G/N	type
36. C₃⋊#564	C₉	direct (N × H)
141. C₃×C₉	#564	semidirect
190. C₉	C₃⋊#564	direct (N × H)
209. C₃	C₉×#564	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×#125	2
C₉×#890	2
C₉×#889	2
C₉×#870	2
C₃×C₃⋊#564	3
C₉×#564	3

Subgroups — C9×C3⋊#564 #13153148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×C₃⋊#564 #13153148