Subgroups — C₉×#5020 #13142769

Order 1728.

358 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₉×#869	C₂	1	1	yes	maximal, characteristic
3	C₉×#835	C₂	1	2	yes	maximal
4	C₉×#808	C₂	1	2	yes	maximal
5	C₉×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	C₃×#5020	C₃	1	1	yes	maximal, characteristic
7	C₉×#475	—	3	1	no	maximal
8	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×Dic₆	—	2	1	no	—
10	C₃×C₉×C₄⋊C₄	—	2	1	no	—
11	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₃×C₄²×C₉	—	2	1	no	—
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₉×C₄⋊Dic₃	—	2	1	no	—
15	C₉×C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₂×C₉×C₃⋊C₈	C₂²	1	2	yes	—
17	C₃²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
18	C₃×#869	C₂×C₃	1	1	yes	characteristic
19	C₃×#835	C₂×C₃	1	2	yes	—
20	C₃×#808	C₂×C₃	1	2	yes	—
21	C₃×C₄⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic
22	C₉×C₄⋊Q₈	—	3	1	no	—
23	C₄×Q₈×C₉	—	2	1	no	—
24	C₄×Q₈×C₉	D₃	1	1	yes	characteristic
25	C₉×#106	—	3	2	no	—
26	C₉×C₄⋊C₈	—	3	1	no	—
27	C₉×#103	—	3	2	no	—
28	C₃×Q₈×C₉	—	2	1	no	—
29	C₃×Q₈×C₉	Q₈	1	2	yes	complemented
30	C₂×C₃×C₄×C₉	—	2	1	no	—
31	C₂×C₃×C₄×C₉	—	2	1	no	—
32	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
35	C₂×C₉×Dic₃	—	2	2	no	—
36	C₉×Dic₆	—	4	2	no	—
37	C₉×C₃⋊C₈	—	2	2	no	—
38	C₃×#475	—	3	1	no	—
39	#5020	C₉	1	1	yes	characteristic, direct product factor
40	C₂×C₃²×Q₈	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₃×Dic₆	—	2	1	no	—
42	C₃²×C₄⋊C₄	—	2	1	no	—
43	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
44	C₃²×C₄²	—	2	1	no	—
45	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
46	C₃×C₄⋊Dic₃	—	2	1	no	—
47	C₃×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
48	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	2	yes	—
49	C₂×Q₈×C₉	—	2	1	no	—
50	C₂×Q₈×C₉	—	6	1	no	—
51	C₂×Q₈×C₉	C₂×D₃	1	1	yes	characteristic
52	C₂×C₈×C₉	—	3	2	no	—
53	C₉×C₄⋊C₄	—	2	2	no	—
54	C₉×C₄⋊C₄	—	2	1	no	—
55	C₉×C₄⋊C₄	—	2	1	no	—
56	C₉×C₄⋊C₄	—	3	2	no	—
57	C₉×C₄⋊C₄	—	6	1	no	—
58	C₉×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
59	C₄²×C₉	—	2	2	no	—
60	C₄²×C₉	—	2	1	no	—
61	C₄²×C₉	—	2	1	no	—
62	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
63	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
64	C₃×C₄×C₉	—	2	2	no	—
65	C₃×C₄×C₉	—	2	1	no	—
66	C₃×C₄×C₉	—	4	1	no	—
67	C₃×C₄×C₉	SD₁₆	1	2	yes	—
68	C₃×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
69	C₃×C₄×C₉	#41	1	1	yes	characteristic
70	C₉×Dic₃	—	4	2	no	—
71	C₃×C₄⋊Q₈	—	3	1	no	—
72	C₃×C₄×Q₈	—	2	1	no	—
73	C₃×C₄×Q₈	C₂×C₉	1	1	yes	characteristic
74	C₃×C₄×Q₈	C₃×D₃	1	1	yes	characteristic
75	#869	C₂×C₉	1	1	yes	characteristic
76	C₃×#106	—	3	2	no	—
77	C₃×C₄⋊C₈	—	3	1	no	—
78	C₃×#103	—	3	2	no	—
79	#835	C₂×C₉	1	2	yes	—
80	#808	C₂×C₉	1	2	yes	—
81	C₄⋊(C₃⋊C₈)	C₂×C₉	1	1	yes	characteristic
82	C₃²×Q₈	—	2	1	no	—
83	C₃²×Q₈	C₃×Q₈	1	2	yes	—
84	C₂×C₃²×C₄	—	2	1	no	—
85	C₂×C₃²×C₄	—	2	1	no	—
86	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
87	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
88	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
89	C₂×C₃×Dic₃	—	2	2	no	—
90	C₃×Dic₆	—	4	2	no	—
91	C₃×C₃⋊C₈	—	2	2	no	—
92	Q₈×C₉	—	2	2	no	—
93	Q₈×C₉	—	2	1	no	—
94	Q₈×C₉	—	2	2	no	—
95	Q₈×C₉	—	12	2	no	—
96	Q₈×C₉	Dic₆	1	2	yes	complemented
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	1	no	—
99	C₂×C₄×C₉	—	2	2	no	—
100	C₂×C₄×C₉	—	2	1	no	—
101	C₂×C₄×C₉	—	2	1	no	—
102	C₂×C₄×C₉	—	2	1	no	—
103	C₂×C₄×C₉	—	2	1	no	—
104	C₂×C₄×C₉	—	6	2	no	—
105	C₂×C₄×C₉	#67	1	1	yes	characteristic
106	C₂×C₄×C₉	#67	1	1	yes	characteristic
107	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
108	C₈×C₉	—	6	2	no	—
109	#475	—	3	1	no	Sylow
110	C₂×C₃×C₉	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₃×C₉	#137	1	1	yes	characteristic
113	C₂×C₃×Q₈	—	2	1	no	—
114	C₂×C₃×Q₈	—	6	1	no	—
115	C₂×C₃×Q₈	C₂²×C₉	1	1	yes	characteristic
116	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
117	C₂×Dic₆	—	2	1	no	—
118	C₂×C₃×C₈	—	3	2	no	—
119	C₃×C₄⋊C₄	—	2	2	no	—
120	C₃×C₄⋊C₄	—	2	1	no	—
121	C₃×C₄⋊C₄	—	2	1	no	—
122	C₃×C₄⋊C₄	—	2	1	no	—
123	C₃×C₄⋊C₄	—	3	2	no	—
124	C₃×C₄⋊C₄	—	6	1	no	—
125	C₃×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
126	C₃×C₄⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
127	C₃×C₄²	—	2	2	no	—
128	C₃×C₄²	—	2	1	no	—
129	C₃×C₄²	—	2	1	no	—
130	C₃×C₄²	—	2	1	no	—
131	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
132	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
133	C₄⋊Dic₃	—	2	1	no	—
134	C₄⋊Dic₃	C₂²×C₉	1	2	yes	—
135	C₂×C₃⋊C₈	C₂²×C₉	1	2	yes	—
136	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
137	C₃²×C₄	—	2	2	no	—
138	C₃²×C₄	—	2	1	no	—
139	C₃²×C₄	—	4	1	no	—
140	C₃²×C₄	C₃×SD₁₆	1	2	yes	—
141	C₃²×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
142	C₃²×C₄	C₃×#41	1	1	yes	characteristic
143	C₃×Dic₃	—	4	2	no	—
144	C₂²×C₉	—	2	1	no	—
145	C₂²×C₉	C₂×#67	1	1	yes	characteristic
146	C₄×C₉	—	2	4	no	—
147	C₄×C₉	—	2	2	no	—
148	C₄×C₉	—	2	1	no	—
149	C₄×C₉	—	2	2	no	—
150	C₄×C₉	—	2	1	no	—
151	C₄×C₉	—	2	1	no	—
152	C₄×C₉	—	4	1	no	—
153	C₄×C₉	—	4	2	no	—
154	C₄×C₉	—	4	1	no	—
155	C₄×C₉	—	12	2	no	—
156	C₄×C₉	#215	1	2	yes	—
157	C₄×C₉	C₂×Dic₆	1	1	yes	characteristic
158	C₄×C₉	#235	1	1	yes	characteristic
159	C₄⋊Q₈	—	3	1	no	—
160	C₄×Q₈	D₃×C₉	1	1	yes	characteristic
161	#106	—	3	2	no	—
162	C₄⋊C₈	—	3	1	no	—
163	#103	—	3	2	no	—
164	C₃×C₉	#475	1	1	yes	characteristic, complemented, Sylow
165	C₃×Q₈	—	2	2	no	—
166	C₃×Q₈	—	2	1	no	—
167	C₃×Q₈	—	2	1	no	—
168	C₃×Q₈	—	2	2	no	—
169	C₃×Q₈	—	12	2	no	—
170	C₃×Q₈	Q₈×C₉	1	2	yes	complemented
171	C₃×Q₈	C₃×Dic₆	1	2	yes	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	1	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	6	2	no	—
182	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
183	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
185	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
186	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
188	C₂×Dic₃	—	2	2	no	—
189	Dic₆	—	4	2	no	—
190	C₃×C₈	—	6	2	no	—
191	C₃⋊C₈	—	2	2	no	—
192	C₂×C₃²	C₃×C₂²⋊Q₈	1	1	yes	characteristic
193	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
194	C₂×C₃²	C₃×#137	1	1	yes	characteristic
195	C₂×C₉	—	2	1	no	—
196	C₂×C₉	—	2	1	no	—
197	C₂×C₉	—	2	1	no	—
198	C₂×C₉	#924	1	1	yes	characteristic
199	C₂×C₉	C₂×#215	1	1	yes	characteristic
200	C₂×C₉	#951	1	1	yes	characteristic
201	C₂×Q₈	—	6	1	no	—
202	C₂×Q₈	C₂×D₃×C₉	1	1	yes	characteristic
203	C₂×C₈	—	3	2	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	3	2	no	—
206	C₄⋊C₄	—	6	1	no	—
207	C₄⋊C₄	C₂×D₃×C₉	1	1	yes	characteristic
208	C₄²	—	2	1	no	—
209	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
212	C₂²×C₃	C₂×C₃×#67	1	1	yes	characteristic
213	C₃×C₄	—	2	4	no	—
214	C₃×C₄	—	2	2	no	—
215	C₃×C₄	—	2	2	no	—
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	2	1	no	—
219	C₃×C₄	—	2	1	no	—
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	—	4	1	no	—
222	C₃×C₄	—	4	2	no	—
223	C₃×C₄	—	4	1	no	—
224	C₃×C₄	—	4	1	no	—
225	C₃×C₄	—	12	2	no	—
226	C₃×C₄	C₉×SD₁₆	1	2	yes	—
227	C₃×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
228	C₃×C₄	C₉×#41	1	1	yes	characteristic
229	C₃×C₄	C₃×#215	1	2	yes	—
230	C₃×C₄	C₂×C₃×Dic₆	1	1	yes	characteristic
231	C₃×C₄	C₃×#235	1	1	yes	characteristic
232	Dic₃	—	4	2	no	—
233	C₃²	C₃×#475	1	1	yes	characteristic
234	C₉	—	2	1	no	—
235	C₉	#5020	1	1	yes	characteristic, direct product factor
236	Q₈	—	2	1	no	—
237	Q₈	—	12	2	no	—
238	Q₈	C₉×Dic₆	1	2	yes	complemented
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	6	2	no	—
242	C₂×C₄	C₉×#67	1	1	yes	characteristic
243	C₂×C₄	C₉×#67	1	1	yes	characteristic
244	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
245	C₈	—	6	2	no	—
246	C₂×C₃	—	2	1	no	—
247	C₂×C₃	—	2	1	no	—
248	C₂×C₃	—	2	1	no	—
249	C₂×C₃	C₉×C₂²⋊Q₈	1	1	yes	characteristic
250	C₂×C₃	C₂×C₉×SD₁₆	1	1	yes	characteristic
251	C₂×C₃	C₉×#137	1	1	yes	characteristic
252	C₂×C₃	C₃×#924	1	1	yes	characteristic
253	C₂×C₃	C₂×C₃×#215	1	1	yes	characteristic
254	C₂×C₃	C₃×#951	1	1	yes	characteristic
255	C₂²	C₂×C₉×#67	1	1	yes	characteristic
256	C₄	—	2	2	no	—
257	C₄	—	2	1	no	—
258	C₄	—	4	1	no	—
259	C₄	—	12	2	no	—
260	C₄	C₉×#215	1	2	yes	—
261	C₄	C₂×C₉×Dic₆	1	1	yes	characteristic
262	C₄	C₉×#235	1	1	yes	characteristic
263	C₃	—	2	1	no	—
264	C₃	C₉×#475	1	1	yes	characteristic, complemented
265	C₃	C₃×#5020	1	1	yes	characteristic
266	C₂	C₉×#924	1	1	yes	characteristic
267	C₂	C₂×C₉×#215	1	1	yes	characteristic
268	C₂	C₉×#951	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×#67
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×Q₈×C₉	C₂
socle, prime radical	C₂²×C₃²	C₂×C₃×D₄
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₄×Q₈×C₉	D₃
hypocenter	C₃	C₉×#475
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ C₄×Q₈×C₉

Complements

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

N	H ≅ G/N	type
29. C₃×Q₈×C₉	Q₈	semidirect
39. #5020	C₉	direct (N × H)
96. Q₈×C₉	Dic₆	semidirect
164. C₃×C₉	#475	semidirect
170. C₃×Q₈	Q₈×C₉	semidirect
235. C₉	#5020	direct (N × H)
238. Q₈	C₉×Dic₆	semidirect
264. C₃	C₉×#475	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×Q₈×C₉	2
C₉×#869	2
C₉×#835	2
C₉×#808	2
C₉×C₄⋊(C₃⋊C₈)	2
C₃×#5020	3
C₉×#475	3

Subgroups — C9×#5020 #13142769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5020 #13142769