Subgroups — C₉×#4659 #13141309

Order 1728.

478 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃×C₉×#95	C₂	1	1	yes	maximal, characteristic
4	C₉×C₃⋊#95	C₂	1	2	yes	maximal
5	C₉×C₃⋊#95	C₂	1	2	yes	maximal
6	C₃×#4659	C₃	1	1	yes	maximal, characteristic
7	C₉×#400	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
9	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₉×Dic₃	C₂²	1	1	yes	characteristic
11	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
12	C₂²×C₉×Dic₃	C₂²	1	1	yes	characteristic
13	C₉×C₄⋊Dic₃	—	2	2	no	—
14	C₉×Dic₃⋊C₄	—	2	2	no	—
15	C₂×C₃×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic
16	C₂×C₃×Dic₃⋊C₄	C₂×C₃	1	1	yes	characteristic
17	C₃²×#95	C₂×C₃	1	1	yes	characteristic
18	C₃×C₃⋊#95	C₂×C₃	1	2	yes	—
19	C₃×C₃⋊#95	C₂×C₃	1	2	yes	—
20	C₂×C₉×C₄⋊C₄	—	3	1	no	—
21	C₂×C₉×C₄⋊C₄	—	3	1	no	—
22	C₉×#95	—	2	1	no	—
23	C₉×#95	—	3	2	no	—
24	C₉×#95	—	3	2	no	—
25	C₉×#95	D₃	1	1	yes	characteristic
26	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
27	C₂×C₃×C₄×C₉	—	2	2	no	—
28	C₂×C₃×C₄×C₉	—	2	2	no	—
29	C₂×C₃×C₄×C₉	—	2	2	no	—
30	C₂×C₃×C₄×C₉	—	2	2	no	—
31	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
32	C₂×C₉×Dic₃	—	2	1	no	—
33	C₂×C₉×Dic₃	—	2	1	no	—
34	C₂×C₉×Dic₃	—	2	1	no	—
35	C₂×C₉×Dic₃	—	2	2	no	—
36	C₂×C₉×Dic₃	—	2	2	no	—
37	C₂×C₉×Dic₃	—	2	4	no	—
38	C₂×C₉×Dic₃	Q₈	1	2	yes	—
39	C₃×#400	—	3	1	no	—
40	#4659	C₉	1	1	yes	characteristic, direct product factor
41	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
42	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
43	C₂²×C₃×Dic₃	C₂²×C₃	1	1	yes	characteristic
44	C₂²×C₃×Dic₃	C₂²×C₃	1	2	yes	—
45	C₂²×C₃×Dic₃	C₂²×C₃	1	1	yes	characteristic
46	C₃×C₄⋊Dic₃	—	2	2	no	—
47	C₃×Dic₃⋊C₄	—	2	2	no	—
48	C₂²×C₄×C₉	—	2	2	no	—
49	C₂²×C₄×C₉	—	2	1	no	—
50	C₂²×C₄×C₉	—	3	1	no	—
51	C₂²×C₄×C₉	—	3	2	no	—
52	C₂²×C₄×C₉	—	3	1	no	—
53	C₂²×C₄×C₉	C₂×D₃	1	2	yes	—
54	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
55	C₉×C₄⋊C₄	—	6	2	no	—
56	C₉×C₄⋊C₄	—	6	2	no	—
57	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₃×C₉	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₃×C₉	#41	1	1	yes	characteristic
60	C₂²×C₃×C₉	#41	1	2	yes	—
61	C₂²×C₃×C₉	#41	1	2	yes	—
62	C₃×C₄×C₉	—	2	2	no	—
63	C₃×C₄×C₉	—	4	2	no	—
64	C₉×Dic₃	—	2	2	no	—
65	C₉×Dic₃	—	4	1	no	—
66	C₉×Dic₃	—	4	2	no	—
67	C₂×C₃×C₄⋊C₄	—	3	1	no	—
68	C₂×C₃×C₄⋊C₄	—	3	1	no	—
69	C₂×C₄⋊Dic₃	C₂×C₉	1	1	yes	characteristic
70	C₂×Dic₃⋊C₄	C₂×C₉	1	1	yes	characteristic
71	C₃×#95	—	2	1	no	—
72	C₃×#95	—	3	2	no	—
73	C₃×#95	—	3	2	no	—
74	C₃×#95	C₂×C₉	1	1	yes	characteristic
75	C₃×#95	C₃×D₃	1	1	yes	characteristic
76	C₃⋊#95	C₂×C₉	1	2	yes	—
77	C₃⋊#95	C₂×C₉	1	2	yes	—
78	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
79	C₂×C₃²×C₄	—	2	2	no	—
80	C₂×C₃²×C₄	—	2	2	no	—
81	C₂×C₃²×C₄	—	2	2	no	—
82	C₂×C₃²×C₄	—	2	2	no	—
83	C₂×C₃²×C₄	C₃×D₄	1	2	yes	—
84	C₂×C₃×Dic₃	—	2	1	no	—
85	C₂×C₃×Dic₃	—	2	1	no	—
86	C₂×C₃×Dic₃	—	2	1	no	—
87	C₂×C₃×Dic₃	—	2	2	no	—
88	C₂×C₃×Dic₃	—	2	4	no	—
89	C₂×C₃×Dic₃	—	2	2	no	—
90	C₂×C₃×Dic₃	C₃×Q₈	1	2	yes	—
91	C₂³×C₉	—	2	1	no	—
92	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
93	C₂×C₄×C₉	—	2	2	no	—
94	C₂×C₄×C₉	—	2	4	no	—
95	C₂×C₄×C₉	—	2	2	no	—
96	C₂×C₄×C₉	—	2	2	no	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	—	2	2	no	—
99	C₂×C₄×C₉	—	3	2	no	—
100	C₂×C₄×C₉	—	4	2	no	—
101	C₂×C₄×C₉	—	4	2	no	—
102	C₂×C₄×C₉	—	4	2	no	—
103	C₂×C₄×C₉	—	6	1	no	—
104	C₂×C₄×C₉	—	6	1	no	—
105	C₂×C₄×C₉	—	6	2	no	—
106	C₂×C₄×C₉	—	6	1	no	—
107	C₂×C₄×C₉	—	6	4	no	—
108	C₂×C₄×C₉	—	6	2	no	—
109	C₂×C₄×C₉	D₁₂	1	2	yes	—
110	#400	—	3	1	no	Sylow
111	C₂×C₃×C₉	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₃×C₉	#123	1	1	yes	characteristic
113	C₂×C₃×C₉	#124	1	1	yes	characteristic
114	C₂×C₃×C₉	#125	1	2	yes	—
115	C₂×C₃×C₉	#126	1	2	yes	—
116	C₂²×C₃×C₄	—	2	2	no	—
117	C₂²×C₃×C₄	—	2	1	no	—
118	C₂²×C₃×C₄	—	3	1	no	—
119	C₂²×C₃×C₄	—	3	2	no	—
120	C₂²×C₃×C₄	—	3	1	no	—
121	C₂²×C₃×C₄	C₂²×C₉	1	2	yes	—
122	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
123	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
124	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
125	C₂²×Dic₃	C₂²×C₉	1	1	yes	characteristic
126	C₂²×Dic₃	C₂²×C₉	1	2	yes	—
127	C₂²×Dic₃	C₂²×C₉	1	1	yes	characteristic
128	C₃×C₄⋊C₄	—	6	2	no	—
129	C₃×C₄⋊C₄	—	6	2	no	—
130	C₄⋊Dic₃	—	2	2	no	—
131	Dic₃⋊C₄	—	2	2	no	—
132	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
133	C₂²×C₃²	C₂×C₃×Q₈	1	1	yes	characteristic
134	C₂²×C₃²	C₃×#41	1	2	yes	—
135	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
136	C₂²×C₃²	C₃×#41	1	2	yes	—
137	C₃²×C₄	—	2	2	no	—
138	C₃²×C₄	—	4	2	no	—
139	C₃×Dic₃	—	2	2	no	—
140	C₃×Dic₃	—	4	1	no	—
141	C₃×Dic₃	—	4	2	no	—
142	C₂²×C₉	—	2	1	no	—
143	C₂²×C₉	—	2	2	no	—
144	C₂²×C₉	—	2	1	no	—
145	C₂²×C₉	—	2	2	no	—
146	C₂²×C₉	—	2	1	no	—
147	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
148	C₂²×C₉	#235	1	2	yes	—
149	C₂²×C₉	#237	1	1	yes	characteristic
150	C₂²×C₉	#237	1	2	yes	—
151	C₂²×C₉	D₃×Q₈	1	1	yes	characteristic
152	C₄×C₉	—	2	2	no	—
153	C₄×C₉	—	4	2	no	—
154	C₄×C₉	—	4	4	no	—
155	C₄×C₉	—	4	2	no	—
156	C₄×C₉	—	6	2	no	—
157	C₄×C₉	—	12	2	no	—
158	C₄×C₉	—	12	1	no	—
159	C₂×C₄⋊C₄	—	3	1	no	—
160	C₂×C₄⋊C₄	—	3	1	no	—
161	#95	—	3	2	no	—
162	#95	—	3	2	no	—
163	#95	D₃×C₉	1	1	yes	characteristic
164	C₃×C₉	#400	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₃	—	2	1	no	—
166	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
167	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	—	2	4	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	3	2	no	—
179	C₂×C₃×C₄	—	4	2	no	—
180	C₂×C₃×C₄	—	4	2	no	—
181	C₂×C₃×C₄	—	4	2	no	—
182	C₂×C₃×C₄	—	6	1	no	—
183	C₂×C₃×C₄	—	6	1	no	—
184	C₂×C₃×C₄	—	6	1	no	—
185	C₂×C₃×C₄	—	6	2	no	—
186	C₂×C₃×C₄	—	6	4	no	—
187	C₂×C₃×C₄	—	6	2	no	—
188	C₂×C₃×C₄	D₄×C₉	1	2	yes	—
189	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
190	C₂×Dic₃	—	2	1	no	—
191	C₂×Dic₃	—	2	1	no	—
192	C₂×Dic₃	—	2	1	no	—
193	C₂×Dic₃	—	2	2	no	—
194	C₂×Dic₃	—	2	4	no	—
195	C₂×Dic₃	—	2	2	no	—
196	C₂×Dic₃	Q₈×C₉	1	2	yes	—
197	C₂×C₃²	C₃×C₂²⋊Q₈	1	1	yes	characteristic
198	C₂×C₃²	C₃×#123	1	1	yes	characteristic
199	C₂×C₃²	C₃×#124	1	1	yes	characteristic
200	C₂×C₃²	C₃×#125	1	2	yes	—
201	C₂×C₃²	C₃×#126	1	2	yes	—
202	C₂×C₉	—	2	2	no	—
203	C₂×C₉	—	2	1	no	—
204	C₂×C₉	—	2	2	no	—
205	C₂×C₉	—	2	1	no	—
206	C₂×C₉	—	2	1	no	—
207	C₂×C₉	#875	1	1	yes	characteristic
208	C₂×C₉	#879	1	2	yes	—
209	C₂×C₉	#886	1	1	yes	characteristic
210	C₂×C₉	#889	1	2	yes	—
211	C₂×C₉	#897	1	1	yes	characteristic
212	C₂²×C₄	—	3	1	no	—
213	C₂²×C₄	—	3	2	no	—
214	C₂²×C₄	—	3	1	no	—
215	C₂²×C₄	C₂×D₃×C₉	1	2	yes	—
216	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
217	C₄⋊C₄	—	6	2	no	—
218	C₄⋊C₄	—	6	2	no	—
219	C₂²×C₃	—	2	1	no	—
220	C₂²×C₃	—	2	2	no	—
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	—	2	2	no	—
223	C₂²×C₃	—	2	1	no	—
224	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
225	C₂²×C₃	C₂×Q₈×C₉	1	1	yes	characteristic
226	C₂²×C₃	C₉×#41	1	2	yes	—
227	C₂²×C₃	C₉×#41	1	1	yes	characteristic
228	C₂²×C₃	C₉×#41	1	2	yes	—
229	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
230	C₂²×C₃	C₃×#235	1	2	yes	—
231	C₂²×C₃	C₃×#237	1	1	yes	characteristic
232	C₂²×C₃	C₃×#237	1	2	yes	—
233	C₂²×C₃	C₃×D₃×Q₈	1	1	yes	characteristic
234	C₃×C₄	—	2	2	no	—
235	C₃×C₄	—	2	2	no	—
236	C₃×C₄	—	4	2	no	—
237	C₃×C₄	—	4	4	no	—
238	C₃×C₄	—	4	2	no	—
239	C₃×C₄	—	4	2	no	—
240	C₃×C₄	—	6	2	no	—
241	C₃×C₄	—	12	1	no	—
242	C₃×C₄	—	12	2	no	—
243	Dic₃	—	2	2	no	—
244	Dic₃	—	4	1	no	—
245	Dic₃	—	4	2	no	—
246	C₃²	C₃×#400	1	1	yes	characteristic
247	C₉	—	2	1	no	—
248	C₉	#4659	1	1	yes	characteristic, direct product factor
249	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	3	2	no	—
255	C₂×C₄	—	6	1	no	—
256	C₂×C₄	—	6	1	no	—
257	C₂×C₄	—	6	2	no	—
258	C₂×C₄	—	6	1	no	—
259	C₂×C₄	—	6	4	no	—
260	C₂×C₄	—	6	2	no	—
261	C₂×C₄	C₉×D₁₂	1	2	yes	—
262	C₂×C₃	—	2	1	no	—
263	C₂×C₃	—	2	2	no	—
264	C₂×C₃	—	2	1	no	—
265	C₂×C₃	—	2	2	no	—
266	C₂×C₃	—	2	1	no	—
267	C₂×C₃	C₉×C₂²⋊Q₈	1	1	yes	characteristic
268	C₂×C₃	C₉×#123	1	1	yes	characteristic
269	C₂×C₃	C₉×#124	1	1	yes	characteristic
270	C₂×C₃	C₉×#125	1	2	yes	—
271	C₂×C₃	C₉×#126	1	2	yes	—
272	C₂×C₃	C₃×#875	1	1	yes	characteristic
273	C₂×C₃	C₃×#879	1	2	yes	—
274	C₂×C₃	C₃×#886	1	1	yes	characteristic
275	C₂×C₃	C₃×#889	1	2	yes	—
276	C₂×C₃	C₃×#897	1	1	yes	characteristic
277	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
278	C₂²	C₉×#235	1	2	yes	—
279	C₂²	C₉×#237	1	1	yes	characteristic
280	C₂²	C₉×#237	1	2	yes	—
281	C₂²	D₃×Q₈×C₉	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	4	2	no	—
284	C₄	—	6	2	no	—
285	C₄	—	12	2	no	—
286	C₄	—	12	1	no	—
287	C₃	—	2	1	no	—
288	C₃	C₉×#400	1	1	yes	characteristic, complemented
289	C₃	C₃×#4659	1	1	yes	characteristic
290	C₂	C₉×#875	1	1	yes	characteristic
291	C₂	C₉×#879	1	2	yes	—
292	C₂	C₉×#886	1	1	yes	characteristic
293	C₂	C₉×#889	1	2	yes	—
294	C₂	C₉×#897	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂²×D₃
commutator	C₂³×C₃	C₂³×C₉
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₃×C₉×#95	C₂
socle, prime radical, cosocle	C₂³×C₃²	C₂³×C₃
hypercenter	C₉×#95	D₃
hypocenter	C₃	C₉×#400
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₉ ◁ C₉×#95

Complements

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

N	H ≅ G/N	type
40. #4659	C₉	direct (N × H)
164. C₃×C₉	#400	semidirect
248. C₉	#4659	direct (N × H)
288. C₃	C₉×#400	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×C₄⋊Dic₃	2
C₂×C₉×Dic₃⋊C₄	2
C₃×C₉×#95	2
C₉×C₃⋊#95	2
C₃×#4659	3
C₉×#400	3

Subgroups — C9×#4659 #13141309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4659 #13141309