Subgroups — C₉×#4738 #13141598

Order 1728.

524 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#930	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#204	C₂	1	2	yes	maximal, complemented
4	C₃×C₉×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#816	C₂	1	2	yes	maximal, complemented
6	C₃×#4738	C₃	1	1	yes	maximal, characteristic
7	C₉×#448	—	3	1	no	maximal
8	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×#67	—	2	1	no	—
10	C₂²×C₉×Dic₃	—	2	1	no	—
11	C₂×C₉×D₁₂	C₂²	1	1	yes	characteristic
12	C₂×C₉×Dic₆	—	2	1	no	—
13	C₂×C₉×Dic₆	—	2	4	no	—
14	C₂×C₉×Dic₆	C₂²	1	2	yes	—
15	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
16	C₉×(C₂×D₃)⋊C₄	—	2	1	no	—
17	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
18	C₉×#204	—	2	4	no	—
19	C₂²×C₃×Dic₆	C₂×C₃	1	1	yes	characteristic
20	C₃×#930	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃×#204	C₂×C₃	1	2	yes	—
22	C₂²×Q₈×C₉	—	3	1	no	—
23	C₃²×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
24	C₃×#816	C₂×C₃	1	2	yes	—
25	C₂×C₉×SD₁₆	—	3	2	no	—
26	C₉×#121	—	3	1	no	—
27	C₉×#103	—	3	2	no	—
28	C₉×C₂²⋊C₈	—	2	1	no	—
29	C₉×C₂²⋊C₈	D₃	1	1	yes	characteristic, complemented
30	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
31	C₂²×D₃×C₉	—	2	1	no	—
32	C₂×C₃×C₄×C₉	—	2	2	no	—
33	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
35	C₉×#67	—	4	2	no	—
36	C₂×C₉×Dic₃	—	2	4	no	—
37	C₂×C₉×Dic₃	—	2	2	no	—
38	C₂×C₉×Dic₃	—	2	1	no	—
39	C₉×D₁₂	—	2	2	no	—
40	C₉×Dic₆	—	2	4	no	—
41	C₉×Dic₆	—	2	2	no	—
42	C₉×Dic₆	D₄	1	4	yes	complemented
43	C₃×C₈×C₉	—	2	2	no	—
44	C₃×#448	—	3	1	no	—
45	#4738	C₉	1	1	yes	characteristic, direct product factor
46	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₃×#67	—	2	1	no	—
48	C₂²×C₃×Dic₃	—	2	1	no	—
49	C₂×C₃×D₁₂	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₃×Dic₆	—	2	1	no	—
51	C₂×C₃×Dic₆	—	2	4	no	—
52	C₂×C₃×Dic₆	C₂²×C₃	1	2	yes	—
53	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
54	C₃×(C₂×D₃)⋊C₄	—	2	1	no	—
55	C₃×C₄⋊Dic₃	C₂²×C₃	1	1	yes	characteristic
56	C₃×#204	—	2	4	no	—
57	C₂×Q₈×C₉	—	3	2	no	—
58	C₂×Q₈×C₉	—	6	4	no	—
59	C₂×Q₈×C₉	—	6	1	no	—
60	C₂×D₄×C₉	—	3	1	no	—
61	C₂×D₄×C₉	—	6	1	no	—
62	C₂²×C₄×C₉	—	2	1	no	—
63	C₂²×C₄×C₉	—	6	1	no	—
64	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
65	C₉×SD₁₆	—	6	4	no	—
66	C₂×C₈×C₉	—	2	2	no	—
67	C₂×C₈×C₉	C₂×D₃	1	2	yes	—
68	C₉×C₄⋊C₄	—	3	1	no	—
69	C₉×C₂²⋊C₄	—	6	1	no	—
70	C₂²×C₃×C₉	—	2	1	no	—
71	C₂²×C₃×C₉	—	2	1	no	—
72	C₂²×C₃×C₉	SD₁₆	1	2	yes	complemented
73	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
74	C₂×D₃×C₉	—	4	1	no	—
75	C₂×D₃×C₉	—	4	1	no	—
76	C₂×D₃×C₉	—	4	1	no	—
77	C₃×C₄×C₉	—	2	1	no	—
78	C₃×C₄×C₉	C₂×D₄	1	2	yes	—
79	C₉×Dic₃	—	2	4	no	—
80	C₉×Dic₃	—	4	1	no	—
81	C₂²×C₃×Q₈	—	3	1	no	—
82	C₂²×Dic₆	C₂×C₉	1	1	yes	characteristic, complemented
83	C₂×C₃×SD₁₆	—	3	2	no	—
84	C₃×#121	—	3	1	no	—
85	#930	C₂×C₉	1	1	yes	characteristic
86	C₂×#204	C₂×C₉	1	2	yes	complemented
87	C₃×#103	—	3	2	no	—
88	C₃×C₂²⋊C₈	—	2	1	no	—
89	C₃×C₂²⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
90	C₃×C₂²⋊C₈	C₃×D₃	1	1	yes	characteristic
91	#816	C₂×C₉	1	2	yes	complemented
92	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
93	C₂²×C₃×D₃	—	2	1	no	—
94	C₂×C₃²×C₄	—	2	2	no	—
95	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
97	C₃×#67	—	4	2	no	—
98	C₂×C₃×Dic₃	—	2	4	no	—
99	C₂×C₃×Dic₃	—	2	2	no	—
100	C₂×C₃×Dic₃	—	2	1	no	—
101	C₃×D₁₂	—	2	2	no	—
102	C₃×Dic₆	—	2	4	no	—
103	C₃×Dic₆	—	2	2	no	—
104	C₃×Dic₆	C₃×D₄	1	4	yes	—
105	C₂³×C₉	—	2	1	no	—
106	C₂³×C₉	—	6	1	no	—
107	C₂³×C₉	D₁₂	1	1	yes	characteristic
108	C₃²×C₈	—	2	2	no	—
109	Q₈×C₉	—	3	4	no	—
110	Q₈×C₉	—	6	4	no	—
111	Q₈×C₉	—	6	2	no	—
112	D₄×C₉	—	6	2	no	—
113	D₄×C₉	—	12	2	no	—
114	C₂×C₄×C₉	—	2	2	no	—
115	C₂×C₄×C₉	—	2	1	no	—
116	C₂×C₄×C₉	—	2	1	no	—
117	C₂×C₄×C₉	—	4	2	no	—
118	C₂×C₄×C₉	—	6	4	no	—
119	C₂×C₄×C₉	—	6	2	no	—
120	C₂×C₄×C₉	—	6	1	no	—
121	C₂×C₄×C₉	D₁₂	1	1	yes	characteristic
122	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
123	C₈×C₉	—	2	2	no	—
124	C₈×C₉	—	4	2	no	—
125	#448	—	3	1	no	Sylow
126	C₂×C₃×C₉	—	2	2	no	—
127	C₂×C₃×C₉	#120	1	1	yes	characteristic
128	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₃×C₉	#137	1	1	yes	characteristic
130	D₃×C₉	—	8	1	no	—
131	C₂×C₃×Q₈	—	3	2	no	—
132	C₂×C₃×Q₈	—	6	4	no	—
133	C₂×C₃×Q₈	—	6	1	no	—
134	C₂×C₃×D₄	—	3	1	no	—
135	C₂×C₃×D₄	—	6	1	no	—
136	C₂²×C₃×C₄	—	2	1	no	—
137	C₂²×C₃×C₄	—	6	1	no	—
138	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
139	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
140	C₂×#67	—	2	1	no	—
141	C₂²×Dic₃	—	2	1	no	—
142	C₂×D₁₂	C₂²×C₉	1	1	yes	characteristic
143	C₂×Dic₆	—	2	1	no	—
144	C₂×Dic₆	—	2	4	no	—
145	C₂×Dic₆	C₂²×C₉	1	2	yes	—
146	C₃×SD₁₆	—	6	4	no	—
147	C₂×C₃×C₈	—	2	2	no	—
148	C₂×C₃×C₈	C₂²×C₉	1	2	yes	—
149	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
150	C₃×C₄⋊C₄	—	3	1	no	—
151	C₃×C₂²⋊C₄	—	6	1	no	—
152	(C₂×D₃)⋊C₄	—	2	1	no	—
153	C₄⋊Dic₃	C₂²×C₉	1	1	yes	characteristic
154	#204	—	2	4	no	—
155	C₂²×C₃²	—	2	1	no	—
156	C₂²×C₃²	—	2	1	no	—
157	C₂²×C₃²	C₃×SD₁₆	1	2	yes	—
158	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
159	C₂×C₃×D₃	—	4	1	no	—
160	C₂×C₃×D₃	—	4	1	no	—
161	C₂×C₃×D₃	—	4	1	no	—
162	C₃²×C₄	—	2	1	no	—
163	C₃²×C₄	C₂×C₃×D₄	1	2	yes	—
164	C₃×Dic₃	—	2	4	no	—
165	C₃×Dic₃	—	4	1	no	—
166	C₂²×C₉	—	2	1	no	—
167	C₂²×C₉	—	2	1	no	—
168	C₂²×C₉	—	2	2	no	—
169	C₂²×C₉	—	2	1	no	—
170	C₂²×C₉	—	4	1	no	—
171	C₂²×C₉	—	4	1	no	—
172	C₂²×C₉	—	12	1	no	—
173	C₂²×C₉	—	12	1	no	—
174	C₂²×C₉	—	12	1	no	—
175	C₂²×C₉	#204	1	2	yes	complemented
176	C₂²×C₉	C₂×D₁₂	1	1	yes	characteristic
177	C₄×C₉	—	2	1	no	—
178	C₄×C₉	—	2	2	no	—
179	C₄×C₉	—	4	1	no	—
180	C₄×C₉	—	6	4	no	—
181	C₄×C₉	—	12	1	no	—
182	C₄×C₉	D₃×D₄	1	2	yes	—
183	C₂²×Q₈	—	3	1	no	—
184	C₂×SD₁₆	—	3	2	no	—
185	#121	—	3	1	no	—
186	#103	—	3	2	no	—
187	C₂²⋊C₈	D₃×C₉	1	1	yes	characteristic, complemented
188	C₃×C₉	#448	1	1	yes	characteristic, complemented, Sylow
189	C₂³×C₃	—	2	1	no	—
190	C₂³×C₃	—	6	1	no	—
191	C₂³×C₃	D₄×C₉	1	1	yes	characteristic
192	C₂³×C₃	C₃×D₁₂	1	1	yes	characteristic
193	C₂²×D₃	—	2	1	no	—
194	C₃×Q₈	—	3	4	no	—
195	C₃×Q₈	—	6	4	no	—
196	C₃×Q₈	—	6	2	no	—
197	C₃×D₄	—	6	2	no	—
198	C₃×D₄	—	12	2	no	—
199	C₂×C₃×C₄	—	2	2	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	4	2	no	—
204	C₂×C₃×C₄	—	6	4	no	—
205	C₂×C₃×C₄	—	6	2	no	—
206	C₂×C₃×C₄	—	6	1	no	—
207	C₂×C₃×C₄	D₄×C₉	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
209	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
211	#67	—	4	2	no	—
212	C₂×Dic₃	—	2	4	no	—
213	C₂×Dic₃	—	2	2	no	—
214	C₂×Dic₃	—	2	1	no	—
215	D₁₂	—	2	2	no	—
216	Dic₆	—	2	4	no	—
217	Dic₆	—	2	2	no	—
218	Dic₆	D₄×C₉	1	4	yes	complemented
219	C₃×C₈	—	2	2	no	—
220	C₃×C₈	—	2	2	no	—
221	C₃×C₈	—	4	2	no	—
222	C₂×C₃²	—	2	2	no	—
223	C₂×C₃²	C₃×#120	1	1	yes	characteristic
224	C₂×C₃²	C₂×C₃×SD₁₆	1	1	yes	characteristic
225	C₂×C₃²	C₃×#137	1	1	yes	characteristic
226	C₃×D₃	—	8	1	no	—
227	C₂×C₉	—	2	2	no	—
228	C₂×C₉	—	2	1	no	—
229	C₂×C₉	—	2	1	no	—
230	C₂×C₉	—	2	1	no	—
231	C₂×C₉	—	4	2	no	—
232	C₂×C₉	—	24	1	no	—
233	C₂×C₉	#882	1	1	yes	characteristic
234	C₂×C₉	C₂×#204	1	1	yes	characteristic
235	C₂×C₉	#909	1	1	yes	characteristic
236	C₂×Q₈	—	3	2	no	—
237	C₂×Q₈	—	6	4	no	—
238	C₂×Q₈	—	6	1	no	—
239	C₂×D₄	—	3	1	no	—
240	C₂×D₄	—	6	1	no	—
241	C₂²×C₄	—	6	1	no	—
242	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
243	SD₁₆	—	6	4	no	—
244	C₂×C₈	C₂×D₃×C₉	1	2	yes	—
245	C₄⋊C₄	—	3	1	no	—
246	C₂²⋊C₄	—	6	1	no	—
247	C₂²×C₃	—	2	1	no	—
248	C₂²×C₃	—	2	1	no	—
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	2	1	no	—
251	C₂²×C₃	—	2	1	no	—
252	C₂²×C₃	—	2	2	no	—
253	C₂²×C₃	—	4	1	no	—
254	C₂²×C₃	—	4	1	no	—
255	C₂²×C₃	—	12	1	no	—
256	C₂²×C₃	—	12	1	no	—
257	C₂²×C₃	—	12	1	no	—
258	C₂²×C₃	C₉×SD₁₆	1	2	yes	complemented
259	C₂²×C₃	C₂×D₄×C₉	1	1	yes	characteristic
260	C₂²×C₃	C₃×#204	1	2	yes	—
261	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
262	C₂×D₃	—	4	1	no	—
263	C₂×D₃	—	4	1	no	—
264	C₂×D₃	—	4	1	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	—	4	1	no	—
269	C₃×C₄	—	6	4	no	—
270	C₃×C₄	—	12	1	no	—
271	C₃×C₄	C₂×D₄×C₉	1	2	yes	—
272	C₃×C₄	C₃×D₃×D₄	1	2	yes	—
273	Dic₃	—	2	4	no	—
274	Dic₃	—	4	1	no	—
275	C₃²	C₃×#448	1	1	yes	characteristic
276	C₉	—	2	1	no	—
277	C₉	#4738	1	1	yes	characteristic, direct product factor
278	C₂³	—	6	1	no	—
279	C₂³	C₉×D₁₂	1	1	yes	characteristic
280	Q₈	—	3	4	no	—
281	Q₈	—	6	4	no	—
282	Q₈	—	6	2	no	—
283	D₄	—	6	2	no	—
284	D₄	—	12	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	6	4	no	—
287	C₂×C₄	—	6	2	no	—
288	C₂×C₄	—	6	1	no	—
289	C₂×C₄	C₉×D₁₂	1	1	yes	characteristic
290	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₂×C₃	—	2	2	no	—
293	C₂×C₃	—	2	2	no	—
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	—	2	1	no	—
296	C₂×C₃	—	2	1	no	—
297	C₂×C₃	—	4	2	no	—
298	C₂×C₃	—	24	1	no	—
299	C₂×C₃	C₉×#120	1	1	yes	characteristic
300	C₂×C₃	C₂×C₉×SD₁₆	1	1	yes	characteristic
301	C₂×C₃	C₉×#137	1	1	yes	characteristic
302	C₂×C₃	C₃×#882	1	1	yes	characteristic
303	C₂×C₃	C₂×C₃×#204	1	1	yes	characteristic
304	C₂×C₃	C₃×#909	1	1	yes	characteristic
305	D₃	—	8	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	12	1	no	—
309	C₂²	—	12	1	no	—
310	C₂²	—	12	1	no	—
311	C₂²	C₉×#204	1	2	yes	complemented
312	C₂²	C₂×C₉×D₁₂	1	1	yes	characteristic
313	C₄	—	2	1	no	—
314	C₄	—	6	4	no	—
315	C₄	—	12	1	no	—
316	C₄	D₃×D₄×C₉	1	2	yes	—
317	C₃	—	2	1	no	—
318	C₃	C₉×#448	1	1	yes	characteristic, complemented
319	C₃	C₃×#4738	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	24	1	no	—
322	C₂	C₉×#882	1	1	yes	characteristic
323	C₂	C₂×C₉×#204	1	1	yes	characteristic
324	C₂	C₉×#909	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂³×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×C₂²⋊C₈	C₂
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₃×#930	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×C₂²⋊C₈	D₃
hypocenter	C₃	C₉×#448
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ C₉×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₉×Dic₆	C₂	semidirect
3. C₂×C₉×#204	C₂	semidirect
4. C₃×C₉×C₂²⋊C₈	C₂	semidirect
5. C₉×#816	C₂	semidirect
29. C₉×C₂²⋊C₈	D₃	semidirect
42. C₉×Dic₆	D₄	semidirect
45. #4738	C₉	direct (N × H)
72. C₂²×C₃×C₉	SD₁₆	semidirect
82. C₂²×Dic₆	C₂×C₉	semidirect
86. C₂×#204	C₂×C₉	semidirect
89. C₃×C₂²⋊C₈	C₂×C₉	semidirect
91. #816	C₂×C₉	semidirect
175. C₂²×C₉	#204	semidirect
187. C₂²⋊C₈	D₃×C₉	semidirect
188. C₃×C₉	#448	semidirect
218. Dic₆	D₄×C₉	semidirect
258. C₂²×C₃	C₉×SD₁₆	semidirect
277. C₉	#4738	direct (N × H)
311. C₂²	C₉×#204	semidirect
318. C₃	C₉×#448	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×Dic₆	2
C₉×#930	2
C₂×C₉×#204	2
C₃×C₉×C₂²⋊C₈	2
C₉×#816	2
C₃×#4738	3
C₉×#448	3

Subgroups — C9×#4738 #13141598

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4738 #13141598