Subgroups — C₉×#4824 #13141976

Order 1728.

362 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#950	C₂	1	1	yes	maximal, characteristic
2	C₉×#948	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#907	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₉×#104	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#822	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#805	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#4824	C₃	1	1	yes	maximal, characteristic
9	C₉×#443	—	3	1	no	maximal
10	C₃×C₉×#41	C₂²	1	1	yes	characteristic
11	C₉×#235	C₂²	1	1	yes	characteristic
12	C₃×C₉×M₁₆	C₂²	1	1	yes	characteristic
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₉×#216	C₄	1	1	yes	characteristic, complemented
15	C₉×#215	C₄	1	1	yes	characteristic, complemented
16	C₉×#214	C₄	1	1	yes	characteristic, complemented
17	C₉×#213	C₄	1	1	yes	characteristic, complemented
18	C₉×#208	—	2	1	no	—
19	C₉×#208	C₂²	1	1	yes	characteristic
20	C₂×C₉×C₃⋊C₈	—	2	1	no	—
21	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
23	C₉×ΓL(1,𝔽₂₅)	—	2	1	no	—
24	D₃×C₈×C₉	—	2	1	no	—
25	C₃×#950	C₂×C₃	1	1	yes	characteristic
26	C₃×#948	C₂×C₃	1	1	yes	characteristic
27	C₃×#907	C₂×C₃	1	1	yes	characteristic
28	C₃²×#104	C₂×C₃	1	1	yes	characteristic
29	C₃×#822	C₂×C₃	1	1	yes	characteristic
30	C₃×#805	C₂×C₃	1	1	yes	characteristic
31	C₃×C₄×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
32	C₉×#135	—	3	1	no	—
33	C₉×#131	—	3	1	no	—
34	C₉×#131	—	3	1	no	—
35	C₉×#108	—	3	1	no	—
36	C₉×#104	—	2	1	no	—
37	C₉×#104	—	3	1	no	—
38	C₉×#104	D₃	1	1	yes	characteristic, complemented
39	C₄×C₈×C₉	—	3	1	no	—
40	C₃×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
41	C₃×D₄×C₉	—	2	1	no	—
42	C₃×D₄×C₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃×C₄×C₉	—	2	1	no	—
44	C₂×C₃×C₄×C₉	—	2	1	no	—
45	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
46	C₉×#67	—	2	1	no	—
47	C₉×D₁₂	C₂×C₄	1	1	yes	characteristic
48	C₄×D₃×C₉	—	2	1	no	—
49	C₉×Dic₆	C₂×C₄	1	1	yes	characteristic
50	C₃×C₈×C₉	—	2	1	no	—
51	C₉×C₃⋊C₈	—	2	1	no	—
52	C₉×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₉×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₉×C₃⋊C₈	D₄	1	1	yes	characteristic
55	C₉×C₃⋊C₈	D₄	1	1	yes	characteristic
56	C₃×#443	—	3	1	no	—
57	#4824	C₉	1	1	yes	characteristic, direct product factor
58	C₃²×#41	C₂²×C₃	1	1	yes	characteristic
59	C₃×#235	C₂²×C₃	1	1	yes	characteristic
60	C₃²×M₁₆	C₂²×C₃	1	1	yes	characteristic
61	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
62	C₃×#216	C₃×C₄	1	1	yes	characteristic
63	C₃×#215	C₃×C₄	1	1	yes	characteristic
64	C₃×#214	C₃×C₄	1	1	yes	characteristic
65	C₃×#213	C₃×C₄	1	1	yes	characteristic
66	C₃×#208	—	2	1	no	—
67	C₃×#208	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₃×C₃⋊C₈	—	2	1	no	—
69	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
70	C₂×C₃×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
71	C₃×ΓL(1,𝔽₂₅)	—	2	1	no	—
72	C₃×D₃×C₈	—	2	1	no	—
73	C₉×#41	—	2	1	no	—
74	C₉×#41	—	3	1	no	—
75	C₉×#41	C₂×D₃	1	1	yes	characteristic
76	C₉×Q₁₆	—	3	1	no	—
77	C₉×SD₁₆	—	3	1	no	—
78	C₉×SD₁₆	—	3	1	no	—
79	C₉×D₈	—	3	1	no	—
80	C₉×M₁₆	—	2	1	no	—
81	C₉×M₁₆	—	3	1	no	—
82	C₉×M₁₆	—	6	1	no	—
83	C₉×M₁₆	—	6	1	no	—
84	C₉×M₁₆	C₂×D₃	1	1	yes	characteristic
85	C₂×C₈×C₉	—	3	1	no	—
86	C₂×C₈×C₉	—	3	1	no	—
87	C₂×C₈×C₉	—	6	1	no	—
88	C₂×C₈×C₉	—	6	1	no	—
89	C₄²×C₉	—	2	1	no	—
90	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
91	C₂²×C₃×C₉	—	2	1	no	—
92	C₂²×C₃×C₉	#41	1	1	yes	characteristic
93	C₂×D₃×C₉	—	2	1	no	—
94	C₃×C₄×C₉	—	2	2	no	—
95	C₃×C₄×C₉	—	2	1	no	—
96	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
97	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
98	C₉×Dic₃	—	2	1	no	—
99	C₃×#135	—	3	1	no	—
100	C₃×#131	—	3	1	no	—
101	C₃×#131	—	3	1	no	—
102	#950	C₂×C₉	1	1	yes	characteristic
103	#948	C₂×C₉	1	1	yes	characteristic, complemented
104	#907	C₂×C₉	1	1	yes	characteristic, complemented
105	C₃×#108	—	3	1	no	—
106	C₃×#104	—	2	1	no	—
107	C₃×#104	—	3	1	no	—
108	C₃×#104	C₂×C₉	1	1	yes	characteristic, complemented
109	C₃×#104	C₃×D₃	1	1	yes	characteristic
110	C₃×C₄×C₈	—	3	1	no	—
111	#822	C₂×C₉	1	1	yes	characteristic, complemented
112	#805	C₂×C₉	1	1	yes	characteristic, complemented
113	C₄×C₃⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
114	C₃²×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
115	C₃²×D₄	—	2	1	no	—
116	C₃²×D₄	C₂×C₃×C₄	1	1	yes	characteristic
117	C₂×C₃²×C₄	—	2	1	no	—
118	C₂×C₃²×C₄	—	2	1	no	—
119	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
120	C₃×#67	—	2	1	no	—
121	C₃×D₁₂	C₂×C₃×C₄	1	1	yes	characteristic
122	C₃×C₄×D₃	—	2	1	no	—
123	C₃×Dic₆	C₂×C₃×C₄	1	1	yes	characteristic
124	C₃²×C₈	—	2	1	no	—
125	C₃×C₃⋊C₈	—	2	1	no	—
126	C₃×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
127	C₃×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
128	C₃×C₃⋊C₈	C₃×D₄	1	1	yes	characteristic
129	C₃×C₃⋊C₈	C₃×D₄	1	1	yes	characteristic
130	Q₈×C₉	—	2	1	no	—
131	Q₈×C₉	—	3	1	no	—
132	Q₈×C₉	C₄×D₃	1	1	yes	characteristic
133	D₄×C₉	—	2	1	no	—
134	D₄×C₉	—	2	1	no	—
135	D₄×C₉	—	3	1	no	—
136	D₄×C₉	—	4	1	no	—
137	D₄×C₉	—	6	1	no	—
138	D₄×C₉	C₄×D₃	1	1	yes	characteristic
139	C₂×C₄×C₉	—	2	1	no	—
140	C₂×C₄×C₉	—	2	1	no	—
141	C₂×C₄×C₉	—	2	1	no	—
142	C₂×C₄×C₉	—	4	1	no	—
143	C₂×C₄×C₉	—	4	1	no	—
144	C₂×C₄×C₉	—	6	1	no	—
145	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
146	C₈×C₉	—	2	1	no	—
147	C₈×C₉	—	3	1	no	—
148	C₈×C₉	—	3	1	no	—
149	C₈×C₉	—	3	1	no	—
150	C₈×C₉	—	3	1	no	—
151	C₈×C₉	—	4	1	no	—
152	C₈×C₉	—	6	1	no	—
153	#443	—	3	1	no	Sylow
154	C₂×C₃×C₉	—	2	1	no	—
155	C₂×C₃×C₉	—	4	1	no	—
156	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
157	D₃×C₉	—	4	1	no	—
158	C₃×#41	—	2	1	no	—
159	C₃×#41	—	3	1	no	—
160	C₃×#41	C₂²×C₉	1	1	yes	characteristic
161	C₃×#41	C₂×C₃×D₃	1	1	yes	characteristic
162	#235	C₂²×C₉	1	1	yes	characteristic
163	C₃×Q₁₆	—	3	1	no	—
164	C₃×SD₁₆	—	3	1	no	—
165	C₃×SD₁₆	—	3	1	no	—
166	C₃×D₈	—	3	1	no	—
167	C₃×M₁₆	—	2	1	no	—
168	C₃×M₁₆	—	3	1	no	—
169	C₃×M₁₆	—	6	1	no	—
170	C₃×M₁₆	—	6	1	no	—
171	C₃×M₁₆	C₂²×C₉	1	1	yes	characteristic
172	C₃×M₁₆	C₂×C₃×D₃	1	1	yes	characteristic
173	C₂×C₃×C₈	—	3	1	no	—
174	C₂×C₃×C₈	—	3	1	no	—
175	C₂×C₃×C₈	—	6	1	no	—
176	C₂×C₃×C₈	—	6	1	no	—
177	C₃×C₄²	—	2	1	no	—
178	C₃×C₄²	C₂²×C₉	1	1	yes	characteristic
179	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
180	#216	C₄×C₉	1	1	yes	characteristic, complemented
181	#215	C₄×C₉	1	1	yes	characteristic, complemented
182	#214	C₄×C₉	1	1	yes	characteristic, complemented
183	#213	C₄×C₉	1	1	yes	characteristic, complemented
184	#208	—	2	1	no	—
185	#208	C₂²×C₉	1	1	yes	characteristic
186	C₂×C₃⋊C₈	—	2	1	no	—
187	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
188	C₂×C₃⋊C₈	C₂²×C₉	1	1	yes	characteristic
189	ΓL(1,𝔽₂₅)	—	2	1	no	—
190	D₃×C₈	—	2	1	no	—
191	C₂²×C₃²	—	2	1	no	—
192	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
193	C₂×C₃×D₃	—	2	1	no	—
194	C₃²×C₄	—	2	2	no	—
195	C₃²×C₄	—	2	1	no	—
196	C₃²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
197	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
198	C₃×Dic₃	—	2	1	no	—
199	C₂²×C₉	—	2	1	no	—
200	C₂²×C₉	—	2	1	no	—
201	C₂²×C₉	—	4	1	no	—
202	C₂²×C₉	—	6	1	no	—
203	C₂²×C₉	#237	1	1	yes	characteristic
204	C₄×C₉	—	2	2	no	—
205	C₄×C₉	—	2	1	no	—
206	C₄×C₉	—	2	1	no	—
207	C₄×C₉	—	2	1	no	—
208	C₄×C₉	—	4	1	no	—
209	C₄×C₉	—	4	2	no	—
210	C₄×C₉	—	6	1	no	—
211	C₄×C₉	C₂×C₄×D₃	1	1	yes	characteristic
212	C₄×C₉	D₃×D₄	1	1	yes	characteristic
213	#135	—	3	1	no	—
214	#131	—	3	1	no	—
215	#131	—	3	1	no	—
216	#108	—	3	1	no	—
217	#104	—	3	1	no	—
218	#104	D₃×C₉	1	1	yes	characteristic, complemented
219	C₄×C₈	—	3	1	no	—
220	C₃×C₉	#443	1	1	yes	characteristic, complemented, Sylow
221	C₃×Q₈	—	2	1	no	—
222	C₃×Q₈	—	3	1	no	—
223	C₃×Q₈	C₂×C₄×C₉	1	1	yes	characteristic
224	C₃×Q₈	C₃×C₄×D₃	1	1	yes	characteristic
225	C₃×D₄	—	2	1	no	—
226	C₃×D₄	—	2	1	no	—
227	C₃×D₄	—	2	1	no	—
228	C₃×D₄	—	3	1	no	—
229	C₃×D₄	—	4	1	no	—
230	C₃×D₄	—	6	1	no	—
231	C₃×D₄	C₂×C₄×C₉	1	1	yes	characteristic
232	C₃×D₄	C₃×C₄×D₃	1	1	yes	characteristic
233	C₂×C₃×C₄	—	2	1	no	—
234	C₂×C₃×C₄	—	2	1	no	—
235	C₂×C₃×C₄	—	2	1	no	—
236	C₂×C₃×C₄	—	2	1	no	—
237	C₂×C₃×C₄	—	2	1	no	—
238	C₂×C₃×C₄	—	4	1	no	—
239	C₂×C₃×C₄	—	4	1	no	—
240	C₂×C₃×C₄	—	6	1	no	—
241	C₂×C₃×C₄	C₂³×C₉	1	1	yes	characteristic
242	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
243	#67	—	2	1	no	—
244	D₁₂	C₂×C₄×C₉	1	1	yes	characteristic
245	C₄×D₃	—	2	1	no	—
246	Dic₆	C₂×C₄×C₉	1	1	yes	characteristic
247	C₃×C₈	—	2	1	no	—
248	C₃×C₈	—	2	1	no	—
249	C₃×C₈	—	3	1	no	—
250	C₃×C₈	—	3	1	no	—
251	C₃×C₈	—	3	1	no	—
252	C₃×C₈	—	3	1	no	—
253	C₃×C₈	—	4	1	no	—
254	C₃×C₈	—	6	1	no	—
255	C₃⋊C₈	—	2	1	no	—
256	C₃⋊C₈	C₂×C₄×C₉	1	1	yes	characteristic
257	C₃⋊C₈	C₂×C₄×C₉	1	1	yes	characteristic
258	C₃⋊C₈	D₄×C₉	1	1	yes	characteristic
259	C₃⋊C₈	D₄×C₉	1	1	yes	characteristic
260	C₂×C₃²	—	2	1	no	—
261	C₂×C₃²	—	4	1	no	—
262	C₂×C₃²	C₃×C₄×D₄	1	1	yes	characteristic
263	C₃×D₃	—	4	1	no	—
264	C₂×C₉	—	2	1	no	—
265	C₂×C₉	—	2	1	no	—
266	C₂×C₉	—	4	1	no	—
267	C₂×C₉	—	4	1	no	—
268	C₂×C₉	—	8	1	no	—
269	C₂×C₉	—	12	1	no	—
270	C₂×C₉	#67⋊C₄	1	1	yes	characteristic
271	#41	—	3	1	no	—
272	#41	C₂×D₃×C₉	1	1	yes	characteristic
273	Q₁₆	—	3	1	no	—
274	SD₁₆	—	3	1	no	—
275	SD₁₆	—	3	1	no	—
276	D₈	—	3	1	no	—
277	M₁₆	—	3	1	no	—
278	M₁₆	—	6	1	no	—
279	M₁₆	—	6	1	no	—
280	M₁₆	C₂×D₃×C₉	1	1	yes	characteristic
281	C₂×C₈	—	3	1	no	—
282	C₂×C₈	—	3	1	no	—
283	C₂×C₈	—	6	1	no	—
284	C₂×C₈	—	6	1	no	—
285	C₄²	C₂×D₃×C₉	1	1	yes	characteristic
286	C₂²×C₃	—	2	1	no	—
287	C₂²×C₃	—	2	1	no	—
288	C₂²×C₃	—	2	1	no	—
289	C₂²×C₃	—	4	1	no	—
290	C₂²×C₃	—	6	1	no	—
291	C₂²×C₃	C₉×#41	1	1	yes	characteristic
292	C₂²×C₃	C₃×#237	1	1	yes	characteristic
293	C₂×D₃	—	2	1	no	—
294	C₃×C₄	—	2	2	no	—
295	C₃×C₄	—	2	2	no	—
296	C₃×C₄	—	2	1	no	—
297	C₃×C₄	—	2	1	no	—
298	C₃×C₄	—	2	1	no	—
299	C₃×C₄	—	2	1	no	—
300	C₃×C₄	—	4	1	no	—
301	C₃×C₄	—	4	2	no	—
302	C₃×C₄	—	6	1	no	—
303	C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
304	C₃×C₄	C₂×D₄×C₉	1	1	yes	characteristic
305	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
306	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
307	Dic₃	—	2	1	no	—
308	C₃²	C₃×#443	1	1	yes	characteristic
309	C₉	—	2	1	no	—
310	C₉	#4824	1	1	yes	characteristic, direct product factor
311	Q₈	—	3	1	no	—
312	Q₈	C₄×D₃×C₉	1	1	yes	characteristic
313	D₄	—	2	1	no	—
314	D₄	—	3	1	no	—
315	D₄	—	6	1	no	—
316	D₄	C₄×D₃×C₉	1	1	yes	characteristic
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	6	1	no	—
320	C₂×C₄	C₂²×D₃×C₉	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	3	1	no	—
323	C₈	—	3	1	no	—
324	C₈	—	3	1	no	—
325	C₈	—	3	1	no	—
326	C₈	—	6	1	no	—
327	C₂×C₃	—	2	1	no	—
328	C₂×C₃	—	2	1	no	—
329	C₂×C₃	—	2	1	no	—
330	C₂×C₃	—	4	1	no	—
331	C₂×C₃	—	4	1	no	—
332	C₂×C₃	—	4	1	no	—
333	C₂×C₃	—	8	1	no	—
334	C₂×C₃	—	12	1	no	—
335	C₂×C₃	C₄×D₄×C₉	1	1	yes	characteristic
336	C₂×C₃	C₃×#67⋊C₄	1	1	yes	characteristic
337	D₃	—	4	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	6	1	no	—
340	C₂²	C₉×#237	1	1	yes	characteristic
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	6	1	no	—
344	C₄	C₂×C₄×D₃×C₉	1	1	yes	characteristic
345	C₄	D₃×D₄×C₉	1	1	yes	characteristic
346	C₃	—	2	1	no	—
347	C₃	C₉×#443	1	1	yes	characteristic, complemented
348	C₃	C₃×#4824	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	4	1	no	—
351	C₂	—	12	1	no	—
352	C₂	C₉×#67⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₉	D₃×D₄
commutator	C₃×C₄	C₂²×C₄×C₉
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₉×#104	C₂
socle	C₂×C₃²	C₃×C₄×D₄
prime radical	C₃×#950	C₂×C₃
cosocle	C₂×C₃²×C₄	C₂³×C₃
hypercenter	C₉×#104	D₃
hypocenter	C₃	C₉×#443
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₉ ◁ C₂×C₄×C₉ ◁ C₉×#104

Complements

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

N	H ≅ G/N	type
2. C₉×#948	C₂	semidirect
3. C₉×#907	C₂	semidirect
4. C₃×C₉×#104	C₂	semidirect
5. C₉×#822	C₂	semidirect
6. C₉×#805	C₂	semidirect
7. C₄×C₉×C₃⋊C₈	C₂	semidirect
14. C₉×#216	C₄	semidirect
15. C₉×#215	C₄	semidirect
16. C₉×#214	C₄	semidirect
17. C₉×#213	C₄	semidirect
38. C₉×#104	D₃	semidirect
57. #4824	C₉	direct (N × H)
103. #948	C₂×C₉	semidirect
104. #907	C₂×C₉	semidirect
108. C₃×#104	C₂×C₉	semidirect
111. #822	C₂×C₉	semidirect
112. #805	C₂×C₉	semidirect
113. C₄×C₃⋊C₈	C₂×C₉	semidirect
180. #216	C₄×C₉	semidirect
181. #215	C₄×C₉	semidirect
182. #214	C₄×C₉	semidirect
183. #213	C₄×C₉	semidirect
218. #104	D₃×C₉	semidirect
220. C₃×C₉	#443	semidirect
310. C₉	#4824	direct (N × H)
347. C₃	C₉×#443	semidirect

Maximal subgroups

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

subgroup	index
C₉×#950	2
C₉×#948	2
C₉×#907	2
C₃×C₉×#104	2
C₉×#822	2
C₉×#805	2
C₄×C₉×C₃⋊C₈	2
C₃×#4824	3
C₉×#443	3

Subgroups — C9×#4824 #13141976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#4824 #13141976