Subgroups — D₉×#835 #13140709

Order 1728.

528 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₈×D₉	C₂	1	1	yes	maximal, characteristic
2	D₉×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×D₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11720582	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#835	C₂	1	1	yes	maximal, characteristic, complemented
6	#11720276	C₂	1	1	yes	maximal, characteristic, complemented
7	#11720268	C₂	1	1	yes	maximal, characteristic, complemented
8	D₃×#835	—	3	1	no	maximal
9	D₉×#103	—	3	1	no	maximal
10	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₃×Q₈×D₉	—	2	2	no	—
12	C₃×Q₈×D₉	C₄	1	4	yes	complemented
13	C₂×C₃×C₄×D₉	—	2	1	no	—
14	C₂×C₃×C₄×D₉	C₂²	1	1	yes	characteristic
15	C₂×C₃×Dic₁₈	—	2	1	no	—
16	C₂×C₃×Dic₁₈	C₂²	1	1	yes	characteristic
17	C₂×Dic₃×D₉	—	2	1	no	—
18	C₄⋊(C₃⋊Dic₉)	C₂²	1	1	yes	characteristic
19	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
20	C₉×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
21	C₂×C₉×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	Dic₉⋊Dic₃	—	2	1	no	—
23	D₉×C₃⋊C₈	—	2	2	no	—
24	C₂×C₃×D₃×Q₈	—	3	1	no	—
25	D₃×C₄⋊Dic₃	—	3	1	no	—
26	C₂×D₃×C₃⋊C₈	—	3	1	no	—
27	C₂×Q₈×D₉	D₃	1	1	yes	characteristic
28	#63662	—	3	1	no	—
29	C₃×#835	D₃	1	1	yes	characteristic
30	#63566	—	3	1	no	—
31	#63565	—	3	1	no	—
32	C₂×C₈×D₉	—	3	1	no	—
33	C₄⋊C₄×D₉	—	3	1	no	—
34	C₉×#103	—	3	1	no	—
35	#63395	—	3	1	no	—
36	#63395	—	6	1	no	—
37	#63376	—	3	1	no	—
38	#63368	—	3	1	no	—
39	C₂²×C₃×D₉	D₄	1	1	yes	characteristic
40	C₃×Q₈×C₉	—	2	1	no	—
41	C₃×Q₈×C₉	C₂×C₄	1	2	yes	complemented
42	C₂×C₃×C₄×C₉	—	2	1	no	—
43	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
44	C₂×C₃⋊Dic₉	—	2	1	no	—
45	C₂×C₉×Dic₃	—	2	1	no	—
46	C₂×C₃×Dic₉	—	2	1	no	—
47	C₂×C₃×Dic₉	D₄	1	1	yes	characteristic
48	C₃×C₄×D₉	—	2	4	no	—
49	C₃×C₄×D₉	C₂×C₄	1	2	yes	—
50	C₃×C₄×D₉	D₄	1	2	yes	—
51	C₃×Dic₁₈	—	2	1	no	—
52	C₃×Dic₁₈	—	2	4	no	—
53	C₃×Dic₁₈	C₂×C₄	1	2	yes	complemented
54	Dic₃×D₉	—	4	2	no	—
55	C₃⋊(C₉⋊C₈)	—	2	1	no	—
56	C₉×C₃⋊C₈	—	2	1	no	—
57	C₂×#835	—	9	1	no	—
58	D₃×#103	—	9	1	no	—
59	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
60	C₃×D₃×Q₈	—	3	4	no	—
61	C₃×D₃×Q₈	—	6	2	no	—
62	C₂×C₃×C₄×D₃	—	3	1	no	—
63	C₂×C₃×C₄×D₃	—	6	1	no	—
64	C₂×C₃×Dic₆	—	3	1	no	—
65	C₂×C₃×Dic₆	—	6	1	no	—
66	C₂×D₃×Dic₃	—	6	1	no	—
67	C₄⋊(C₃⋊Dic₃)	—	3	1	no	—
68	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
69	C₃×C₄⋊Dic₃	C₂×D₃	1	1	yes	characteristic
70	C₂×C₃×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
71	Dic₃⋊Dic₃	—	6	1	no	—
72	D₃×C₃⋊C₈	—	6	2	no	—
73	C₂×Q₈×C₉	—	2	1	no	—
74	C₂×Q₈×C₉	C₂×D₃	1	1	yes	characteristic
75	Q₈×D₉	—	2	2	no	—
76	Q₈×D₉	Dic₃	1	4	yes	complemented
77	C₂×C₄×D₉	—	2	1	no	—
78	C₂×C₄×D₉	—	6	1	no	—
79	C₂×C₄×D₉	C₂×D₃	1	1	yes	characteristic
80	C₂×Dic₁₈	—	2	1	no	—
81	C₂×Dic₁₈	C₂×D₃	1	1	yes	characteristic
82	C₂×C₈×C₉	—	3	1	no	—
83	C₉×C₄⋊C₄	—	3	1	no	—
84	C₄⋊Dic₉	—	3	1	no	—
85	C₄⋊Dic₉	—	6	1	no	—
86	Dic₉⋊C₄	—	6	1	no	—
87	C₂×C₉⋊C₈	—	3	1	no	—
88	C₂×C₉⋊C₈	—	6	1	no	—
89	C₈×D₉	—	6	2	no	—
90	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×D₉	C₂²⋊C₄	1	2	yes	—
92	C₂×C₃×D₉	SD₁₆	1	2	yes	—
93	C₂×C₃×D₉	Q₁₆	1	2	yes	—
94	C₃×C₄×C₉	—	2	2	no	—
95	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
96	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
97	C₃⋊Dic₉	—	4	1	no	—
98	C₉×Dic₃	—	4	1	no	—
99	C₃×Dic₉	—	2	2	no	—
100	C₃×Dic₉	C₂²⋊C₄	1	2	yes	—
101	C₂²×C₃×Q₈	—	9	1	no	—
102	C₂×D₃×Q₈	—	3	1	no	—
103	C₂×C₄⋊Dic₃	—	9	1	no	—
104	C₂²×C₃⋊C₈	—	9	1	no	—
105	C₂×D₃×C₈	—	9	1	no	—
106	D₃×C₄⋊C₄	—	9	1	no	—
107	C₃×#103	—	3	1	no	—
108	#835	—	9	1	no	—
109	#835	—	9	1	no	—
110	#835	—	9	1	no	—
111	#835	—	9	1	no	—
112	#835	—	18	1	no	—
113	#835	D₉	1	1	yes	characteristic, direct product factor
114	#816	—	9	1	no	—
115	#810	—	9	1	no	—
116	C₂²×C₃×D₃	—	3	1	no	—
117	C₃²×Q₈	—	2	1	no	—
118	C₃²×Q₈	C₄×D₃	1	2	yes	—
119	C₂×C₃²×C₄	—	2	1	no	—
120	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
121	C₂×C₃⋊Dic₃	—	6	1	no	—
122	C₂×C₃×Dic₃	—	2	1	no	—
123	C₂×C₃×Dic₃	—	3	1	no	—
124	C₂×C₃×Dic₃	—	6	1	no	—
125	C₃×C₄×D₃	—	3	2	no	—
126	C₃×C₄×D₃	—	3	2	no	—
127	C₃×C₄×D₃	—	6	4	no	—
128	C₃×Dic₆	—	3	2	no	—
129	C₃×Dic₆	—	6	4	no	—
130	C₃×Dic₆	—	6	1	no	—
131	D₃×Dic₃	—	12	2	no	—
132	C₂²×D₉	#67	1	1	yes	characteristic
133	C₃⋊(C₃⋊C₈)	—	6	1	no	—
134	C₃×C₃⋊C₈	—	2	1	no	—
135	Q₈×C₉	—	2	1	no	—
136	Q₈×C₉	—	2	2	no	—
137	Q₈×C₉	—	4	1	no	—
138	Q₈×C₉	C₂×Dic₃	1	2	yes	complemented
139	C₂×C₄×C₉	—	2	1	no	—
140	C₂×C₄×C₉	—	2	1	no	—
141	C₂×C₄×C₉	—	4	1	no	—
142	C₂×C₄×C₉	—	6	1	no	—
143	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
144	C₂×Dic₉	—	2	1	no	—
145	C₂×Dic₉	—	6	1	no	—
146	C₂×Dic₉	—	12	1	no	—
147	C₂×Dic₉	#67	1	1	yes	characteristic
148	C₄×D₉	—	2	4	no	—
149	C₄×D₉	—	12	2	no	—
150	C₄×D₉	C₂×Dic₃	1	2	yes	—
151	C₄×D₉	#67	1	2	yes	—
152	Dic₁₈	—	2	1	no	—
153	Dic₁₈	—	2	4	no	—
154	Dic₁₈	C₂×Dic₃	1	2	yes	complemented
155	C₈×C₉	—	6	1	no	—
156	C₉⋊C₈	—	6	1	no	—
157	C₉⋊C₈	—	12	1	no	—
158	C₂×#103	—	27	1	no	Sylow
159	C₂×C₃×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
161	C₂×C₃×C₉	C₂×Q₁₆	1	1	yes	characteristic
162	C₃×D₉	#103	1	4	yes	complemented
163	C₂×C₃×Q₈	—	2	1	no	—
164	C₂×C₃×Q₈	—	9	4	no	—
165	C₂×C₃×Q₈	—	9	1	no	—
166	C₂×C₃×Q₈	—	18	1	no	—
167	C₂×C₃×Q₈	—	18	2	no	—
168	C₂×C₃×Q₈	C₂×D₉	1	1	yes	characteristic
169	C₂×C₃×Q₈	D₃²	1	1	yes	characteristic
170	C₂²×C₃×C₄	—	9	1	no	—
171	C₂²×C₃×C₄	—	18	1	no	—
172	C₂²×Dic₃	—	18	1	no	—
173	D₃×Q₈	—	3	4	no	—
174	D₃×Q₈	—	6	2	no	—
175	C₂×C₄×D₃	—	3	1	no	—
176	C₂×C₄×D₃	—	6	1	no	—
177	C₂×C₄×D₃	—	18	1	no	—
178	C₂×Dic₆	—	3	1	no	—
179	C₂×Dic₆	—	6	1	no	—
180	C₂×C₃×C₈	—	3	1	no	—
181	C₃×C₄⋊C₄	—	3	1	no	—
182	C₄⋊Dic₃	—	9	1	no	—
183	C₄⋊Dic₃	—	9	1	no	—
184	C₄⋊Dic₃	—	18	1	no	—
185	C₄⋊Dic₃	—	18	1	no	—
186	C₄⋊Dic₃	C₂×D₉	1	1	yes	characteristic
187	Dic₃⋊C₄	—	18	1	no	—
188	C₂×C₃⋊C₈	—	9	1	no	—
189	C₂×C₃⋊C₈	—	9	1	no	—
190	C₂×C₃⋊C₈	—	18	1	no	—
191	C₂×C₃⋊C₈	—	18	2	no	—
192	C₂×C₃⋊C₈	C₂×D₉	1	1	yes	characteristic
193	D₃×C₈	—	18	2	no	—
194	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
195	C₂×C₃×D₃	—	3	2	no	—
196	C₂×C₃×D₃	—	3	2	no	—
197	C₂×C₃×D₃	—	3	2	no	—
198	C₃²×C₄	—	2	2	no	—
199	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
200	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
201	C₃⋊Dic₃	—	12	1	no	—
202	C₃×Dic₃	—	3	2	no	—
203	C₃×Dic₃	—	4	1	no	—
204	C₃×Dic₃	—	6	2	no	—
205	C₂²×C₉	—	2	1	no	—
206	C₂²×C₉	C₂×#67	1	1	yes	characteristic
207	C₂×D₉	#215	1	2	yes	—
208	C₂×D₉	#216	1	2	yes	—
209	C₂×D₉	#217	1	2	yes	—
210	C₄×C₉	—	2	2	no	—
211	C₄×C₉	—	2	1	no	—
212	C₄×C₉	—	2	1	no	—
213	C₄×C₉	—	4	2	no	—
214	C₄×C₉	—	12	1	no	—
215	C₄×C₉	C₂²×Dic₃	1	1	yes	characteristic
216	C₄×C₉	C₂×#67	1	1	yes	characteristic
217	Dic₉	—	2	2	no	—
218	Dic₉	—	12	1	no	—
219	Dic₉	—	24	1	no	—
220	Dic₉	#217	1	2	yes	—
221	C₂²×Q₈	—	9	1	no	—
222	C₂²×C₈	—	27	1	no	—
223	C₂×C₄⋊C₄	—	27	1	no	—
224	#103	—	3	1	no	—
225	#103	—	27	1	no	—
226	#103	—	27	1	no	—
227	#103	—	27	1	no	—
228	C₃×C₉	C₂×#103	1	1	yes	characteristic, complemented, Sylow
229	C₂³×C₃	—	9	1	no	—
230	C₂²×D₃	—	3	1	no	—
231	C₃×Q₈	—	2	1	no	—
232	C₃×Q₈	—	2	1	no	—
233	C₃×Q₈	—	2	2	no	—
234	C₃×Q₈	—	4	1	no	—
235	C₃×Q₈	—	9	2	no	—
236	C₃×Q₈	—	18	4	no	—
237	C₃×Q₈	—	18	1	no	—
238	C₃×Q₈	C₄×D₉	1	2	yes	complemented
239	C₃×Q₈	D₃×Dic₃	1	2	yes	—
240	C₂×C₃×C₄	—	2	1	no	—
241	C₂×C₃×C₄	—	2	1	no	—
242	C₂×C₃×C₄	—	2	1	no	—
243	C₂×C₃×C₄	—	4	1	no	—
244	C₂×C₃×C₄	—	6	1	no	—
245	C₂×C₃×C₄	—	9	1	no	—
246	C₂×C₃×C₄	—	9	2	no	—
247	C₂×C₃×C₄	—	9	2	no	—
248	C₂×C₃×C₄	—	18	4	no	—
249	C₂×C₃×C₄	—	18	1	no	—
250	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
251	C₂×C₃×C₄	C₂×D₃²	1	1	yes	characteristic
252	C₂×Dic₃	—	2	1	no	—
253	C₂×Dic₃	—	3	1	no	—
254	C₂×Dic₃	—	6	1	no	—
255	C₂×Dic₃	—	18	1	no	—
256	C₂×Dic₃	—	18	1	no	—
257	C₂×Dic₃	—	36	1	no	—
258	C₂×Dic₃	—	36	2	no	—
259	C₄×D₃	—	3	2	no	—
260	C₄×D₃	—	3	2	no	—
261	C₄×D₃	—	6	4	no	—
262	C₄×D₃	—	36	2	no	—
263	Dic₆	—	3	2	no	—
264	Dic₆	—	6	4	no	—
265	Dic₆	—	6	1	no	—
266	C₃×C₈	—	6	1	no	—
267	C₃⋊C₈	—	2	1	no	—
268	C₃⋊C₈	—	18	1	no	—
269	C₃⋊C₈	—	18	1	no	—
270	C₃⋊C₈	—	36	1	no	—
271	C₂×C₃²	D₃×C₂²⋊C₄	1	1	yes	characteristic
272	C₂×C₃²	D₃×SD₁₆	1	1	yes	characteristic
273	C₂×C₃²	D₃×Q₁₆	1	1	yes	characteristic
274	C₃×D₃	—	3	4	no	—
275	C₂×C₉	—	2	1	no	—
276	C₂×C₉	—	2	1	no	—
277	C₂×C₉	—	2	1	no	—
278	C₂×C₉	C₂×#215	1	1	yes	characteristic
279	C₂×C₉	C₂×#216	1	1	yes	characteristic
280	C₂×C₉	C₂×#217	1	1	yes	characteristic
281	D₉	#835	1	4	yes	direct product factor
282	C₂×Q₈	—	9	4	no	—
283	C₂×Q₈	—	9	1	no	—
284	C₂×Q₈	—	18	1	no	—
285	C₂×Q₈	—	18	2	no	—
286	C₂×Q₈	D₃×D₉	1	1	yes	characteristic
287	C₂²×C₄	—	9	1	no	—
288	C₂²×C₄	—	18	1	no	—
289	C₂²×C₄	—	54	1	no	—
290	C₂×C₈	—	3	1	no	—
291	C₂×C₈	—	27	1	no	—
292	C₂×C₈	—	54	2	no	—
293	C₄⋊C₄	—	3	1	no	—
294	C₄⋊C₄	—	27	1	no	—
295	C₄⋊C₄	—	54	1	no	—
296	C₂²×C₃	—	2	1	no	—
297	C₂²×C₃	—	9	2	no	—
298	C₂²×C₃	—	9	2	no	—
299	C₂²×C₃	—	9	2	no	—
300	C₂²×C₃	D₄×D₉	1	1	yes	characteristic
301	C₂²×C₃	D₃×#67	1	1	yes	characteristic
302	C₂×D₃	—	3	2	no	—
303	C₂×D₃	—	3	2	no	—
304	C₂×D₃	—	3	2	no	—
305	C₃×C₄	—	2	2	no	—
306	C₃×C₄	—	2	2	no	—
307	C₃×C₄	—	2	1	no	—
308	C₃×C₄	—	2	1	no	—
309	C₃×C₄	—	4	2	no	—
310	C₃×C₄	—	9	2	no	—
311	C₃×C₄	—	12	1	no	—
312	C₃×C₄	—	18	2	no	—
313	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
314	C₃×C₄	D₄×D₉	1	1	yes	characteristic
315	C₃×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
316	C₃×C₄	D₃×#67	1	1	yes	characteristic
317	Dic₃	—	3	2	no	—
318	Dic₃	—	4	1	no	—
319	Dic₃	—	6	2	no	—
320	Dic₃	—	36	1	no	—
321	Dic₃	—	36	1	no	—
322	Dic₃	—	72	1	no	—
323	C₃²	D₃×#103	1	1	yes	characteristic
324	C₉	—	2	1	no	—
325	C₉	C₂×#835	1	1	yes	characteristic, complemented
326	C₂³	—	9	1	no	—
327	Q₈	—	2	1	no	—
328	Q₈	—	9	2	no	—
329	Q₈	—	18	4	no	—
330	Q₈	—	18	1	no	—
331	Q₈	Dic₃×D₉	1	2	yes	complemented
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	6	1	no	—
334	C₂×C₄	—	9	1	no	—
335	C₂×C₄	—	9	2	no	—
336	C₂×C₄	—	9	2	no	—
337	C₂×C₄	—	18	4	no	—
338	C₂×C₄	—	18	1	no	—
339	C₂×C₄	—	54	1	no	—
340	C₂×C₄	—	108	2	no	—
341	C₂×C₄	C₂×D₃×D₉	1	1	yes	characteristic
342	C₈	—	6	1	no	—
343	C₈	—	54	1	no	—
344	C₂×C₃	—	2	1	no	—
345	C₂×C₃	—	2	1	no	—
346	C₂×C₃	—	2	1	no	—
347	C₂×C₃	—	9	4	no	—
348	C₂×C₃	C₂²⋊C₄×D₉	1	1	yes	characteristic
349	C₂×C₃	SD₁₆×D₉	1	1	yes	characteristic
350	C₂×C₃	Q₁₆×D₉	1	1	yes	characteristic
351	C₂×C₃	D₃×#215	1	1	yes	characteristic
352	C₂×C₃	D₃×#216	1	1	yes	characteristic
353	C₂×C₃	D₃×#217	1	1	yes	characteristic
354	D₃	—	3	4	no	—
355	C₂²	—	9	2	no	—
356	C₂²	—	9	2	no	—
357	C₂²	—	9	2	no	—
358	C₂²	D₉×#67	1	1	yes	characteristic
359	C₄	—	2	2	no	—
360	C₄	—	9	2	no	—
361	C₄	—	12	1	no	—
362	C₄	—	18	2	no	—
363	C₄	—	108	1	no	—
364	C₄	C₂×Dic₃×D₉	1	1	yes	characteristic
365	C₄	D₉×#67	1	1	yes	characteristic
366	C₃	—	2	1	no	—
367	C₃	D₉×#103	1	1	yes	characteristic, complemented
368	C₃	D₃×#835	1	1	yes	characteristic
369	C₂	—	9	4	no	—
370	C₂	D₉×#215	1	1	yes	characteristic
371	C₂	D₉×#216	1	1	yes	characteristic
372	C₂	D₉×#217	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₉×#67
commutator	C₃×C₄×C₉	C₂²×C₄
Frattini	C₂×C₃×C₄	C₂×D₃²
Fitting	C₂×C₃×Q₈×C₉	C₂²
socle	C₂²×C₃²	D₃×D₄
prime radical	C₂²×C₃×D₉	D₄
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	C₂×Q₈	D₃×D₉
hypocenter	C₃×C₉	C₂×#103
LCS	G ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
4. #11720582	C₂	semidirect
5. C₉×#835	C₂	semidirect
6. #11720276	C₂	semidirect
7. #11720268	C₂	semidirect
12. C₃×Q₈×D₉	C₄	semidirect
41. C₃×Q₈×C₉	C₂×C₄	semidirect
53. C₃×Dic₁₈	C₂×C₄	semidirect
76. Q₈×D₉	Dic₃	semidirect
113. #835	D₉	direct (N × H)
138. Q₈×C₉	C₂×Dic₃	semidirect
154. Dic₁₈	C₂×Dic₃	semidirect
162. C₃×D₉	#103	semidirect
228. C₃×C₉	C₂×#103	semidirect
238. C₃×Q₈	C₄×D₉	semidirect
281. D₉	#835	direct (N × H)
325. C₉	C₂×#835	semidirect
331. Q₈	Dic₃×D₉	semidirect
367. C₃	D₉×#103	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×Q₈×D₉	2
D₉×C₄⋊Dic₃	2
C₂×D₉×C₃⋊C₈	2
#11720582	2
C₉×#835	2
#11720276	2
#11720268	2
D₃×#835	3
D₉×#103	3

Subgroups — D9×#835 #13140709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₉×#835 #13140709