Subgroups — C₉×#3204 #11767925

Order 1152.

603 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×#537	C₂	1	2	yes	maximal, complemented
2	C₉×#483	C₂	1	2	yes	maximal, complemented
3	C₉×#481	C₂	1	2	yes	maximal, complemented
4	C₉×#470	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#469	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#465	C₂	1	2	yes	maximal, complemented
7	C₉×#452	C₂	1	2	yes	maximal, complemented
8	C₉×#449	C₂	1	2	yes	maximal, complemented
9	C₉×#408	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#3204	C₃	1	1	yes	maximal, characteristic
11	C₂×C₉×#41	C₂²	1	2	yes	—
12	C₂×C₉×#41	C₂²	1	1	yes	characteristic
13	C₂×C₉×Q₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₉×SD₁₆	C₂²	1	2	yes	—
15	C₂×C₉×D₈	C₂²	1	1	yes	characteristic
16	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
18	C₉×C₂²⋊Q₈	—	2	2	no	—
19	C₉×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₉×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₉×#121	—	2	2	no	—
22	C₉×#121	—	2	2	no	—
23	C₉×#121	—	2	2	no	—
24	C₉×#121	C₂²	1	2	yes	—
25	C₉×#121	C₂²	1	2	yes	—
26	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
27	C₉×#106	C₂²	1	2	yes	—
28	C₉×#106	C₂²	1	1	yes	characteristic
29	C₉×#106	C₂²	1	1	yes	characteristic
30	C₉×#103	C₂²	1	2	yes	—
31	C₉×#103	C₂²	1	2	yes	—
32	C₉×#102	C₂²	1	2	yes	—
33	C₉×#102	C₂²	1	2	yes	—
34	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
35	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
36	C₃×#537	C₂×C₃	1	2	yes	—
37	C₃×#483	C₂×C₃	1	2	yes	—
38	C₃×#481	C₂×C₃	1	2	yes	—
39	C₃×#470	C₂×C₃	1	1	yes	characteristic
40	C₃×#469	C₂×C₃	1	1	yes	characteristic
41	C₃×#465	C₂×C₃	1	2	yes	—
42	C₃×#452	C₂×C₃	1	2	yes	—
43	C₃×#449	C₂×C₃	1	2	yes	—
44	C₃×#408	C₂×C₃	1	1	yes	characteristic
45	C₉×#41	—	4	1	no	—
46	C₉×#41	—	4	1	no	—
47	C₉×#41	—	4	2	no	—
48	C₉×#41	—	4	2	no	—
49	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
50	C₂×Q₈×C₉	C₂³	1	2	yes	—
51	C₂×D₄×C₉	—	2	2	no	—
52	C₂×D₄×C₉	—	2	2	no	—
53	C₂×D₄×C₉	—	2	2	no	—
54	C₂×D₄×C₉	D₄	1	1	yes	characteristic
55	C₂×D₄×C₉	D₄	1	2	yes	—
56	C₂×D₄×C₉	C₂³	1	2	yes	—
57	C₂²×C₄×C₉	—	2	2	no	—
58	C₂²×C₄×C₉	—	2	2	no	—
59	C₂²×C₄×C₉	C₂³	1	2	yes	—
60	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
61	C₉×Q₁₆	—	4	1	no	—
62	C₉×SD₁₆	—	4	2	no	—
63	C₉×D₈	—	4	1	no	—
64	C₉×M₁₆	—	4	1	no	—
65	C₂×C₈×C₉	—	4	1	no	—
66	C₂×C₈×C₉	C₂³	1	2	yes	—
67	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
69	C₉×C₄⋊C₄	—	2	2	no	—
70	C₉×C₄⋊C₄	—	2	2	no	—
71	C₉×C₄⋊C₄	—	2	2	no	—
72	C₉×C₄⋊C₄	C₂³	1	2	yes	—
73	C₉×C₄⋊C₄	C₂³	1	2	yes	—
74	C₉×C₂²⋊C₄	—	2	2	no	—
75	C₉×C₂²⋊C₄	—	2	2	no	—
76	C₉×C₂²⋊C₄	—	2	2	no	—
77	C₉×C₂²⋊C₄	—	2	2	no	—
78	#3204	C₉	1	1	yes	characteristic, direct product factor, Sylow
79	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
80	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
81	C₂×C₃×Q₁₆	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₃×SD₁₆	C₂²×C₃	1	2	yes	—
83	C₂×C₃×D₈	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
85	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
86	C₃×C₂²⋊Q₈	—	2	2	no	—
87	C₃×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
88	C₃×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
89	C₃×#121	—	2	2	no	—
90	C₃×#121	—	2	2	no	—
91	C₃×#121	—	2	2	no	—
92	C₃×#121	C₂²×C₃	1	2	yes	—
93	C₃×#121	C₂²×C₃	1	2	yes	—
94	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
95	C₃×#106	C₂²×C₃	1	2	yes	—
96	C₃×#106	C₂²×C₃	1	1	yes	characteristic
97	C₃×#106	C₂²×C₃	1	1	yes	characteristic
98	C₃×#103	C₂²×C₃	1	2	yes	—
99	C₃×#103	C₂²×C₃	1	2	yes	—
100	C₃×#102	C₂²×C₃	1	2	yes	—
101	C₃×#102	C₂²×C₃	1	2	yes	—
102	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
103	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
104	C₂³×C₉	—	2	2	no	—
105	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
106	C₂³×C₉	C₂×D₄	1	2	yes	—
107	Q₈×C₉	—	2	2	no	—
108	Q₈×C₉	—	2	2	no	—
109	Q₈×C₉	—	4	1	no	—
110	D₄×C₉	—	2	2	no	—
111	D₄×C₉	—	2	2	no	—
112	D₄×C₉	—	4	2	no	—
113	D₄×C₉	—	4	1	no	—
114	D₄×C₉	—	4	4	no	—
115	D₄×C₉	—	8	2	no	—
116	D₄×C₉	—	8	2	no	—
117	C₂×C₄×C₉	—	2	1	no	—
118	C₂×C₄×C₉	—	2	1	no	—
119	C₂×C₄×C₉	—	2	2	no	—
120	C₂×C₄×C₉	—	2	2	no	—
121	C₂×C₄×C₉	—	2	2	no	—
122	C₂×C₄×C₉	—	2	2	no	—
123	C₂×C₄×C₉	—	2	2	no	—
124	C₂×C₄×C₉	—	4	4	no	—
125	C₂×C₄×C₉	—	8	2	no	—
126	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
128	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
129	C₈×C₉	—	2	2	no	—
130	C₈×C₉	—	2	1	no	—
131	C₈×C₉	—	2	1	no	—
132	#537	C₂×C₉	1	2	yes	complemented
133	#483	C₂×C₉	1	2	yes	complemented
134	#481	C₂×C₉	1	2	yes	complemented
135	#470	C₂×C₉	1	1	yes	characteristic, complemented
136	#469	C₂×C₉	1	1	yes	characteristic, complemented
137	#465	C₂×C₉	1	2	yes	complemented
138	#452	C₂×C₉	1	2	yes	complemented
139	#449	C₂×C₉	1	2	yes	complemented
140	#408	C₂×C₉	1	1	yes	characteristic, complemented
141	C₃×#41	—	4	1	no	—
142	C₃×#41	—	4	1	no	—
143	C₃×#41	—	4	2	no	—
144	C₃×#41	—	4	2	no	—
145	C₂×C₃×Q₈	C₃×D₄	1	1	yes	characteristic
146	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
147	C₂×C₃×D₄	—	2	2	no	—
148	C₂×C₃×D₄	—	2	2	no	—
149	C₂×C₃×D₄	—	2	2	no	—
150	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
151	C₂×C₃×D₄	C₃×D₄	1	2	yes	—
152	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
153	C₂²×C₃×C₄	—	2	2	no	—
154	C₂²×C₃×C₄	—	2	2	no	—
155	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
156	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
157	C₃×Q₁₆	—	4	1	no	—
158	C₃×SD₁₆	—	4	2	no	—
159	C₃×D₈	—	4	1	no	—
160	C₃×M₁₆	—	4	1	no	—
161	C₂×C₃×C₈	—	4	1	no	—
162	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
163	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
164	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
165	C₃×C₄⋊C₄	—	2	2	no	—
166	C₃×C₄⋊C₄	—	2	2	no	—
167	C₃×C₄⋊C₄	—	2	2	no	—
168	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
169	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
170	C₃×C₂²⋊C₄	—	2	2	no	—
171	C₃×C₂²⋊C₄	—	2	2	no	—
172	C₃×C₂²⋊C₄	—	2	2	no	—
173	C₃×C₂²⋊C₄	—	2	2	no	—
174	C₂²×C₉	—	2	2	no	—
175	C₂²×C₉	—	2	1	no	—
176	C₂²×C₉	—	2	2	no	—
177	C₂²×C₉	—	2	2	no	—
178	C₂²×C₉	—	2	2	no	—
179	C₂²×C₉	—	4	2	no	—
180	C₂²×C₉	—	4	2	no	—
181	C₂²×C₉	—	4	2	no	—
182	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
183	C₄×C₉	—	2	2	no	—
184	C₄×C₉	—	2	1	no	—
185	C₄×C₉	—	4	2	no	—
186	C₄×C₉	—	4	2	no	—
187	C₄×C₉	—	4	2	no	—
188	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
189	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
190	C₂×#41	C₂²×C₉	1	2	yes	—
191	C₂×#41	C₂²×C₉	1	1	yes	characteristic
192	C₂×Q₁₆	C₂²×C₉	1	1	yes	characteristic
193	C₂×SD₁₆	C₂²×C₉	1	2	yes	—
194	C₂×D₈	C₂²×C₉	1	1	yes	characteristic
195	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
196	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
197	C₂²⋊Q₈	—	2	2	no	—
198	C₂²⋊Q₈	C₂²×C₉	1	2	yes	—
199	C₂²⋊Q₈	C₂²×C₉	1	2	yes	—
200	#121	—	2	2	no	—
201	#121	—	2	2	no	—
202	#121	—	2	2	no	—
203	#121	C₂²×C₉	1	2	yes	—
204	#121	C₂²×C₉	1	2	yes	—
205	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
206	#106	C₂²×C₉	1	2	yes	—
207	#106	C₂²×C₉	1	1	yes	characteristic
208	#106	C₂²×C₉	1	1	yes	characteristic
209	#103	C₂²×C₉	1	2	yes	—
210	#103	C₂²×C₉	1	2	yes	—
211	#102	C₂²×C₉	1	2	yes	—
212	#102	C₂²×C₉	1	2	yes	—
213	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
214	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
215	C₂³×C₃	—	2	2	no	—
216	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
217	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
218	C₃×Q₈	—	2	2	no	—
219	C₃×Q₈	—	2	2	no	—
220	C₃×Q₈	—	4	1	no	—
221	C₃×D₄	—	2	2	no	—
222	C₃×D₄	—	2	2	no	—
223	C₃×D₄	—	4	2	no	—
224	C₃×D₄	—	4	1	no	—
225	C₃×D₄	—	4	4	no	—
226	C₃×D₄	—	8	2	no	—
227	C₃×D₄	—	8	2	no	—
228	C₂×C₃×C₄	—	2	1	no	—
229	C₂×C₃×C₄	—	2	1	no	—
230	C₂×C₃×C₄	—	2	2	no	—
231	C₂×C₃×C₄	—	2	2	no	—
232	C₂×C₃×C₄	—	2	2	no	—
233	C₂×C₃×C₄	—	2	2	no	—
234	C₂×C₃×C₄	—	2	2	no	—
235	C₂×C₃×C₄	—	4	4	no	—
236	C₂×C₃×C₄	—	8	2	no	—
237	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
238	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
239	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
240	C₃×C₈	—	2	2	no	—
241	C₃×C₈	—	2	1	no	—
242	C₃×C₈	—	2	1	no	—
243	C₂×C₉	—	4	2	no	—
244	C₂×C₉	—	4	1	no	—
245	C₂×C₉	—	8	2	no	—
246	C₂×C₉	#534	1	1	yes	characteristic
247	C₂×C₉	#577	1	2	yes	—
248	#41	—	4	1	no	—
249	#41	—	4	1	no	—
250	#41	—	4	2	no	—
251	#41	—	4	2	no	—
252	C₂×Q₈	D₄×C₉	1	1	yes	characteristic
253	C₂×Q₈	C₂³×C₉	1	2	yes	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	D₄×C₉	1	1	yes	characteristic
258	C₂×D₄	D₄×C₉	1	2	yes	—
259	C₂×D₄	C₂³×C₉	1	2	yes	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	C₂³×C₉	1	2	yes	—
263	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
264	Q₁₆	—	4	1	no	—
265	SD₁₆	—	4	2	no	—
266	D₈	—	4	1	no	—
267	M₁₆	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
270	C₂×C₈	C₂³×C₉	1	2	yes	—
271	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	C₂³×C₉	1	2	yes	—
276	C₄⋊C₄	C₂³×C₉	1	2	yes	—
277	C₂²⋊C₄	—	2	2	no	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²×C₃	—	2	2	no	—
282	C₂²×C₃	—	2	2	no	—
283	C₂²×C₃	—	2	2	no	—
284	C₂²×C₃	—	2	2	no	—
285	C₂²×C₃	—	2	1	no	—
286	C₂²×C₃	—	4	2	no	—
287	C₂²×C₃	—	4	2	no	—
288	C₂²×C₃	—	4	2	no	—
289	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
290	C₃×C₄	—	2	2	no	—
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	—	4	2	no	—
293	C₃×C₄	—	4	2	no	—
294	C₃×C₄	—	4	2	no	—
295	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
296	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
297	C₉	#3204	1	1	yes	characteristic, direct product factor, Sylow
298	C₂³	—	2	2	no	—
299	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
300	C₂³	C₂×D₄×C₉	1	2	yes	—
301	Q₈	—	2	2	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	4	1	no	—
304	D₄	—	2	2	no	—
305	D₄	—	2	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	1	no	—
308	D₄	—	4	4	no	—
309	D₄	—	8	2	no	—
310	D₄	—	8	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	8	2	no	—
320	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
321	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
322	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
323	C₈	—	2	2	no	—
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₂×C₃	—	4	2	no	—
327	C₂×C₃	—	4	1	no	—
328	C₂×C₃	—	8	2	no	—
329	C₂×C₃	C₃×#534	1	1	yes	characteristic
330	C₂×C₃	C₃×#577	1	2	yes	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	1	no	—
342	C₄	—	4	2	no	—
343	C₄	—	4	2	no	—
344	C₄	—	4	2	no	—
345	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
346	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
347	C₃	C₃×#3204	1	1	yes	characteristic
348	C₂	—	4	2	no	—
349	C₂	—	4	1	no	—
350	C₂	—	8	2	no	—
351	C₂	C₉×#534	1	1	yes	characteristic
352	C₂	C₉×#577	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₃×D₄
prime radical	C₃×#3204	C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₉×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₉×#537	C₂	semidirect
2. C₉×#483	C₂	semidirect
3. C₉×#481	C₂	semidirect
4. C₉×#470	C₂	semidirect
5. C₉×#469	C₂	semidirect
6. C₉×#465	C₂	semidirect
7. C₉×#452	C₂	semidirect
8. C₉×#449	C₂	semidirect
9. C₉×#408	C₂	semidirect
78. #3204	C₉	direct (N × H)
132. #537	C₂×C₉	semidirect
133. #483	C₂×C₉	semidirect
134. #481	C₂×C₉	semidirect
135. #470	C₂×C₉	semidirect
136. #469	C₂×C₉	semidirect
137. #465	C₂×C₉	semidirect
138. #452	C₂×C₉	semidirect
139. #449	C₂×C₉	semidirect
140. #408	C₂×C₉	semidirect
297. C₉	#3204	direct (N × H)

Maximal subgroups

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

subgroup	index
C₉×#537	2
C₉×#483	2
C₉×#481	2
C₉×#470	2
C₉×#469	2
C₉×#465	2
C₉×#452	2
C₉×#449	2
C₉×#408	2
C₃×#3204	3

Subgroups — C9×#3204 #11767925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3204 #11767925