Subgroups — C₉×#3216 #11767937

Order 1152.

585 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×#532	C₂	1	1	yes	maximal, characteristic
3	C₉×#497	C₂	1	2	yes	maximal, complemented
4	C₉×#489	C₂	1	2	yes	maximal, complemented
5	C₉×#462	C₂	1	2	yes	maximal, complemented
6	C₉×#460	C₂	1	2	yes	maximal, complemented
7	C₉×#452	C₂	1	4	yes	maximal, complemented
8	C₉×#432	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#3216	C₃	1	1	yes	maximal, characteristic
10	C₂×C₉×#41	C₂²	1	2	yes	—
11	C₂×C₉×Q₁₆	C₂²	1	4	yes	—
12	C₂×C₉×SD₁₆	C₂²	1	4	yes	—
13	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₉×#127	C₂²	1	1	yes	characteristic
17	C₉×#125	—	2	1	no	—
18	C₉×#124	C₂²	1	2	yes	—
19	C₉×C₂²⋊Q₈	—	2	1	no	—
20	C₉×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₉×#121	—	2	1	no	—
22	C₉×#121	—	2	1	no	—
23	C₄×Q₈×C₉	—	2	1	no	—
24	C₄×Q₈×C₉	C₂²	1	2	yes	—
25	C₄×D₄×C₉	—	2	1	no	—
26	C₄×D₄×C₉	—	2	1	no	—
27	C₉×#117	—	2	1	no	—
28	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
29	C₉×C₄⋊C₈	C₂²	1	2	yes	—
30	C₉×#103	C₂²	1	4	yes	—
31	C₉×#102	C₂²	1	4	yes	—
32	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₉×#97	C₂²	1	2	yes	—
34	C₃×#533	C₂×C₃	1	1	yes	characteristic
35	C₃×#532	C₂×C₃	1	1	yes	characteristic
36	C₃×#497	C₂×C₃	1	2	yes	—
37	C₃×#489	C₂×C₃	1	2	yes	—
38	C₃×#462	C₂×C₃	1	2	yes	—
39	C₃×#460	C₂×C₃	1	2	yes	—
40	C₃×#452	C₂×C₃	1	4	yes	—
41	C₃×#432	C₂×C₃	1	1	yes	characteristic
42	C₉×#41	—	4	4	no	—
43	C₂×Q₈×C₉	—	2	1	no	—
44	C₂×Q₈×C₉	C₂³	1	2	yes	—
45	C₂×Q₈×C₉	C₂³	1	2	yes	—
46	C₂×D₄×C₉	—	2	2	no	—
47	C₂×D₄×C₉	—	2	1	no	—
48	C₂×D₄×C₉	—	2	1	no	—
49	C₂×D₄×C₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₉	—	2	2	no	—
51	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
54	C₉×Q₁₆	—	4	4	no	—
55	C₉×SD₁₆	—	4	4	no	—
56	C₂×C₈×C₉	C₂³	1	4	yes	—
57	C₉×C₄⋊C₄	—	2	1	no	—
58	C₉×C₄⋊C₄	—	2	1	no	—
59	C₉×C₄⋊C₄	—	2	1	no	—
60	C₉×C₄⋊C₄	—	2	1	no	—
61	C₉×C₄⋊C₄	—	2	1	no	—
62	C₉×C₄⋊C₄	—	2	2	no	—
63	C₉×C₄⋊C₄	—	2	2	no	—
64	C₉×C₄⋊C₄	C₂³	1	2	yes	—
65	C₉×C₂²⋊C₄	—	2	2	no	—
66	C₉×C₂²⋊C₄	—	2	2	no	—
67	C₉×C₂²⋊C₄	—	2	2	no	—
68	C₄²×C₉	—	2	2	no	—
69	C₄²×C₉	D₄	1	1	yes	characteristic
70	C₄²×C₉	D₄	1	1	yes	characteristic
71	C₄²×C₉	C₂³	1	1	yes	characteristic
72	C₄²×C₉	C₂³	1	1	yes	characteristic
73	#3216	C₉	1	1	yes	characteristic, direct product factor, Sylow
74	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
75	C₂×C₃×Q₁₆	C₂²×C₃	1	4	yes	—
76	C₂×C₃×SD₁₆	C₂²×C₃	1	4	yes	—
77	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
78	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
79	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
80	C₃×#127	C₂²×C₃	1	1	yes	characteristic
81	C₃×#125	—	2	1	no	—
82	C₃×#124	C₂²×C₃	1	2	yes	—
83	C₃×C₂²⋊Q₈	—	2	1	no	—
84	C₃×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
85	C₃×#121	—	2	1	no	—
86	C₃×#121	—	2	1	no	—
87	C₃×C₄×Q₈	—	2	1	no	—
88	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
89	C₃×C₄×D₄	—	2	1	no	—
90	C₃×C₄×D₄	—	2	1	no	—
91	C₃×#117	—	2	1	no	—
92	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
93	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
94	C₃×#103	C₂²×C₃	1	4	yes	—
95	C₃×#102	C₂²×C₃	1	4	yes	—
96	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
97	C₃×#97	C₂²×C₃	1	2	yes	—
98	C₂³×C₉	—	2	2	no	—
99	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
100	Q₈×C₉	—	2	2	no	—
101	Q₈×C₉	—	2	2	no	—
102	Q₈×C₉	—	2	4	no	—
103	Q₈×C₉	—	4	2	no	—
104	D₄×C₉	—	2	4	no	—
105	D₄×C₉	—	4	2	no	—
106	D₄×C₉	—	4	4	no	—
107	D₄×C₉	—	4	2	no	—
108	C₂×C₄×C₉	—	2	2	no	—
109	C₂×C₄×C₉	—	2	2	no	—
110	C₂×C₄×C₉	—	2	2	no	—
111	C₂×C₄×C₉	—	2	2	no	—
112	C₂×C₄×C₉	—	2	2	no	—
113	C₂×C₄×C₉	—	4	1	no	—
114	C₂×C₄×C₉	—	4	4	no	—
115	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
121	C₈×C₉	—	2	4	no	—
122	#533	C₂×C₉	1	1	yes	characteristic, complemented
123	#532	C₂×C₉	1	1	yes	characteristic
124	#497	C₂×C₉	1	2	yes	complemented
125	#489	C₂×C₉	1	2	yes	complemented
126	#462	C₂×C₉	1	2	yes	complemented
127	#460	C₂×C₉	1	2	yes	complemented
128	#452	C₂×C₉	1	4	yes	complemented
129	#432	C₂×C₉	1	1	yes	characteristic, complemented
130	C₃×#41	—	4	4	no	—
131	C₂×C₃×Q₈	—	2	1	no	—
132	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
133	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
134	C₂×C₃×D₄	—	2	2	no	—
135	C₂×C₃×D₄	—	2	1	no	—
136	C₂×C₃×D₄	—	2	1	no	—
137	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
138	C₂²×C₃×C₄	—	2	2	no	—
139	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
140	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
141	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
142	C₃×Q₁₆	—	4	4	no	—
143	C₃×SD₁₆	—	4	4	no	—
144	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
145	C₃×C₄⋊C₄	—	2	1	no	—
146	C₃×C₄⋊C₄	—	2	1	no	—
147	C₃×C₄⋊C₄	—	2	1	no	—
148	C₃×C₄⋊C₄	—	2	1	no	—
149	C₃×C₄⋊C₄	—	2	1	no	—
150	C₃×C₄⋊C₄	—	2	2	no	—
151	C₃×C₄⋊C₄	—	2	2	no	—
152	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
153	C₃×C₂²⋊C₄	—	2	2	no	—
154	C₃×C₂²⋊C₄	—	2	2	no	—
155	C₃×C₂²⋊C₄	—	2	2	no	—
156	C₃×C₄²	—	2	2	no	—
157	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
158	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
159	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
160	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
161	C₂²×C₉	—	2	1	no	—
162	C₂²×C₉	—	2	1	no	—
163	C₂²×C₉	—	2	1	no	—
164	C₂²×C₉	—	4	2	no	—
165	C₂²×C₉	—	4	2	no	—
166	C₂²×C₉	—	4	2	no	—
167	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
168	C₄×C₉	—	2	1	no	—
169	C₄×C₉	—	2	1	no	—
170	C₄×C₉	—	2	1	no	—
171	C₄×C₉	—	2	1	no	—
172	C₄×C₉	—	4	2	no	—
173	C₄×C₉	—	4	2	no	—
174	C₄×C₉	—	4	2	no	—
175	C₄×C₉	#137	1	2	yes	—
176	C₄×C₉	H(5,𝔽₂)	1	2	yes	—
177	C₂×#41	C₂²×C₉	1	2	yes	—
178	C₂×Q₁₆	C₂²×C₉	1	4	yes	—
179	C₂×SD₁₆	C₂²×C₉	1	4	yes	—
180	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
181	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
182	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic
183	#127	C₂²×C₉	1	1	yes	characteristic
184	#125	—	2	1	no	—
185	#124	C₂²×C₉	1	2	yes	—
186	C₂²⋊Q₈	—	2	1	no	—
187	C₂²⋊Q₈	C₂²×C₉	1	2	yes	—
188	#121	—	2	1	no	—
189	#121	—	2	1	no	—
190	C₄×Q₈	—	2	1	no	—
191	C₄×Q₈	C₂²×C₉	1	2	yes	—
192	C₄×D₄	—	2	1	no	—
193	C₄×D₄	—	2	1	no	—
194	#117	—	2	1	no	—
195	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
196	C₄⋊C₈	C₂²×C₉	1	2	yes	—
197	#103	C₂²×C₉	1	4	yes	—
198	#102	C₂²×C₉	1	4	yes	—
199	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
200	#97	C₂²×C₉	1	2	yes	—
201	C₂³×C₃	—	2	2	no	—
202	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
203	C₃×Q₈	—	2	2	no	—
204	C₃×Q₈	—	2	2	no	—
205	C₃×Q₈	—	2	4	no	—
206	C₃×Q₈	—	4	2	no	—
207	C₃×D₄	—	2	4	no	—
208	C₃×D₄	—	4	2	no	—
209	C₃×D₄	—	4	4	no	—
210	C₃×D₄	—	4	2	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	2	2	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	4	1	no	—
217	C₂×C₃×C₄	—	4	4	no	—
218	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
224	C₃×C₈	—	2	4	no	—
225	C₂×C₉	—	4	1	no	—
226	C₂×C₉	—	8	2	no	—
227	C₂×C₉	#535	1	1	yes	characteristic
228	C₂×C₉	C₂×#137	1	1	yes	characteristic
229	C₂×C₉	#575	1	1	yes	characteristic
230	#41	—	4	4	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×Q₈	C₂³×C₉	1	2	yes	—
233	C₂×Q₈	C₂³×C₉	1	2	yes	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	C₂³×C₉	1	2	yes	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
240	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
241	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
242	Q₁₆	—	4	4	no	—
243	SD₁₆	—	4	4	no	—
244	C₂×C₈	C₂³×C₉	1	4	yes	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	C₂³×C₉	1	2	yes	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	D₄×C₉	1	1	yes	characteristic
258	C₄²	D₄×C₉	1	1	yes	characteristic
259	C₄²	C₂³×C₉	1	1	yes	characteristic
260	C₄²	C₂³×C₉	1	1	yes	characteristic
261	C₂²×C₃	—	2	1	no	—
262	C₂²×C₃	—	2	1	no	—
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	4	2	no	—
265	C₂²×C₃	—	4	2	no	—
266	C₂²×C₃	—	4	2	no	—
267	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	4	2	no	—
273	C₃×C₄	—	4	2	no	—
274	C₃×C₄	—	4	2	no	—
275	C₃×C₄	C₃×#137	1	2	yes	—
276	C₃×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
277	C₉	#3216	1	1	yes	characteristic, direct product factor, Sylow
278	C₂³	—	2	2	no	—
279	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
280	Q₈	—	2	2	no	—
281	Q₈	—	2	2	no	—
282	Q₈	—	2	4	no	—
283	Q₈	—	4	2	no	—
284	D₄	—	2	4	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	4	no	—
287	D₄	—	4	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
297	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
298	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
299	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
300	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
301	C₈	—	2	4	no	—
302	C₂×C₃	—	4	1	no	—
303	C₂×C₃	—	8	2	no	—
304	C₂×C₃	C₃×#535	1	1	yes	characteristic
305	C₂×C₃	C₂×C₃×#137	1	1	yes	characteristic
306	C₂×C₃	C₃×#575	1	1	yes	characteristic
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	2	no	—
321	C₄	C₉×#137	1	2	yes	—
322	C₄	C₉×H(5,𝔽₂)	1	2	yes	—
323	C₃	C₃×#3216	1	1	yes	characteristic
324	C₂	—	4	1	no	—
325	C₂	—	8	2	no	—
326	C₂	C₉×#535	1	1	yes	characteristic
327	C₂	C₂×C₉×#137	1	1	yes	characteristic
328	C₂	C₉×#575	1	1	yes	characteristic

	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×#532	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₉×#533	C₂	semidirect
3. C₉×#497	C₂	semidirect
4. C₉×#489	C₂	semidirect
5. C₉×#462	C₂	semidirect
6. C₉×#460	C₂	semidirect
7. C₉×#452	C₂	semidirect
8. C₉×#432	C₂	semidirect
73. #3216	C₉	direct (N × H)
122. #533	C₂×C₉	semidirect
124. #497	C₂×C₉	semidirect
125. #489	C₂×C₉	semidirect
126. #462	C₂×C₉	semidirect
127. #460	C₂×C₉	semidirect
128. #452	C₂×C₉	semidirect
129. #432	C₂×C₉	semidirect
277. C₉	#3216	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×#533	2
C₉×#532	2
C₉×#497	2
C₉×#489	2
C₉×#462	2
C₉×#460	2
C₉×#452	2
C₉×#432	2
C₃×#3216	3

Subgroups — C9×#3216 #11767937

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3216 #11767937