Subgroups — C₉×#3277 #11767998

Order 1152.

714 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	C₂×C₉×#137	C₂	1	2	yes	maximal, complemented
2	C₉×#548	C₂	1	2	yes	maximal, complemented
3	C₉×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#468	C₂	1	2	yes	maximal, complemented
5	C₉×#464	C₂	1	2	yes	maximal, complemented
6	C₉×#451	C₂	1	2	yes	maximal, complemented
7	C₉×#449	C₂	1	2	yes	maximal, complemented
8	C₄×C₉×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₉×#436	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#3277	C₃	1	1	yes	maximal, characteristic
11	C₂×C₉×#41	C₂²	1	2	yes	—
12	C₂²×Q₈×C₉	C₂²	1	2	yes	—
13	C₉×#137	—	2	8	no	—
14	C₂×C₉×Q₁₆	C₂²	1	2	yes	—
15	C₂×C₉×Q₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₉×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₉×SD₁₆	C₂²	1	2	yes	—
18	C₂×C₉×SD₁₆	C₂²	1	2	yes	—
19	C₂×C₉×D₈	C₂²	1	1	yes	characteristic
20	C₂×C₉×M₁₆	C₂²	1	2	yes	—
21	C₉×#124	—	2	2	no	—
22	C₉×#124	C₂²	1	2	yes	—
23	C₉×C₂²⋊Q₈	—	2	2	no	—
24	C₉×C₂²⋊Q₈	C₂²	1	2	yes	—
25	C₉×#121	—	2	2	no	—
26	C₉×#121	C₂²	1	2	yes	—
27	C₄×Q₈×C₉	C₂²	1	2	yes	—
28	C₄×D₄×C₉	—	2	2	no	—
29	C₄×D₄×C₉	C₂²	1	1	yes	characteristic
30	C₉×#107	C₂²	1	1	yes	characteristic
31	C₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₉×#103	C₂²	1	2	yes	—
33	C₉×#103	C₂²	1	2	yes	—
34	C₉×#102	C₂²	1	2	yes	—
35	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
36	C₄×C₈×C₉	C₂²	1	1	yes	characteristic
37	C₂×C₃×#137	C₂×C₃	1	2	yes	—
38	C₃×#548	C₂×C₃	1	2	yes	—
39	C₃×#495	C₂×C₃	1	1	yes	characteristic
40	C₃×#468	C₂×C₃	1	2	yes	—
41	C₃×#464	C₂×C₃	1	2	yes	—
42	C₃×#451	C₂×C₃	1	2	yes	—
43	C₃×#449	C₂×C₃	1	2	yes	—
44	C₃×C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic
45	C₃×#436	C₂×C₃	1	1	yes	characteristic
46	C₉×#41	—	2	4	no	—
47	C₉×#41	—	4	2	no	—
48	C₂×Q₈×C₉	—	2	2	no	—
49	C₂×Q₈×C₉	—	2	4	no	—
50	C₂×Q₈×C₉	—	4	2	no	—
51	C₂×Q₈×C₉	C₂³	1	2	yes	—
52	C₂×Q₈×C₉	C₂³	1	2	yes	—
53	C₂×D₄×C₉	—	2	2	no	—
54	C₂×D₄×C₉	—	2	2	no	—
55	C₂×D₄×C₉	D₄	1	1	yes	characteristic
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	C₉×Q₁₆	—	2	4	no	—
61	C₉×Q₁₆	—	4	1	no	—
62	C₉×Q₁₆	D₄	1	4	yes	complemented
63	C₉×SD₁₆	—	2	4	no	—
64	C₉×SD₁₆	—	2	4	no	—
65	C₉×D₈	—	4	1	no	—
66	C₉×M₁₆	—	2	4	no	—
67	C₂×C₈×C₉	C₂³	1	2	yes	—
68	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₉	C₂³	1	1	yes	characteristic
70	C₉×C₄⋊C₄	—	2	2	no	—
71	C₉×C₄⋊C₄	—	2	2	no	—
72	C₉×C₄⋊C₄	D₄	1	1	yes	characteristic
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	C₉×C₂²⋊C₄	D₄	1	2	yes	—
79	C₄²×C₉	—	2	2	no	—
80	C₄²×C₉	C₂³	1	1	yes	characteristic
81	#3277	C₉	1	1	yes	characteristic, direct product factor, Sylow
82	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
83	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
84	C₃×#137	—	2	8	no	—
85	C₂×C₃×Q₁₆	C₂²×C₃	1	2	yes	—
86	C₂×C₃×Q₁₆	C₂²×C₃	1	1	yes	characteristic
87	C₂×C₃×Q₁₆	C₂²×C₃	1	1	yes	characteristic
88	C₂×C₃×SD₁₆	C₂²×C₃	1	2	yes	—
89	C₂×C₃×SD₁₆	C₂²×C₃	1	2	yes	—
90	C₂×C₃×D₈	C₂²×C₃	1	1	yes	characteristic
91	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
92	C₃×#124	—	2	2	no	—
93	C₃×#124	C₂²×C₃	1	2	yes	—
94	C₃×C₂²⋊Q₈	—	2	2	no	—
95	C₃×C₂²⋊Q₈	C₂²×C₃	1	2	yes	—
96	C₃×#121	—	2	2	no	—
97	C₃×#121	C₂²×C₃	1	2	yes	—
98	C₃×C₄×Q₈	C₂²×C₃	1	2	yes	—
99	C₃×C₄×D₄	—	2	2	no	—
100	C₃×C₄×D₄	C₂²×C₃	1	1	yes	characteristic
101	C₃×#107	C₂²×C₃	1	1	yes	characteristic
102	C₃×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
103	C₃×#103	C₂²×C₃	1	2	yes	—
104	C₃×#103	C₂²×C₃	1	2	yes	—
105	C₃×#102	C₂²×C₃	1	2	yes	—
106	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
107	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
108	C₂³×C₉	—	2	2	no	—
109	C₂³×C₉	C₂×D₄	1	2	yes	—
110	Q₈×C₉	—	2	2	no	—
111	Q₈×C₉	—	2	2	no	—
112	Q₈×C₉	—	2	2	no	—
113	Q₈×C₉	—	4	4	no	—
114	Q₈×C₉	C₂×D₄	1	4	yes	—
115	D₄×C₉	—	2	2	no	—
116	D₄×C₉	—	2	2	no	—
117	D₄×C₉	—	4	1	no	—
118	D₄×C₉	—	4	4	no	—
119	D₄×C₉	—	4	4	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	—	2	2	no	—
125	C₂×C₄×C₉	—	4	4	no	—
126	C₂×C₄×C₉	—	4	4	no	—
127	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
130	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
131	C₈×C₉	—	2	2	no	—
132	C₈×C₉	—	2	1	no	—
133	C₈×C₉	C₂×D₄	1	2	yes	—
134	C₂×#137	C₂×C₉	1	2	yes	complemented
135	#548	C₂×C₉	1	2	yes	complemented
136	#495	C₂×C₉	1	1	yes	characteristic, complemented
137	#468	C₂×C₉	1	2	yes	complemented
138	#464	C₂×C₉	1	2	yes	complemented
139	#451	C₂×C₉	1	2	yes	complemented
140	#449	C₂×C₉	1	2	yes	complemented
141	C₄×Q₁₆	C₂×C₉	1	1	yes	characteristic, complemented
142	#436	C₂×C₉	1	1	yes	characteristic, complemented
143	C₃×#41	—	2	4	no	—
144	C₃×#41	—	4	2	no	—
145	C₂×C₃×Q₈	—	2	2	no	—
146	C₂×C₃×Q₈	—	2	4	no	—
147	C₂×C₃×Q₈	—	4	2	no	—
148	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
149	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
150	C₂×C₃×D₄	—	2	2	no	—
151	C₂×C₃×D₄	—	2	2	no	—
152	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
153	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
154	C₂²×C₃×C₄	—	2	2	no	—
155	C₂²×C₃×C₄	—	2	2	no	—
156	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
157	C₃×Q₁₆	—	2	4	no	—
158	C₃×Q₁₆	—	4	1	no	—
159	C₃×Q₁₆	C₃×D₄	1	4	yes	—
160	C₃×SD₁₆	—	2	4	no	—
161	C₃×SD₁₆	—	2	4	no	—
162	C₃×D₈	—	4	1	no	—
163	C₃×M₁₆	—	2	4	no	—
164	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
165	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
166	C₂×C₃×C₈	C₂³×C₃	1	1	yes	characteristic
167	C₃×C₄⋊C₄	—	2	2	no	—
168	C₃×C₄⋊C₄	—	2	2	no	—
169	C₃×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
170	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
171	C₃×C₂²⋊C₄	—	2	2	no	—
172	C₃×C₂²⋊C₄	—	2	2	no	—
173	C₃×C₂²⋊C₄	—	2	2	no	—
174	C₃×C₂²⋊C₄	—	2	2	no	—
175	C₃×C₂²⋊C₄	C₃×D₄	1	2	yes	—
176	C₃×C₄²	—	2	2	no	—
177	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
178	C₂²×C₉	—	2	2	no	—
179	C₂²×C₉	—	2	2	no	—
180	C₂²×C₉	—	2	2	no	—
181	C₂²×C₉	—	4	2	no	—
182	C₂²×C₉	—	4	2	no	—
183	C₂²×C₉	—	4	2	no	—
184	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
185	C₄×C₉	—	2	2	no	—
186	C₄×C₉	—	2	4	no	—
187	C₄×C₉	—	2	1	no	—
188	C₄×C₉	—	2	1	no	—
189	C₄×C₉	—	4	2	no	—
190	C₄×C₉	—	4	2	no	—
191	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
192	C₄×C₉	H(5,𝔽₂)	1	1	yes	characteristic
193	C₂×#41	C₂²×C₉	1	2	yes	—
194	C₂²×Q₈	C₂²×C₉	1	2	yes	—
195	#137	—	2	8	no	—
196	C₂×Q₁₆	C₂²×C₉	1	2	yes	—
197	C₂×Q₁₆	C₂²×C₉	1	1	yes	characteristic
198	C₂×Q₁₆	C₂²×C₉	1	1	yes	characteristic
199	C₂×SD₁₆	C₂²×C₉	1	2	yes	—
200	C₂×SD₁₆	C₂²×C₉	1	2	yes	—
201	C₂×D₈	C₂²×C₉	1	1	yes	characteristic
202	C₂×M₁₆	C₂²×C₉	1	2	yes	—
203	#124	—	2	2	no	—
204	#124	C₂²×C₉	1	2	yes	—
205	C₂²⋊Q₈	—	2	2	no	—
206	C₂²⋊Q₈	C₂²×C₉	1	2	yes	—
207	#121	—	2	2	no	—
208	#121	C₂²×C₉	1	2	yes	—
209	C₄×Q₈	C₂²×C₉	1	2	yes	—
210	C₄×D₄	—	2	2	no	—
211	C₄×D₄	C₂²×C₉	1	1	yes	characteristic
212	#107	C₂²×C₉	1	1	yes	characteristic
213	C₄⋊C₈	C₂²×C₉	1	1	yes	characteristic
214	#103	C₂²×C₉	1	2	yes	—
215	#103	C₂²×C₉	1	2	yes	—
216	#102	C₂²×C₉	1	2	yes	—
217	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
218	C₄×C₈	C₂²×C₉	1	1	yes	characteristic
219	C₂³×C₃	—	2	2	no	—
220	C₂³×C₃	C₂×C₃×D₄	1	2	yes	—
221	C₃×Q₈	—	2	2	no	—
222	C₃×Q₈	—	2	2	no	—
223	C₃×Q₈	—	2	2	no	—
224	C₃×Q₈	—	4	4	no	—
225	C₃×Q₈	C₂×C₃×D₄	1	4	yes	—
226	C₃×D₄	—	2	2	no	—
227	C₃×D₄	—	2	2	no	—
228	C₃×D₄	—	4	1	no	—
229	C₃×D₄	—	4	4	no	—
230	C₃×D₄	—	4	4	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	—	2	2	no	—
236	C₂×C₃×C₄	—	4	4	no	—
237	C₂×C₃×C₄	—	4	4	no	—
238	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
239	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
241	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
242	C₃×C₈	—	2	2	no	—
243	C₃×C₈	—	2	1	no	—
244	C₃×C₈	C₂×C₃×D₄	1	2	yes	—
245	C₂×C₉	—	4	2	no	—
246	C₂×C₉	—	8	2	no	—
247	C₂×C₉	D₄²	1	1	yes	characteristic
248	C₂×C₉	#575	1	1	yes	characteristic
249	C₂×C₉	#578	1	1	yes	characteristic
250	#41	—	2	4	no	—
251	#41	—	4	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	4	no	—
254	C₂×Q₈	—	4	2	no	—
255	C₂×Q₈	C₂³×C₉	1	2	yes	—
256	C₂×Q₈	C₂³×C₉	1	2	yes	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	D₄×C₉	1	1	yes	characteristic
260	C₂×D₄	C₂³×C₉	1	2	yes	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	C₂³×C₉	1	2	yes	—
264	Q₁₆	—	2	4	no	—
265	Q₁₆	—	4	1	no	—
266	Q₁₆	D₄×C₉	1	4	yes	complemented
267	SD₁₆	—	2	4	no	—
268	SD₁₆	—	2	4	no	—
269	D₈	—	4	1	no	—
270	M₁₆	—	2	4	no	—
271	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
272	C₂×C₈	C₂³×C₉	1	2	yes	—
273	C₂×C₈	C₂³×C₉	1	1	yes	characteristic
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	D₄×C₉	1	1	yes	characteristic
277	C₄⋊C₄	C₂³×C₉	1	2	yes	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₂²⋊C₄	D₄×C₉	1	2	yes	—
283	C₄²	—	2	2	no	—
284	C₄²	C₂³×C₉	1	1	yes	characteristic
285	C₂²×C₃	—	2	2	no	—
286	C₂²×C₃	—	2	2	no	—
287	C₂²×C₃	—	2	2	no	—
288	C₂²×C₃	—	4	2	no	—
289	C₂²×C₃	—	4	2	no	—
290	C₂²×C₃	—	4	2	no	—
291	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	2	1	no	—
294	C₃×C₄	—	2	2	no	—
295	C₃×C₄	—	2	4	no	—
296	C₃×C₄	—	4	2	no	—
297	C₃×C₄	—	4	2	no	—
298	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
299	C₃×C₄	C₃×H(5,𝔽₂)	1	1	yes	characteristic
300	C₉	#3277	1	1	yes	characteristic, direct product factor, Sylow
301	C₂³	—	2	2	no	—
302	C₂³	C₂×D₄×C₉	1	2	yes	—
303	Q₈	—	2	2	no	—
304	Q₈	—	2	2	no	—
305	Q₈	—	2	2	no	—
306	Q₈	—	4	4	no	—
307	Q₈	C₂×D₄×C₉	1	4	yes	—
308	D₄	—	2	2	no	—
309	D₄	—	2	2	no	—
310	D₄	—	4	1	no	—
311	D₄	—	4	4	no	—
312	D₄	—	4	4	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	—	4	4	no	—
320	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
321	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
322	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
323	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	2	1	no	—
326	C₈	C₂×D₄×C₉	1	2	yes	—
327	C₂×C₃	—	4	2	no	—
328	C₂×C₃	—	8	2	no	—
329	C₂×C₃	C₃×D₄²	1	1	yes	characteristic
330	C₂×C₃	C₃×#575	1	1	yes	characteristic
331	C₂×C₃	C₃×#578	1	1	yes	characteristic
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	4	2	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	4	2	no	—
338	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
339	C₄	—	2	2	no	—
340	C₄	—	2	4	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	4	2	no	—
344	C₄	—	4	2	no	—
345	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
346	C₄	C₉×H(5,𝔽₂)	1	1	yes	characteristic
347	C₃	C₃×#3277	1	1	yes	characteristic
348	C₂	—	4	2	no	—
349	C₂	—	8	2	no	—
350	C₂	D₄²×C₉	1	1	yes	characteristic
351	C₂	C₉×#575	1	1	yes	characteristic
352	C₂	C₉×#578	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×#3277	C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₄×D₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₉×#137	C₂	semidirect
2. C₉×#548	C₂	semidirect
3. C₉×#495	C₂	semidirect
4. C₉×#468	C₂	semidirect
5. C₉×#464	C₂	semidirect
6. C₉×#451	C₂	semidirect
7. C₉×#449	C₂	semidirect
8. C₄×C₉×Q₁₆	C₂	semidirect
9. C₉×#436	C₂	semidirect
62. C₉×Q₁₆	D₄	semidirect
81. #3277	C₉	direct (N × H)
134. C₂×#137	C₂×C₉	semidirect
135. #548	C₂×C₉	semidirect
136. #495	C₂×C₉	semidirect
137. #468	C₂×C₉	semidirect
138. #464	C₂×C₉	semidirect
139. #451	C₂×C₉	semidirect
140. #449	C₂×C₉	semidirect
141. C₄×Q₁₆	C₂×C₉	semidirect
142. #436	C₂×C₉	semidirect
266. Q₁₆	D₄×C₉	semidirect
300. C₉	#3277	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₉×#137	2
C₉×#548	2
C₉×#495	2
C₉×#468	2
C₉×#464	2
C₉×#451	2
C₉×#449	2
C₄×C₉×Q₁₆	2
C₉×#436	2
C₃×#3277	3

Subgroups — C9×#3277 #11767998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3277 #11767998