Subgroups — C₉×#3205 #11767926

Order 1152.

798 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₉×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#493	C₂	1	2	yes	maximal, complemented
4	C₉×#486	C₂	1	2	yes	maximal, complemented
5	C₉×#459	C₂	1	4	yes	maximal, complemented
6	C₉×#447	C₂	1	4	yes	maximal, complemented
7	C₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#3205	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×#41	—	2	1	no	—
10	C₂²×D₄×C₉	—	2	1	no	—
11	C₂²×D₄×C₉	—	2	1	no	—
12	C₂²×D₄×C₉	C₂²	1	2	yes	—
13	C₂×C₉×D₈	C₂²	1	8	yes	complemented
14	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
15	C₉×#127	—	2	1	no	—
16	C₉×#127	—	2	1	no	—
17	C₉×#127	C₂²	1	2	yes	—
18	C₉×#127	C₂²	1	2	yes	—
19	C₉×#127	C₂²	1	1	yes	characteristic, complemented
20	C₉×#124	—	2	1	no	—
21	C₉×#121	—	2	1	no	—
22	C₉×#121	C₂²	1	2	yes	—
23	C₄×D₄×C₉	—	2	1	no	—
24	C₄×D₄×C₉	C₂²	1	2	yes	complemented
25	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
26	C₉×C₄⋊C₈	C₂²	1	2	yes	complemented
27	C₉×#102	C₂²	1	8	yes	complemented
28	C₉×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₄×C₈×C₉	C₂²	1	2	yes	complemented
30	C₃×#532	C₂×C₃	1	1	yes	characteristic
31	C₂×C₃×#127	C₂×C₃	1	1	yes	characteristic
32	C₃×#493	C₂×C₃	1	2	yes	—
33	C₃×#486	C₂×C₃	1	2	yes	—
34	C₃×#459	C₂×C₃	1	4	yes	—
35	C₃×#447	C₂×C₃	1	4	yes	—
36	C₃×#431	C₂×C₃	1	1	yes	characteristic
37	C₂⁴×C₉	—	2	2	no	—
38	C₉×#41	—	4	4	no	—
39	C₂×Q₈×C₉	—	2	1	no	—
40	C₂×D₄×C₉	—	2	2	no	—
41	C₂×D₄×C₉	—	2	1	no	—
42	C₂×D₄×C₉	—	2	1	no	—
43	C₂×D₄×C₉	—	2	2	no	—
44	C₂×D₄×C₉	—	2	2	no	—
45	C₂×D₄×C₉	—	2	2	no	—
46	C₂×D₄×C₉	—	2	2	no	—
47	C₂×D₄×C₉	—	2	2	no	—
48	C₂×D₄×C₉	—	4	4	no	—
49	C₂×D₄×C₉	—	4	4	no	—
50	C₂×D₄×C₉	—	4	4	no	—
51	C₂×D₄×C₉	C₂³	1	2	yes	—
52	C₂×D₄×C₉	C₂³	1	4	yes	—
53	C₂²×C₄×C₉	—	2	2	no	—
54	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
55	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
57	C₉×D₈	—	4	8	no	—
58	C₂×C₈×C₉	C₂³	1	4	yes	—
59	C₉×C₄⋊C₄	—	2	1	no	—
60	C₉×C₄⋊C₄	C₂³	1	2	yes	—
61	C₉×C₂²⋊C₄	—	2	2	no	—
62	C₉×C₂²⋊C₄	—	2	2	no	—
63	C₄²×C₉	D₄	1	1	yes	characteristic
64	C₄²×C₉	D₄	1	1	yes	characteristic
65	C₄²×C₉	C₂³	1	1	yes	characteristic
66	C₄²×C₉	C₂³	1	1	yes	characteristic
67	#3205	C₉	1	1	yes	characteristic, direct product factor, Sylow
68	C₂×C₃×#41	—	2	1	no	—
69	C₂²×C₃×D₄	—	2	1	no	—
70	C₂²×C₃×D₄	—	2	1	no	—
71	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
72	C₂×C₃×D₈	C₂²×C₃	1	8	yes	—
73	C₃×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
74	C₃×#127	—	2	1	no	—
75	C₃×#127	—	2	1	no	—
76	C₃×#127	C₂²×C₃	1	2	yes	—
77	C₃×#127	C₂²×C₃	1	2	yes	—
78	C₃×#127	C₂²×C₃	1	1	yes	characteristic
79	C₃×#124	—	2	1	no	—
80	C₃×#121	—	2	1	no	—
81	C₃×#121	C₂²×C₃	1	2	yes	—
82	C₃×C₄×D₄	—	2	1	no	—
83	C₃×C₄×D₄	C₂²×C₃	1	2	yes	—
84	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
85	C₃×C₄⋊C₈	C₂²×C₃	1	2	yes	—
86	C₃×#102	C₂²×C₃	1	8	yes	—
87	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
88	C₃×C₄×C₈	C₂²×C₃	1	2	yes	—
89	C₂³×C₉	—	2	2	no	—
90	C₂³×C₉	—	2	4	no	—
91	C₂³×C₉	—	4	4	no	—
92	C₂³×C₉	—	4	4	no	—
93	C₂³×C₉	—	4	4	no	—
94	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
95	Q₈×C₉	—	4	2	no	—
96	D₄×C₉	—	2	4	no	—
97	D₄×C₉	—	2	8	no	—
98	D₄×C₉	—	4	4	no	—
99	D₄×C₉	—	4	4	no	—
100	D₄×C₉	—	4	4	no	—
101	D₄×C₉	—	4	4	no	—
102	D₄×C₉	—	4	4	no	—
103	D₄×C₉	—	4	2	no	—
104	D₄×C₉	—	4	2	no	—
105	D₄×C₉	—	4	4	no	—
106	C₂×C₄×C₉	—	2	2	no	—
107	C₂×C₄×C₉	—	2	2	no	—
108	C₂×C₄×C₉	—	2	2	no	—
109	C₂×C₄×C₉	—	4	4	no	—
110	C₂×C₄×C₉	D₈	1	4	yes	complemented
111	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
117	C₈×C₉	—	2	4	no	—
118	#532	C₂×C₉	1	1	yes	characteristic, complemented
119	C₂×#127	C₂×C₉	1	1	yes	characteristic, complemented
120	#493	C₂×C₉	1	2	yes	complemented
121	#486	C₂×C₉	1	2	yes	complemented
122	#459	C₂×C₉	1	4	yes	complemented
123	#447	C₂×C₉	1	4	yes	complemented
124	#431	C₂×C₉	1	1	yes	characteristic, complemented
125	C₂⁴×C₃	—	2	2	no	—
126	C₃×#41	—	4	4	no	—
127	C₂×C₃×Q₈	—	2	1	no	—
128	C₂×C₃×D₄	—	2	2	no	—
129	C₂×C₃×D₄	—	2	1	no	—
130	C₂×C₃×D₄	—	2	1	no	—
131	C₂×C₃×D₄	—	2	2	no	—
132	C₂×C₃×D₄	—	2	2	no	—
133	C₂×C₃×D₄	—	2	2	no	—
134	C₂×C₃×D₄	—	2	2	no	—
135	C₂×C₃×D₄	—	2	2	no	—
136	C₂×C₃×D₄	—	4	4	no	—
137	C₂×C₃×D₄	—	4	4	no	—
138	C₂×C₃×D₄	—	4	4	no	—
139	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
140	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
141	C₂²×C₃×C₄	—	2	2	no	—
142	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
143	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
144	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
145	C₃×D₈	—	4	8	no	—
146	C₂×C₃×C₈	C₂³×C₃	1	4	yes	—
147	C₃×C₄⋊C₄	—	2	1	no	—
148	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
149	C₃×C₂²⋊C₄	—	2	2	no	—
150	C₃×C₂²⋊C₄	—	2	2	no	—
151	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
152	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
153	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
154	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
155	C₂²×C₉	—	2	1	no	—
156	C₂²×C₉	—	2	1	no	—
157	C₂²×C₉	—	4	4	no	—
158	C₂²×C₉	—	4	2	no	—
159	C₂²×C₉	—	4	2	no	—
160	C₂²×C₉	—	4	4	no	—
161	C₂²×C₉	—	4	4	no	—
162	C₂²×C₉	—	4	2	no	—
163	C₂²×C₉	—	8	8	no	—
164	C₂²×C₉	C₂×D₈	1	2	yes	—
165	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
166	C₄×C₉	—	2	1	no	—
167	C₄×C₉	—	2	1	no	—
168	C₄×C₉	—	4	2	no	—
169	C₄×C₉	C₂×D₈	1	2	yes	—
170	C₄×C₉	C₂×D₈	1	2	yes	—
171	C₄×C₉	AGL(1,ℤ/8)	1	2	yes	—
172	C₄×C₉	H(5,𝔽₂)	1	2	yes	—
173	C₂×#41	—	2	1	no	—
174	C₂²×D₄	—	2	1	no	—
175	C₂²×D₄	—	2	1	no	—
176	C₂²×D₄	C₂²×C₉	1	2	yes	—
177	C₂×D₈	C₂²×C₉	1	8	yes	complemented
178	C₄⋊Q₈	C₂²×C₉	1	1	yes	characteristic, complemented
179	#127	—	2	1	no	—
180	#127	—	2	1	no	—
181	#127	C₂²×C₉	1	2	yes	—
182	#127	C₂²×C₉	1	2	yes	—
183	#127	C₂²×C₉	1	1	yes	characteristic, complemented
184	#124	—	2	1	no	—
185	#121	—	2	1	no	—
186	#121	C₂²×C₉	1	2	yes	—
187	C₄×D₄	—	2	1	no	—
188	C₄×D₄	C₂²×C₉	1	2	yes	complemented
189	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
190	C₄⋊C₈	C₂²×C₉	1	2	yes	complemented
191	#102	C₂²×C₉	1	8	yes	complemented
192	C₂²⋊C₈	C₂²×C₉	1	2	yes	—
193	C₄×C₈	C₂²×C₉	1	2	yes	complemented
194	C₂³×C₃	—	2	2	no	—
195	C₂³×C₃	—	2	4	no	—
196	C₂³×C₃	—	4	4	no	—
197	C₂³×C₃	—	4	4	no	—
198	C₂³×C₃	—	4	4	no	—
199	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
200	C₃×Q₈	—	4	2	no	—
201	C₃×D₄	—	2	4	no	—
202	C₃×D₄	—	2	8	no	—
203	C₃×D₄	—	4	4	no	—
204	C₃×D₄	—	4	4	no	—
205	C₃×D₄	—	4	4	no	—
206	C₃×D₄	—	4	4	no	—
207	C₃×D₄	—	4	4	no	—
208	C₃×D₄	—	4	2	no	—
209	C₃×D₄	—	4	2	no	—
210	C₃×D₄	—	4	4	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	—	4	4	no	—
215	C₂×C₃×C₄	C₃×D₈	1	4	yes	—
216	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
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	C₂⁴×C₃	1	1	yes	characteristic
222	C₃×C₈	—	2	4	no	—
223	C₂×C₉	—	2	2	no	—
224	C₂×C₉	—	8	2	no	—
225	C₂×C₉	—	8	4	no	—
226	C₂×C₉	#535	1	1	yes	characteristic
227	C₂×C₉	C₂²×D₈	1	1	yes	characteristic
228	C₂×C₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
229	C₂⁴	—	2	2	no	—
230	#41	—	4	4	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	4	4	no	—
241	C₂×D₄	—	4	4	no	—
242	C₂×D₄	—	4	4	no	—
243	C₂×D₄	C₂³×C₉	1	4	yes	—
244	C₂×D₄	C₂³×C₉	1	2	yes	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
247	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
248	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
249	D₈	—	4	8	no	—
250	C₂×C₈	C₂³×C₉	1	4	yes	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	C₂³×C₉	1	2	yes	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	2	2	no	—
255	C₄²	D₄×C₉	1	1	yes	characteristic
256	C₄²	D₄×C₉	1	1	yes	characteristic
257	C₄²	C₂³×C₉	1	1	yes	characteristic
258	C₄²	C₂³×C₉	1	1	yes	characteristic
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	—	4	2	no	—
262	C₂²×C₃	—	4	4	no	—
263	C₂²×C₃	—	4	4	no	—
264	C₂²×C₃	—	4	2	no	—
265	C₂²×C₃	—	4	4	no	—
266	C₂²×C₃	—	4	2	no	—
267	C₂²×C₃	—	8	8	no	—
268	C₂²×C₃	C₂×C₃×D₈	1	2	yes	—
269	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	4	2	no	—
273	C₃×C₄	C₂×C₃×D₈	1	2	yes	—
274	C₃×C₄	C₂×C₃×D₈	1	2	yes	—
275	C₃×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
276	C₃×C₄	C₃×H(5,𝔽₂)	1	2	yes	—
277	C₉	#3205	1	1	yes	characteristic, direct product factor, Sylow
278	C₂³	—	2	2	no	—
279	C₂³	—	2	4	no	—
280	C₂³	—	4	4	no	—
281	C₂³	—	4	4	no	—
282	C₂³	—	4	4	no	—
283	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
284	Q₈	—	4	2	no	—
285	D₄	—	2	4	no	—
286	D₄	—	2	8	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	4	no	—
289	D₄	—	4	4	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	4	no	—
294	D₄	—	4	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	C₉×D₈	1	4	yes	complemented
300	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
301	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
302	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
303	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
304	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
305	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
306	C₈	—	2	4	no	—
307	C₂×C₃	—	2	2	no	—
308	C₂×C₃	—	8	4	no	—
309	C₂×C₃	—	8	2	no	—
310	C₂×C₃	C₃×#535	1	1	yes	characteristic
311	C₂×C₃	C₂²×C₃×D₈	1	1	yes	characteristic
312	C₂×C₃	C₂×C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	4	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	4	4	no	—
319	C₂²	—	4	4	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	8	8	no	—
322	C₂²	C₂×C₉×D₈	1	2	yes	—
323	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	2	no	—
327	C₄	C₂×C₉×D₈	1	2	yes	—
328	C₄	C₂×C₉×D₈	1	2	yes	—
329	C₄	C₉×AGL(1,ℤ/8)	1	2	yes	—
330	C₄	C₉×H(5,𝔽₂)	1	2	yes	—
331	C₃	C₃×#3205	1	1	yes	characteristic
332	C₂	—	2	2	no	—
333	C₂	—	8	4	no	—
334	C₂	—	8	2	no	—
335	C₂	C₉×#535	1	1	yes	characteristic
336	C₂	C₂²×C₉×D₈	1	1	yes	characteristic
337	C₂	C₂×C₉×AGL(1,ℤ/8)	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×#3205	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×#532	C₂	semidirect
2. C₂×C₉×#127	C₂	semidirect
3. C₉×#493	C₂	semidirect
4. C₉×#486	C₂	semidirect
5. C₉×#459	C₂	semidirect
6. C₉×#447	C₂	semidirect
7. C₉×#431	C₂	semidirect
13. C₂×C₉×D₈	C₂²	semidirect
14. C₉×C₄⋊Q₈	C₂²	semidirect
19. C₉×#127	C₂²	semidirect
24. C₄×D₄×C₉	C₂²	semidirect
26. C₉×C₄⋊C₈	C₂²	semidirect
27. C₉×#102	C₂²	semidirect
29. C₄×C₈×C₉	C₂²	semidirect
67. #3205	C₉	direct (N × H)
110. C₂×C₄×C₉	D₈	semidirect
118. #532	C₂×C₉	semidirect
119. C₂×#127	C₂×C₉	semidirect
120. #493	C₂×C₉	semidirect
121. #486	C₂×C₉	semidirect
122. #459	C₂×C₉	semidirect
123. #447	C₂×C₉	semidirect
124. #431	C₂×C₉	semidirect
177. C₂×D₈	C₂²×C₉	semidirect
178. C₄⋊Q₈	C₂²×C₉	semidirect
183. #127	C₂²×C₉	semidirect
188. C₄×D₄	C₂²×C₉	semidirect
190. C₄⋊C₈	C₂²×C₉	semidirect
191. #102	C₂²×C₉	semidirect
193. C₄×C₈	C₂²×C₉	semidirect
277. C₉	#3205	direct (N × H)
299. C₂×C₄	C₉×D₈	semidirect

Maximal subgroups

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

subgroup	index
C₉×#532	2
C₂×C₉×#127	2
C₉×#493	2
C₉×#486	2
C₉×#459	2
C₉×#447	2
C₉×#431	2
C₃×#3205	3

Subgroups — C9×#3205 #11767926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3205 #11767926