Subgroups — C₉×#1904 #11766625

Order 1152.

519 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
2	C₉×#518	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#421	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#418	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#410	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#344	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#1904	C₃	1	1	yes	maximal, characteristic
9	C₂×C₉×#41	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
11	C₉×AGL(1,ℤ/8)	—	2	2	no	—
12	C₉×AGL(1,ℤ/8)	—	2	2	no	—
13	C₂×C₉×SD₁₆	C₄	1	2	yes	complemented
14	C₂×C₉×D₈	C₄	1	2	yes	complemented
15	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₉	—	2	2	no	—
18	C₄×D₄×C₉	—	2	2	no	—
19	C₉×#117	C₂²	1	1	yes	characteristic
20	C₉×#117	C₂²	1	1	yes	characteristic
21	C₉×#117	C₂²	1	1	yes	characteristic
22	C₂×C₉×C₂²⋊C₄	—	2	2	no	—
23	C₉×#104	—	2	2	no	—
24	C₉×#102	—	2	2	no	—
25	C₉×#99	—	2	2	no	—
26	C₂×C₃×AGL(1,ℤ/8)	C₂×C₃	1	1	yes	characteristic
27	C₃×#518	C₂×C₃	1	1	yes	characteristic
28	C₃×#421	C₂×C₃	1	1	yes	characteristic
29	C₃×#418	C₂×C₃	1	1	yes	characteristic
30	C₃×#410	C₂×C₃	1	1	yes	characteristic
31	C₃×#344	C₂×C₃	1	1	yes	characteristic
32	C₃×H(3,ℤ/4)	C₂×C₃	1	1	yes	characteristic
33	C₂⁴×C₉	—	2	1	no	—
34	C₉×#41	—	2	1	no	—
35	C₉×#41	—	2	1	no	—
36	C₉×#41	—	2	1	no	—
37	C₉×#41	—	2	1	no	—
38	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂×D₄×C₉	—	2	1	no	—
40	C₂×D₄×C₉	—	2	1	no	—
41	C₂×D₄×C₉	—	2	1	no	—
42	C₂×D₄×C₉	—	2	1	no	—
43	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
44	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic, complemented
45	C₂×D₄×C₉	D₄	1	2	yes	—
46	C₂×D₄×C₉	D₄	1	2	yes	—
47	C₂²×C₄×C₉	—	2	4	no	—
48	C₂²×C₄×C₉	—	2	1	no	—
49	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
50	C₉×SD₁₆	—	2	4	no	—
51	C₉×D₈	—	2	4	no	—
52	C₉×M₁₆	—	2	1	no	—
53	C₉×M₁₆	—	2	1	no	—
54	C₉×M₁₆	—	2	1	no	—
55	C₉×M₁₆	—	2	1	no	—
56	C₂×C₈×C₉	—	2	1	no	—
57	C₂×C₈×C₉	C₂×C₄	1	2	yes	complemented
58	C₉×C₄⋊C₄	—	2	1	no	—
59	C₉×C₄⋊C₄	—	2	1	no	—
60	C₉×C₄⋊C₄	—	2	1	no	—
61	C₉×C₂²⋊C₄	—	2	4	no	—
62	C₉×C₂²⋊C₄	—	2	1	no	—
63	C₉×C₂²⋊C₄	—	2	1	no	—
64	C₉×C₂²⋊C₄	D₄	1	2	yes	—
65	C₄²×C₉	—	2	1	no	—
66	C₄²×C₉	—	2	1	no	—
67	C₄²×C₉	D₄	1	2	yes	—
68	#1904	C₉	1	1	yes	characteristic, direct product factor, Sylow
69	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
70	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
71	C₃×AGL(1,ℤ/8)	—	2	2	no	—
72	C₃×AGL(1,ℤ/8)	—	2	2	no	—
73	C₂×C₃×SD₁₆	C₃×C₄	1	2	yes	—
74	C₂×C₃×D₈	C₃×C₄	1	2	yes	—
75	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
76	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
77	C₃×C₄×D₄	—	2	2	no	—
78	C₃×C₄×D₄	—	2	2	no	—
79	C₃×#117	C₂²×C₃	1	1	yes	characteristic
80	C₃×#117	C₂²×C₃	1	1	yes	characteristic
81	C₃×#117	C₂²×C₃	1	1	yes	characteristic
82	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
83	C₃×#104	—	2	2	no	—
84	C₃×#102	—	2	2	no	—
85	C₃×#99	—	2	2	no	—
86	C₂³×C₉	—	2	2	no	—
87	C₂³×C₉	—	2	2	no	—
88	C₂³×C₉	—	2	2	no	—
89	C₂³×C₉	—	2	1	no	—
90	C₂³×C₉	—	8	2	no	—
91	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
92	Q₈×C₉	—	2	1	no	—
93	Q₈×C₉	—	2	1	no	—
94	D₄×C₉	—	2	1	no	—
95	D₄×C₉	—	2	2	no	—
96	D₄×C₉	—	2	1	no	—
97	D₄×C₉	—	2	2	no	—
98	D₄×C₉	—	2	2	no	—
99	D₄×C₉	—	4	1	no	—
100	D₄×C₉	—	4	1	no	—
101	D₄×C₉	C₂²⋊C₄	1	4	yes	—
102	C₂×C₄×C₉	—	2	1	no	—
103	C₂×C₄×C₉	—	2	2	no	—
104	C₂×C₄×C₉	—	2	2	no	—
105	C₂×C₄×C₉	—	4	1	no	—
106	C₂×C₄×C₉	—	4	1	no	—
107	C₂×C₄×C₉	—	4	1	no	—
108	C₂×C₄×C₉	—	8	4	no	—
109	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₉	#41	1	1	yes	characteristic
114	C₂×C₄×C₉	#41	1	1	yes	characteristic
115	C₈×C₉	—	2	2	no	—
116	C₈×C₉	—	4	1	no	—
117	C₂×AGL(1,ℤ/8)	C₂×C₉	1	1	yes	characteristic
118	#518	C₂×C₉	1	1	yes	characteristic, complemented
119	#421	C₂×C₉	1	1	yes	characteristic, complemented
120	#418	C₂×C₉	1	1	yes	characteristic, complemented
121	#410	C₂×C₉	1	1	yes	characteristic, complemented
122	#344	C₂×C₉	1	1	yes	characteristic, complemented
123	H(3,ℤ/4)	C₂×C₉	1	1	yes	characteristic, complemented
124	C₂⁴×C₃	—	2	1	no	—
125	C₃×#41	—	2	1	no	—
126	C₃×#41	—	2	1	no	—
127	C₃×#41	—	2	1	no	—
128	C₃×#41	—	2	1	no	—
129	C₂×C₃×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
130	C₂×C₃×D₄	—	2	1	no	—
131	C₂×C₃×D₄	—	2	1	no	—
132	C₂×C₃×D₄	—	2	1	no	—
133	C₂×C₃×D₄	—	2	1	no	—
134	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
135	C₂×C₃×D₄	C₂×C₃×C₄	1	1	yes	characteristic
136	C₂×C₃×D₄	C₃×D₄	1	2	yes	—
137	C₂×C₃×D₄	C₃×D₄	1	2	yes	—
138	C₂²×C₃×C₄	—	2	4	no	—
139	C₂²×C₃×C₄	—	2	1	no	—
140	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
141	C₃×SD₁₆	—	2	4	no	—
142	C₃×D₈	—	2	4	no	—
143	C₃×M₁₆	—	2	1	no	—
144	C₃×M₁₆	—	2	1	no	—
145	C₃×M₁₆	—	2	1	no	—
146	C₃×M₁₆	—	2	1	no	—
147	C₂×C₃×C₈	—	2	1	no	—
148	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
149	C₃×C₄⋊C₄	—	2	1	no	—
150	C₃×C₄⋊C₄	—	2	1	no	—
151	C₃×C₄⋊C₄	—	2	1	no	—
152	C₃×C₂²⋊C₄	—	2	4	no	—
153	C₃×C₂²⋊C₄	—	2	1	no	—
154	C₃×C₂²⋊C₄	—	2	1	no	—
155	C₃×C₂²⋊C₄	C₃×D₄	1	2	yes	—
156	C₃×C₄²	—	2	1	no	—
157	C₃×C₄²	—	2	1	no	—
158	C₃×C₄²	C₃×D₄	1	2	yes	—
159	C₂²×C₉	—	2	4	no	—
160	C₂²×C₉	—	4	1	no	—
161	C₂²×C₉	—	4	4	no	—
162	C₂²×C₉	—	4	1	no	—
163	C₂²×C₉	—	8	2	no	—
164	C₂²×C₉	—	8	2	no	—
165	C₂²×C₉	—	8	1	no	—
166	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
167	C₂²×C₉	#120	1	1	yes	characteristic
168	C₂²×C₉	#121	1	1	yes	characteristic
169	C₄×C₉	—	4	2	no	—
170	C₄×C₉	—	4	2	no	—
171	C₄×C₉	—	4	1	no	—
172	C₄×C₉	—	8	1	no	—
173	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
174	C₄×C₉	C₄×D₄	1	1	yes	characteristic
175	C₄×C₉	#121	1	1	yes	characteristic
176	C₄×C₉	#123	1	1	yes	characteristic
177	C₂×#41	C₂²×C₉	1	1	yes	characteristic
178	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
179	AGL(1,ℤ/8)	—	2	2	no	—
180	AGL(1,ℤ/8)	—	2	2	no	—
181	C₂×SD₁₆	C₄×C₉	1	2	yes	complemented
182	C₂×D₈	C₄×C₉	1	2	yes	complemented
183	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
184	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
185	C₄×D₄	—	2	2	no	—
186	C₄×D₄	—	2	2	no	—
187	#117	C₂²×C₉	1	1	yes	characteristic
188	#117	C₂²×C₉	1	1	yes	characteristic
189	#117	C₂²×C₉	1	1	yes	characteristic
190	C₂×C₂²⋊C₄	—	2	2	no	—
191	#104	—	2	2	no	—
192	#102	—	2	2	no	—
193	#99	—	2	2	no	—
194	C₂³×C₃	—	2	2	no	—
195	C₂³×C₃	—	2	2	no	—
196	C₂³×C₃	—	2	2	no	—
197	C₂³×C₃	—	2	1	no	—
198	C₂³×C₃	—	8	2	no	—
199	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
200	C₃×Q₈	—	2	1	no	—
201	C₃×Q₈	—	2	1	no	—
202	C₃×D₄	—	2	1	no	—
203	C₃×D₄	—	2	2	no	—
204	C₃×D₄	—	2	1	no	—
205	C₃×D₄	—	2	2	no	—
206	C₃×D₄	—	2	2	no	—
207	C₃×D₄	—	4	1	no	—
208	C₃×D₄	—	4	1	no	—
209	C₃×D₄	C₃×C₂²⋊C₄	1	4	yes	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	4	1	no	—
214	C₂×C₃×C₄	—	4	1	no	—
215	C₂×C₃×C₄	—	4	1	no	—
216	C₂×C₃×C₄	—	8	4	no	—
217	C₂×C₃×C₄	C₂²×C₃×C₄	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	C₃×#41	1	1	yes	characteristic
222	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
223	C₃×C₈	—	2	2	no	—
224	C₃×C₈	—	4	1	no	—
225	C₂×C₉	—	2	1	no	—
226	C₂×C₉	—	2	1	no	—
227	C₂×C₉	—	2	1	no	—
228	C₂×C₉	—	4	4	no	—
229	C₂×C₉	—	8	1	no	—
230	C₂×C₉	#386	1	1	yes	characteristic
231	C₂⁴	—	2	1	no	—
232	#41	—	2	1	no	—
233	#41	—	2	1	no	—
234	#41	—	2	1	no	—
235	#41	—	2	1	no	—
236	C₂×Q₈	C₂×C₄×C₉	1	1	yes	characteristic, complemented
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
242	C₂×D₄	C₂×C₄×C₉	1	1	yes	characteristic, complemented
243	C₂×D₄	D₄×C₉	1	2	yes	—
244	C₂×D₄	D₄×C₉	1	2	yes	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
248	SD₁₆	—	2	4	no	—
249	D₈	—	2	4	no	—
250	M₁₆	—	2	1	no	—
251	M₁₆	—	2	1	no	—
252	M₁₆	—	2	1	no	—
253	M₁₆	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	C₂×C₄×C₉	1	2	yes	complemented
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	4	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	D₄×C₉	1	2	yes	—
263	C₄²	—	2	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	D₄×C₉	1	2	yes	—
266	C₂²×C₃	—	2	4	no	—
267	C₂²×C₃	—	4	1	no	—
268	C₂²×C₃	—	4	4	no	—
269	C₂²×C₃	—	4	1	no	—
270	C₂²×C₃	—	8	2	no	—
271	C₂²×C₃	—	8	2	no	—
272	C₂²×C₃	—	8	1	no	—
273	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
274	C₂²×C₃	C₃×#120	1	1	yes	characteristic
275	C₂²×C₃	C₃×#121	1	1	yes	characteristic
276	C₃×C₄	—	4	2	no	—
277	C₃×C₄	—	4	2	no	—
278	C₃×C₄	—	4	1	no	—
279	C₃×C₄	—	8	1	no	—
280	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
281	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
282	C₃×C₄	C₃×#121	1	1	yes	characteristic
283	C₃×C₄	C₃×#123	1	1	yes	characteristic
284	C₉	#1904	1	1	yes	characteristic, direct product factor, Sylow
285	C₂³	—	2	2	no	—
286	C₂³	—	2	2	no	—
287	C₂³	—	2	2	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	8	2	no	—
290	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
291	Q₈	—	2	1	no	—
292	Q₈	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	2	2	no	—
295	D₄	—	2	1	no	—
296	D₄	—	2	2	no	—
297	D₄	—	2	2	no	—
298	D₄	—	4	1	no	—
299	D₄	—	4	1	no	—
300	D₄	C₉×C₂²⋊C₄	1	4	yes	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	8	4	no	—
308	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
310	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
311	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
312	C₂×C₄	C₉×#41	1	1	yes	characteristic
313	C₂×C₄	C₉×#41	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	—	4	1	no	—
316	C₂×C₃	—	2	1	no	—
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	4	4	no	—
320	C₂×C₃	—	8	1	no	—
321	C₂×C₃	C₃×#386	1	1	yes	characteristic
322	C₂²	—	2	4	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	1	no	—
329	C₂²	C₄×D₄×C₉	1	1	yes	characteristic
330	C₂²	C₉×#120	1	1	yes	characteristic
331	C₂²	C₉×#121	1	1	yes	characteristic
332	C₄	—	4	2	no	—
333	C₄	—	4	2	no	—
334	C₄	—	4	1	no	—
335	C₄	—	8	1	no	—
336	C₄	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
337	C₄	C₄×D₄×C₉	1	1	yes	characteristic
338	C₄	C₉×#121	1	1	yes	characteristic
339	C₄	C₉×#123	1	1	yes	characteristic
340	C₃	C₃×#1904	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	2	1	no	—
344	C₂	—	4	4	no	—
345	C₂	—	8	1	no	—
346	C₂	C₉×#386	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	#386
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂×C₃	C₃×#386
prime radical	C₂×C₃×AGL(1,ℤ/8)	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₉ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×#518	C₂	semidirect
3. C₉×#421	C₂	semidirect
4. C₉×#418	C₂	semidirect
5. C₉×#410	C₂	semidirect
6. C₉×#344	C₂	semidirect
7. C₉×H(3,ℤ/4)	C₂	semidirect
13. C₂×C₉×SD₁₆	C₄	semidirect
14. C₂×C₉×D₈	C₄	semidirect
38. C₂×Q₈×C₉	C₂×C₄	semidirect
44. C₂×D₄×C₉	C₂×C₄	semidirect
57. C₂×C₈×C₉	C₂×C₄	semidirect
68. #1904	C₉	direct (N × H)
118. #518	C₂×C₉	semidirect
119. #421	C₂×C₉	semidirect
120. #418	C₂×C₉	semidirect
121. #410	C₂×C₉	semidirect
122. #344	C₂×C₉	semidirect
123. H(3,ℤ/4)	C₂×C₉	semidirect
181. C₂×SD₁₆	C₄×C₉	semidirect
182. C₂×D₈	C₄×C₉	semidirect
236. C₂×Q₈	C₂×C₄×C₉	semidirect
242. C₂×D₄	C₂×C₄×C₉	semidirect
255. C₂×C₈	C₂×C₄×C₉	semidirect
284. C₉	#1904	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×AGL(1,ℤ/8)	2
C₉×#518	2
C₉×#421	2
C₉×#418	2
C₉×#410	2
C₉×#344	2
C₉×H(3,ℤ/4)	2
C₃×#1904	3

Subgroups — C9×#1904 #11766625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1904 #11766625