Subgroups — C₉×#1907 #11766628

Order 1152.

498 conjugacy classes of subgroups.

The subgroup lattice has 372 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	C₂×C₉×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#417	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₉×#102	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₉×#100	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#343	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#341	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#1907	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)	C₄	1	4	yes	complemented
13	C₂×C₉×SD₁₆	—	2	1	no	—
14	C₂×C₉×D₈	—	2	1	no	—
15	C₉×#131	—	2	2	no	—
16	C₉×#131	—	2	2	no	—
17	C₂×C₉×M₁₆	—	2	1	no	—
18	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₉	—	2	1	no	—
22	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
23	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₉×#103	—	2	1	no	—
25	C₉×#102	—	2	1	no	—
26	C₉×#102	—	2	1	no	—
27	C₉×#102	—	2	1	no	—
28	C₉×#100	—	2	2	no	—
29	C₂×C₃×AGL(1,ℤ/8)	C₂×C₃	1	1	yes	characteristic
30	C₂×C₃×#131	C₂×C₃	1	1	yes	characteristic
31	C₃×#417	C₂×C₃	1	1	yes	characteristic
32	C₂×C₃×#102	C₂×C₃	1	1	yes	characteristic
33	C₂×C₃×#100	C₂×C₃	1	1	yes	characteristic
34	C₃×#343	C₂×C₃	1	1	yes	characteristic
35	C₃×#341	C₂×C₃	1	1	yes	characteristic
36	C₂⁴×C₉	—	2	1	no	—
37	C₉×#41	—	2	1	no	—
38	C₉×#41	—	2	1	no	—
39	C₉×#41	C₂×C₄	1	2	yes	—
40	C₉×#41	D₄	1	2	yes	—
41	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
42	C₂×D₄×C₉	—	2	1	no	—
43	C₂×D₄×C₉	—	2	1	no	—
44	C₂×D₄×C₉	—	2	1	no	—
45	C₂×D₄×C₉	—	2	1	no	—
46	C₂×D₄×C₉	—	2	1	no	—
47	C₂×D₄×C₉	—	2	1	no	—
48	C₂×D₄×C₉	C₂×C₄	1	2	yes	—
49	C₂×D₄×C₉	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₉	—	2	1	no	—
51	C₂²×C₄×C₉	—	2	1	no	—
52	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
53	C₉×SD₁₆	—	2	4	no	—
54	C₉×D₈	—	2	4	no	—
55	C₉×M₁₆	—	2	4	no	—
56	C₉×M₁₆	—	2	1	no	—
57	C₉×M₁₆	—	2	1	no	—
58	C₉×M₁₆	—	2	1	no	—
59	C₉×M₁₆	—	2	1	no	—
60	C₉×M₁₆	C₂×C₄	1	2	yes	—
61	C₉×M₁₆	D₄	1	2	yes	—
62	C₂×C₈×C₉	—	2	4	no	—
63	C₂×C₈×C₉	—	2	1	no	—
64	C₂×C₈×C₉	—	2	1	no	—
65	C₂×C₈×C₉	—	2	1	no	—
66	C₂×C₈×C₉	—	2	1	no	—
67	C₂×C₈×C₉	—	2	1	no	—
68	C₂×C₈×C₉	D₄	1	1	yes	characteristic
69	C₂×C₈×C₉	D₄	1	1	yes	characteristic
70	C₉×C₄⋊C₄	—	2	1	no	—
71	C₉×C₄⋊C₄	—	2	1	no	—
72	#1907	C₉	1	1	yes	characteristic, direct product factor, Sylow
73	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
74	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
75	C₃×AGL(1,ℤ/8)	—	2	2	no	—
76	C₃×AGL(1,ℤ/8)	C₃×C₄	1	4	yes	—
77	C₂×C₃×SD₁₆	—	2	1	no	—
78	C₂×C₃×D₈	—	2	1	no	—
79	C₃×#131	—	2	2	no	—
80	C₃×#131	—	2	2	no	—
81	C₂×C₃×M₁₆	—	2	1	no	—
82	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
84	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
85	C₂²×C₃×C₈	—	2	1	no	—
86	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
87	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
88	C₃×#103	—	2	1	no	—
89	C₃×#102	—	2	1	no	—
90	C₃×#102	—	2	1	no	—
91	C₃×#102	—	2	1	no	—
92	C₃×#100	—	2	2	no	—
93	C₂³×C₉	—	2	1	no	—
94	C₂³×C₉	—	2	2	no	—
95	C₂³×C₉	—	4	1	no	—
96	C₂³×C₉	—	4	1	no	—
97	C₂³×C₉	—	8	1	no	—
98	C₂³×C₉	—	8	1	no	—
99	C₂³×C₉	#41	1	1	yes	characteristic
100	Q₈×C₉	—	2	1	no	—
101	Q₈×C₉	C₂²⋊C₄	1	2	yes	—
102	D₄×C₉	—	2	2	no	—
103	D₄×C₉	—	2	2	no	—
104	D₄×C₉	—	2	2	no	—
105	D₄×C₉	—	2	2	no	—
106	D₄×C₉	—	2	1	no	—
107	D₄×C₉	—	4	1	no	—
108	D₄×C₉	—	4	1	no	—
109	D₄×C₉	C₂²⋊C₄	1	2	yes	—
110	C₂×C₄×C₉	—	2	1	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	—	8	1	no	—
115	C₂×C₄×C₉	C₂²×C₄	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	#41	1	1	yes	characteristic
121	C₈×C₉	—	2	2	no	—
122	C₈×C₉	—	2	1	no	—
123	C₈×C₉	—	2	2	no	—
124	C₈×C₉	—	2	1	no	—
125	C₈×C₉	—	4	1	no	—
126	C₂×AGL(1,ℤ/8)	C₂×C₉	1	1	yes	characteristic
127	C₂×#131	C₂×C₉	1	1	yes	characteristic, complemented
128	#417	C₂×C₉	1	1	yes	characteristic, complemented
129	C₂×#102	C₂×C₉	1	1	yes	characteristic, complemented
130	C₂×#100	C₂×C₉	1	1	yes	characteristic, complemented
131	#343	C₂×C₉	1	1	yes	characteristic, complemented
132	#341	C₂×C₉	1	1	yes	characteristic, complemented
133	C₂⁴×C₃	—	2	1	no	—
134	C₃×#41	—	2	1	no	—
135	C₃×#41	—	2	1	no	—
136	C₃×#41	C₂×C₃×C₄	1	2	yes	—
137	C₃×#41	C₃×D₄	1	2	yes	—
138	C₂×C₃×Q₈	C₃×D₄	1	1	yes	characteristic
139	C₂×C₃×D₄	—	2	1	no	—
140	C₂×C₃×D₄	—	2	1	no	—
141	C₂×C₃×D₄	—	2	1	no	—
142	C₂×C₃×D₄	—	2	1	no	—
143	C₂×C₃×D₄	—	2	1	no	—
144	C₂×C₃×D₄	—	2	1	no	—
145	C₂×C₃×D₄	C₂×C₃×C₄	1	2	yes	—
146	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
147	C₂²×C₃×C₄	—	2	1	no	—
148	C₂²×C₃×C₄	—	2	1	no	—
149	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
150	C₃×SD₁₆	—	2	4	no	—
151	C₃×D₈	—	2	4	no	—
152	C₃×M₁₆	—	2	4	no	—
153	C₃×M₁₆	—	2	1	no	—
154	C₃×M₁₆	—	2	1	no	—
155	C₃×M₁₆	—	2	1	no	—
156	C₃×M₁₆	—	2	1	no	—
157	C₃×M₁₆	C₂×C₃×C₄	1	2	yes	—
158	C₃×M₁₆	C₃×D₄	1	2	yes	—
159	C₂×C₃×C₈	—	2	4	no	—
160	C₂×C₃×C₈	—	2	1	no	—
161	C₂×C₃×C₈	—	2	1	no	—
162	C₂×C₃×C₈	—	2	1	no	—
163	C₂×C₃×C₈	—	2	1	no	—
164	C₂×C₃×C₈	—	2	1	no	—
165	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
166	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
167	C₃×C₄⋊C₄	—	2	1	no	—
168	C₃×C₄⋊C₄	—	2	1	no	—
169	C₂²×C₉	—	2	1	no	—
170	C₂²×C₉	—	2	1	no	—
171	C₂²×C₉	—	2	2	no	—
172	C₂²×C₉	—	4	1	no	—
173	C₂²×C₉	—	4	1	no	—
174	C₂²×C₉	—	4	2	no	—
175	C₂²×C₉	—	8	2	no	—
176	C₂²×C₉	—	8	2	no	—
177	C₂²×C₉	—	8	1	no	—
178	C₂²×C₉	—	8	1	no	—
179	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
180	C₂²×C₉	#121	1	1	yes	characteristic
181	C₂²×C₉	#123	1	1	yes	characteristic
182	C₄×C₉	—	2	2	no	—
183	C₄×C₉	—	8	1	no	—
184	C₄×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₄×C₉	C₄×D₄	1	1	yes	characteristic
186	C₄×C₉	#120	1	1	yes	characteristic
187	C₄×C₉	#121	1	1	yes	characteristic
188	C₂×#41	C₂²×C₉	1	1	yes	characteristic
189	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
190	AGL(1,ℤ/8)	—	2	2	no	—
191	AGL(1,ℤ/8)	C₄×C₉	1	4	yes	complemented
192	C₂×SD₁₆	—	2	1	no	—
193	C₂×D₈	—	2	1	no	—
194	#131	—	2	2	no	—
195	#131	—	2	2	no	—
196	C₂×M₁₆	—	2	1	no	—
197	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
198	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
199	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
202	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
203	#103	—	2	1	no	—
204	#102	—	2	1	no	—
205	#102	—	2	1	no	—
206	#102	—	2	1	no	—
207	#100	—	2	2	no	—
208	C₂³×C₃	—	2	1	no	—
209	C₂³×C₃	—	2	2	no	—
210	C₂³×C₃	—	4	1	no	—
211	C₂³×C₃	—	4	1	no	—
212	C₂³×C₃	—	8	1	no	—
213	C₂³×C₃	—	8	1	no	—
214	C₂³×C₃	C₃×#41	1	1	yes	characteristic
215	C₃×Q₈	—	2	1	no	—
216	C₃×Q₈	C₃×C₂²⋊C₄	1	2	yes	—
217	C₃×D₄	—	2	2	no	—
218	C₃×D₄	—	2	2	no	—
219	C₃×D₄	—	2	2	no	—
220	C₃×D₄	—	2	2	no	—
221	C₃×D₄	—	2	1	no	—
222	C₃×D₄	—	4	1	no	—
223	C₃×D₄	—	4	1	no	—
224	C₃×D₄	C₃×C₂²⋊C₄	1	2	yes	—
225	C₂×C₃×C₄	—	2	1	no	—
226	C₂×C₃×C₄	—	2	2	no	—
227	C₂×C₃×C₄	—	2	2	no	—
228	C₂×C₃×C₄	—	4	1	no	—
229	C₂×C₃×C₄	—	8	1	no	—
230	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
235	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
236	C₃×C₈	—	2	2	no	—
237	C₃×C₈	—	2	1	no	—
238	C₃×C₈	—	2	2	no	—
239	C₃×C₈	—	2	1	no	—
240	C₃×C₈	—	4	1	no	—
241	C₂×C₉	—	2	1	no	—
242	C₂×C₉	—	2	1	no	—
243	C₂×C₉	—	4	2	no	—
244	C₂×C₉	—	8	2	no	—
245	C₂×C₉	#386	1	1	yes	characteristic
246	C₂×C₉	#471	1	1	yes	characteristic
247	C₂×C₉	#472	1	1	yes	characteristic
248	C₂⁴	—	2	1	no	—
249	#41	—	2	1	no	—
250	#41	—	2	1	no	—
251	#41	C₂×C₄×C₉	1	2	yes	—
252	#41	D₄×C₉	1	2	yes	—
253	C₂×Q₈	D₄×C₉	1	1	yes	characteristic
254	C₂×D₄	—	2	1	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	C₂×C₄×C₉	1	2	yes	—
261	C₂×D₄	D₄×C₉	1	1	yes	characteristic
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
265	SD₁₆	—	2	4	no	—
266	D₈	—	2	4	no	—
267	M₁₆	—	2	4	no	—
268	M₁₆	—	2	1	no	—
269	M₁₆	—	2	1	no	—
270	M₁₆	—	2	1	no	—
271	M₁₆	—	2	1	no	—
272	M₁₆	C₂×C₄×C₉	1	2	yes	—
273	M₁₆	D₄×C₉	1	2	yes	—
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	2	1	no	—
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	D₄×C₉	1	1	yes	characteristic
281	C₂×C₈	D₄×C₉	1	1	yes	characteristic
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	1	no	—
284	C₂²×C₃	—	2	1	no	—
285	C₂²×C₃	—	2	1	no	—
286	C₂²×C₃	—	2	2	no	—
287	C₂²×C₃	—	4	1	no	—
288	C₂²×C₃	—	4	1	no	—
289	C₂²×C₃	—	4	2	no	—
290	C₂²×C₃	—	8	2	no	—
291	C₂²×C₃	—	8	2	no	—
292	C₂²×C₃	—	8	1	no	—
293	C₂²×C₃	—	8	1	no	—
294	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
295	C₂²×C₃	C₃×#121	1	1	yes	characteristic
296	C₂²×C₃	C₃×#123	1	1	yes	characteristic
297	C₃×C₄	—	2	2	no	—
298	C₃×C₄	—	8	1	no	—
299	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
300	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
301	C₃×C₄	C₃×#120	1	1	yes	characteristic
302	C₃×C₄	C₃×#121	1	1	yes	characteristic
303	C₉	#1907	1	1	yes	characteristic, direct product factor, Sylow
304	C₂³	—	2	1	no	—
305	C₂³	—	2	2	no	—
306	C₂³	—	4	1	no	—
307	C₂³	—	4	1	no	—
308	C₂³	—	8	1	no	—
309	C₂³	—	8	1	no	—
310	C₂³	C₉×#41	1	1	yes	characteristic
311	Q₈	—	2	1	no	—
312	Q₈	C₉×C₂²⋊C₄	1	2	yes	—
313	D₄	—	2	2	no	—
314	D₄	—	2	2	no	—
315	D₄	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	2	1	no	—
318	D₄	—	4	1	no	—
319	D₄	—	4	1	no	—
320	D₄	C₉×C₂²⋊C₄	1	2	yes	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	8	1	no	—
326	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
327	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
329	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
330	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
331	C₂×C₄	C₉×#41	1	1	yes	characteristic
332	C₈	—	2	2	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	2	no	—
335	C₈	—	2	1	no	—
336	C₈	—	4	1	no	—
337	C₂×C₃	—	2	1	no	—
338	C₂×C₃	—	2	1	no	—
339	C₂×C₃	—	4	2	no	—
340	C₂×C₃	—	8	2	no	—
341	C₂×C₃	C₃×#386	1	1	yes	characteristic
342	C₂×C₃	C₃×#471	1	1	yes	characteristic
343	C₂×C₃	C₃×#472	1	1	yes	characteristic
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	2	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	2	no	—
350	C₂²	—	8	2	no	—
351	C₂²	—	8	2	no	—
352	C₂²	—	8	1	no	—
353	C₂²	—	8	1	no	—
354	C₂²	C₄×D₄×C₉	1	1	yes	characteristic
355	C₂²	C₉×#121	1	1	yes	characteristic
356	C₂²	C₉×#123	1	1	yes	characteristic
357	C₄	—	2	2	no	—
358	C₄	—	8	1	no	—
359	C₄	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
360	C₄	C₄×D₄×C₉	1	1	yes	characteristic
361	C₄	C₉×#120	1	1	yes	characteristic
362	C₄	C₉×#121	1	1	yes	characteristic
363	C₃	C₃×#1907	1	1	yes	characteristic
364	C₂	—	2	1	no	—
365	C₂	—	2	1	no	—
366	C₂	—	4	2	no	—
367	C₂	—	8	2	no	—
368	C₂	C₉×#386	1	1	yes	characteristic
369	C₂	C₉×#471	1	1	yes	characteristic
370	C₂	C₉×#472	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	#121
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂²×C₃	C₃×#121
prime radical	C₂×C₃×AGL(1,ℤ/8)	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×#131	C₂	semidirect
3. C₉×#417	C₂	semidirect
4. C₂×C₉×#102	C₂	semidirect
5. C₂×C₉×#100	C₂	semidirect
6. C₉×#343	C₂	semidirect
7. C₉×#341	C₂	semidirect
12. C₉×AGL(1,ℤ/8)	C₄	semidirect
72. #1907	C₉	direct (N × H)
127. C₂×#131	C₂×C₉	semidirect
128. #417	C₂×C₉	semidirect
129. C₂×#102	C₂×C₉	semidirect
130. C₂×#100	C₂×C₉	semidirect
131. #343	C₂×C₉	semidirect
132. #341	C₂×C₉	semidirect
191. AGL(1,ℤ/8)	C₄×C₉	semidirect
303. C₉	#1907	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
C₂×C₉×#131	2
C₉×#417	2
C₂×C₉×#102	2
C₂×C₉×#100	2
C₉×#343	2
C₉×#341	2
C₃×#1907	3

Subgroups — C9×#1907 #11766628

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1907 #11766628