Subgroups — C₉×#2051 #11766772

Order 1152.

423 conjugacy classes of subgroups.

The subgroup lattice has 351 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×#108	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×#408	C₂	1	1	yes	maximal, characteristic, complemented
7	C₉×#343	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#2051	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	—	2	1	no	—
14	C₂×C₉×D₈	—	2	1	no	—
15	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
19	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₉×#108	—	2	2	no	—
21	C₉×#103	—	2	1	no	—
22	C₉×#102	—	2	1	no	—
23	C₉×#102	—	2	2	no	—
24	C₉×#100	—	2	2	no	—
25	C₉×C₂²⋊C₈	—	2	2	no	—
26	C₂×C₃×AGL(1,ℤ/8)	C₂×C₃	1	1	yes	characteristic
27	C₂×C₃×#108	C₂×C₃	1	1	yes	characteristic
28	C₃×#417	C₂×C₃	1	1	yes	characteristic
29	C₂×C₃×#102	C₂×C₃	1	1	yes	characteristic
30	C₂×C₃×#100	C₂×C₃	1	1	yes	characteristic
31	C₃×#408	C₂×C₃	1	1	yes	characteristic
32	C₃×#343	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	D₄	1	1	yes	characteristic
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	—	2	1	no	—
44	C₂×D₄×C₉	—	2	1	no	—
45	C₂×D₄×C₉	D₄	1	1	yes	characteristic
46	C₂×D₄×C₉	D₄	1	2	yes	—
47	C₂²×C₄×C₉	—	2	1	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	C₉×M₁₆	—	2	1	no	—
57	C₉×M₁₆	D₄	1	2	yes	—
58	C₂×C₈×C₉	—	2	1	no	—
59	C₂×C₈×C₉	—	2	1	no	—
60	C₂×C₈×C₉	—	2	1	no	—
61	C₂×C₈×C₉	—	2	1	no	—
62	C₂×C₈×C₉	—	2	1	no	—
63	C₂×C₈×C₉	D₄	1	2	yes	—
64	C₉×C₄⋊C₄	—	2	1	no	—
65	C₉×C₄⋊C₄	—	2	1	no	—
66	#2051	C₉	1	1	yes	characteristic, direct product factor, Sylow
67	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
68	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
69	C₃×AGL(1,ℤ/8)	—	2	2	no	—
70	C₃×AGL(1,ℤ/8)	—	2	2	no	—
71	C₂×C₃×SD₁₆	—	2	1	no	—
72	C₂×C₃×D₈	—	2	1	no	—
73	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
74	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
75	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
76	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
78	C₃×#108	—	2	2	no	—
79	C₃×#103	—	2	1	no	—
80	C₃×#102	—	2	1	no	—
81	C₃×#102	—	2	2	no	—
82	C₃×#100	—	2	2	no	—
83	C₃×C₂²⋊C₈	—	2	2	no	—
84	C₂³×C₉	—	2	1	no	—
85	C₂³×C₉	—	2	2	no	—
86	C₂³×C₉	—	4	1	no	—
87	C₂³×C₉	—	4	1	no	—
88	C₂³×C₉	—	8	1	no	—
89	C₂³×C₉	—	8	1	no	—
90	C₂³×C₉	#41	1	1	yes	characteristic
91	Q₈×C₉	—	2	1	no	—
92	Q₈×C₉	—	2	1	no	—
93	D₄×C₉	—	2	1	no	—
94	D₄×C₉	—	2	2	no	—
95	D₄×C₉	—	2	2	no	—
96	D₄×C₉	—	2	1	no	—
97	D₄×C₉	—	4	1	no	—
98	D₄×C₉	—	4	1	no	—
99	D₄×C₉	—	4	1	no	—
100	D₄×C₉	—	4	1	no	—
101	C₂×C₄×C₉	—	2	1	no	—
102	C₂×C₄×C₉	—	4	1	no	—
103	C₂×C₄×C₉	—	4	1	no	—
104	C₂×C₄×C₉	—	4	1	no	—
105	C₂×C₄×C₉	—	8	1	no	—
106	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₉	#41	1	1	yes	characteristic
111	C₂×C₄×C₉	#41	1	1	yes	characteristic
112	C₈×C₉	—	2	2	no	—
113	C₈×C₉	—	2	2	no	—
114	C₈×C₉	—	4	1	no	—
115	C₈×C₉	—	4	1	no	—
116	C₂×AGL(1,ℤ/8)	C₂×C₉	1	1	yes	characteristic
117	C₂×#108	C₂×C₉	1	1	yes	characteristic, complemented
118	#417	C₂×C₉	1	1	yes	characteristic, complemented
119	C₂×#102	C₂×C₉	1	1	yes	characteristic, complemented
120	C₂×#100	C₂×C₉	1	1	yes	characteristic, complemented
121	#408	C₂×C₉	1	1	yes	characteristic, complemented
122	#343	C₂×C₉	1	1	yes	characteristic, complemented
123	C₂⁴×C₃	—	2	1	no	—
124	C₃×#41	—	2	1	no	—
125	C₃×#41	—	2	1	no	—
126	C₃×#41	—	2	1	no	—
127	C₃×#41	—	2	1	no	—
128	C₂×C₃×Q₈	C₃×D₄	1	1	yes	characteristic
129	C₂×C₃×D₄	—	2	1	no	—
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	—	2	1	no	—
135	C₂×C₃×D₄	C₃×D₄	1	2	yes	—
136	C₂×C₃×D₄	C₃×D₄	1	1	yes	characteristic
137	C₂²×C₃×C₄	—	2	1	no	—
138	C₂²×C₃×C₄	—	2	1	no	—
139	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
140	C₃×SD₁₆	—	2	4	no	—
141	C₃×D₈	—	2	4	no	—
142	C₃×M₁₆	—	2	1	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	C₃×M₁₆	C₃×D₄	1	2	yes	—
148	C₂×C₃×C₈	—	2	1	no	—
149	C₂×C₃×C₈	—	2	1	no	—
150	C₂×C₃×C₈	—	2	1	no	—
151	C₂×C₃×C₈	—	2	1	no	—
152	C₂×C₃×C₈	—	2	1	no	—
153	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
154	C₃×C₄⋊C₄	—	2	1	no	—
155	C₃×C₄⋊C₄	—	2	1	no	—
156	C₂²×C₉	—	2	1	no	—
157	C₂²×C₉	—	2	1	no	—
158	C₂²×C₉	—	4	1	no	—
159	C₂²×C₉	—	4	2	no	—
160	C₂²×C₉	—	4	1	no	—
161	C₂²×C₉	—	4	1	no	—
162	C₂²×C₉	—	8	2	no	—
163	C₂²×C₉	—	8	2	no	—
164	C₂²×C₉	—	8	1	no	—
165	C₂²×C₉	—	8	1	no	—
166	C₂²×C₉	#121	1	1	yes	characteristic
167	C₂²×C₉	#123	1	1	yes	characteristic
168	C₂²×C₉	#124	1	1	yes	characteristic
169	C₄×C₉	—	4	1	no	—
170	C₄×C₉	—	8	1	no	—
171	C₄×C₉	#120	1	1	yes	characteristic
172	C₄×C₉	#121	1	1	yes	characteristic
173	C₄×C₉	#121	1	1	yes	characteristic
174	C₄×C₉	#123	1	1	yes	characteristic
175	C₂×#41	C₂²×C₉	1	1	yes	characteristic
176	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
177	AGL(1,ℤ/8)	—	2	2	no	—
178	AGL(1,ℤ/8)	—	2	2	no	—
179	C₂×SD₁₆	—	2	1	no	—
180	C₂×D₈	—	2	1	no	—
181	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
182	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
183	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
184	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
186	#108	—	2	2	no	—
187	#103	—	2	1	no	—
188	#102	—	2	1	no	—
189	#102	—	2	2	no	—
190	#100	—	2	2	no	—
191	C₂²⋊C₈	—	2	2	no	—
192	C₂³×C₃	—	2	1	no	—
193	C₂³×C₃	—	2	2	no	—
194	C₂³×C₃	—	4	1	no	—
195	C₂³×C₃	—	4	1	no	—
196	C₂³×C₃	—	8	1	no	—
197	C₂³×C₃	—	8	1	no	—
198	C₂³×C₃	C₃×#41	1	1	yes	characteristic
199	C₃×Q₈	—	2	1	no	—
200	C₃×Q₈	—	2	1	no	—
201	C₃×D₄	—	2	1	no	—
202	C₃×D₄	—	2	2	no	—
203	C₃×D₄	—	2	2	no	—
204	C₃×D₄	—	2	1	no	—
205	C₃×D₄	—	4	1	no	—
206	C₃×D₄	—	4	1	no	—
207	C₃×D₄	—	4	1	no	—
208	C₃×D₄	—	4	1	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	4	1	no	—
211	C₂×C₃×C₄	—	4	1	no	—
212	C₂×C₃×C₄	—	4	1	no	—
213	C₂×C₃×C₄	—	8	1	no	—
214	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
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	C₃×#41	1	1	yes	characteristic
219	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
220	C₃×C₈	—	2	2	no	—
221	C₃×C₈	—	2	2	no	—
222	C₃×C₈	—	4	1	no	—
223	C₃×C₈	—	4	1	no	—
224	C₂×C₉	—	2	1	no	—
225	C₂×C₉	—	2	1	no	—
226	C₂×C₉	—	8	1	no	—
227	C₂×C₉	—	8	2	no	—
228	C₂×C₉	#394	1	1	yes	characteristic
229	C₂×C₉	#471	1	1	yes	characteristic
230	C₂×C₉	#472	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	D₄×C₉	1	1	yes	characteristic
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	—	2	1	no	—
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	D₄×C₉	1	2	yes	—
244	C₂×D₄	D₄×C₉	1	1	yes	characteristic
245	C₂²×C₄	—	2	1	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	M₁₆	—	2	1	no	—
255	M₁₆	D₄×C₉	1	2	yes	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	D₄×C₉	1	2	yes	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	—	2	1	no	—
266	C₂²×C₃	—	4	1	no	—
267	C₂²×C₃	—	4	2	no	—
268	C₂²×C₃	—	4	1	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	—	8	1	no	—
274	C₂²×C₃	C₃×#121	1	1	yes	characteristic
275	C₂²×C₃	C₃×#123	1	1	yes	characteristic
276	C₂²×C₃	C₃×#124	1	1	yes	characteristic
277	C₃×C₄	—	4	1	no	—
278	C₃×C₄	—	8	1	no	—
279	C₃×C₄	C₃×#120	1	1	yes	characteristic
280	C₃×C₄	C₃×#121	1	1	yes	characteristic
281	C₃×C₄	C₃×#121	1	1	yes	characteristic
282	C₃×C₄	C₃×#123	1	1	yes	characteristic
283	C₉	#2051	1	1	yes	characteristic, direct product factor, Sylow
284	C₂³	—	2	1	no	—
285	C₂³	—	2	2	no	—
286	C₂³	—	4	1	no	—
287	C₂³	—	4	1	no	—
288	C₂³	—	8	1	no	—
289	C₂³	—	8	1	no	—
290	C₂³	C₉×#41	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	2	no	—
296	D₄	—	2	1	no	—
297	D₄	—	4	1	no	—
298	D₄	—	4	1	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	8	1	no	—
306	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
310	C₂×C₄	C₉×#41	1	1	yes	characteristic
311	C₂×C₄	C₉×#41	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	2	2	no	—
314	C₈	—	4	1	no	—
315	C₈	—	4	1	no	—
316	C₂×C₃	—	2	1	no	—
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	8	1	no	—
319	C₂×C₃	—	8	2	no	—
320	C₂×C₃	C₃×#394	1	1	yes	characteristic
321	C₂×C₃	C₃×#471	1	1	yes	characteristic
322	C₂×C₃	C₃×#472	1	1	yes	characteristic
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	1	no	—
332	C₂²	—	8	1	no	—
333	C₂²	C₉×#121	1	1	yes	characteristic
334	C₂²	C₉×#123	1	1	yes	characteristic
335	C₂²	C₉×#124	1	1	yes	characteristic
336	C₄	—	4	1	no	—
337	C₄	—	8	1	no	—
338	C₄	C₉×#120	1	1	yes	characteristic
339	C₄	C₉×#121	1	1	yes	characteristic
340	C₄	C₉×#121	1	1	yes	characteristic
341	C₄	C₉×#123	1	1	yes	characteristic
342	C₃	C₃×#2051	1	1	yes	characteristic
343	C₂	—	2	1	no	—
344	C₂	—	2	1	no	—
345	C₂	—	8	1	no	—
346	C₂	—	8	2	no	—
347	C₂	C₉×#394	1	1	yes	characteristic
348	C₂	C₉×#471	1	1	yes	characteristic
349	C₂	C₉×#472	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	#121
commutator	C₂²×C₄	C₂³×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 ▷ C₂²×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×#108	C₂	semidirect
3. C₉×#417	C₂	semidirect
4. C₂×C₉×#102	C₂	semidirect
5. C₂×C₉×#100	C₂	semidirect
6. C₉×#408	C₂	semidirect
7. C₉×#343	C₂	semidirect
66. #2051	C₉	direct (N × H)
117. C₂×#108	C₂×C₉	semidirect
118. #417	C₂×C₉	semidirect
119. C₂×#102	C₂×C₉	semidirect
120. C₂×#100	C₂×C₉	semidirect
121. #408	C₂×C₉	semidirect
122. #343	C₂×C₉	semidirect
283. C₉	#2051	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×#108	2
C₉×#417	2
C₂×C₉×#102	2
C₂×C₉×#100	2
C₉×#408	2
C₉×#343	2
C₃×#2051	3

Subgroups — C9×#2051 #11766772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2051 #11766772