Subgroups — C₉×#1872 #11766593

Order 1152.

606 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₉×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₉×#103	C₂	1	2	yes	maximal, complemented
4	C₂×C₉×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₉×#340	C₂	1	2	yes	maximal, complemented
6	C₃×#1872	C₃	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₉	—	2	1	no	—
9	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉	C₂²	1	2	yes	—
11	C₉×C₂²⋊Q₈	—	2	2	no	—
12	C₉×C₂²⋊Q₈	C₄	1	4	yes	complemented
13	C₂×C₉×C₄⋊C₄	—	2	1	no	—
14	C₂×C₉×C₄⋊C₄	—	2	1	no	—
15	C₂×C₉×C₄⋊C₄	—	2	4	no	—
16	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₉×C₂²⋊C₄	—	2	1	no	—
19	C₉×#103	—	2	4	no	—
20	C₉×C₂²⋊C₈	—	2	2	no	—
21	C₂×C₃×C₂²⋊Q₈	C₂×C₃	1	1	yes	characteristic
22	C₂²×C₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×#103	C₂×C₃	1	2	yes	—
24	C₂×C₃×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
25	C₃×#340	C₂×C₃	1	2	yes	—
26	C₂⁴×C₉	D₄	1	1	yes	characteristic
27	C₂×Q₈×C₉	—	2	1	no	—
28	C₂×Q₈×C₉	—	2	2	no	—
29	C₂×Q₈×C₉	—	2	2	no	—
30	C₂×Q₈×C₉	C₂×C₄	1	2	yes	complemented
31	C₂²×C₄×C₉	—	2	2	no	—
32	C₂²×C₄×C₉	—	2	2	no	—
33	C₂²×C₄×C₉	—	2	4	no	—
34	C₂²×C₄×C₉	—	2	1	no	—
35	C₂²×C₄×C₉	—	2	1	no	—
36	C₂²×C₄×C₉	—	2	2	no	—
37	C₂²×C₄×C₉	—	4	2	no	—
38	C₂²×C₄×C₉	—	4	2	no	—
39	C₂²×C₄×C₉	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
41	C₂²×C₄×C₉	D₄	1	2	yes	—
42	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₉	—	2	2	no	—
44	C₂×C₈×C₉	—	2	2	no	—
45	C₂×C₈×C₉	—	2	2	no	—
46	C₉×C₄⋊C₄	—	2	4	no	—
47	C₉×C₄⋊C₄	—	2	4	no	—
48	C₉×C₄⋊C₄	—	2	2	no	—
49	C₉×C₄⋊C₄	—	2	1	no	—
50	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	complemented
51	C₉×C₄⋊C₄	D₄	1	4	yes	—
52	C₉×C₂²⋊C₄	—	2	4	no	—
53	#1872	C₉	1	1	yes	characteristic, direct product factor, Sylow
54	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
55	C₂³×C₃×C₄	—	2	1	no	—
56	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
57	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
58	C₃×C₂²⋊Q₈	—	2	2	no	—
59	C₃×C₂²⋊Q₈	C₃×C₄	1	4	yes	—
60	C₂×C₃×C₄⋊C₄	—	2	1	no	—
61	C₂×C₃×C₄⋊C₄	—	2	1	no	—
62	C₂×C₃×C₄⋊C₄	—	2	4	no	—
63	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
65	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
66	C₃×#103	—	2	4	no	—
67	C₃×C₂²⋊C₈	—	2	2	no	—
68	C₂³×C₉	—	2	1	no	—
69	C₂³×C₉	—	2	1	no	—
70	C₂³×C₉	—	2	1	no	—
71	C₂³×C₉	—	2	1	no	—
72	C₂³×C₉	C₂²⋊C₄	1	2	yes	—
73	C₂³×C₉	SD₁₆	1	2	yes	—
74	C₂³×C₉	Q₁₆	1	2	yes	—
75	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
76	Q₈×C₉	—	2	4	no	—
77	Q₈×C₉	—	4	2	no	—
78	C₂×C₄×C₉	—	2	1	no	—
79	C₂×C₄×C₉	—	2	1	no	—
80	C₂×C₄×C₉	—	2	4	no	—
81	C₂×C₄×C₉	—	2	4	no	—
82	C₂×C₄×C₉	—	2	2	no	—
83	C₂×C₄×C₉	—	2	2	no	—
84	C₂×C₄×C₉	—	2	4	no	—
85	C₂×C₄×C₉	—	4	8	no	—
86	C₂×C₄×C₉	—	4	1	no	—
87	C₂×C₄×C₉	—	4	1	no	—
88	C₂×C₄×C₉	—	4	2	no	—
89	C₂×C₄×C₉	—	4	1	no	—
90	C₂×C₄×C₉	—	4	2	no	—
91	C₂×C₄×C₉	—	4	1	no	—
92	C₂×C₄×C₉	C₂²⋊C₄	1	2	yes	—
93	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₉	#41	1	2	yes	—
97	C₈×C₉	—	4	2	no	—
98	C₂×C₂²⋊Q₈	C₂×C₉	1	1	yes	characteristic
99	C₂²×C₄⋊C₄	C₂×C₉	1	1	yes	characteristic
100	C₂×#103	C₂×C₉	1	2	yes	complemented
101	C₂×C₂²⋊C₈	C₂×C₉	1	1	yes	characteristic
102	#340	C₂×C₉	1	2	yes	complemented
103	C₂⁴×C₃	C₃×D₄	1	1	yes	characteristic
104	C₂×C₃×Q₈	—	2	1	no	—
105	C₂×C₃×Q₈	—	2	2	no	—
106	C₂×C₃×Q₈	—	2	2	no	—
107	C₂×C₃×Q₈	C₂×C₃×C₄	1	2	yes	—
108	C₂²×C₃×C₄	—	2	2	no	—
109	C₂²×C₃×C₄	—	2	2	no	—
110	C₂²×C₃×C₄	—	2	4	no	—
111	C₂²×C₃×C₄	—	2	1	no	—
112	C₂²×C₃×C₄	—	2	1	no	—
113	C₂²×C₃×C₄	—	2	2	no	—
114	C₂²×C₃×C₄	—	4	2	no	—
115	C₂²×C₃×C₄	—	4	2	no	—
116	C₂²×C₃×C₄	C₂×C₃×C₄	1	2	yes	—
117	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
119	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
120	C₂×C₃×C₈	—	2	2	no	—
121	C₂×C₃×C₈	—	2	2	no	—
122	C₂×C₃×C₈	—	2	2	no	—
123	C₃×C₄⋊C₄	—	2	4	no	—
124	C₃×C₄⋊C₄	—	2	4	no	—
125	C₃×C₄⋊C₄	—	2	2	no	—
126	C₃×C₄⋊C₄	—	2	1	no	—
127	C₃×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
128	C₃×C₄⋊C₄	C₃×D₄	1	4	yes	—
129	C₃×C₂²⋊C₄	—	2	4	no	—
130	C₂²×C₉	—	2	2	no	—
131	C₂²×C₉	—	2	2	no	—
132	C₂²×C₉	—	2	2	no	—
133	C₂²×C₉	—	2	2	no	—
134	C₂²×C₉	—	2	2	no	—
135	C₂²×C₉	—	2	2	no	—
136	C₂²×C₉	#103	1	4	yes	complemented
137	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₉	#120	1	1	yes	characteristic
139	C₂²×C₉	#123	1	1	yes	characteristic
140	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
141	C₂²×C₉	C₂×Q₁₆	1	1	yes	characteristic
142	C₂²×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂²×C₉	#137	1	1	yes	characteristic
144	C₄×C₉	—	2	2	no	—
145	C₄×C₉	—	4	2	no	—
146	C₄×C₉	—	4	2	no	—
147	C₄×C₉	—	4	4	no	—
148	C₄×C₉	C₄×D₄	1	2	yes	—
149	C₄×C₉	#121	1	2	yes	—
150	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
151	C₂³×C₄	—	2	1	no	—
152	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
153	C₂²×C₈	C₂²×C₉	1	2	yes	—
154	C₂²⋊Q₈	—	2	2	no	—
155	C₂²⋊Q₈	C₄×C₉	1	4	yes	complemented
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	—	2	4	no	—
159	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
160	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
161	C₂×C₂²⋊C₄	—	2	1	no	—
162	#103	—	2	4	no	—
163	C₂²⋊C₈	—	2	2	no	—
164	C₂³×C₃	—	2	1	no	—
165	C₂³×C₃	—	2	1	no	—
166	C₂³×C₃	—	2	1	no	—
167	C₂³×C₃	—	2	1	no	—
168	C₂³×C₃	C₃×C₂²⋊C₄	1	2	yes	—
169	C₂³×C₃	C₃×SD₁₆	1	2	yes	—
170	C₂³×C₃	C₃×Q₁₆	1	2	yes	—
171	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
172	C₃×Q₈	—	2	4	no	—
173	C₃×Q₈	—	4	2	no	—
174	C₂×C₃×C₄	—	2	1	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	2	4	no	—
177	C₂×C₃×C₄	—	2	4	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	4	no	—
181	C₂×C₃×C₄	—	4	8	no	—
182	C₂×C₃×C₄	—	4	1	no	—
183	C₂×C₃×C₄	—	4	1	no	—
184	C₂×C₃×C₄	—	4	2	no	—
185	C₂×C₃×C₄	—	4	1	no	—
186	C₂×C₃×C₄	—	4	2	no	—
187	C₂×C₃×C₄	—	4	1	no	—
188	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	2	yes	—
189	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
191	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
192	C₂×C₃×C₄	C₃×#41	1	2	yes	—
193	C₃×C₈	—	4	2	no	—
194	C₂×C₉	—	2	4	no	—
195	C₂×C₉	#386	1	1	yes	characteristic
196	C₂×C₉	C₂×#103	1	1	yes	characteristic
197	C₂×C₉	#417	1	1	yes	characteristic
198	C₂×C₉	#450	1	1	yes	characteristic
199	C₂×C₉	#451	1	1	yes	characteristic
200	C₂×C₉	#483	1	1	yes	characteristic
201	C₂×C₉	#484	1	1	yes	characteristic
202	C₂⁴	D₄×C₉	1	1	yes	characteristic
203	C₂×Q₈	—	2	1	no	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	C₂×C₄×C₉	1	2	yes	complemented
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	C₂×C₄×C₉	1	2	yes	—
216	C₂²×C₄	D₄×C₉	1	1	yes	characteristic
217	C₂²×C₄	D₄×C₉	1	2	yes	—
218	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	C₂×C₄×C₉	1	2	yes	complemented
227	C₄⋊C₄	D₄×C₉	1	4	yes	—
228	C₂²⋊C₄	—	2	4	no	—
229	C₂²×C₃	—	2	2	no	—
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	C₃×#103	1	4	yes	—
236	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
237	C₂²×C₃	C₃×#120	1	1	yes	characteristic
238	C₂²×C₃	C₃×#123	1	1	yes	characteristic
239	C₂²×C₃	C₂×C₃×SD₁₆	1	1	yes	characteristic
240	C₂²×C₃	C₂×C₃×Q₁₆	1	1	yes	characteristic
241	C₂²×C₃	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₂²×C₃	C₃×#137	1	1	yes	characteristic
243	C₃×C₄	—	2	2	no	—
244	C₃×C₄	—	4	2	no	—
245	C₃×C₄	—	4	2	no	—
246	C₃×C₄	—	4	4	no	—
247	C₃×C₄	C₃×C₄×D₄	1	2	yes	—
248	C₃×C₄	C₃×#121	1	2	yes	—
249	C₉	#1872	1	1	yes	characteristic, direct product factor, Sylow
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	2	1	no	—
253	C₂³	—	2	1	no	—
254	C₂³	C₉×C₂²⋊C₄	1	2	yes	—
255	C₂³	C₉×SD₁₆	1	2	yes	—
256	C₂³	C₉×Q₁₆	1	2	yes	—
257	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
258	Q₈	—	2	4	no	—
259	Q₈	—	4	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	8	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	C₉×C₂²⋊C₄	1	2	yes	—
275	C₂×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
276	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
277	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
278	C₂×C₄	C₉×#41	1	2	yes	—
279	C₈	—	4	2	no	—
280	C₂×C₃	—	2	4	no	—
281	C₂×C₃	C₃×#386	1	1	yes	characteristic
282	C₂×C₃	C₂×C₃×#103	1	1	yes	characteristic
283	C₂×C₃	C₃×#417	1	1	yes	characteristic
284	C₂×C₃	C₃×#450	1	1	yes	characteristic
285	C₂×C₃	C₃×#451	1	1	yes	characteristic
286	C₂×C₃	C₃×#483	1	1	yes	characteristic
287	C₂×C₃	C₃×#484	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	2	no	—
294	C₂²	C₉×#103	1	4	yes	complemented
295	C₂²	C₂×C₉×C₂²⋊C₄	1	1	yes	characteristic
296	C₂²	C₉×#120	1	1	yes	characteristic
297	C₂²	C₉×#123	1	1	yes	characteristic
298	C₂²	C₂×C₉×SD₁₆	1	1	yes	characteristic
299	C₂²	C₂×C₉×Q₁₆	1	1	yes	characteristic
300	C₂²	C₉×AGL(1,ℤ/8)	1	1	yes	characteristic
301	C₂²	C₉×#137	1	1	yes	characteristic
302	C₄	—	2	2	no	—
303	C₄	—	4	2	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	4	no	—
306	C₄	C₄×D₄×C₉	1	2	yes	—
307	C₄	C₉×#121	1	2	yes	—
308	C₃	C₃×#1872	1	1	yes	characteristic
309	C₂	—	2	4	no	—
310	C₂	C₉×#386	1	1	yes	characteristic
311	C₂	C₂×C₉×#103	1	1	yes	characteristic
312	C₂	C₉×#417	1	1	yes	characteristic
313	C₂	C₉×#450	1	1	yes	characteristic
314	C₂	C₉×#451	1	1	yes	characteristic
315	C₂	C₉×#483	1	1	yes	characteristic
316	C₂	C₉×#484	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle	C₂³×C₃	C₂×C₃×D₄
prime radical	C₂⁴×C₃	C₃×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂³×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₉×#103	C₂	semidirect
5. C₉×#340	C₂	semidirect
12. C₉×C₂²⋊Q₈	C₄	semidirect
30. C₂×Q₈×C₉	C₂×C₄	semidirect
50. C₉×C₄⋊C₄	C₂×C₄	semidirect
53. #1872	C₉	direct (N × H)
100. C₂×#103	C₂×C₉	semidirect
102. #340	C₂×C₉	semidirect
136. C₂²×C₉	#103	semidirect
155. C₂²⋊Q₈	C₄×C₉	semidirect
206. C₂×Q₈	C₂×C₄×C₉	semidirect
226. C₄⋊C₄	C₂×C₄×C₉	semidirect
249. C₉	#1872	direct (N × H)
294. C₂²	C₉×#103	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×C₂²⋊Q₈	2
C₂²×C₉×C₄⋊C₄	2
C₂×C₉×#103	2
C₂×C₉×C₂²⋊C₈	2
C₉×#340	2
C₃×#1872	3

Subgroups — C9×#1872 #11766593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#1872 #11766593