Subgroups — C₂×C₉×#536 #11768168

Order 1152.

2268 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	C₂²×C₉×#41	C₂	1	1	yes	maximal, characteristic
2	C₂³×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#536	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₉×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₉×#124	C₂	1	2	yes	maximal, complemented
6	C₂×C₉×#123	C₂	1	4	yes	maximal, complemented
7	C₂×C₉×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
8	C₂×C₉×#117	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×#536	C₃	1	1	yes	maximal, characteristic
10	C₂×C₉×#41	—	2	8	no	—
11	C₂×C₉×#41	C₂²	1	8	yes	—
12	C₂²×Q₈×C₉	—	2	8	no	—
13	C₂²×Q₈×C₉	C₂²	1	2	yes	—
14	C₂²×Q₈×C₉	C₂²	1	2	yes	—
15	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
16	C₂²×Q₈×C₉	C₂²	1	8	yes	complemented
17	C₂²×D₄×C₉	C₂²	1	2	yes	—
18	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
19	C₂³×C₄×C₉	C₂²	1	2	yes	—
20	C₂³×C₄×C₉	C₂²	1	2	yes	—
21	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
22	C₉×C₄⋊Q₈	C₂²	1	16	yes	complemented
23	C₉×#124	C₂²	1	16	yes	complemented
24	C₉×#123	C₂²	1	32	yes	complemented
25	C₉×C₂²⋊Q₈	C₂²	1	32	yes	complemented
26	C₉×#117	C₂²	1	8	yes	complemented
27	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
28	C₂×C₉×C₄⋊C₄	C₂²	1	8	yes	—
29	C₂×C₉×C₂²⋊C₄	C₂²	1	8	yes	—
30	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
31	C₂×C₄²×C₉	C₂²	1	2	yes	—
32	C₂²×C₃×#41	C₂×C₃	1	1	yes	characteristic
33	C₂³×C₃×Q₈	C₂×C₃	1	1	yes	characteristic
34	C₃×#536	C₂×C₃	1	16	yes	—
35	C₂×C₃×C₄⋊Q₈	C₂×C₃	1	2	yes	—
36	C₂×C₃×#124	C₂×C₃	1	2	yes	—
37	C₂×C₃×#123	C₂×C₃	1	4	yes	—
38	C₂×C₃×C₂²⋊Q₈	C₂×C₃	1	4	yes	—
39	C₂×C₃×#117	C₂×C₃	1	1	yes	characteristic
40	C₂⁴×C₉	C₂³	1	2	yes	—
41	C₂⁴×C₉	C₂³	1	1	yes	characteristic
42	C₉×#41	—	2	32	no	—
43	C₂×Q₈×C₉	—	2	8	no	—
44	C₂×Q₈×C₉	—	2	8	no	—
45	C₂×Q₈×C₉	—	2	32	no	—
46	C₂×Q₈×C₉	—	2	4	no	—
47	C₂×Q₈×C₉	C₂³	1	8	yes	—
48	C₂×Q₈×C₉	C₂³	1	8	yes	—
49	C₂×Q₈×C₉	C₂³	1	4	yes	—
50	C₂×D₄×C₉	—	2	8	no	—
51	C₂×D₄×C₉	—	2	4	no	—
52	C₂×D₄×C₉	C₂³	1	8	yes	—
53	C₂×D₄×C₉	C₂³	1	4	yes	—
54	C₂²×C₄×C₉	—	2	8	no	—
55	C₂²×C₄×C₉	—	2	8	no	—
56	C₂²×C₄×C₉	D₄	1	8	yes	—
57	C₂²×C₄×C₉	C₂³	1	8	yes	—
58	C₂²×C₄×C₉	C₂³	1	8	yes	—
59	C₂²×C₄×C₉	C₂³	1	4	yes	—
60	C₂²×C₄×C₉	C₂³	1	4	yes	—
61	C₂²×C₄×C₉	C₂³	1	2	yes	—
62	C₂²×C₄×C₉	C₂³	1	4	yes	—
63	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
65	C₉×C₄⋊C₄	C₂³	1	8	yes	—
66	C₉×C₄⋊C₄	C₂³	1	32	yes	—
67	C₉×C₂²⋊C₄	C₂³	1	8	yes	—
68	C₉×C₂²⋊C₄	C₂³	1	32	yes	—
69	C₄²×C₉	C₂³	1	8	yes	—
70	C₂×#536	C₉	1	1	yes	characteristic, direct product factor, Sylow
71	C₂×C₃×#41	—	2	8	no	—
72	C₂×C₃×#41	C₂²×C₃	1	8	yes	—
73	C₂²×C₃×Q₈	—	2	8	no	—
74	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
75	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
76	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
77	C₂²×C₃×Q₈	C₂²×C₃	1	8	yes	—
78	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
79	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
80	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
81	C₂³×C₃×C₄	C₂²×C₃	1	2	yes	—
82	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
83	C₃×C₄⋊Q₈	C₂²×C₃	1	16	yes	—
84	C₃×#124	C₂²×C₃	1	16	yes	—
85	C₃×#123	C₂²×C₃	1	32	yes	—
86	C₃×C₂²⋊Q₈	C₂²×C₃	1	32	yes	—
87	C₃×#117	C₂²×C₃	1	8	yes	—
88	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
89	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	8	yes	—
90	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
91	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
92	C₂×C₃×C₄²	C₂²×C₃	1	2	yes	—
93	C₂³×C₉	—	2	4	no	—
94	C₂³×C₉	—	2	4	no	—
95	C₂³×C₉	—	2	8	no	—
96	C₂³×C₉	C₂×D₄	1	4	yes	—
97	C₂³×C₉	C₂⁴	1	4	yes	—
98	C₂³×C₉	C₂⁴	1	2	yes	—
99	C₂³×C₉	C₂⁴	1	1	yes	characteristic
100	Q₈×C₉	—	2	16	no	—
101	Q₈×C₉	—	2	16	no	—
102	Q₈×C₉	—	2	8	no	—
103	D₄×C₉	—	2	16	no	—
104	D₄×C₉	—	2	8	no	—
105	C₂×C₄×C₉	—	2	4	no	—
106	C₂×C₄×C₉	—	2	16	no	—
107	C₂×C₄×C₉	—	2	8	no	—
108	C₂×C₄×C₉	—	2	8	no	—
109	C₂×C₄×C₉	—	2	16	no	—
110	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
111	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
112	C₂×C₄×C₉	C₂×D₄	1	16	yes	—
113	C₂×C₄×C₉	C₂⁴	1	8	yes	—
114	C₂×C₄×C₉	C₂⁴	1	8	yes	—
115	C₂×C₄×C₉	C₂⁴	1	4	yes	—
116	C₂×C₄×C₉	C₂⁴	1	2	yes	—
117	C₂×C₄×C₉	C₂⁴	1	2	yes	—
118	C₂²×#41	C₂×C₉	1	1	yes	characteristic
119	C₂³×Q₈	C₂×C₉	1	1	yes	characteristic, complemented
120	#536	C₂×C₉	1	16	yes	direct product factor
121	C₂×C₄⋊Q₈	C₂×C₉	1	2	yes	complemented
122	C₂×#124	C₂×C₉	1	2	yes	complemented
123	C₂×#123	C₂×C₉	1	4	yes	complemented
124	C₂×C₂²⋊Q₈	C₂×C₉	1	4	yes	complemented
125	C₂×#117	C₂×C₉	1	1	yes	characteristic, complemented
126	C₂⁴×C₃	C₂³×C₃	1	2	yes	—
127	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
128	C₃×#41	—	2	32	no	—
129	C₂×C₃×Q₈	—	2	8	no	—
130	C₂×C₃×Q₈	—	2	8	no	—
131	C₂×C₃×Q₈	—	2	32	no	—
132	C₂×C₃×Q₈	—	2	4	no	—
133	C₂×C₃×Q₈	C₂³×C₃	1	8	yes	—
134	C₂×C₃×Q₈	C₂³×C₃	1	8	yes	—
135	C₂×C₃×Q₈	C₂³×C₃	1	4	yes	—
136	C₂×C₃×D₄	—	2	8	no	—
137	C₂×C₃×D₄	—	2	4	no	—
138	C₂×C₃×D₄	C₂³×C₃	1	8	yes	—
139	C₂×C₃×D₄	C₂³×C₃	1	4	yes	—
140	C₂²×C₃×C₄	—	2	8	no	—
141	C₂²×C₃×C₄	—	2	8	no	—
142	C₂²×C₃×C₄	C₃×D₄	1	8	yes	—
143	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
144	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
145	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
146	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
147	C₂²×C₃×C₄	C₂³×C₃	1	8	yes	—
148	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
149	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
150	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
151	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
152	C₃×C₄⋊C₄	C₂³×C₃	1	32	yes	—
153	C₃×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
154	C₃×C₂²⋊C₄	C₂³×C₃	1	32	yes	—
155	C₃×C₄²	C₂³×C₃	1	8	yes	—
156	C₂²×C₉	—	2	8	no	—
157	C₂²×C₉	—	2	4	no	—
158	C₂²×C₉	—	2	4	no	—
159	C₂²×C₉	—	2	8	no	—
160	C₂²×C₉	—	4	8	no	—
161	C₂²×C₉	C₂²×D₄	1	4	yes	—
162	C₂²×C₉	C₂²×D₄	1	2	yes	—
163	C₂²×C₉	2₋¹⁺⁴	1	4	yes	—
164	C₂²×C₉	C₂⁵	1	1	yes	characteristic
165	C₄×C₉	—	2	4	no	—
166	C₄×C₉	—	2	8	no	—
167	C₄×C₉	—	2	8	no	—
168	C₄×C₉	C₂²×D₄	1	4	yes	—
169	C₄×C₉	C₂²×D₄	1	4	yes	—
170	C₂×#41	—	2	8	no	—
171	C₂×#41	C₂²×C₉	1	8	yes	—
172	C₂²×Q₈	—	2	8	no	—
173	C₂²×Q₈	C₂²×C₉	1	2	yes	—
174	C₂²×Q₈	C₂²×C₉	1	2	yes	—
175	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
176	C₂²×Q₈	C₂²×C₉	1	8	yes	complemented
177	C₂²×D₄	C₂²×C₉	1	2	yes	—
178	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
179	C₂³×C₄	C₂²×C₉	1	2	yes	—
180	C₂³×C₄	C₂²×C₉	1	2	yes	—
181	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
182	C₄⋊Q₈	C₂²×C₉	1	16	yes	complemented
183	#124	C₂²×C₉	1	16	yes	complemented
184	#123	C₂²×C₉	1	32	yes	complemented
185	C₂²⋊Q₈	C₂²×C₉	1	32	yes	complemented
186	#117	C₂²×C₉	1	8	yes	complemented
187	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
188	C₂×C₄⋊C₄	C₂²×C₉	1	8	yes	—
189	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
190	C₂×C₂²⋊C₄	C₂²×C₉	1	8	yes	—
191	C₂×C₄²	C₂²×C₉	1	2	yes	—
192	C₂³×C₃	—	2	4	no	—
193	C₂³×C₃	—	2	4	no	—
194	C₂³×C₃	—	2	8	no	—
195	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
196	C₂³×C₃	C₂⁴×C₃	1	4	yes	—
197	C₂³×C₃	C₂⁴×C₃	1	2	yes	—
198	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
199	C₃×Q₈	—	2	16	no	—
200	C₃×Q₈	—	2	16	no	—
201	C₃×Q₈	—	2	8	no	—
202	C₃×D₄	—	2	16	no	—
203	C₃×D₄	—	2	8	no	—
204	C₂×C₃×C₄	—	2	4	no	—
205	C₂×C₃×C₄	—	2	16	no	—
206	C₂×C₃×C₄	—	2	8	no	—
207	C₂×C₃×C₄	—	2	8	no	—
208	C₂×C₃×C₄	—	2	16	no	—
209	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
210	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
211	C₂×C₃×C₄	C₂×C₃×D₄	1	16	yes	—
212	C₂×C₃×C₄	C₂⁴×C₃	1	8	yes	—
213	C₂×C₃×C₄	C₂⁴×C₃	1	8	yes	—
214	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
215	C₂×C₃×C₄	C₂⁴×C₃	1	2	yes	—
216	C₂×C₃×C₄	C₂⁴×C₃	1	2	yes	—
217	C₂×C₉	—	2	4	no	—
218	C₂×C₉	—	4	4	no	—
219	C₂×C₉	#536	1	4	yes	direct product factor
220	C₂×C₉	C₂³×D₄	1	1	yes	characteristic
221	C₂×C₉	C₂×2₋¹⁺⁴	1	2	yes	—
222	C₂⁴	C₂³×C₉	1	2	yes	—
223	C₂⁴	C₂³×C₉	1	1	yes	characteristic
224	#41	—	2	32	no	—
225	C₂×Q₈	—	2	8	no	—
226	C₂×Q₈	—	2	8	no	—
227	C₂×Q₈	—	2	32	no	—
228	C₂×Q₈	—	2	4	no	—
229	C₂×Q₈	C₂³×C₉	1	8	yes	—
230	C₂×Q₈	C₂³×C₉	1	8	yes	—
231	C₂×Q₈	C₂³×C₉	1	4	yes	—
232	C₂×D₄	—	2	8	no	—
233	C₂×D₄	—	2	4	no	—
234	C₂×D₄	C₂³×C₉	1	8	yes	—
235	C₂×D₄	C₂³×C₉	1	4	yes	—
236	C₂²×C₄	—	2	8	no	—
237	C₂²×C₄	—	2	8	no	—
238	C₂²×C₄	D₄×C₉	1	8	yes	—
239	C₂²×C₄	C₂³×C₉	1	4	yes	—
240	C₂²×C₄	C₂³×C₉	1	4	yes	—
241	C₂²×C₄	C₂³×C₉	1	2	yes	—
242	C₂²×C₄	C₂³×C₉	1	8	yes	—
243	C₂²×C₄	C₂³×C₉	1	8	yes	—
244	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
245	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
246	C₂²×C₄	C₂³×C₉	1	4	yes	—
247	C₄⋊C₄	C₂³×C₉	1	8	yes	—
248	C₄⋊C₄	C₂³×C₉	1	32	yes	—
249	C₂²⋊C₄	C₂³×C₉	1	8	yes	—
250	C₂²⋊C₄	C₂³×C₉	1	32	yes	—
251	C₄²	C₂³×C₉	1	8	yes	—
252	C₂²×C₃	—	2	4	no	—
253	C₂²×C₃	—	2	8	no	—
254	C₂²×C₃	—	2	4	no	—
255	C₂²×C₃	—	2	8	no	—
256	C₂²×C₃	—	4	8	no	—
257	C₂²×C₃	C₂²×C₃×D₄	1	4	yes	—
258	C₂²×C₃	C₂²×C₃×D₄	1	2	yes	—
259	C₂²×C₃	C₃×2₋¹⁺⁴	1	4	yes	—
260	C₂²×C₃	C₂⁵×C₃	1	1	yes	characteristic
261	C₃×C₄	—	2	4	no	—
262	C₃×C₄	—	2	8	no	—
263	C₃×C₄	—	2	8	no	—
264	C₃×C₄	C₂²×C₃×D₄	1	4	yes	—
265	C₃×C₄	C₂²×C₃×D₄	1	4	yes	—
266	C₉	C₂×#536	1	1	yes	characteristic, direct product factor, Sylow
267	C₂³	—	2	4	no	—
268	C₂³	—	2	4	no	—
269	C₂³	—	2	8	no	—
270	C₂³	C₂×D₄×C₉	1	4	yes	—
271	C₂³	C₂⁴×C₉	1	4	yes	—
272	C₂³	C₂⁴×C₉	1	2	yes	—
273	C₂³	C₂⁴×C₉	1	1	yes	characteristic
274	Q₈	—	2	16	no	—
275	Q₈	—	2	16	no	—
276	Q₈	—	2	8	no	—
277	D₄	—	2	16	no	—
278	D₄	—	2	8	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	16	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	16	no	—
284	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
285	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
286	C₂×C₄	C₂×D₄×C₉	1	16	yes	—
287	C₂×C₄	C₂⁴×C₉	1	8	yes	—
288	C₂×C₄	C₂⁴×C₉	1	8	yes	—
289	C₂×C₄	C₂⁴×C₉	1	4	yes	—
290	C₂×C₄	C₂⁴×C₉	1	2	yes	—
291	C₂×C₄	C₂⁴×C₉	1	2	yes	—
292	C₂×C₃	—	2	4	no	—
293	C₂×C₃	—	4	4	no	—
294	C₂×C₃	C₃×#536	1	4	yes	—
295	C₂×C₃	C₂³×C₃×D₄	1	1	yes	characteristic
296	C₂×C₃	C₂×C₃×2₋¹⁺⁴	1	2	yes	—
297	C₂²	—	2	8	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	8	no	—
301	C₂²	—	4	8	no	—
302	C₂²	C₂²×D₄×C₉	1	4	yes	—
303	C₂²	C₂²×D₄×C₉	1	2	yes	—
304	C₂²	C₉×2₋¹⁺⁴	1	4	yes	—
305	C₂²	C₂⁵×C₉	1	1	yes	characteristic
306	C₄	—	2	4	no	—
307	C₄	—	2	8	no	—
308	C₄	—	2	8	no	—
309	C₄	C₂²×D₄×C₉	1	4	yes	—
310	C₄	C₂²×D₄×C₉	1	4	yes	—
311	C₃	C₂×C₃×#536	1	1	yes	characteristic
312	C₂	—	2	4	no	—
313	C₂	—	4	4	no	—
314	C₂	C₉×#536	1	4	yes	direct product factor
315	C₂	C₂³×D₄×C₉	1	1	yes	characteristic
316	C₂	C₂×C₉×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂²	C₂⁵×C₉
Frattini, cosocle	C₂²×C₃	C₂⁵×C₃
socle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂²×C₃×#41	C₂×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×Q₈×C₉	C₂	semidirect
3. C₉×#536	C₂	direct (N × H)
4. C₂×C₉×C₄⋊Q₈	C₂	semidirect
5. C₂×C₉×#124	C₂	semidirect
6. C₂×C₉×#123	C₂	semidirect
7. C₂×C₉×C₂²⋊Q₈	C₂	semidirect
8. C₂×C₉×#117	C₂	semidirect
16. C₂²×Q₈×C₉	C₂²	semidirect
22. C₉×C₄⋊Q₈	C₂²	semidirect
23. C₉×#124	C₂²	semidirect
24. C₉×#123	C₂²	semidirect
25. C₉×C₂²⋊Q₈	C₂²	semidirect
26. C₉×#117	C₂²	semidirect
70. C₂×#536	C₉	direct (N × H)
119. C₂³×Q₈	C₂×C₉	semidirect
120. #536	C₂×C₉	direct (N × H)
121. C₂×C₄⋊Q₈	C₂×C₉	semidirect
122. C₂×#124	C₂×C₉	semidirect
123. C₂×#123	C₂×C₉	semidirect
124. C₂×C₂²⋊Q₈	C₂×C₉	semidirect
125. C₂×#117	C₂×C₉	semidirect
176. C₂²×Q₈	C₂²×C₉	semidirect
182. C₄⋊Q₈	C₂²×C₉	semidirect
183. #124	C₂²×C₉	semidirect
184. #123	C₂²×C₉	semidirect
185. C₂²⋊Q₈	C₂²×C₉	semidirect
186. #117	C₂²×C₉	semidirect
219. C₂×C₉	#536	direct (N × H)
266. C₉	C₂×#536	direct (N × H)
314. C₂	C₉×#536	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₉×#41	2
C₂³×Q₈×C₉	2
C₉×#536	2
C₂×C₉×C₄⋊Q₈	2
C₂×C₉×#124	2
C₂×C₉×#123	2
C₂×C₉×C₂²⋊Q₈	2
C₂×C₉×#117	2
C₂×C₃×#536	3

Subgroups — C2×C9×#536 #11768168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#536 #11768168