Subgroups — C₉×#3042 #11767763

Order 1152.

741 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	C₂²×C₉×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#536	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#518	C₂	1	1	yes	maximal, characteristic, complemented
5	C₉×#463	C₂	1	4	yes	maximal, complemented
6	C₉×#461	C₂	1	4	yes	maximal, complemented
7	C₉×#424	C₂	1	1	yes	maximal, characteristic, complemented
8	C₉×#419	C₂	1	1	yes	maximal, characteristic, complemented
9	C₉×#416	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#3042	C₃	1	1	yes	maximal, characteristic
11	C₂×C₉×#41	C₂²	1	1	yes	characteristic
12	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
14	C₉×AGL(1,ℤ/8)	—	2	4	no	—
15	C₂×C₉×SD₁₆	—	2	4	no	—
16	C₂×C₉×SD₁₆	C₂²	1	4	yes	—
17	C₂×C₉×SD₁₆	C₂²	1	2	yes	—
18	C₂×C₉×D₈	C₂²	1	2	yes	—
19	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
21	C₉×C₄⋊Q₈	C₂²	1	2	yes	complemented
22	C₉×#124	C₂²	1	2	yes	complemented
23	C₉×#123	—	2	2	no	—
24	C₉×C₂²⋊Q₈	—	2	2	no	—
25	C₄×D₄×C₉	—	2	2	no	—
26	C₄×D₄×C₉	C₂²	1	4	yes	—
27	C₉×#117	C₂²	1	1	yes	characteristic
28	C₉×#117	C₂²	1	1	yes	characteristic
29	C₂×C₉×C₂²⋊C₄	—	2	2	no	—
30	C₉×C₄⋊C₈	C₂²	1	4	yes	complemented
31	C₉×#103	C₂²	1	2	yes	complemented
32	C₉×#103	C₂²	1	4	yes	complemented
33	C₉×#102	C₂²	1	2	yes	complemented
34	C₂×C₃×AGL(1,ℤ/8)	C₂×C₃	1	1	yes	characteristic
35	C₂²×C₃×SD₁₆	C₂×C₃	1	1	yes	characteristic
36	C₃×#536	C₂×C₃	1	1	yes	characteristic
37	C₃×#518	C₂×C₃	1	1	yes	characteristic
38	C₃×#463	C₂×C₃	1	4	yes	—
39	C₃×#461	C₂×C₃	1	4	yes	—
40	C₃×#424	C₂×C₃	1	1	yes	characteristic
41	C₃×#419	C₂×C₃	1	1	yes	characteristic
42	C₃×#416	C₂×C₃	1	1	yes	characteristic
43	C₂⁴×C₉	—	2	1	no	—
44	C₉×#41	—	2	2	no	—
45	C₉×#41	—	2	2	no	—
46	C₂×Q₈×C₉	—	2	1	no	—
47	C₂×Q₈×C₉	—	2	2	no	—
48	C₂×Q₈×C₉	—	2	2	no	—
49	C₂×Q₈×C₉	C₂³	1	2	yes	—
50	C₂×Q₈×C₉	C₂³	1	1	yes	characteristic
51	C₂×D₄×C₉	—	2	1	no	—
52	C₂×D₄×C₉	—	2	1	no	—
53	C₂×D₄×C₉	—	2	2	no	—
54	C₂×D₄×C₉	D₄	1	4	yes	—
55	C₂×D₄×C₉	C₂³	1	1	yes	characteristic
56	C₂×D₄×C₉	C₂³	1	2	yes	—
57	C₂²×C₄×C₉	—	2	1	no	—
58	C₂²×C₄×C₉	—	2	1	no	—
59	C₂²×C₄×C₉	—	2	4	no	—
60	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
61	C₉×SD₁₆	—	2	8	no	—
62	C₉×SD₁₆	—	2	4	no	—
63	C₉×D₈	—	2	4	no	—
64	C₉×M₁₆	—	2	2	no	—
65	C₂×C₈×C₉	—	2	2	no	—
66	C₂×C₈×C₉	C₂³	1	2	yes	—
67	C₂×C₈×C₉	C₂³	1	2	yes	—
68	C₉×C₄⋊C₄	—	2	4	no	—
69	C₉×C₄⋊C₄	D₄	1	2	yes	—
70	C₉×C₄⋊C₄	C₂³	1	2	yes	—
71	C₉×C₂²⋊C₄	—	2	1	no	—
72	C₉×C₂²⋊C₄	—	2	4	no	—
73	C₉×C₂²⋊C₄	—	2	4	no	—
74	C₉×C₂²⋊C₄	D₄	1	2	yes	—
75	C₄²×C₉	—	2	1	no	—
76	C₄²×C₉	C₂³	1	2	yes	—
77	#3042	C₉	1	1	yes	characteristic, direct product factor, Sylow
78	C₂×C₃×#41	C₂²×C₃	1	1	yes	characteristic
79	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
80	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
81	C₃×AGL(1,ℤ/8)	—	2	4	no	—
82	C₂×C₃×SD₁₆	—	2	4	no	—
83	C₂×C₃×SD₁₆	C₂²×C₃	1	4	yes	—
84	C₂×C₃×SD₁₆	C₂²×C₃	1	2	yes	—
85	C₂×C₃×D₈	C₂²×C₃	1	2	yes	—
86	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
87	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
88	C₃×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
89	C₃×#124	C₂²×C₃	1	2	yes	—
90	C₃×#123	—	2	2	no	—
91	C₃×C₂²⋊Q₈	—	2	2	no	—
92	C₃×C₄×D₄	—	2	2	no	—
93	C₃×C₄×D₄	C₂²×C₃	1	4	yes	—
94	C₃×#117	C₂²×C₃	1	1	yes	characteristic
95	C₃×#117	C₂²×C₃	1	1	yes	characteristic
96	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
97	C₃×C₄⋊C₈	C₂²×C₃	1	4	yes	—
98	C₃×#103	C₂²×C₃	1	2	yes	—
99	C₃×#103	C₂²×C₃	1	4	yes	—
100	C₃×#102	C₂²×C₃	1	2	yes	—
101	C₂³×C₉	—	2	2	no	—
102	C₂³×C₉	—	2	4	no	—
103	C₂³×C₉	—	2	1	no	—
104	C₂³×C₉	—	8	2	no	—
105	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
106	Q₈×C₉	—	2	2	no	—
107	Q₈×C₉	—	2	1	no	—
108	Q₈×C₉	—	2	2	no	—
109	Q₈×C₉	—	2	1	no	—
110	Q₈×C₉	—	4	2	no	—
111	D₄×C₉	—	2	4	no	—
112	D₄×C₉	—	2	1	no	—
113	D₄×C₉	—	2	2	no	—
114	D₄×C₉	—	2	1	no	—
115	D₄×C₉	—	4	2	no	—
116	D₄×C₉	C₂×D₄	1	4	yes	—
117	C₂×C₄×C₉	—	2	2	no	—
118	C₂×C₄×C₉	—	2	2	no	—
119	C₂×C₄×C₉	—	2	1	no	—
120	C₂×C₄×C₉	—	4	2	no	—
121	C₂×C₄×C₉	—	4	2	no	—
122	C₂×C₄×C₉	—	8	4	no	—
123	C₂×C₄×C₉	C₂×D₄	1	4	yes	—
124	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
126	C₂×C₄×C₉	#41	1	2	yes	—
127	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
128	C₈×C₉	—	2	2	no	—
129	C₈×C₉	—	2	2	no	—
130	C₂×AGL(1,ℤ/8)	C₂×C₉	1	1	yes	characteristic
131	C₂²×SD₁₆	C₂×C₉	1	1	yes	characteristic, complemented
132	#536	C₂×C₉	1	1	yes	characteristic, complemented
133	#518	C₂×C₉	1	1	yes	characteristic, complemented
134	#463	C₂×C₉	1	4	yes	complemented
135	#461	C₂×C₉	1	4	yes	complemented
136	#424	C₂×C₉	1	1	yes	characteristic, complemented
137	#419	C₂×C₉	1	1	yes	characteristic, complemented
138	#416	C₂×C₉	1	1	yes	characteristic, complemented
139	C₂⁴×C₃	—	2	1	no	—
140	C₃×#41	—	2	2	no	—
141	C₃×#41	—	2	2	no	—
142	C₂×C₃×Q₈	—	2	1	no	—
143	C₂×C₃×Q₈	—	2	2	no	—
144	C₂×C₃×Q₈	—	2	2	no	—
145	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
146	C₂×C₃×Q₈	C₂³×C₃	1	1	yes	characteristic
147	C₂×C₃×D₄	—	2	1	no	—
148	C₂×C₃×D₄	—	2	1	no	—
149	C₂×C₃×D₄	—	2	2	no	—
150	C₂×C₃×D₄	C₃×D₄	1	4	yes	—
151	C₂×C₃×D₄	C₂³×C₃	1	1	yes	characteristic
152	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
153	C₂²×C₃×C₄	—	2	1	no	—
154	C₂²×C₃×C₄	—	2	1	no	—
155	C₂²×C₃×C₄	—	2	4	no	—
156	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
157	C₃×SD₁₆	—	2	8	no	—
158	C₃×SD₁₆	—	2	4	no	—
159	C₃×D₈	—	2	4	no	—
160	C₃×M₁₆	—	2	2	no	—
161	C₂×C₃×C₈	—	2	2	no	—
162	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
163	C₂×C₃×C₈	C₂³×C₃	1	2	yes	—
164	C₃×C₄⋊C₄	—	2	4	no	—
165	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
166	C₃×C₄⋊C₄	C₂³×C₃	1	2	yes	—
167	C₃×C₂²⋊C₄	—	2	1	no	—
168	C₃×C₂²⋊C₄	—	2	4	no	—
169	C₃×C₂²⋊C₄	—	2	4	no	—
170	C₃×C₂²⋊C₄	C₃×D₄	1	2	yes	—
171	C₃×C₄²	—	2	1	no	—
172	C₃×C₄²	C₂³×C₃	1	2	yes	—
173	C₂²×C₉	—	2	2	no	—
174	C₂²×C₉	—	2	4	no	—
175	C₂²×C₉	—	4	4	no	—
176	C₂²×C₉	—	4	1	no	—
177	C₂²×C₉	—	4	2	no	—
178	C₂²×C₉	—	8	4	no	—
179	C₂²×C₉	#121	1	2	yes	—
180	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
181	C₄×C₉	—	2	4	no	—
182	C₄×C₉	—	4	1	no	—
183	C₄×C₉	—	4	2	no	—
184	C₄×C₉	—	4	2	no	—
185	C₄×C₉	#121	1	2	yes	—
186	C₄×C₉	C₂²×D₄	1	1	yes	characteristic
187	C₄×C₉	C₂×#41	1	1	yes	characteristic
188	C₂×#41	C₂²×C₉	1	1	yes	characteristic
189	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
190	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
191	AGL(1,ℤ/8)	—	2	4	no	—
192	C₂×SD₁₆	—	2	4	no	—
193	C₂×SD₁₆	C₂²×C₉	1	4	yes	—
194	C₂×SD₁₆	C₂²×C₉	1	2	yes	—
195	C₂×D₈	C₂²×C₉	1	2	yes	—
196	C₂×M₁₆	C₂²×C₉	1	1	yes	characteristic
197	C₂²×C₈	C₂²×C₉	1	1	yes	characteristic
198	C₄⋊Q₈	C₂²×C₉	1	2	yes	complemented
199	#124	C₂²×C₉	1	2	yes	complemented
200	#123	—	2	2	no	—
201	C₂²⋊Q₈	—	2	2	no	—
202	C₄×D₄	—	2	2	no	—
203	C₄×D₄	C₂²×C₉	1	4	yes	—
204	#117	C₂²×C₉	1	1	yes	characteristic
205	#117	C₂²×C₉	1	1	yes	characteristic
206	C₂×C₂²⋊C₄	—	2	2	no	—
207	C₄⋊C₈	C₂²×C₉	1	4	yes	complemented
208	#103	C₂²×C₉	1	2	yes	complemented
209	#103	C₂²×C₉	1	4	yes	complemented
210	#102	C₂²×C₉	1	2	yes	complemented
211	C₂³×C₃	—	2	2	no	—
212	C₂³×C₃	—	2	4	no	—
213	C₂³×C₃	—	2	1	no	—
214	C₂³×C₃	—	8	2	no	—
215	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
216	C₃×Q₈	—	2	2	no	—
217	C₃×Q₈	—	2	1	no	—
218	C₃×Q₈	—	2	2	no	—
219	C₃×Q₈	—	2	1	no	—
220	C₃×Q₈	—	4	2	no	—
221	C₃×D₄	—	2	4	no	—
222	C₃×D₄	—	2	1	no	—
223	C₃×D₄	—	2	2	no	—
224	C₃×D₄	—	2	1	no	—
225	C₃×D₄	—	4	2	no	—
226	C₃×D₄	C₂×C₃×D₄	1	4	yes	—
227	C₂×C₃×C₄	—	2	2	no	—
228	C₂×C₃×C₄	—	2	2	no	—
229	C₂×C₃×C₄	—	2	1	no	—
230	C₂×C₃×C₄	—	4	2	no	—
231	C₂×C₃×C₄	—	4	2	no	—
232	C₂×C₃×C₄	—	8	4	no	—
233	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
234	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
236	C₂×C₃×C₄	C₃×#41	1	2	yes	—
237	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
238	C₃×C₈	—	2	2	no	—
239	C₃×C₈	—	2	2	no	—
240	C₂×C₉	—	2	2	no	—
241	C₂×C₉	—	4	4	no	—
242	C₂×C₉	—	8	1	no	—
243	C₂×C₉	C₂×#121	1	1	yes	characteristic
244	C₂×C₉	#577	1	2	yes	—
245	C₂⁴	—	2	1	no	—
246	#41	—	2	2	no	—
247	#41	—	2	2	no	—
248	C₂×Q₈	—	2	1	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	C₂³×C₉	1	2	yes	—
252	C₂×Q₈	C₂³×C₉	1	1	yes	characteristic
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	2	1	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	D₄×C₉	1	4	yes	—
257	C₂×D₄	C₂³×C₉	1	1	yes	characteristic
258	C₂×D₄	C₂³×C₉	1	2	yes	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	2	4	no	—
262	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
263	SD₁₆	—	2	8	no	—
264	SD₁₆	—	2	4	no	—
265	D₈	—	2	4	no	—
266	M₁₆	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	C₂³×C₉	1	2	yes	—
269	C₂×C₈	C₂³×C₉	1	2	yes	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	D₄×C₉	1	2	yes	—
272	C₄⋊C₄	C₂³×C₉	1	2	yes	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	2	4	no	—
275	C₂²⋊C₄	—	2	4	no	—
276	C₂²⋊C₄	D₄×C₉	1	2	yes	—
277	C₄²	—	2	1	no	—
278	C₄²	C₂³×C₉	1	2	yes	—
279	C₂²×C₃	—	2	2	no	—
280	C₂²×C₃	—	2	4	no	—
281	C₂²×C₃	—	4	4	no	—
282	C₂²×C₃	—	4	2	no	—
283	C₂²×C₃	—	4	1	no	—
284	C₂²×C₃	—	8	4	no	—
285	C₂²×C₃	C₃×#121	1	2	yes	—
286	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
287	C₃×C₄	—	2	4	no	—
288	C₃×C₄	—	4	2	no	—
289	C₃×C₄	—	4	1	no	—
290	C₃×C₄	—	4	2	no	—
291	C₃×C₄	C₃×#121	1	2	yes	—
292	C₃×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
293	C₃×C₄	C₂×C₃×#41	1	1	yes	characteristic
294	C₉	#3042	1	1	yes	characteristic, direct product factor, Sylow
295	C₂³	—	2	2	no	—
296	C₂³	—	2	4	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	8	2	no	—
299	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
300	Q₈	—	2	2	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	2	1	no	—
304	Q₈	—	4	2	no	—
305	D₄	—	2	4	no	—
306	D₄	—	2	1	no	—
307	D₄	—	2	2	no	—
308	D₄	—	2	1	no	—
309	D₄	—	4	2	no	—
310	D₄	C₂×D₄×C₉	1	4	yes	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	4	no	—
317	C₂×C₄	C₂×D₄×C₉	1	4	yes	—
318	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
319	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
320	C₂×C₄	C₉×#41	1	2	yes	—
321	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	—	2	2	no	—
324	C₂×C₃	—	2	2	no	—
325	C₂×C₃	—	4	4	no	—
326	C₂×C₃	—	8	1	no	—
327	C₂×C₃	C₂×C₃×#121	1	1	yes	characteristic
328	C₂×C₃	C₃×#577	1	2	yes	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	4	no	—
331	C₂²	—	4	4	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	8	4	no	—
335	C₂²	C₉×#121	1	2	yes	—
336	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
337	C₄	—	2	4	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	2	no	—
340	C₄	—	4	2	no	—
341	C₄	C₉×#121	1	2	yes	—
342	C₄	C₂²×D₄×C₉	1	1	yes	characteristic
343	C₄	C₂×C₉×#41	1	1	yes	characteristic
344	C₃	C₃×#3042	1	1	yes	characteristic
345	C₂	—	2	2	no	—
346	C₂	—	4	4	no	—
347	C₂	—	8	1	no	—
348	C₂	C₂×C₉×#121	1	1	yes	characteristic
349	C₂	C₉×#577	1	2	yes	—

	subgroup	G/H
center	C₂²×C₉	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₂²×C₃×D₄
prime radical	C₂×C₃×AGL(1,ℤ/8)	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₉×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₉×SD₁₆	C₂	semidirect
3. C₉×#536	C₂	semidirect
4. C₉×#518	C₂	semidirect
5. C₉×#463	C₂	semidirect
6. C₉×#461	C₂	semidirect
7. C₉×#424	C₂	semidirect
8. C₉×#419	C₂	semidirect
9. C₉×#416	C₂	semidirect
21. C₉×C₄⋊Q₈	C₂²	semidirect
22. C₉×#124	C₂²	semidirect
30. C₉×C₄⋊C₈	C₂²	semidirect
31. C₉×#103	C₂²	semidirect
32. C₉×#103	C₂²	semidirect
33. C₉×#102	C₂²	semidirect
77. #3042	C₉	direct (N × H)
131. C₂²×SD₁₆	C₂×C₉	semidirect
132. #536	C₂×C₉	semidirect
133. #518	C₂×C₉	semidirect
134. #463	C₂×C₉	semidirect
135. #461	C₂×C₉	semidirect
136. #424	C₂×C₉	semidirect
137. #419	C₂×C₉	semidirect
138. #416	C₂×C₉	semidirect
198. C₄⋊Q₈	C₂²×C₉	semidirect
199. #124	C₂²×C₉	semidirect
207. C₄⋊C₈	C₂²×C₉	semidirect
208. #103	C₂²×C₉	semidirect
209. #103	C₂²×C₉	semidirect
210. #102	C₂²×C₉	semidirect
294. C₉	#3042	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
C₂²×C₉×SD₁₆	2
C₉×#536	2
C₉×#518	2
C₉×#463	2
C₉×#461	2
C₉×#424	2
C₉×#419	2
C₉×#416	2
C₃×#3042	3

Subgroups — C9×#3042 #11767763

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#3042 #11767763