Subgroups — #11868535

Order 1152.

582 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#3697	C₂	1	1	yes	maximal, characteristic
3	#10588772	C₂	1	1	yes	maximal, characteristic, complemented
4	#10588769	C₂	1	1	yes	maximal, characteristic, complemented
5	#10588767	C₂	1	1	yes	maximal, characteristic, complemented
6	#10588764	C₂	1	1	yes	maximal, characteristic, complemented
7	#10588747	C₂	1	2	yes	maximal, complemented
8	#10588745	C₂	1	2	yes	maximal, complemented
9	#10588683	C₂	1	2	yes	maximal
10	#10588676	C₂	1	2	yes	maximal, complemented
11	C₂×C₄⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
12	#83349	—	3	1	no	maximal
13	C₂×#3697	C₂²	1	1	yes	characteristic
14	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₉×#125	C₂²	1	2	yes	—
17	C₉×C₂²⋊Q₈	—	2	1	no	—
18	C₉×C₂²⋊Q₈	—	2	1	no	—
19	C₄×Q₈×C₉	—	2	1	no	—
20	C₄×Q₈×C₉	—	2	1	no	—
21	C₉×#117	C₂²	1	2	yes	—
22	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
23	C₄×#611	—	2	1	no	—
24	C₄×#611	—	2	1	no	—
25	#63488	C₂²	1	1	yes	characteristic
26	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
27	C₄×D₃₆	C₂²	1	1	yes	characteristic
28	C₄×D₃₆	C₂²	1	1	yes	characteristic
29	#63434	—	2	1	no	—
30	C₄²×D₉	—	2	1	no	—
31	C₄×Dic₁₈	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₈	C₂²	1	1	yes	characteristic
33	#63369	C₂²	1	2	yes	—
34	#63369	C₂²	1	2	yes	—
35	#63368	C₂²	1	2	yes	—
36	#63368	C₂²	1	2	yes	—
37	#63367	C₂²	1	2	yes	—
38	#63367	C₂²	1	2	yes	—
39	#63366	C₂²	1	2	yes	—
40	#63366	C₂²	1	2	yes	—
41	C₄⋊(C₉⋊C₈)	C₂²	1	2	yes	—
42	C₄⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
43	C₄⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
44	C₃×#533	D₃	1	1	yes	characteristic
45	C₄×#235	—	3	1	no	—
46	#5094	—	3	1	no	—
47	#5091	—	3	1	no	—
48	#5087	—	3	1	no	—
49	#5084	—	3	1	no	—
50	#5066	—	3	2	no	—
51	#5063	—	3	2	no	—
52	#5002	—	3	2	no	—
53	#4998	—	3	2	no	—
54	C₂×C₄⋊(C₃⋊C₈)	—	3	1	no	—
55	C₂×Q₈×C₉	—	2	1	no	—
56	C₂×Q₈×C₉	—	2	1	no	—
57	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
58	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
59	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
60	C₂×#611	—	2	1	no	—
61	#3697	—	2	2	no	—
62	#3697	Q₈	1	4	yes	complemented
63	C₂×D₃₆	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₉	—	2	1	no	—
65	C₂×C₄×D₉	—	2	1	no	—
66	C₂×C₄×D₉	—	2	1	no	—
67	C₂×Dic₁₈	C₂³	1	1	yes	characteristic
68	C₉×C₄⋊C₄	—	2	2	no	—
69	C₉×C₄⋊C₄	—	2	2	no	—
70	C₉×C₄⋊C₄	—	2	1	no	—
71	C₉×C₄⋊C₄	—	2	1	no	—
72	C₉×C₄⋊C₄	C₂³	1	2	yes	—
73	C₉×C₄⋊C₄	C₂³	1	2	yes	—
74	C₉×C₂²⋊C₄	—	2	2	no	—
75	C₄²×C₉	—	2	2	no	—
76	C₄²×C₉	D₄	1	1	yes	characteristic
77	C₄²×C₉	D₄	1	1	yes	characteristic
78	C₄²×C₉	C₂³	1	1	yes	characteristic
79	C₄²×C₉	C₂³	1	1	yes	characteristic
80	#3676	—	2	1	no	—
81	(C₂×D₉)⋊C₄	—	2	1	no	—
82	(C₂×D₉)⋊C₄	—	2	1	no	—
83	C₄⋊Dic₉	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₉	C₂³	1	1	yes	characteristic
85	Dic₉⋊C₄	—	2	1	no	—
86	Dic₉⋊C₄	—	2	1	no	—
87	C₄×Dic₉	—	2	1	no	—
88	C₄×Dic₉	—	2	1	no	—
89	C₄×Dic₉	—	2	1	no	—
90	C₂×C₉⋊C₈	—	2	4	no	—
91	C₂×C₉⋊C₈	C₂³	1	2	yes	—
92	C₂×C₉⋊C₈	C₂³	1	2	yes	—
93	#3076	—	9	1	no	Sylow
94	C₂×#235	—	3	1	no	—
95	C₃×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
96	C₃×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
97	C₃×#125	C₂×D₃	1	2	yes	—
98	C₃×C₂²⋊Q₈	—	2	1	no	—
99	C₃×C₂²⋊Q₈	—	2	1	no	—
100	C₃×C₄×Q₈	—	2	1	no	—
101	C₃×C₄×Q₈	—	2	1	no	—
102	C₃×#117	C₂×D₃	1	2	yes	—
103	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
104	C₄×#67	—	6	1	no	—
105	C₄×#67	—	6	1	no	—
106	#926	—	3	1	no	—
107	C₂²×C₃⋊C₈	—	3	2	no	—
108	C₄×D₁₂	—	3	1	no	—
109	C₄×D₁₂	—	3	1	no	—
110	#872	—	6	1	no	—
111	C₄²×D₃	—	6	1	no	—
112	C₄×Dic₆	—	3	1	no	—
113	C₄×Dic₆	—	3	1	no	—
114	#810	—	3	2	no	—
115	#810	—	3	2	no	—
116	#809	—	3	2	no	—
117	#809	—	3	2	no	—
118	#808	—	3	2	no	—
119	#808	—	3	2	no	—
120	#807	—	3	2	no	—
121	#807	—	3	2	no	—
122	C₄⋊(C₃⋊C₈)	—	3	2	no	—
123	C₄⋊(C₃⋊C₈)	—	3	1	no	—
124	C₄⋊(C₃⋊C₈)	—	3	1	no	—
125	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
126	C₂²×D₉	—	2	1	no	—
127	Q₈×C₉	—	4	2	no	—
128	Q₈×C₉	—	4	2	no	—
129	C₂×C₄×C₉	—	2	2	no	—
130	C₂×C₄×C₉	—	2	2	no	—
131	C₂×C₄×C₉	—	2	2	no	—
132	C₂×C₄×C₉	—	4	1	no	—
133	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₉	C₂×Q₈	1	2	yes	—
139	C₂×C₄×C₉	#41	1	2	yes	—
140	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
141	#611	—	2	4	no	—
142	C₂×Dic₉	—	2	1	no	—
143	C₂×Dic₉	—	2	1	no	—
144	C₂×Dic₉	—	2	1	no	—
145	D₃₆	—	2	1	no	—
146	D₃₆	C₂×Q₈	1	2	yes	—
147	C₄×D₉	—	2	4	no	—
148	C₄×D₉	—	4	2	no	—
149	C₄×D₉	—	4	2	no	—
150	Dic₁₈	—	2	1	no	—
151	Dic₁₈	C₂×Q₈	1	2	yes	—
152	C₉⋊C₈	—	2	2	no	—
153	C₉⋊C₈	—	2	2	no	—
154	#533	D₉	1	1	yes	characteristic, complemented
155	C₄×#41	—	9	1	no	—
156	#479	—	9	2	no	—
157	#478	—	9	2	no	—
158	#477	—	9	1	no	—
159	#476	—	9	1	no	—
160	#475	—	9	1	no	—
161	#474	—	9	1	no	—
162	#427	—	9	2	no	—
163	C₂×C₄⋊C₈	—	9	1	no	—
164	#416	—	9	2	no	—
165	C₂×C₃×Q₈	—	2	1	no	—
166	C₂×C₃×Q₈	—	2	1	no	—
167	C₂²×C₃×C₄	#67	1	1	yes	characteristic
168	C₂²×C₃×C₄	#67	1	1	yes	characteristic
169	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
170	C₂×#67	—	6	1	no	—
171	#235	—	3	4	no	—
172	#235	—	6	2	no	—
173	C₂×D₁₂	—	3	1	no	—
174	C₂×C₄×D₃	—	6	1	no	—
175	C₂×C₄×D₃	—	6	1	no	—
176	C₂×C₄×D₃	—	6	1	no	—
177	C₂×Dic₆	—	3	1	no	—
178	C₃×C₄⋊C₄	—	2	2	no	—
179	C₃×C₄⋊C₄	—	2	2	no	—
180	C₃×C₄⋊C₄	—	2	1	no	—
181	C₃×C₄⋊C₄	—	2	1	no	—
182	C₃×C₄⋊C₄	C₂²×D₃	1	2	yes	—
183	C₃×C₄⋊C₄	C₂²×D₃	1	2	yes	—
184	C₃×C₂²⋊C₄	—	2	2	no	—
185	C₃×C₄²	—	2	2	no	—
186	C₃×C₄²	#67	1	1	yes	characteristic
187	C₃×C₄²	#67	1	1	yes	characteristic
188	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
189	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
190	#217	—	6	1	no	—
191	(C₂×D₃)⋊C₄	—	6	1	no	—
192	(C₂×D₃)⋊C₄	—	6	1	no	—
193	C₄⋊Dic₃	—	3	1	no	—
194	C₄⋊Dic₃	—	3	1	no	—
195	Dic₃⋊C₄	—	6	1	no	—
196	Dic₃⋊C₄	—	6	1	no	—
197	C₄×Dic₃	—	6	1	no	—
198	C₄×Dic₃	—	6	1	no	—
199	C₄×Dic₃	—	6	1	no	—
200	C₂×C₃⋊C₈	—	3	2	no	—
201	C₂×C₃⋊C₈	—	3	2	no	—
202	C₂×C₃⋊C₈	—	6	4	no	—
203	C₂²×C₉	—	2	1	no	—
204	C₂²×C₉	—	2	1	no	—
205	C₂²×C₉	C₂²⋊Q₈	1	2	yes	—
206	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
207	C₂×D₉	—	2	2	no	—
208	C₂×D₉	—	4	1	no	—
209	C₂×D₉	—	4	1	no	—
210	C₄×C₉	—	2	1	no	—
211	C₄×C₉	—	2	1	no	—
212	C₄×C₉	—	2	1	no	—
213	C₄×C₉	—	4	2	no	—
214	C₄×C₉	—	4	2	no	—
215	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
216	C₄×C₉	#135	1	2	yes	—
217	C₄×C₉	C₂²×Q₈	1	1	yes	characteristic
218	C₄×C₉	C₂×#41	1	1	yes	characteristic
219	Dic₉	—	2	2	no	—
220	Dic₉	—	4	1	no	—
221	Dic₉	—	4	1	no	—
222	C₂×#41	—	9	1	no	—
223	C₂²×C₈	—	9	2	no	—
224	C₄⋊Q₈	C₂×D₉	1	1	yes	characteristic
225	C₄⋊Q₈	C₂×D₉	1	1	yes	characteristic
226	#125	C₂×D₉	1	2	yes	—
227	C₂²⋊Q₈	—	2	1	no	—
228	C₂²⋊Q₈	—	2	1	no	—
229	C₄×Q₈	—	2	1	no	—
230	C₄×Q₈	—	2	1	no	—
231	C₄×Q₈	—	9	1	no	—
232	C₄×Q₈	—	9	1	no	—
233	C₄×D₄	—	9	1	no	—
234	C₄×D₄	—	9	1	no	—
235	C₄×D₄	—	18	1	no	—
236	C₄×D₄	—	18	1	no	—
237	#117	—	9	1	no	—
238	#117	—	18	1	no	—
239	#117	C₂×D₉	1	2	yes	—
240	C₂×C₄²	—	18	1	no	—
241	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
242	#107	—	9	2	no	—
243	#107	—	9	2	no	—
244	#106	—	9	2	no	—
245	#106	—	9	2	no	—
246	C₄⋊C₈	—	9	1	no	—
247	C₄⋊C₈	—	9	1	no	—
248	C₄⋊C₈	—	9	2	no	—
249	#103	—	9	2	no	—
250	#103	—	9	2	no	—
251	#102	—	9	2	no	—
252	#102	—	9	2	no	—
253	C₂³×C₃	C₂×#67	1	1	yes	characteristic
254	C₂²×D₃	—	6	1	no	—
255	C₃×Q₈	—	4	2	no	—
256	C₃×Q₈	—	4	2	no	—
257	C₂×C₃×C₄	—	2	2	no	—
258	C₂×C₃×C₄	—	2	2	no	—
259	C₂×C₃×C₄	—	2	2	no	—
260	C₂×C₃×C₄	—	4	1	no	—
261	C₂×C₃×C₄	D₃×Q₈	1	2	yes	—
262	C₂×C₃×C₄	#239	1	2	yes	—
263	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
264	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
265	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
266	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
267	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
268	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
269	#67	—	6	4	no	—
270	C₂×Dic₃	—	6	1	no	—
271	C₂×Dic₃	—	6	1	no	—
272	C₂×Dic₃	—	6	1	no	—
273	D₁₂	—	3	2	no	—
274	D₁₂	—	6	1	no	—
275	C₄×D₃	—	6	4	no	—
276	C₄×D₃	—	12	2	no	—
277	C₄×D₃	—	12	2	no	—
278	Dic₆	—	3	2	no	—
279	Dic₆	—	6	1	no	—
280	C₃⋊C₈	—	6	2	no	—
281	C₃⋊C₈	—	6	2	no	—
282	C₂×C₉	—	2	2	no	—
283	C₂×C₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
284	C₂×C₉	C₂×#135	1	1	yes	characteristic
285	C₂×C₉	#575	1	1	yes	characteristic
286	D₉	—	4	2	no	—
287	#41	—	9	4	no	—
288	#41	—	18	2	no	—
289	C₂×Q₈	—	2	1	no	—
290	C₂×Q₈	—	2	1	no	—
291	C₂×Q₈	—	9	1	no	—
292	C₂×D₄	—	9	1	no	—
293	C₂×D₄	—	18	1	no	—
294	C₂²×C₄	—	18	1	no	—
295	C₂²×C₄	—	18	1	no	—
296	C₂²×C₄	—	18	1	no	—
297	C₂²×C₄	#611	1	1	yes	characteristic
298	C₂²×C₄	#611	1	1	yes	characteristic
299	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
300	C₂×C₈	—	9	2	no	—
301	C₂×C₈	—	9	2	no	—
302	C₂×C₈	—	18	4	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	—	2	1	no	—
307	C₄⋊C₄	—	9	1	no	—
308	C₄⋊C₄	—	9	1	no	—
309	C₄⋊C₄	—	18	1	no	—
310	C₄⋊C₄	—	18	1	no	—
311	C₄⋊C₄	C₂²×D₉	1	2	yes	—
312	C₄⋊C₄	C₂²×D₉	1	2	yes	—
313	C₂²⋊C₄	—	2	2	no	—
314	C₂²⋊C₄	—	18	1	no	—
315	C₂²⋊C₄	—	18	1	no	—
316	C₂²⋊C₄	—	18	1	no	—
317	C₄²	—	2	2	no	—
318	C₄²	—	18	1	no	—
319	C₄²	—	18	1	no	—
320	C₄²	—	18	1	no	—
321	C₄²	#611	1	1	yes	characteristic
322	C₄²	#611	1	1	yes	characteristic
323	C₄²	C₂²×D₉	1	1	yes	characteristic
324	C₄²	C₂²×D₉	1	1	yes	characteristic
325	C₂²×C₃	—	2	1	no	—
326	C₂²×C₃	—	2	1	no	—
327	C₂²×C₃	#946	1	2	yes	—
328	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
329	C₂×D₃	—	6	2	no	—
330	C₂×D₃	—	12	1	no	—
331	C₂×D₃	—	12	1	no	—
332	C₃×C₄	—	2	1	no	—
333	C₃×C₄	—	2	1	no	—
334	C₃×C₄	—	2	1	no	—
335	C₃×C₄	—	4	2	no	—
336	C₃×C₄	—	4	2	no	—
337	C₃×C₄	#946	1	2	yes	—
338	C₃×C₄	#950	1	2	yes	—
339	C₃×C₄	C₂×D₃×Q₈	1	1	yes	characteristic
340	C₃×C₄	C₂×#239	1	1	yes	characteristic
341	Dic₃	—	6	2	no	—
342	Dic₃	—	12	1	no	—
343	Dic₃	—	12	1	no	—
344	C₉	#3076	1	1	yes	characteristic, complemented, Sylow
345	C₂³	—	18	1	no	—
346	C₂³	C₂×#611	1	1	yes	characteristic
347	Q₈	—	4	2	no	—
348	Q₈	—	4	2	no	—
349	Q₈	—	9	2	no	—
350	Q₈	—	18	1	no	—
351	D₄	—	9	2	no	—
352	D₄	—	18	4	no	—
353	D₄	—	18	1	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	18	1	no	—
359	C₂×C₄	—	18	4	no	—
360	C₂×C₄	—	18	1	no	—
361	C₂×C₄	—	18	1	no	—
362	C₂×C₄	—	36	2	no	—
363	C₂×C₄	—	36	2	no	—
364	C₂×C₄	Q₈×D₉	1	2	yes	—
365	C₂×C₄	#3701	1	2	yes	—
366	C₂×C₄	C₂×#611	1	1	yes	characteristic
367	C₂×C₄	C₂×#611	1	1	yes	characteristic
368	C₂×C₄	C₂×#611	1	1	yes	characteristic
369	C₂×C₄	C₂×#611	1	1	yes	characteristic
370	C₂×C₄	C₂×#611	1	1	yes	characteristic
371	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
372	C₈	—	18	2	no	—
373	C₈	—	18	2	no	—
374	C₂×C₃	—	2	2	no	—
375	C₂×C₃	C₂×#946	1	1	yes	characteristic
376	C₂×C₃	C₂×#950	1	1	yes	characteristic
377	C₂×C₃	#5822	1	1	yes	characteristic
378	D₃	—	12	2	no	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	18	2	no	—
382	C₂²	—	36	1	no	—
383	C₂²	—	36	1	no	—
384	C₂²	#63508	1	2	yes	—
385	C₂²	C₂²×#611	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	4	2	no	—
390	C₄	—	4	2	no	—
391	C₄	—	18	2	no	—
392	C₄	—	36	1	no	—
393	C₄	—	36	1	no	—
394	C₄	#63508	1	2	yes	—
395	C₄	#63513	1	2	yes	—
396	C₄	C₂×Q₈×D₉	1	1	yes	characteristic
397	C₄	C₂×#3701	1	1	yes	characteristic
398	C₃	#83349	1	1	yes	characteristic
399	C₂	—	2	2	no	—
400	C₂	—	36	2	no	—
401	C₂	C₂×#63508	1	1	yes	characteristic
402	C₂	C₂×#63513	1	1	yes	characteristic
403	C₂	#10589925	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#611
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₉×#533	C₂
socle	C₂²×C₃	C₂²×#67
prime radical	C₂×#3697	C₂²
hypercenter	#533	D₉
hypocenter	C₉	#3076
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
1. C₉×#533	C₂	semidirect
3. #10588772	C₂	semidirect
4. #10588769	C₂	semidirect
5. #10588767	C₂	semidirect
6. #10588764	C₂	semidirect
7. #10588747	C₂	semidirect
8. #10588745	C₂	semidirect
10. #10588676	C₂	semidirect
11. C₂×C₄⋊(C₉⋊C₈)	C₂	semidirect
62. #3697	Q₈	semidirect
154. #533	D₉	semidirect
344. C₉	#3076	semidirect

Maximal subgroups

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

subgroup	index
C₉×#533	2
C₄×#3697	2
#10588772	2
#10588769	2
#10588767	2
#10588764	2
#10588747	2
#10588745	2
#10588683	2
#10588676	2
C₂×C₄⋊(C₉⋊C₈)	2
#83349	3