Subgroups — #11791417

Order 1152.

486 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63506	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#103	C₂	1	1	yes	maximal, characteristic
3	C₂×#63395	C₂	1	1	yes	maximal, characteristic
4	C₂×#63376	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₉⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₄⋊C₄×Dic₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#63368	C₂	1	1	yes	maximal, characteristic
8	#70229	—	3	1	no	maximal
9	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₈	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	#63506	—	2	2	no	—
14	#63506	C₄	1	4	yes	complemented
15	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₉⋊C₄	—	2	1	no	—
17	C₂×Dic₉⋊C₄	—	2	1	no	—
18	C₂×C₄×Dic₉	—	2	1	no	—
19	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
20	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
21	C₉×#103	—	2	2	no	—
22	#63395	—	2	2	no	—
23	C₉⋊#95	—	2	1	no	—
24	#63376	—	2	2	no	—
25	Dic₉⋊C₈	—	2	2	no	—
26	#63368	—	2	2	no	—
27	C₂×#944	—	3	1	no	—
28	C₂×C₃×#103	D₃	1	1	yes	characteristic
29	C₂×#835	—	3	1	no	—
30	C₂×#816	—	3	1	no	—
31	C₂×Dic₃⋊C₈	—	3	1	no	—
32	Dic₃×C₄⋊C₄	—	3	1	no	—
33	C₂×#810	—	3	1	no	—
34	C₂×Q₈×C₉	—	2	1	no	—
35	C₂×Q₈×C₉	—	2	1	no	—
36	C₂×Q₈×C₉	—	2	1	no	—
37	C₂×Q₈×C₉	—	2	1	no	—
38	C₂×Q₈×C₉	—	2	1	no	—
39	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
40	C₂²×C₄×C₉	—	2	1	no	—
41	C₂²×C₄×C₉	—	2	1	no	—
42	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
43	C₂²×Dic₉	—	2	1	no	—
44	C₂²×Dic₉	—	2	1	no	—
45	C₂²×Dic₉	D₄	1	1	yes	characteristic
46	C₂²×Dic₉	D₄	1	1	yes	characteristic
47	C₂×Dic₁₈	—	2	1	no	—
48	C₂×Dic₁₈	—	2	1	no	—
49	C₂×Dic₁₈	—	2	1	no	—
50	C₂×Dic₁₈	—	2	1	no	—
51	C₂×Dic₁₈	—	2	1	no	—
52	C₂×Dic₁₈	C₂×C₄	1	2	yes	—
53	C₂×C₈×C₉	—	2	1	no	—
54	C₂×C₈×C₉	—	2	1	no	—
55	C₂×C₈×C₉	—	2	1	no	—
56	C₉×C₄⋊C₄	—	2	1	no	—
57	C₉×C₄⋊C₄	D₄	1	2	yes	—
58	C₄⋊Dic₉	—	2	1	no	—
59	C₄⋊Dic₉	D₄	1	2	yes	—
60	Dic₉⋊C₄	—	2	4	no	—
61	Dic₉⋊C₄	—	2	4	no	—
62	C₄×Dic₉	—	4	2	no	—
63	C₄×Dic₉	C₂×C₄	1	2	yes	—
64	C₄×Dic₉	D₄	1	2	yes	—
65	C₂×C₉⋊C₈	—	2	1	no	—
66	C₂×C₉⋊C₈	—	2	1	no	—
67	C₂×C₉⋊C₈	—	2	1	no	—
68	#1981	—	9	1	no	Sylow
69	C₂²×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
70	C₂²×Dic₆	—	3	1	no	—
71	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
72	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
73	#944	—	3	4	no	—
74	#944	—	6	2	no	—
75	C₂×C₄⋊Dic₃	—	3	1	no	—
76	C₂×Dic₃⋊C₄	—	6	1	no	—
77	C₂×Dic₃⋊C₄	—	6	1	no	—
78	C₂×C₄×Dic₃	—	3	1	no	—
79	C₂×C₄×Dic₃	—	6	1	no	—
80	C₂²×C₃⋊C₈	—	3	1	no	—
81	C₃×#103	—	2	2	no	—
82	#835	—	6	2	no	—
83	C₃⋊#95	—	6	1	no	—
84	#816	—	6	2	no	—
85	Dic₃⋊C₈	—	6	2	no	—
86	#810	—	6	2	no	—
87	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
88	Q₈×C₉	—	2	2	no	—
89	Q₈×C₉	—	2	2	no	—
90	Q₈×C₉	—	4	1	no	—
91	Q₈×C₉	—	4	1	no	—
92	C₂×C₄×C₉	—	2	2	no	—
93	C₂×C₄×C₉	—	2	2	no	—
94	C₂×C₄×C₉	—	4	1	no	—
95	C₂×C₄×C₉	—	4	1	no	—
96	C₂×C₄×C₉	—	4	1	no	—
97	C₂×C₄×C₉	—	4	1	no	—
98	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₉	#41	1	1	yes	characteristic
103	C₂×C₄×C₉	#41	1	1	yes	characteristic
104	C₂×Dic₉	—	2	1	no	—
105	C₂×Dic₉	—	2	1	no	—
106	C₂×Dic₉	—	2	2	no	—
107	C₂×Dic₉	—	2	2	no	—
108	C₂×Dic₉	—	4	1	no	—
109	C₂×Dic₉	—	4	1	no	—
110	C₂×Dic₉	—	4	1	no	—
111	C₂×Dic₉	—	4	1	no	—
112	C₂×Dic₉	C₂²⋊C₄	1	2	yes	—
113	C₂×Dic₉	C₂²⋊C₄	1	2	yes	—
114	C₂×Dic₉	SD₁₆	1	2	yes	—
115	C₂×Dic₉	Q₁₆	1	2	yes	—
116	Dic₁₈	—	2	2	no	—
117	Dic₁₈	—	2	2	no	—
118	Dic₁₈	—	4	1	no	—
119	Dic₁₈	—	4	1	no	—
120	C₈×C₉	—	4	1	no	—
121	C₉⋊C₈	—	4	1	no	—
122	C₂×C₄⋊Q₈	—	9	1	no	—
123	C₂×C₄⋊C₈	—	9	1	no	—
124	C₂×#103	—	9	1	no	—
125	C₂×#103	—	9	1	no	—
126	C₂×#103	—	9	1	no	—
127	C₂×#103	D₉	1	1	yes	characteristic
128	C₄×C₄⋊C₄	—	9	1	no	—
129	C₂×C₃×Q₈	—	2	1	no	—
130	C₂×C₃×Q₈	—	2	1	no	—
131	C₂×C₃×Q₈	—	2	1	no	—
132	C₂×C₃×Q₈	—	2	1	no	—
133	C₂×C₃×Q₈	—	2	1	no	—
134	C₂×C₃×Q₈	C₄×D₃	1	2	yes	—
135	C₂²×C₃×C₄	—	2	1	no	—
136	C₂²×C₃×C₄	—	2	1	no	—
137	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
138	C₂²×Dic₃	—	3	1	no	—
139	C₂²×Dic₃	—	3	1	no	—
140	C₂²×Dic₃	—	6	1	no	—
141	C₂²×Dic₃	—	6	1	no	—
142	C₂×Dic₆	—	3	2	no	—
143	C₂×Dic₆	—	6	1	no	—
144	C₂×Dic₆	—	6	1	no	—
145	C₂×Dic₆	—	6	1	no	—
146	C₂×Dic₆	—	6	1	no	—
147	C₂×Dic₆	—	6	1	no	—
148	C₂×C₃×C₈	—	2	1	no	—
149	C₂×C₃×C₈	—	2	1	no	—
150	C₂×C₃×C₈	—	2	1	no	—
151	C₃×C₄⋊C₄	—	2	1	no	—
152	C₃×C₄⋊C₄	#67	1	2	yes	—
153	C₄⋊Dic₃	—	3	2	no	—
154	C₄⋊Dic₃	—	6	1	no	—
155	Dic₃⋊C₄	—	6	4	no	—
156	Dic₃⋊C₄	—	6	4	no	—
157	C₄×Dic₃	—	3	2	no	—
158	C₄×Dic₃	—	3	2	no	—
159	C₄×Dic₃	—	12	2	no	—
160	C₂×C₃⋊C₈	—	6	1	no	—
161	C₂×C₃⋊C₈	—	6	1	no	—
162	C₂×C₃⋊C₈	—	6	1	no	—
163	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
164	C₂²×C₉	#121	1	1	yes	characteristic
165	C₂²×C₉	#121	1	1	yes	characteristic
166	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
167	C₂²×C₉	C₂×Q₁₆	1	1	yes	characteristic
168	C₂²×C₉	#137	1	1	yes	characteristic
169	C₂²×C₉	#137	1	1	yes	characteristic
170	C₄×C₉	—	4	2	no	—
171	C₄×C₉	—	4	2	no	—
172	C₄×C₉	C₄×D₄	1	1	yes	characteristic
173	C₄×C₉	C₄×D₄	1	1	yes	characteristic
174	C₄×C₉	#124	1	1	yes	characteristic
175	C₄×C₉	#127	1	1	yes	characteristic
176	Dic₉	—	2	2	no	—
177	Dic₉	—	4	2	no	—
178	Dic₉	—	4	2	no	—
179	Dic₉	#103	1	4	yes	complemented
180	C₂²×Q₈	—	9	1	no	—
181	C₂²×Q₈	C₂×D₉	1	1	yes	characteristic
182	C₂²×C₈	—	9	1	no	—
183	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
184	C₄⋊Q₈	—	9	4	no	—
185	C₄⋊Q₈	—	18	2	no	—
186	C₂×C₄⋊C₄	—	9	1	no	—
187	C₂×C₄⋊C₄	—	18	1	no	—
188	C₂×C₄⋊C₄	—	18	1	no	—
189	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
190	C₂×C₄²	—	9	1	no	—
191	C₂×C₄²	—	18	1	no	—
192	C₄⋊C₈	—	18	2	no	—
193	#103	—	2	2	no	—
194	#103	—	18	2	no	—
195	#103	—	18	2	no	—
196	#103	—	18	2	no	—
197	#95	—	18	1	no	—
198	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
199	C₃×Q₈	—	2	2	no	—
200	C₃×Q₈	—	2	2	no	—
201	C₃×Q₈	—	4	1	no	—
202	C₃×Q₈	—	4	1	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	4	1	no	—
206	C₂×C₃×C₄	—	4	1	no	—
207	C₂×C₃×C₄	—	4	1	no	—
208	C₂×C₃×C₄	—	4	1	no	—
209	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
210	C₂×C₃×C₄	#235	1	1	yes	characteristic
211	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
212	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
213	C₂×C₃×C₄	#237	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
215	C₂×Dic₃	—	3	2	no	—
216	C₂×Dic₃	—	3	2	no	—
217	C₂×Dic₃	—	3	2	no	—
218	C₂×Dic₃	—	3	2	no	—
219	C₂×Dic₃	—	6	1	no	—
220	C₂×Dic₃	—	6	1	no	—
221	C₂×Dic₃	—	6	2	no	—
222	C₂×Dic₃	—	6	2	no	—
223	C₂×Dic₃	—	12	1	no	—
224	C₂×Dic₃	—	12	1	no	—
225	C₂×Dic₃	—	12	1	no	—
226	C₂×Dic₃	—	12	1	no	—
227	Dic₆	—	6	2	no	—
228	Dic₆	—	6	2	no	—
229	Dic₆	—	12	1	no	—
230	Dic₆	—	12	1	no	—
231	C₃×C₈	—	4	1	no	—
232	C₃⋊C₈	—	12	1	no	—
233	C₂×C₉	#390	1	1	yes	characteristic
234	C₂×C₉	C₂×#103	1	1	yes	characteristic
235	C₂×C₉	#419	1	1	yes	characteristic
236	C₂×C₉	#461	1	1	yes	characteristic
237	C₂×C₉	#461	1	1	yes	characteristic
238	C₂×C₉	#462	1	1	yes	characteristic
239	C₂×C₉	#462	1	1	yes	characteristic
240	C₂×Q₈	—	2	1	no	—
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	9	2	no	—
246	C₂×Q₈	—	18	1	no	—
247	C₂×Q₈	—	18	1	no	—
248	C₂×Q₈	—	18	1	no	—
249	C₂×Q₈	—	18	1	no	—
250	C₂×Q₈	—	18	1	no	—
251	C₂×Q₈	C₄×D₉	1	2	yes	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	9	1	no	—
255	C₂²×C₄	—	9	1	no	—
256	C₂²×C₄	—	18	1	no	—
257	C₂²×C₄	—	18	1	no	—
258	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	18	1	no	—
263	C₂×C₈	—	18	1	no	—
264	C₂×C₈	—	18	1	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	9	2	no	—
267	C₄⋊C₄	—	18	4	no	—
268	C₄⋊C₄	—	18	4	no	—
269	C₄⋊C₄	—	18	1	no	—
270	C₄⋊C₄	#611	1	2	yes	—
271	C₄²	—	9	2	no	—
272	C₄²	—	9	2	no	—
273	C₄²	—	36	2	no	—
274	C₂²×C₃	D₃×C₂²⋊C₄	1	1	yes	characteristic
275	C₂²×C₃	#884	1	1	yes	characteristic
276	C₂²×C₃	D₃×SD₁₆	1	1	yes	characteristic
277	C₂²×C₃	#915	1	1	yes	characteristic
278	C₂²×C₃	D₃×Q₁₆	1	1	yes	characteristic
279	C₂²×C₃	#918	1	1	yes	characteristic
280	C₂²×C₃	#939	1	1	yes	characteristic
281	C₃×C₄	—	4	2	no	—
282	C₃×C₄	—	4	2	no	—
283	C₃×C₄	#67⋊C₄	1	1	yes	characteristic
284	C₃×C₄	#885	1	1	yes	characteristic
285	C₃×C₄	C₄×#67	1	1	yes	characteristic
286	C₃×C₄	#940	1	1	yes	characteristic
287	Dic₃	—	3	4	no	—
288	Dic₃	—	6	2	no	—
289	Dic₃	—	12	2	no	—
290	Dic₃	—	12	2	no	—
291	C₉	#1981	1	1	yes	characteristic, complemented, Sylow
292	C₂³	D₄×D₉	1	1	yes	characteristic
293	Q₈	—	2	2	no	—
294	Q₈	—	2	2	no	—
295	Q₈	—	4	1	no	—
296	Q₈	—	4	1	no	—
297	Q₈	—	18	2	no	—
298	Q₈	—	18	2	no	—
299	Q₈	—	36	1	no	—
300	Q₈	—	36	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	9	2	no	—
308	C₂×C₄	—	9	2	no	—
309	C₂×C₄	—	9	2	no	—
310	C₂×C₄	—	9	2	no	—
311	C₂×C₄	—	18	1	no	—
312	C₂×C₄	—	18	1	no	—
313	C₂×C₄	—	18	2	no	—
314	C₂×C₄	—	18	2	no	—
315	C₂×C₄	—	36	1	no	—
316	C₂×C₄	—	36	1	no	—
317	C₂×C₄	—	36	1	no	—
318	C₂×C₄	—	36	1	no	—
319	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
320	C₂×C₄	#3697	1	1	yes	characteristic
321	C₂×C₄	D₄×D₉	1	1	yes	characteristic
322	C₂×C₄	D₄×D₉	1	1	yes	characteristic
323	C₂×C₄	#3699	1	1	yes	characteristic
324	C₂×C₄	C₂×#611	1	1	yes	characteristic
325	C₈	—	4	1	no	—
326	C₈	—	36	1	no	—
327	C₂×C₃	#4795	1	1	yes	characteristic
328	C₂×C₃	D₃×#103	1	1	yes	characteristic
329	C₂×C₃	#4802	1	1	yes	characteristic
330	C₂×C₃	#4818	1	1	yes	characteristic
331	C₂×C₃	#4957	1	1	yes	characteristic
332	C₂×C₃	#5162	1	1	yes	characteristic
333	C₂×C₃	#5182	1	1	yes	characteristic
334	C₂²	C₂²⋊C₄×D₉	1	1	yes	characteristic
335	C₂²	#63446	1	1	yes	characteristic
336	C₂²	SD₁₆×D₉	1	1	yes	characteristic
337	C₂²	#63477	1	1	yes	characteristic
338	C₂²	Q₁₆×D₉	1	1	yes	characteristic
339	C₂²	#63480	1	1	yes	characteristic
340	C₂²	#63501	1	1	yes	characteristic
341	C₄	—	4	2	no	—
342	C₄	—	4	2	no	—
343	C₄	—	9	4	no	—
344	C₄	—	18	2	no	—
345	C₄	—	36	2	no	—
346	C₄	—	36	2	no	—
347	C₄	#611⋊C₄	1	1	yes	characteristic
348	C₄	#63447	1	1	yes	characteristic
349	C₄	C₄×#611	1	1	yes	characteristic
350	C₄	#63502	1	1	yes	characteristic
351	C₃	#70229	1	1	yes	characteristic
352	C₂	#10588475	1	1	yes	characteristic
353	C₂	D₉×#103	1	1	yes	characteristic
354	C₂	#10588482	1	1	yes	characteristic
355	C₂	#10588498	1	1	yes	characteristic
356	C₂	#10588637	1	1	yes	characteristic
357	C₂	#10588842	1	1	yes	characteristic
358	C₂	#10588862	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₉
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×#103	C₂
socle, prime radical	C₂³×C₃	D₃×D₄
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×#103	D₉
hypocenter	C₉	#1981
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

Complements

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

N	H ≅ G/N	type
14. #63506	C₄	semidirect
179. Dic₉	#103	semidirect
291. C₉	#1981	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63506	2
C₂×C₉×#103	2
C₂×#63395	2
C₂×#63376	2
C₂×Dic₉⋊C₈	2
C₄⋊C₄×Dic₉	2
C₂×#63368	2
#70229	3