Subgroups — #11774774

Order 1152.

527 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5470	C₂	1	1	yes	maximal, characteristic
2	#10592623	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#10589084	C₂	1	4	yes	maximal, complemented
5	#71826	—	3	1	no	maximal
6	#69738	—	3	1	no	maximal
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₂×#3748	—	2	1	no	—
9	C₃×#926	—	2	1	no	—
10	C₃×#924	—	2	1	no	—
11	C₃×#924	C₄	1	2	yes	—
12	C₂²×C₃×C₃⋊C₈	—	2	1	no	—
13	C₃×#870	—	2	1	no	—
14	C₃×#869	C₄	1	1	yes	characteristic
15	C₃×#869	C₄	1	1	yes	characteristic
16	C₃×C₄×Dic₆	—	2	1	no	—
17	C₃×C₄×Dic₆	C₂²	1	2	yes	—
18	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
19	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	2	yes	—
20	C₃×C₄×C₃⋊C₈	—	2	1	no	—
21	C₃×C₄×C₃⋊C₈	C₂²	1	2	yes	—
22	C₃×#533	—	3	1	no	—
23	#5470	D₃	1	1	yes	characteristic
24	C₂×C₃×C₄×C₈	—	3	1	no	—
25	#4924	—	3	1	no	—
26	#4924	—	3	1	no	—
27	#4924	—	6	1	no	—
28	C₂×C₄×C₃⋊C₈	D₃	1	1	yes	characteristic
29	#4535	—	3	4	no	—
30	#4456	—	3	4	no	—
31	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
33	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
34	C₂×C₃×Dic₆	—	2	1	no	—
35	C₂×C₃×Dic₆	C₂×C₄	1	2	yes	—
36	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₃²×C₄²	D₄	1	1	yes	characteristic
38	C₃²×C₄²	D₄	1	1	yes	characteristic
39	C₃²×C₄²	C₂³	1	1	yes	characteristic
40	#3748	—	4	2	no	—
41	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
42	C₃×#217	—	2	2	no	—
43	C₃×C₄⋊Dic₃	—	2	1	no	—
44	C₃×C₄⋊Dic₃	—	2	1	no	—
45	C₃×C₄⋊Dic₃	—	2	1	no	—
46	C₃×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
47	C₃×Dic₃⋊C₄	—	2	2	no	—
48	C₃×Dic₃⋊C₄	—	2	2	no	—
49	C₃×C₄×Dic₃	—	2	2	no	—
50	C₂×C₃×C₃⋊C₈	—	2	4	no	—
51	C₂×C₃×C₃⋊C₈	—	2	2	no	—
52	#1493	—	9	1	no	Sylow
53	C₂²×C₃×C₈	—	6	1	no	—
54	C₃×C₄⋊Q₈	—	3	1	no	—
55	C₃×C₄⋊Q₈	—	3	1	no	—
56	C₃×#125	—	6	1	no	—
57	C₃×C₂²⋊Q₈	—	3	2	no	—
58	C₃×C₂²⋊Q₈	—	6	1	no	—
59	C₃×C₄×Q₈	—	3	2	no	—
60	C₃×C₄×Q₈	—	6	1	no	—
61	C₃×#117	—	6	1	no	—
62	C₂×C₃×C₄²	—	2	1	no	—
63	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
64	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
65	#926	—	2	1	no	—
66	#924	—	2	1	no	—
67	#924	Dic₃	1	2	yes	—
68	C₂×#208	—	6	2	no	—
69	C₂×#208	—	6	1	no	—
70	C₂×#208	—	6	1	no	—
71	C₂²×C₃⋊C₈	—	2	1	no	—
72	#870	—	2	1	no	—
73	#869	Dic₃	1	1	yes	characteristic
74	#869	Dic₃	1	1	yes	characteristic
75	C₄×Dic₆	—	2	1	no	—
76	C₄×Dic₆	C₂×D₃	1	2	yes	—
77	C₃×C₄×C₈	—	3	2	no	—
78	C₃×C₄×C₈	—	6	1	no	—
79	C₄⋊(C₃⋊C₈)	—	3	2	no	—
80	C₄⋊(C₃⋊C₈)	—	3	2	no	—
81	C₄⋊(C₃⋊C₈)	—	6	2	no	—
82	C₄⋊(C₃⋊C₈)	—	6	2	no	—
83	C₄⋊(C₃⋊C₈)	—	6	1	no	—
84	C₄⋊(C₃⋊C₈)	—	6	1	no	—
85	C₄×C₃⋊C₈	—	2	1	no	—
86	C₄×C₃⋊C₈	C₂×D₃	1	2	yes	—
87	C₂³×C₃²	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₃²×C₄	—	2	2	no	—
89	C₂×C₃²×C₄	—	2	2	no	—
90	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
94	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
95	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
98	C₂×C₃×Dic₃	—	2	2	no	—
99	C₂×C₃×Dic₃	—	2	2	no	—
100	C₃×Dic₆	—	2	4	no	—
101	C₃×Dic₆	—	4	2	no	—
102	C₃⋊(C₃⋊C₈)	—	4	2	no	—
103	C₃×C₃⋊C₈	—	2	4	no	—
104	#533	—	3	1	no	—
105	#423	—	9	1	no	—
106	C₂×C₄×C₈	—	3	1	no	—
107	Q₈⋊C₈	—	9	4	no	—
108	C₂×C₃×Q₈	—	3	2	no	—
109	C₂×C₃×Q₈	—	6	1	no	—
110	C₂²×C₃×C₄	—	2	1	no	—
111	C₂²×C₃×C₄	—	2	1	no	—
112	C₂²×C₃×C₄	—	2	1	no	—
113	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
114	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
115	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
116	C₂²×C₃×C₄	#67	1	1	yes	characteristic
117	C₂²×C₃×C₄	#67	1	1	yes	characteristic
118	C₂²×C₃×C₄	#67	1	1	yes	characteristic
119	C₂×Dic₆	—	2	1	no	—
120	C₂×Dic₆	C₂×Dic₃	1	2	yes	—
121	C₂×C₃×C₈	—	6	4	no	—
122	C₂×C₃×C₈	—	6	2	no	—
123	C₃×C₄⋊C₄	—	3	2	no	—
124	C₃×C₄⋊C₄	—	6	2	no	—
125	C₃×C₄⋊C₄	—	6	1	no	—
126	C₃×C₄⋊C₄	—	6	1	no	—
127	C₃×C₄⋊C₄	—	6	1	no	—
128	C₃×C₄⋊C₄	—	6	2	no	—
129	C₃×C₂²⋊C₄	—	6	2	no	—
130	C₃×C₄²	—	2	1	no	—
131	C₃×C₄²	—	2	1	no	—
132	C₃×C₄²	—	2	1	no	—
133	C₃×C₄²	—	2	1	no	—
134	C₃×C₄²	—	6	2	no	—
135	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
136	C₃×C₄²	D₁₂	1	1	yes	characteristic
137	C₃×C₄²	C₂×Dic₃	1	1	yes	characteristic
138	C₃×C₄²	#67	1	1	yes	characteristic
139	C₃×C₄²	#67	1	1	yes	characteristic
140	C₃×C₄²	#67	1	1	yes	characteristic
141	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
142	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
143	#217	—	2	2	no	—
144	C₄⋊Dic₃	—	2	1	no	—
145	C₄⋊Dic₃	—	2	1	no	—
146	C₄⋊Dic₃	—	2	1	no	—
147	C₄⋊Dic₃	C₂×Dic₃	1	2	yes	—
148	Dic₃⋊C₄	—	2	2	no	—
149	Dic₃⋊C₄	—	2	2	no	—
150	C₄×Dic₃	—	2	2	no	—
151	#208	—	12	4	no	—
152	#208	—	12	2	no	—
153	#208	—	12	2	no	—
154	C₂×C₃⋊C₈	—	2	4	no	—
155	C₂×C₃⋊C₈	—	2	2	no	—
156	C₂×C₃⋊C₈	—	6	4	no	—
157	C₂×C₃⋊C₈	—	6	2	no	—
158	C₂×C₃⋊C₈	—	6	2	no	—
159	C₂²×C₃²	—	2	1	no	—
160	C₂²×C₃²	—	2	1	no	—
161	C₂²×C₃²	#103	1	2	yes	—
162	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₃²×C₄	—	2	1	no	—
164	C₃²×C₄	—	2	1	no	—
165	C₃²×C₄	#103	1	2	yes	—
166	C₃²×C₄	#104	1	2	yes	—
167	C₃²×C₄	#131	1	2	yes	—
168	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
169	C₃²×C₄	C₂×Q₁₆	1	1	yes	characteristic
170	C₃×Dic₃	—	4	2	no	—
171	C₃×Dic₃	—	4	2	no	—
172	C₂×M₁₆	—	18	1	no	—
173	C₂²×C₈	—	6	1	no	—
174	C₄⋊Q₈	—	3	1	no	—
175	C₄⋊Q₈	—	3	1	no	—
176	#125	—	6	1	no	—
177	C₂²⋊Q₈	—	3	2	no	—
178	C₂²⋊Q₈	—	6	1	no	—
179	C₄×Q₈	—	3	2	no	—
180	C₄×Q₈	—	6	1	no	—
181	#117	—	6	1	no	—
182	C₂×C₄²	D₃²	1	1	yes	characteristic
183	C₄⋊C₈	—	9	2	no	—
184	C₄⋊C₈	—	18	1	no	—
185	C₄×C₈	—	3	2	no	—
186	C₄×C₈	—	6	1	no	—
187	C₂³×C₃	—	2	1	no	—
188	C₂³×C₃	(C₂×D₃)⋊C₄	1	1	yes	characteristic
189	C₂³×C₃	#217	1	1	yes	characteristic
190	C₃×Q₈	—	6	4	no	—
191	C₃×Q₈	—	12	2	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	—	2	2	no	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	4	2	no	—
205	C₂×C₃×C₄	—	4	2	no	—
206	C₂×C₃×C₄	—	6	2	no	—
207	C₂×C₃×C₄	—	6	2	no	—
208	C₂×C₃×C₄	#204	1	2	yes	—
209	C₂×C₃×C₄	Dic₁₂	1	2	yes	—
210	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
211	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
212	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
213	C₂×C₃×C₄	#215	1	2	yes	—
214	C₂×C₃×C₄	#216	1	2	yes	—
215	C₂×C₃×C₄	#217	1	1	yes	characteristic
216	C₂×C₃×C₄	#217	1	1	yes	characteristic
217	C₂×C₃×C₄	#217	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
224	C₂×Dic₃	—	2	2	no	—
225	C₂×Dic₃	—	2	2	no	—
226	Dic₆	—	2	4	no	—
227	Dic₆	—	4	2	no	—
228	C₃×C₈	—	6	4	no	—
229	C₃⋊C₈	—	2	4	no	—
230	C₃⋊C₈	—	12	4	no	—
231	C₃⋊C₈	—	12	2	no	—
232	C₃⋊C₈	—	12	2	no	—
233	C₂×C₃²	—	2	2	no	—
234	C₂×C₃²	#408	1	1	yes	characteristic
235	C₂×C₃²	C₂×#103	1	1	yes	characteristic
236	C₂×C₃²	C₂×#104	1	1	yes	characteristic
237	C₂×Q₈	—	3	2	no	—
238	C₂×Q₈	—	6	1	no	—
239	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
240	C₂²×C₄	#625	1	1	yes	characteristic
241	C₂²×C₄	#626	1	1	yes	characteristic
242	M₁₆	—	36	2	no	—
243	C₂×C₈	—	6	4	no	—
244	C₂×C₈	—	6	2	no	—
245	C₂×C₈	—	18	2	no	—
246	C₄⋊C₄	—	3	2	no	—
247	C₄⋊C₄	—	6	1	no	—
248	C₄⋊C₄	—	6	1	no	—
249	C₄⋊C₄	—	6	2	no	—
250	C₄⋊C₄	—	6	1	no	—
251	C₄⋊C₄	—	6	2	no	—
252	C₂²⋊C₄	—	6	2	no	—
253	C₄²	—	6	2	no	—
254	C₄²	D₃×Dic₃	1	1	yes	characteristic
255	C₄²	#625	1	1	yes	characteristic
256	C₄²	#626	1	1	yes	characteristic
257	C₄²	C₂×D₃²	1	1	yes	characteristic
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	—	2	1	no	—
262	C₂²×C₃	—	2	2	no	—
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	4	1	no	—
265	C₂²×C₃	—	4	1	no	—
266	C₂²×C₃	#816	1	2	yes	—
267	C₂²×C₃	#835	1	2	yes	—
268	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
269	C₂²×C₃	C₂×#217	1	1	yes	characteristic
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	2	1	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	2	2	no	—
275	C₃×C₄	—	2	2	no	—
276	C₃×C₄	—	2	2	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	4	1	no	—
280	C₃×C₄	—	4	1	no	—
281	C₃×C₄	—	12	2	no	—
282	C₃×C₄	—	12	2	no	—
283	C₃×C₄	#805	1	2	yes	—
284	C₃×C₄	#816	1	2	yes	—
285	C₃×C₄	#835	1	2	yes	—
286	C₃×C₄	#837	1	2	yes	—
287	C₃×C₄	#901	1	2	yes	—
288	C₃×C₄	C₂×#204	1	1	yes	characteristic
289	C₃×C₄	C₂×Dic₁₂	1	1	yes	characteristic
290	C₃×C₄	C₂×#215	1	1	yes	characteristic
291	C₃×C₄	C₂×#216	1	1	yes	characteristic
292	C₃×C₄	#948	1	2	yes	—
293	Dic₃	—	4	2	no	—
294	Dic₃	—	4	2	no	—
295	C₃²	#1493	1	1	yes	characteristic, complemented, Sylow
296	C₂³	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
297	Q₈	—	6	4	no	—
298	Q₈	—	12	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	6	2	no	—
302	C₂×C₄	—	6	2	no	—
303	C₂×C₄	#3717	1	2	yes	—
304	C₂×C₄	#3719	1	2	yes	—
305	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
306	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
307	C₂×C₄	(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
308	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
309	C₂×C₄	C₂×#625	1	1	yes	characteristic
310	C₂×C₄	C₂×#626	1	1	yes	characteristic
311	C₈	—	6	4	no	—
312	C₈	—	36	2	no	—
313	C₂×C₃	—	2	2	no	—
314	C₂×C₃	—	2	2	no	—
315	C₂×C₃	—	2	1	no	—
316	C₂×C₃	—	2	1	no	—
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	4	2	no	—
319	C₂×C₃	C₂×#805	1	1	yes	characteristic
320	C₂×C₃	C₂×#816	1	1	yes	characteristic
321	C₂×C₃	#5110	1	1	yes	characteristic
322	C₂×C₃	C₂×#835	1	1	yes	characteristic
323	C₂×C₃	#5234	1	1	yes	characteristic
324	C₂×C₃	C₂×#837	1	1	yes	characteristic
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	#63566	1	2	yes	—
328	C₂²	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	12	2	no	—
332	C₄	—	12	2	no	—
333	C₄	#63566	1	2	yes	—
334	C₄	#63569	1	2	yes	—
335	C₄	#63814	1	2	yes	—
336	C₄	C₂×#3717	1	1	yes	characteristic
337	C₄	C₂×#3719	1	1	yes	characteristic
338	C₃	—	2	1	no	—
339	C₃	#69738	1	1	yes	characteristic, complemented
340	C₃	#71826	1	1	yes	characteristic, complemented
341	C₂	—	2	2	no	—
342	C₂	#10591070	1	1	yes	characteristic
343	C₂	C₂×#63566	1	1	yes	characteristic
344	C₂	C₂×#63569	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#626
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₄²	C₂×D₃²
Fitting	C₂×C₃²×C₄²	C₂²
socle	C₂²×C₃²	C₂×C₂²⋊C₄
prime radical	C₂³×C₃²	C₂²⋊C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1493
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
4. #10589084	C₂	semidirect
295. C₃²	#1493	semidirect
339. C₃	#69738	semidirect
340. C₃	#71826	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5470	2
#10592623	2
C₂×C₃×C₄×C₃⋊C₈	2
#10589084	2
#71826	3
#69738	3