Subgroups — #11786131

Order 1152.

539 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic
2	#10589390	C₂	1	1	yes	maximal, characteristic
3	C₃×#4524	C₂	1	1	yes	maximal, characteristic
4	#10589155	C₂	1	2	yes	maximal
5	#10589155	C₂	1	2	yes	maximal
6	#69393	—	3	1	no	maximal
7	#68607	—	3	1	no	maximal
8	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₃×C₄×Dic₃	C₄	1	1	yes	characteristic
11	C₂×C₃×C₄×Dic₃	C₄	1	1	yes	characteristic
12	C₂×C₃×C₄×Dic₃	C₄	1	1	yes	characteristic
13	C₂×C₃×C₄×Dic₃	C₄	1	1	yes	characteristic
14	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
16	C₂²×C₃×C₃⋊C₈	C₂²	1	2	yes	—
17	C₃×C₄³	—	3	1	no	—
18	C₄²×Dic₃	D₃	1	1	yes	characteristic
19	C₃×#336	—	3	1	no	—
20	#4550	—	3	2	no	—
21	#4550	—	3	2	no	—
22	#4524	—	3	1	no	—
23	#4524	—	3	1	no	—
24	#4524	—	3	2	no	—
25	#4524	—	3	2	no	—
26	#4524	—	6	1	no	—
27	#4524	D₃	1	1	yes	characteristic
28	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
31	C₂²×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃×Dic₃	C₂×C₄	1	1	yes	characteristic
35	C₃²×C₄²	—	2	2	no	—
36	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
37	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
38	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
39	C₃×C₄×Dic₃	—	2	2	no	—
40	C₃×C₄×Dic₃	—	2	2	no	—
41	C₃×C₄×Dic₃	—	2	1	no	—
42	C₃×C₄×Dic₃	—	2	1	no	—
43	C₃×C₄×Dic₃	—	2	2	no	—
44	C₃×C₄×Dic₃	—	2	2	no	—
45	C₃×C₄×Dic₃	C₈	1	4	yes	complemented
46	C₂×C₃×C₃⋊C₈	—	2	2	no	—
47	C₂×C₃×C₃⋊C₈	—	2	2	no	—
48	C₂×C₃×C₃⋊C₈	—	2	2	no	—
49	#1839	—	9	1	no	Sylow
50	C₂²×C₃×C₈	—	3	2	no	—
51	C₂×C₃×C₄²	—	2	1	no	—
52	C₂×C₃×C₄²	—	3	1	no	—
53	C₂×C₃×C₄²	—	3	1	no	—
54	C₂×C₃×C₄²	—	3	1	no	—
55	C₂×C₃×C₄²	—	3	1	no	—
56	C₂×C₃×C₄²	—	3	1	no	—
57	C₂×C₃×C₄²	—	3	1	no	—
58	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
59	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
60	C₂×C₄×Dic₃	Dic₃	1	1	yes	characteristic
61	C₂×C₄×Dic₃	Dic₃	1	1	yes	characteristic
62	C₂×C₄×Dic₃	Dic₃	1	1	yes	characteristic
63	C₂×C₄×Dic₃	Dic₃	1	1	yes	characteristic
64	C₂×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
65	C₂×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
66	C₂²×C₃⋊C₈	—	3	2	no	—
67	C₂²×C₃⋊C₈	—	3	2	no	—
68	C₂²×C₃⋊C₈	—	6	2	no	—
69	C₂²×C₃⋊C₈	C₂×D₃	1	2	yes	—
70	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃²×C₄	—	2	1	no	—
72	C₂×C₃²×C₄	—	2	1	no	—
73	C₂×C₃²×C₄	—	2	1	no	—
74	C₂×C₃²×C₄	—	2	1	no	—
75	C₂×C₃²×C₄	C₂×C₈	1	2	yes	—
76	C₂×C₃²×C₄	M₁₆	1	2	yes	—
77	C₂×C₃²×C₄	#41	1	2	yes	—
78	C₂×C₃²×C₄	#41	1	1	yes	characteristic
79	C₂×C₃²×C₄	#41	1	2	yes	—
80	C₂×C₃²×C₄	#41	1	1	yes	characteristic
81	C₂×C₃×Dic₃	—	2	1	no	—
82	C₂×C₃×Dic₃	—	2	1	no	—
83	C₂×C₃×Dic₃	—	2	1	no	—
84	C₂×C₃×Dic₃	—	2	1	no	—
85	C₂×C₃×Dic₃	—	2	1	no	—
86	C₂×C₃×Dic₃	—	2	1	no	—
87	C₂×C₃×Dic₃	—	2	1	no	—
88	C₂×C₃×Dic₃	—	2	1	no	—
89	C₂×C₃×Dic₃	C₂×C₈	1	2	yes	—
90	C₂×C₃×Dic₃	C₂×C₈	1	2	yes	—
91	C₂×C₃×Dic₃	M₁₆	1	2	yes	—
92	C₂×C₃×Dic₃	M₁₆	1	2	yes	—
93	C₃⋊(C₃⋊C₈)	—	4	2	no	—
94	C₃×C₃⋊C₈	—	4	2	no	—
95	C₄³	—	3	1	no	—
96	#336	—	3	1	no	—
97	#336	—	9	2	no	—
98	#336	—	9	2	no	—
99	#336	—	9	1	no	—
100	C₂²×C₃×C₄	—	2	1	no	—
101	C₂²×C₃×C₄	—	2	1	no	—
102	C₂²×C₃×C₄	—	2	1	no	—
103	C₂²×C₃×C₄	—	3	1	no	—
104	C₂²×C₃×C₄	—	3	1	no	—
105	C₂²×C₃×C₄	—	3	1	no	—
106	C₂²×C₃×C₄	—	3	1	no	—
107	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
113	C₂²×Dic₃	C₂×Dic₃	1	1	yes	characteristic
114	C₂²×Dic₃	C₂×Dic₃	1	1	yes	characteristic
115	C₂²×Dic₃	C₂×Dic₃	1	1	yes	characteristic
116	C₂²×Dic₃	C₂×Dic₃	1	1	yes	characteristic
117	C₂×C₃×C₈	—	6	2	no	—
118	C₂×C₃×C₈	—	6	2	no	—
119	C₂×C₃×C₈	—	6	2	no	—
120	C₃×C₄²	—	2	2	no	—
121	C₃×C₄²	—	2	2	no	—
122	C₃×C₄²	—	3	4	no	—
123	C₃×C₄²	—	4	2	no	—
124	C₃×C₄²	—	6	2	no	—
125	C₃×C₄²	—	6	2	no	—
126	C₃×C₄²	—	6	2	no	—
127	C₃×C₄²	—	6	2	no	—
128	C₃×C₄²	—	6	1	no	—
129	C₃×C₄²	—	6	1	no	—
130	C₄×Dic₃	—	2	2	no	—
131	C₄×Dic₃	—	2	1	no	—
132	C₄×Dic₃	—	2	2	no	—
133	C₄×Dic₃	—	2	1	no	—
134	C₄×Dic₃	—	2	2	no	—
135	C₄×Dic₃	—	2	2	no	—
136	C₄×Dic₃	C₃⋊C₈	1	4	yes	complemented
137	C₂×C₃⋊C₈	—	2	2	no	—
138	C₂×C₃⋊C₈	—	2	2	no	—
139	C₂×C₃⋊C₈	—	2	2	no	—
140	C₂×C₃⋊C₈	—	6	4	no	—
141	C₂×C₃⋊C₈	—	6	2	no	—
142	C₂×C₃⋊C₈	—	6	2	no	—
143	C₂×C₃⋊C₈	—	6	2	no	—
144	C₂×C₃⋊C₈	—	6	2	no	—
145	C₂×C₃⋊C₈	—	6	2	no	—
146	C₂×C₃⋊C₈	—	6	2	no	—
147	C₂×C₃⋊C₈	—	12	2	no	—
148	C₂×C₃⋊C₈	—	12	2	no	—
149	C₂²×C₃²	#117	1	1	yes	characteristic
150	C₂²×C₃²	#117	1	1	yes	characteristic
151	C₂²×C₃²	#117	1	1	yes	characteristic
152	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
153	C₂²×C₃²	C₂×M₁₆	1	1	yes	characteristic
154	C₂²×C₃²	C₂×M₁₆	1	1	yes	characteristic
155	C₂²×C₃²	C₂×M₁₆	1	1	yes	characteristic
156	C₃²×C₄	—	2	2	no	—
157	C₃²×C₄	—	2	2	no	—
158	C₃²×C₄	#126	1	2	yes	—
159	C₃²×C₄	#126	1	2	yes	—
160	C₃×Dic₃	—	2	2	no	—
161	C₃×Dic₃	—	2	2	no	—
162	C₃×Dic₃	—	2	2	no	—
163	C₃×Dic₃	—	2	2	no	—
164	C₂²×C₈	—	3	2	no	—
165	C₂²×C₈	—	9	2	no	—
166	C₂×C₄²	—	3	1	no	—
167	C₂×C₄²	—	3	1	no	—
168	C₂×C₄²	—	3	1	no	—
169	C₂×C₄²	—	3	1	no	—
170	C₂×C₄²	—	3	1	no	—
171	C₂×C₄²	—	3	1	no	—
172	C₂×C₄²	D₃²	1	1	yes	characteristic
173	C₂³×C₃	—	2	1	no	—
174	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
175	C₂³×C₃	C₂²×Dic₃	1	1	yes	characteristic
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	2	1	no	—
179	C₂×C₃×C₄	—	2	1	no	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	2	1	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	3	2	no	—
191	C₂×C₃×C₄	—	3	2	no	—
192	C₂×C₃×C₄	—	3	2	no	—
193	C₂×C₃×C₄	—	3	2	no	—
194	C₂×C₃×C₄	—	4	1	no	—
195	C₂×C₃×C₄	—	4	1	no	—
196	C₂×C₃×C₄	—	4	1	no	—
197	C₂×C₃×C₄	—	4	1	no	—
198	C₂×C₃×C₄	—	6	1	no	—
199	C₂×C₃×C₄	—	6	1	no	—
200	C₂×C₃×C₄	—	6	1	no	—
201	C₂×C₃×C₄	—	6	1	no	—
202	C₂×C₃×C₄	—	6	1	no	—
203	C₂×C₃×C₄	—	6	1	no	—
204	C₂×C₃×C₄	—	6	1	no	—
205	C₂×C₃×C₄	—	6	1	no	—
206	C₂×C₃×C₄	D₃×C₈	1	2	yes	—
207	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
208	C₂×C₃×C₄	C₂×C₃⋊C₈	1	2	yes	—
209	C₂×C₃×C₄	#208	1	2	yes	—
210	C₂×C₃×C₄	#235	1	1	yes	characteristic
211	C₂×C₃×C₄	#235	1	2	yes	—
212	C₂×C₃×C₄	#235	1	2	yes	—
213	C₂×C₃×C₄	#235	1	1	yes	characteristic
214	C₂×C₃×C₄	#235	1	2	yes	—
215	C₂×C₃×C₄	#237	1	1	yes	characteristic
216	C₂×C₃×C₄	#237	1	2	yes	—
217	C₂×C₃×C₄	#239	1	1	yes	characteristic
218	C₂×Dic₃	—	2	1	no	—
219	C₂×Dic₃	—	2	1	no	—
220	C₂×Dic₃	—	2	1	no	—
221	C₂×Dic₃	—	2	1	no	—
222	C₂×Dic₃	—	2	1	no	—
223	C₂×Dic₃	—	2	1	no	—
224	C₂×Dic₃	—	2	1	no	—
225	C₂×Dic₃	—	2	1	no	—
226	C₂×Dic₃	C₂×C₃⋊C₈	1	2	yes	—
227	C₂×Dic₃	C₂×C₃⋊C₈	1	2	yes	—
228	C₂×Dic₃	#208	1	2	yes	—
229	C₂×Dic₃	#208	1	2	yes	—
230	C₃×C₈	—	12	2	no	—
231	C₃⋊C₈	—	4	2	no	—
232	C₃⋊C₈	—	12	4	no	—
233	C₃⋊C₈	—	12	2	no	—
234	C₃⋊C₈	—	12	2	no	—
235	C₂×C₃²	#383	1	1	yes	characteristic
236	C₂×C₃²	#431	1	1	yes	characteristic
237	C₂×C₃²	#431	1	1	yes	characteristic
238	C₂×C₃²	#431	1	1	yes	characteristic
239	C₂×C₃²	#432	1	1	yes	characteristic
240	C₂×C₃²	#432	1	1	yes	characteristic
241	C₂×C₃²	#432	1	1	yes	characteristic
242	C₂²×C₄	—	3	1	no	—
243	C₂²×C₄	—	3	1	no	—
244	C₂²×C₄	—	3	1	no	—
245	C₂²×C₄	—	3	1	no	—
246	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
247	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
249	C₂×C₈	—	6	2	no	—
250	C₂×C₈	—	6	2	no	—
251	C₂×C₈	—	6	2	no	—
252	C₂×C₈	—	18	2	no	—
253	C₂×C₈	—	18	2	no	—
254	C₂×C₈	—	18	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	3	4	no	—
257	C₄²	—	6	2	no	—
258	C₄²	—	6	2	no	—
259	C₄²	—	6	2	no	—
260	C₄²	—	6	1	no	—
261	C₄²	—	6	2	no	—
262	C₄²	—	6	1	no	—
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	—	2	1	no	—
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	—	2	1	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	#872	1	1	yes	characteristic
271	C₂²×C₃	#877	1	1	yes	characteristic
272	C₂²×C₃	#892	1	1	yes	characteristic
273	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
274	C₂²×C₃	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
275	C₂²×C₃	D₃×M₁₆	1	1	yes	characteristic
276	C₂²×C₃	D₃×M₁₆	1	1	yes	characteristic
277	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
278	C₂²×C₃	C₂×#208	1	1	yes	characteristic
279	C₂²×C₃	C₂×#208	1	1	yes	characteristic
280	C₂²×C₃	C₂×#208	1	1	yes	characteristic
281	C₂²×C₃	#926	1	1	yes	characteristic
282	C₂²×C₃	#926	1	1	yes	characteristic
283	C₂²×C₃	#926	1	1	yes	characteristic
284	C₃×C₄	—	2	2	no	—
285	C₃×C₄	—	2	2	no	—
286	C₃×C₄	—	2	2	no	—
287	C₃×C₄	—	2	2	no	—
288	C₃×C₄	—	2	2	no	—
289	C₃×C₄	—	2	2	no	—
290	C₃×C₄	—	4	2	no	—
291	C₃×C₄	—	4	2	no	—
292	C₃×C₄	—	6	2	no	—
293	C₃×C₄	—	6	2	no	—
294	C₃×C₄	—	6	2	no	—
295	C₃×C₄	—	6	2	no	—
296	C₃×C₄	#876	1	2	yes	—
297	C₃×C₄	#876	1	2	yes	—
298	C₃×C₄	#879	1	2	yes	—
299	C₃×C₄	#898	1	2	yes	—
300	Dic₃	—	2	2	no	—
301	Dic₃	—	2	2	no	—
302	Dic₃	—	2	2	no	—
303	Dic₃	—	2	2	no	—
304	C₃²	#1839	1	1	yes	characteristic, complemented, Sylow
305	C₂³	C₂×D₃×Dic₃	1	1	yes	characteristic
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	3	2	no	—
311	C₂×C₄	—	3	2	no	—
312	C₂×C₄	—	3	2	no	—
313	C₂×C₄	—	3	2	no	—
314	C₂×C₄	—	6	1	no	—
315	C₂×C₄	—	6	1	no	—
316	C₂×C₄	—	6	1	no	—
317	C₂×C₄	—	6	1	no	—
318	C₂×C₄	—	6	1	no	—
319	C₂×C₄	—	6	1	no	—
320	C₂×C₄	—	6	1	no	—
321	C₂×C₄	—	6	1	no	—
322	C₂×C₄	D₃×C₃⋊C₈	1	2	yes	—
323	C₂×C₄	#3711	1	2	yes	—
324	C₂×C₄	#3795	1	1	yes	characteristic
325	C₂×C₄	#3798	1	2	yes	—
326	C₂×C₄	#3799	1	1	yes	characteristic
327	C₂×C₄	#3804	1	2	yes	—
328	C₈	—	12	2	no	—
329	C₈	—	36	2	no	—
330	C₂×C₃	—	2	1	no	—
331	C₂×C₃	—	2	1	no	—
332	C₂×C₃	—	2	1	no	—
333	C₂×C₃	—	2	1	no	—
334	C₂×C₃	—	2	1	no	—
335	C₂×C₃	—	2	1	no	—
336	C₂×C₃	—	2	1	no	—
337	C₂×C₃	#4651	1	1	yes	characteristic
338	C₂×C₃	#4685	1	1	yes	characteristic
339	C₂×C₃	#4705	1	1	yes	characteristic
340	C₂×C₃	#4718	1	1	yes	characteristic
341	C₂×C₃	#4719	1	1	yes	characteristic
342	C₂×C₃	#4833	1	1	yes	characteristic
343	C₂×C₃	#4835	1	1	yes	characteristic
344	C₂×C₃	#4925	1	1	yes	characteristic
345	C₂×C₃	#4925	1	1	yes	characteristic
346	C₂×C₃	#4925	1	1	yes	characteristic
347	C₂×C₃	#4926	1	1	yes	characteristic
348	C₂×C₃	#4926	1	1	yes	characteristic
349	C₂×C₃	#4926	1	1	yes	characteristic
350	C₂×C₃	#4937	1	1	yes	characteristic
351	C₂²	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
352	C₂²	D₃×#208	1	1	yes	characteristic
353	C₂²	D₃×#208	1	1	yes	characteristic
354	C₂²	C₂×#3711	1	1	yes	characteristic
355	C₂²	#63841	1	1	yes	characteristic
356	C₂²	#63855	1	1	yes	characteristic
357	C₂²	#63955	1	1	yes	characteristic
358	C₄	—	2	2	no	—
359	C₄	—	2	2	no	—
360	C₄	—	6	2	no	—
361	C₄	—	6	2	no	—
362	C₄	—	6	2	no	—
363	C₄	—	6	2	no	—
364	C₄	#63859	1	2	yes	—
365	C₄	#63868	1	2	yes	—
366	C₃	—	2	1	no	—
367	C₃	#68607	1	1	yes	characteristic, complemented
368	C₃	#69393	1	1	yes	characteristic, complemented
369	C₂	#10590248	1	1	yes	characteristic
370	C₂	#10590261	1	1	yes	characteristic
371	C₂	#10590262	1	1	yes	characteristic
372	C₂	#10590263	1	1	yes	characteristic
373	C₂	#10591053	1	1	yes	characteristic
374	C₂	#10591063	1	1	yes	characteristic
375	C₂	#10591223	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃²
commutator	C₂²×C₃²	C₂²×C₈
Fitting	C₂×C₃²×C₄²	C₂²
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂×C₄²	D₃²
hypocenter	C₃²	#1839
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
45. C₃×C₄×Dic₃	C₈	semidirect
136. C₄×Dic₃	C₃⋊C₈	semidirect
304. C₃²	#1839	semidirect
367. C₃	#68607	semidirect
368. C₃	#69393	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄²×Dic₃	2
#10589390	2
C₃×#4524	2
#10589155	2
#69393	3
#68607	3