Subgroups — #11791007

Order 1152.

689 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Dic₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#63645	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
4	#10591220	C₂	1	2	yes	maximal
5	C₂×#63566	C₂	1	2	yes	maximal
6	#72066	—	3	1	no	maximal
7	#69775	—	3	1	no	maximal
8	C₂²×C₃×Dic₆	C₂²	1	2	yes	—
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
11	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×Dic₁₂	—	2	2	no	—
14	C₂×C₃×Dic₁₂	—	2	2	no	—
15	C₂×C₃×Dic₁₂	C₄	1	4	yes	complemented
16	#63645	—	2	2	no	—
17	C₃×C₈×Dic₃	—	2	2	no	—
18	#63579	—	2	4	no	—
19	#63566	—	2	4	no	—
20	C₂²×C₃×Q₁₆	—	3	1	no	—
21	C₂²×Dic₁₂	D₃	1	1	yes	characteristic
22	C₂×C₃×C₄×C₈	—	3	1	no	—
23	#5229	—	3	2	no	—
24	C₂×#835	—	3	2	no	—
25	C₂×#818	—	3	1	no	—
26	C₂×#818	—	3	1	no	—
27	C₂×#818	—	6	1	no	—
28	C₂×#816	—	3	2	no	—
29	C₂×C₈×Dic₃	D₃	1	1	yes	characteristic
30	#4929	—	3	2	no	—
31	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
32	C₂²×C₃⋊Dic₃	—	2	2	no	—
33	C₂²×C₃×Dic₃	—	2	2	no	—
34	C₂²×C₃×Dic₃	D₄	1	2	yes	—
35	C₂×C₃×Dic₆	—	2	2	no	—
36	C₂×C₃×Dic₆	—	2	2	no	—
37	C₂×C₃×Dic₆	—	2	2	no	—
38	C₂×C₃×Dic₆	—	2	2	no	—
39	C₂×C₃×Dic₆	—	2	2	no	—
40	C₂×C₃×Dic₆	C₂×C₄	1	4	yes	—
41	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
42	C₂×C₃²×C₈	D₄	1	2	yes	—
43	C₂×C₃²×C₈	D₄	1	2	yes	—
44	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
45	C₄⋊(C₃⋊Dic₃)	—	2	2	no	—
46	C₃×C₄×Dic₃	—	2	1	no	—
47	C₃×C₄×Dic₃	—	2	1	no	—
48	C₂×C₃×C₃⋊C₈	—	2	1	no	—
49	C₂×C₃×C₃⋊C₈	—	2	1	no	—
50	C₂×C₃×C₃⋊C₈	D₄	1	2	yes	—
51	C₃×Dic₁₂	—	2	8	no	—
52	#1969	—	9	1	no	Sylow
53	C₂²×C₃×Q₈	—	3	2	no	—
54	C₂²×Dic₆	C₂×D₃	1	2	yes	—
55	C₂×C₃×Q₁₆	—	3	4	no	—
56	C₂×C₃×Q₁₆	—	6	2	no	—
57	C₂×C₃×Q₁₆	—	6	2	no	—
58	C₂²×C₃×C₈	—	2	1	no	—
59	C₂²×C₃×C₈	—	3	1	no	—
60	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
62	C₂×C₃×C₄²	—	3	1	no	—
63	C₂×C₄⋊Dic₃	—	3	2	no	—
64	C₂×C₄⋊Dic₃	—	3	2	no	—
65	C₂×C₄⋊Dic₃	—	6	2	no	—
66	C₂×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
67	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
68	C₂×Dic₁₂	—	2	2	no	—
69	C₂×Dic₁₂	—	2	2	no	—
70	C₂×Dic₁₂	Dic₃	1	4	yes	complemented
71	C₃×C₄×C₈	—	6	2	no	—
72	#835	—	6	4	no	—
73	C₃⋊#95	—	6	4	no	—
74	C₃⋊#95	—	6	4	no	—
75	#818	—	6	4	no	—
76	#818	—	6	2	no	—
77	#818	—	6	2	no	—
78	#816	—	6	4	no	—
79	C₈×Dic₃	—	2	2	no	—
80	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃²×C₄	#41	1	1	yes	characteristic
86	C₂×C₃²×C₄	#41	1	1	yes	characteristic
87	C₂×C₃⋊Dic₃	—	4	2	no	—
88	C₂×C₃⋊Dic₃	—	4	4	no	—
89	C₂×C₃×Dic₃	—	2	2	no	—
90	C₂×C₃×Dic₃	—	2	2	no	—
91	C₂×C₃×Dic₃	—	2	4	no	—
92	C₂×C₃×Dic₃	—	4	2	no	—
93	C₂×C₃×Dic₃	—	4	2	no	—
94	C₂×C₃×Dic₃	Q₁₆	1	4	yes	complemented
95	C₃×Dic₆	—	2	4	no	—
96	C₃×Dic₆	—	2	4	no	—
97	C₃×Dic₆	—	4	2	no	—
98	C₃×Dic₆	—	4	2	no	—
99	C₃²×C₈	C₂²⋊C₄	1	4	yes	—
100	C₃×C₃⋊C₈	—	2	2	no	—
101	C₂²×Q₁₆	—	3	1	no	—
102	C₂×#107	—	9	1	no	—
103	C₂×#103	—	9	2	no	—
104	C₂×C₄×C₈	—	3	1	no	—
105	#391	—	9	2	no	—
106	C₂×C₃×Q₈	—	3	4	no	—
107	C₂×C₃×Q₈	—	6	2	no	—
108	C₂×C₃×Q₈	—	6	2	no	—
109	C₂×C₃×Q₈	—	6	2	no	—
110	C₂×C₃×Q₈	—	6	2	no	—
111	C₂×C₃×Q₈	—	6	2	no	—
112	C₂²×C₃×C₄	—	2	1	no	—
113	C₂²×C₃×C₄	—	3	2	no	—
114	C₂²×C₃×C₄	—	6	2	no	—
115	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
117	C₂²×Dic₃	—	2	2	no	—
118	C₂²×Dic₃	—	6	2	no	—
119	C₂²×Dic₃	—	6	2	no	—
120	C₂²×Dic₃	—	12	2	no	—
121	C₂²×Dic₃	D₁₂	1	2	yes	—
122	C₂×Dic₆	—	2	2	no	—
123	C₂×Dic₆	—	2	2	no	—
124	C₂×Dic₆	—	2	2	no	—
125	C₂×Dic₆	—	2	2	no	—
126	C₂×Dic₆	—	2	2	no	—
127	C₂×Dic₆	C₂×Dic₃	1	4	yes	—
128	C₃×Q₁₆	—	6	8	no	—
129	C₂×C₃×C₈	—	2	2	no	—
130	C₂×C₃×C₈	—	2	2	no	—
131	C₂×C₃×C₈	—	2	2	no	—
132	C₂×C₃×C₈	—	3	2	no	—
133	C₂×C₃×C₈	—	6	1	no	—
134	C₂×C₃×C₈	—	6	1	no	—
135	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
136	C₂×C₃×C₈	D₁₂	1	2	yes	—
137	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
138	C₂×C₃×C₈	#67	1	2	yes	—
139	C₂×C₃×C₈	#67	1	2	yes	—
140	C₂×C₃×C₈	#67	1	2	yes	—
141	C₃×C₄²	—	6	1	no	—
142	C₃×C₄²	—	6	1	no	—
143	C₄⋊Dic₃	—	6	4	no	—
144	C₄⋊Dic₃	—	6	2	no	—
145	C₄⋊Dic₃	—	6	2	no	—
146	C₄⋊Dic₃	—	6	2	no	—
147	C₄⋊Dic₃	—	6	2	no	—
148	C₄⋊Dic₃	—	12	2	no	—
149	C₄×Dic₃	—	2	1	no	—
150	C₄×Dic₃	—	2	1	no	—
151	C₂×C₃⋊C₈	—	2	1	no	—
152	C₂×C₃⋊C₈	—	2	1	no	—
153	C₂×C₃⋊C₈	D₁₂	1	2	yes	—
154	Dic₁₂	—	2	8	no	—
155	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
156	C₂²×C₃²	#121	1	1	yes	characteristic
157	C₂²×C₃²	#127	1	1	yes	characteristic
158	C₂²×C₃²	C₂×Q₁₆	1	2	yes	—
159	C₂²×C₃²	#135	1	2	yes	—
160	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
162	C₃²×C₄	#121	1	1	yes	characteristic
163	C₃²×C₄	#124	1	1	yes	characteristic
164	C₃⋊Dic₃	—	8	2	no	—
165	C₃×Dic₃	—	2	4	no	—
166	C₃×Dic₃	—	4	4	no	—
167	C₂²×Q₈	—	3	2	no	—
168	C₂×Q₁₆	—	3	4	no	—
169	C₂×Q₁₆	—	6	2	no	—
170	C₂×Q₁₆	—	6	2	no	—
171	C₂²×C₈	—	3	1	no	—
172	C₂²×C₈	D₃²	1	1	yes	characteristic
173	C₂×C₄⋊C₄	—	9	2	no	—
174	C₂×C₄²	—	3	1	no	—
175	#107	—	18	2	no	—
176	#103	—	18	4	no	—
177	C₄×C₈	—	6	2	no	—
178	#95	—	18	4	no	—
179	C₂³×C₃	—	2	1	no	—
180	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
181	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
182	C₃×Q₈	—	6	4	no	—
183	C₃×Q₈	—	6	4	no	—
184	C₃×Q₈	—	12	2	no	—
185	C₃×Q₈	—	12	2	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	2	1	no	—
192	C₂×C₃×C₄	—	3	4	no	—
193	C₂×C₃×C₄	—	6	2	no	—
194	C₂×C₃×C₄	—	6	2	no	—
195	C₂×C₃×C₄	—	6	4	no	—
196	C₂×C₃×C₄	—	12	2	no	—
197	C₂×C₃×C₄	—	12	2	no	—
198	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
199	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
201	C₂×C₃×C₄	#235	1	1	yes	characteristic
202	C₂×C₃×C₄	#235	1	1	yes	characteristic
203	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
204	C₂×C₃×C₄	#237	1	1	yes	characteristic
205	C₂×C₃×C₄	#237	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
210	C₂×Dic₃	—	2	2	no	—
211	C₂×Dic₃	—	2	2	no	—
212	C₂×Dic₃	—	2	4	no	—
213	C₂×Dic₃	—	4	2	no	—
214	C₂×Dic₃	—	4	2	no	—
215	C₂×Dic₃	—	12	4	no	—
216	C₂×Dic₃	—	12	2	no	—
217	C₂×Dic₃	—	12	4	no	—
218	C₂×Dic₃	—	12	4	no	—
219	C₂×Dic₃	—	12	2	no	—
220	C₂×Dic₃	—	24	4	no	—
221	C₂×Dic₃	Dic₁₂	1	4	yes	complemented
222	Dic₆	—	2	4	no	—
223	Dic₆	—	2	4	no	—
224	Dic₆	—	4	2	no	—
225	Dic₆	—	4	2	no	—
226	C₃×C₈	—	2	4	no	—
227	C₃×C₈	—	6	2	no	—
228	C₃×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
229	C₃×C₈	#217	1	4	yes	—
230	C₃⋊C₈	—	2	2	no	—
231	C₂×C₃²	#390	1	1	yes	characteristic
232	C₂×C₃²	C₄×Q₁₆	1	2	yes	—
233	C₂×C₃²	#467	1	2	yes	—
234	C₂×C₃²	#494	1	1	yes	characteristic
235	C₂×C₃²	#495	1	1	yes	characteristic
236	C₂×Q₈	—	3	4	no	—
237	C₂×Q₈	—	6	2	no	—
238	C₂×Q₈	—	6	2	no	—
239	C₂×Q₈	—	6	2	no	—
240	C₂×Q₈	—	6	2	no	—
241	C₂×Q₈	—	6	2	no	—
242	C₂²×C₄	—	3	2	no	—
243	C₂²×C₄	—	6	2	no	—
244	C₂²×C₄	—	18	2	no	—
245	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
246	Q₁₆	—	6	8	no	—
247	C₂×C₈	—	3	2	no	—
248	C₂×C₈	—	6	1	no	—
249	C₂×C₈	—	6	1	no	—
250	C₂×C₈	D₃×Dic₃	1	2	yes	—
251	C₂×C₈	#625	1	2	yes	—
252	C₂×C₈	#626	1	2	yes	—
253	C₄⋊C₄	—	18	2	no	—
254	C₄⋊C₄	—	18	2	no	—
255	C₄²	—	6	1	no	—
256	C₄²	—	6	1	no	—
257	C₂²×C₃	—	2	2	no	—
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	—	2	1	no	—
262	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
263	C₂²×C₃	#874	1	1	yes	characteristic
264	C₂²×C₃	#904	1	2	yes	—
265	C₂²×C₃	C₂×Dic₁₂	1	2	yes	—
266	C₂²×C₃	D₃×Q₁₆	1	2	yes	—
267	C₂²×C₃	#919	1	2	yes	—
268	C₂²×C₃	#930	1	1	yes	characteristic
269	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
270	C₂²×C₃	#938	1	1	yes	characteristic
271	C₂²×C₃	#940	1	1	yes	characteristic
272	C₃×C₄	—	2	1	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	6	4	no	—
277	C₃×C₄	—	12	4	no	—
278	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
279	C₃×C₄	#875	1	1	yes	characteristic
280	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
281	C₃×C₄	#930	1	1	yes	characteristic
282	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
283	C₃×C₄	#936	1	1	yes	characteristic
284	C₃×C₄	#938	1	1	yes	characteristic
285	C₃×C₄	C₂×#217	1	1	yes	characteristic
286	Dic₃	—	2	4	no	—
287	Dic₃	—	4	4	no	—
288	Dic₃	—	24	4	no	—
289	Dic₃	—	24	2	no	—
290	Dic₃	—	24	2	no	—
291	C₃²	#1969	1	1	yes	characteristic, complemented, Sylow
292	C₂³	D₃×D₁₂	1	1	yes	characteristic
293	Q₈	—	6	4	no	—
294	Q₈	—	6	4	no	—
295	Q₈	—	12	2	no	—
296	Q₈	—	12	2	no	—
297	C₂×C₄	—	3	4	no	—
298	C₂×C₄	—	6	2	no	—
299	C₂×C₄	—	6	2	no	—
300	C₂×C₄	—	6	4	no	—
301	C₂×C₄	—	12	2	no	—
302	C₂×C₄	—	12	2	no	—
303	C₂×C₄	—	36	4	no	—
304	C₂×C₄	—	36	2	no	—
305	C₂×C₄	#3795	1	1	yes	characteristic
306	C₂×C₄	#3795	1	1	yes	characteristic
307	C₂×C₄	D₃×D₁₂	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	2	no	—
312	C₈	(C₂×D₃)⋊Dic₃	1	4	yes	—
313	C₂×C₃	—	2	2	no	—
314	C₂×C₃	—	2	1	no	—
315	C₂×C₃	—	2	2	no	—
316	C₂×C₃	—	2	1	no	—
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	#4696	1	1	yes	characteristic
319	C₂×C₃	C₄×Dic₁₂	1	2	yes	—
320	C₂×C₃	#4699	1	1	yes	characteristic
321	C₂×C₃	#4939	1	1	yes	characteristic
322	C₂×C₃	#5116	1	2	yes	—
323	C₂×C₃	Dic₃×Q₁₆	1	2	yes	—
324	C₂×C₃	#5183	1	1	yes	characteristic
325	C₂×C₃	#5188	1	2	yes	—
326	C₂×C₃	#5191	1	1	yes	characteristic
327	C₂×C₃	#5221	1	1	yes	characteristic
328	C₂²	D₃×Dic₁₂	1	2	yes	—
329	C₂²	#63808	1	2	yes	—
330	C₂²	Dic₃×D₁₂	1	1	yes	characteristic
331	C₂²	#63908	1	1	yes	characteristic
332	C₂²	#63911	1	1	yes	characteristic
333	C₄	—	6	4	no	—
334	C₄	—	12	4	no	—
335	C₄	—	72	2	no	—
336	C₄	#63860	1	1	yes	characteristic
337	C₄	Dic₃×D₁₂	1	1	yes	characteristic
338	C₄	#63908	1	1	yes	characteristic
339	C₄	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
340	C₃	—	2	1	no	—
341	C₃	#69775	1	1	yes	characteristic, complemented
342	C₃	#72066	1	1	yes	characteristic, complemented
343	C₂	Dic₃×Dic₁₂	1	2	yes	—
344	C₂	#10590851	1	2	yes	—
345	C₂	#10590865	1	1	yes	characteristic
346	C₂	#10590896	1	1	yes	characteristic
347	C₂	#10591318	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₃×D₁₂
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#1969
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
15. C₂×C₃×Dic₁₂	C₄	semidirect
70. C₂×Dic₁₂	Dic₃	semidirect
94. C₂×C₃×Dic₃	Q₁₆	semidirect
221. C₂×Dic₃	Dic₁₂	semidirect
291. C₃²	#1969	semidirect
341. C₃	#69775	semidirect
342. C₃	#72066	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×Dic₁₂	2
C₂×#63645	2
C₂×C₃×C₈×Dic₃	2
#10591220	2
C₂×#63566	2
#72066	3
#69775	3