Subgroups — #11791188

Order 1152.

653 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#904	C₂	1	1	yes	maximal, characteristic
2	C₂×#63646	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#63569	C₂	1	2	yes	maximal, complemented
5	#10591070	C₂	1	2	yes	maximal, complemented
6	#72084	—	3	1	no	maximal
7	#69795	—	3	1	no	maximal
8	C₂×C₃×#235	C₂²	1	2	yes	—
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₂×#3748	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	C₄	1	1	yes	characteristic, complemented
14	C₃×#904	—	2	2	no	—
15	C₃×#904	—	2	2	no	—
16	C₂×C₃×D₂₄	C₄	1	1	yes	characteristic, complemented
17	C₂×C₃×#204	C₄	1	2	yes	complemented
18	#63646	—	2	2	no	—
19	C₃×C₈×Dic₃	—	2	2	no	—
20	#63569	—	2	4	no	—
21	(C₂×D₃)⋊(C₃⋊C₈)	—	2	4	no	—
22	C₂×C₃×#135	—	3	1	no	—
23	C₂×#904	D₃	1	1	yes	characteristic
24	C₂×C₃×C₄×C₈	—	3	1	no	—
25	C₂×#837	—	3	2	no	—
26	#5234	—	3	2	no	—
27	#5110	—	3	2	no	—
28	C₂×#819	—	3	1	no	—
29	C₂×#819	—	3	1	no	—
30	C₂×#819	—	6	1	no	—
31	C₂×C₈×Dic₃	D₃	1	1	yes	characteristic, complemented
32	C₂×#805	—	3	2	no	—
33	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
34	C₂×C₃×#67	—	2	2	no	—
35	C₂²×C₃×Dic₃	—	2	1	no	—
36	C₃×#235	—	2	2	no	—
37	C₃×#235	—	2	2	no	—
38	C₃×#235	—	2	2	no	—
39	C₃×#235	—	2	2	no	—
40	C₂×C₃×D₁₂	C₂×C₄	1	2	yes	—
41	C₂×C₃×C₄×D₃	—	2	2	no	—
42	C₂×C₃×Dic₆	C₂×C₄	1	2	yes	—
43	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃²×C₈	D₄	1	2	yes	—
46	C₂×C₃²×C₈	D₄	1	2	yes	—
47	#3748	—	2	2	no	—
48	#3748	—	2	2	no	—
49	C₂×C₃⋊(C₃⋊C₈)	—	2	2	no	—
50	C₃×C₄×Dic₃	—	2	1	no	—
51	C₃×C₄×Dic₃	D₄	1	2	yes	—
52	C₂×C₃×C₃⋊C₈	—	2	1	no	—
53	C₂×C₃×C₃⋊C₈	—	2	1	no	—
54	C₂×C₃×C₃⋊C₈	D₄	1	2	yes	—
55	C₃×Dic₁₂	—	2	2	no	—
56	C₃×D₂₄	—	2	2	no	—
57	C₃×#204	—	2	4	no	—
58	#1974	—	9	1	no	Sylow
59	C₂×C₃×#41	—	3	2	no	—
60	C₂×#235	C₂×D₃	1	2	yes	—
61	C₃×#135	—	6	2	no	—
62	C₃×#135	—	6	2	no	—
63	C₂×C₃×Q₁₆	—	3	1	no	—
64	C₂×C₃×SD₁₆	—	3	2	no	—
65	C₂×C₃×D₈	—	3	1	no	—
66	C₂²×C₃×C₈	—	2	1	no	—
67	C₂²×C₃×C₈	—	3	1	no	—
68	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
69	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
70	C₂×C₃×C₄²	—	3	1	no	—
71	C₂×C₄×Dic₃	C₂×D₃	1	1	yes	characteristic
72	C₂×#208	—	3	2	no	—
73	C₂×#208	—	3	2	no	—
74	C₂×#208	—	6	2	no	—
75	C₂²×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
76	C₂×Dic₁₂	Dic₃	1	1	yes	characteristic, complemented
77	#904	—	2	2	no	—
78	#904	—	2	2	no	—
79	C₂×D₂₄	Dic₃	1	1	yes	characteristic, complemented
80	C₂×#204	Dic₃	1	2	yes	complemented
81	C₃×C₄×C₈	—	6	2	no	—
82	#837	—	6	4	no	—
83	#830	—	6	4	no	—
84	(C₂×D₃)⋊C₈	—	6	4	no	—
85	#819	—	6	4	no	—
86	#819	—	6	2	no	—
87	#819	—	6	2	no	—
88	C₈×Dic₃	—	2	2	no	—
89	#805	—	6	4	no	—
90	C₂³×C₃²	#41	1	1	yes	characteristic
91	C₂²×C₃×D₃	—	2	2	no	—
92	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃²×C₄	#41	1	1	yes	characteristic
98	C₃×#67	—	4	2	no	—
99	C₃×#67	—	4	2	no	—
100	C₂×C₃×Dic₃	—	2	4	no	—
101	C₂×C₃×Dic₃	—	2	2	no	—
102	C₂×C₃×Dic₃	—	2	2	no	—
103	C₃×D₁₂	—	2	2	no	—
104	C₃×D₁₂	—	2	2	no	—
105	C₃×C₄×D₃	—	4	2	no	—
106	C₃×C₄×D₃	—	4	2	no	—
107	C₃×Dic₆	—	2	2	no	—
108	C₃×Dic₆	—	2	2	no	—
109	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
110	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
111	C₃⋊(C₃⋊C₈)	—	4	2	no	—
112	C₃×C₃⋊C₈	—	2	2	no	—
113	C₂×#135	—	3	1	no	—
114	C₂×#108	—	9	1	no	—
115	C₂×#104	—	9	2	no	—
116	#408	—	9	2	no	—
117	C₂×C₄×C₈	—	3	1	no	—
118	C₃×#41	—	6	2	no	—
119	C₃×#41	—	6	2	no	—
120	C₃×#41	—	6	2	no	—
121	C₃×#41	—	6	2	no	—
122	C₂×C₃×Q₈	—	3	2	no	—
123	C₂×C₃×D₄	—	3	2	no	—
124	C₂×C₃×D₄	—	6	2	no	—
125	C₂²×C₃×C₄	—	2	1	no	—
126	C₂²×C₃×C₄	—	6	1	no	—
127	C₂²×C₃×C₄	—	6	2	no	—
128	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
130	C₂×#67	—	2	2	no	—
131	C₂²×Dic₃	—	2	1	no	—
132	#235	—	2	2	no	—
133	#235	—	2	2	no	—
134	#235	—	2	2	no	—
135	#235	—	2	2	no	—
136	C₂×D₁₂	C₂×Dic₃	1	2	yes	—
137	C₂×C₄×D₃	—	2	2	no	—
138	C₂×Dic₆	C₂×Dic₃	1	2	yes	—
139	C₃×Q₁₆	—	6	2	no	—
140	C₃×SD₁₆	—	6	4	no	—
141	C₃×D₈	—	6	2	no	—
142	C₂×C₃×C₈	—	2	1	no	—
143	C₂×C₃×C₈	—	2	1	no	—
144	C₂×C₃×C₈	—	2	2	no	—
145	C₂×C₃×C₈	—	2	2	no	—
146	C₂×C₃×C₈	—	3	2	no	—
147	C₂×C₃×C₈	—	6	1	no	—
148	C₂×C₃×C₈	—	6	1	no	—
149	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
150	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
151	C₂×C₃×C₈	D₁₂	1	2	yes	—
152	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
153	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
154	C₂×C₃×C₈	#67	1	2	yes	—
155	C₂×C₃×C₈	#67	1	2	yes	—
156	C₂×C₃×C₈	#67	1	2	yes	—
157	C₃×C₄²	—	3	2	no	—
158	C₃×C₄²	—	6	1	no	—
159	C₄×Dic₃	—	2	1	no	—
160	C₄×Dic₃	D₁₂	1	2	yes	—
161	#208	—	6	4	no	—
162	#208	—	6	2	no	—
163	#208	—	6	2	no	—
164	#208	—	6	2	no	—
165	#208	—	6	2	no	—
166	#208	—	12	2	no	—
167	C₂×C₃⋊C₈	—	2	1	no	—
168	C₂×C₃⋊C₈	—	2	1	no	—
169	C₂×C₃⋊C₈	—	6	2	no	—
170	C₂×C₃⋊C₈	—	6	2	no	—
171	C₂×C₃⋊C₈	—	12	2	no	—
172	C₂×C₃⋊C₈	D₁₂	1	2	yes	—
173	Dic₁₂	—	2	2	no	—
174	D₂₄	—	2	2	no	—
175	#204	—	2	4	no	—
176	C₂²×C₃²	—	2	2	no	—
177	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
178	C₂²×C₃²	#121	1	1	yes	characteristic
179	C₂²×C₃²	#124	1	1	yes	characteristic
180	C₂×C₃×D₃	—	4	2	no	—
181	C₂×C₃×D₃	—	4	4	no	—
182	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
183	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
184	C₃²×C₄	#121	1	1	yes	characteristic
185	C₃²×C₄	#127	1	1	yes	characteristic
186	C₃×Dic₃	—	2	4	no	—
187	C₃×Dic₃	—	4	2	no	—
188	C₂×#41	—	3	2	no	—
189	#135	—	6	2	no	—
190	#135	—	6	2	no	—
191	C₂×Q₁₆	—	3	1	no	—
192	C₂×SD₁₆	—	3	2	no	—
193	C₂×D₈	—	3	1	no	—
194	C₂×M₁₆	—	9	2	no	—
195	C₂²×C₈	—	3	1	no	—
196	C₂²×C₈	D₃²	1	1	yes	characteristic
197	C₂×C₄²	—	3	1	no	—
198	#108	—	18	2	no	—
199	#104	—	18	4	no	—
200	C₂²⋊C₈	—	18	4	no	—
201	C₄×C₈	—	6	2	no	—
202	C₂³×C₃	—	2	1	no	—
203	C₂³×C₃	—	6	2	no	—
204	C₂³×C₃	#235	1	1	yes	characteristic
205	C₂³×C₃	#237	1	1	yes	characteristic
206	C₂²×D₃	—	2	2	no	—
207	C₃×Q₈	—	6	2	no	—
208	C₃×Q₈	—	6	2	no	—
209	C₃×D₄	—	6	2	no	—
210	C₃×D₄	—	6	2	no	—
211	C₃×D₄	—	12	2	no	—
212	C₃×D₄	—	12	2	no	—
213	C₂×C₃×C₄	—	2	1	no	—
214	C₂×C₃×C₄	—	2	1	no	—
215	C₂×C₃×C₄	—	2	1	no	—
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	2	1	no	—
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	6	4	no	—
220	C₂×C₃×C₄	—	6	2	no	—
221	C₂×C₃×C₄	—	6	2	no	—
222	C₂×C₃×C₄	—	12	2	no	—
223	C₂×C₃×C₄	—	12	2	no	—
224	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
228	C₂×C₃×C₄	#235	1	1	yes	characteristic
229	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
230	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
231	C₂×C₃×C₄	#237	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
236	#67	—	4	2	no	—
237	#67	—	4	2	no	—
238	C₂×Dic₃	—	2	4	no	—
239	C₂×Dic₃	—	2	2	no	—
240	C₂×Dic₃	—	2	2	no	—
241	D₁₂	—	2	2	no	—
242	D₁₂	—	2	2	no	—
243	C₄×D₃	—	4	2	no	—
244	C₄×D₃	—	4	2	no	—
245	Dic₆	—	2	2	no	—
246	Dic₆	—	2	2	no	—
247	C₃×C₈	—	2	2	no	—
248	C₃×C₈	—	2	2	no	—
249	C₃×C₈	—	6	2	no	—
250	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
251	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
252	C₃×C₈	#217	1	2	yes	—
253	C₃×C₈	#217	1	2	yes	—
254	C₃⋊C₈	—	2	2	no	—
255	C₃⋊C₈	—	12	4	no	—
256	C₃⋊C₈	—	12	2	no	—
257	C₃⋊C₈	—	12	2	no	—
258	C₂×C₃²	—	2	1	no	—
259	C₂×C₃²	—	2	1	no	—
260	C₂×C₃²	#390	1	1	yes	characteristic
261	C₂×C₃²	#443	1	2	yes	—
262	C₃×D₃	—	8	2	no	—
263	#41	—	6	2	no	—
264	#41	—	6	2	no	—
265	#41	—	6	2	no	—
266	#41	—	6	2	no	—
267	C₂×Q₈	—	3	2	no	—
268	C₂×D₄	—	3	2	no	—
269	C₂×D₄	—	6	2	no	—
270	C₂²×C₄	—	6	1	no	—
271	C₂²×C₄	—	6	2	no	—
272	C₂²×C₄	C₂×D₃²	1	1	yes	characteristic
273	Q₁₆	—	6	2	no	—
274	SD₁₆	—	6	4	no	—
275	D₈	—	6	2	no	—
276	M₁₆	—	18	2	no	—
277	M₁₆	—	18	2	no	—
278	C₂×C₈	—	3	2	no	—
279	C₂×C₈	—	6	1	no	—
280	C₂×C₈	—	6	1	no	—
281	C₂×C₈	—	18	2	no	—
282	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
283	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
284	C₂×C₈	#625	1	2	yes	—
285	C₂×C₈	#626	1	2	yes	—
286	C₄²	—	3	2	no	—
287	C₄²	—	6	1	no	—
288	C₂²×C₃	—	2	2	no	—
289	C₂²×C₃	—	2	2	no	—
290	C₂²×C₃	—	2	1	no	—
291	C₂²×C₃	—	2	1	no	—
292	C₂²×C₃	—	2	1	no	—
293	C₂²×C₃	—	4	2	no	—
294	C₂²×C₃	—	12	2	no	—
295	C₂²×C₃	—	12	4	no	—
296	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
297	C₂²×C₃	#875	1	1	yes	characteristic
298	C₂²×C₃	#930	1	1	yes	characteristic
299	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
300	C₂²×C₃	#936	1	1	yes	characteristic
301	C₂²×C₃	#938	1	1	yes	characteristic
302	C₂×D₃	—	4	4	no	—
303	C₂×D₃	—	4	2	no	—
304	C₃×C₄	—	2	1	no	—
305	C₃×C₄	—	2	1	no	—
306	C₃×C₄	—	2	1	no	—
307	C₃×C₄	—	2	1	no	—
308	C₃×C₄	—	6	4	no	—
309	C₃×C₄	—	12	2	no	—
310	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
311	C₃×C₄	#874	1	1	yes	characteristic
312	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
313	C₃×C₄	#930	1	1	yes	characteristic
314	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
315	C₃×C₄	#938	1	1	yes	characteristic
316	C₃×C₄	#940	1	1	yes	characteristic
317	C₃×C₄	C₂×#217	1	1	yes	characteristic
318	Dic₃	—	2	4	no	—
319	Dic₃	—	4	2	no	—
320	C₃²	#1974	1	1	yes	characteristic, complemented, Sylow
321	C₂³	—	6	2	no	—
322	C₂³	#3795	1	1	yes	characteristic
323	Q₈	—	6	2	no	—
324	Q₈	—	6	2	no	—
325	D₄	—	6	2	no	—
326	D₄	—	6	2	no	—
327	D₄	—	12	2	no	—
328	D₄	—	12	2	no	—
329	C₂×C₄	—	6	4	no	—
330	C₂×C₄	—	6	2	no	—
331	C₂×C₄	—	6	2	no	—
332	C₂×C₄	—	12	2	no	—
333	C₂×C₄	—	12	2	no	—
334	C₂×C₄	#3795	1	1	yes	characteristic
335	C₂×C₄	D₃×D₁₂	1	1	yes	characteristic
336	C₂×C₄	D₃×D₁₂	1	1	yes	characteristic
337	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
338	C₂×C₄	C₂×#625	1	1	yes	characteristic
339	C₂×C₄	C₂×#626	1	1	yes	characteristic
340	C₈	—	6	2	no	—
341	C₈	—	36	2	no	—
342	C₈	(C₂×D₃)⋊Dic₃	1	2	yes	—
343	C₈	(C₂×D₃)⋊Dic₃	1	2	yes	—
344	C₂×C₃	—	2	1	no	—
345	C₂×C₃	—	2	1	no	—
346	C₂×C₃	—	2	1	no	—
347	C₂×C₃	—	2	1	no	—
348	C₂×C₃	—	2	1	no	—
349	C₂×C₃	—	2	2	no	—
350	C₂×C₃	—	4	1	no	—
351	C₂×C₃	—	4	1	no	—
352	C₂×C₃	—	24	2	no	—
353	C₂×C₃	#4700	1	2	yes	—
354	C₂×C₃	#4939	1	1	yes	characteristic
355	C₂×C₃	#5196	1	2	yes	—
356	C₂×C₃	#5221	1	1	yes	characteristic
357	D₃	—	8	2	no	—
358	C₂²	—	2	2	no	—
359	C₂²	—	12	2	no	—
360	C₂²	—	12	4	no	—
361	C₂²	#63860	1	1	yes	characteristic
362	C₂²	Dic₃×D₁₂	1	1	yes	characteristic
363	C₂²	#63908	1	1	yes	characteristic
364	C₄	—	6	4	no	—
365	C₄	—	12	2	no	—
366	C₄	Dic₃×D₁₂	1	1	yes	characteristic
367	C₄	#63908	1	1	yes	characteristic
368	C₄	#63911	1	1	yes	characteristic
369	C₄	C₂×(C₂×D₃)⋊Dic₃	1	1	yes	characteristic
370	C₃	—	2	1	no	—
371	C₃	#69795	1	1	yes	characteristic, complemented
372	C₃	#72084	1	1	yes	characteristic, complemented
373	C₂	—	2	1	no	—
374	C₂	—	2	1	no	—
375	C₂	—	24	2	no	—
376	C₂	#10590915	1	2	yes	—
377	C₂	#10591318	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×D₁₂
commutator	C₂×C₃²×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃²
Fitting	C₂²×C₃²×C₈	C₂²
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×C₃×#904	C₂
cosocle	C₂²×C₃²×C₄	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	#1974
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#63646	C₂	semidirect
3. C₂×C₃×C₈×Dic₃	C₂	semidirect
4. C₂×#63569	C₂	semidirect
5. #10591070	C₂	semidirect
13. C₂×C₃×Dic₁₂	C₄	semidirect
16. C₂×C₃×D₂₄	C₄	semidirect
17. C₂×C₃×#204	C₄	semidirect
31. C₂×C₈×Dic₃	D₃	semidirect
76. C₂×Dic₁₂	Dic₃	semidirect
79. C₂×D₂₄	Dic₃	semidirect
80. C₂×#204	Dic₃	semidirect
320. C₃²	#1974	semidirect
371. C₃	#69795	semidirect
372. C₃	#72084	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#904	2
C₂×#63646	2
C₂×C₃×C₈×Dic₃	2
C₂×#63569	2
#10591070	2
#72084	3
#69795	3