Subgroups — #11776812

Order 1152.

419 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10592532	C₂	1	1	yes	maximal, characteristic
2	C₄×#3710	C₂	1	1	yes	maximal, characteristic
3	#10590256	C₂	1	1	yes	maximal, characteristic
4	C₃×#4480	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#4479	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589099	C₂	1	1	yes	maximal, characteristic, complemented
7	#10589097	C₂	1	1	yes	maximal, characteristic, complemented
8	#68567	—	3	1	no	maximal
9	#68527	—	3	1	no	maximal
10	C₄²×C₃⋊D₃	C₂²	1	1	yes	characteristic
11	C₂×#3710	C₄	1	1	yes	characteristic
12	C₂×#3710	C₄	1	1	yes	characteristic
13	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	#63647	—	2	1	no	—
15	C₈×C₃⋊Dic₃	—	2	1	no	—
16	C₄⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
17	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
19	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	#63557	—	2	1	no	—
22	#63553	—	2	1	no	—
23	#63553	C₄	1	1	yes	characteristic
24	#63553	C₄	1	1	yes	characteristic
25	#4833	—	3	1	no	—
26	#4833	—	3	1	no	—
27	#4833	—	3	1	no	—
28	#4833	—	3	1	no	—
29	#4833	—	6	1	no	—
30	C₄×D₃×C₈	—	3	1	no	—
31	C₄×D₃×C₈	—	3	1	no	—
32	C₃×#335	—	3	1	no	—
33	C₃×#334	—	3	1	no	—
34	#4480	—	3	1	no	—
35	#4480	D₃	1	1	yes	characteristic, complemented
36	#4479	—	3	1	no	—
37	#4479	D₃	1	1	yes	characteristic, complemented
38	#4458	—	3	1	no	—
39	#4457	—	3	1	no	—
40	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₃⋊D₃	D₄	1	1	yes	characteristic
42	C₂×C₄×C₃⋊D₃	Q₈	1	1	yes	characteristic
43	C₂×C₃²×C₈	—	2	1	no	—
44	C₃²×C₄²	C₂³	1	1	yes	characteristic
45	C₄×C₃⋊Dic₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₃⋊Dic₃	D₄	1	1	yes	characteristic
47	C₄×C₃⋊Dic₃	Q₈	1	1	yes	characteristic
48	C₂×C₃⋊(C₃⋊C₈)	—	2	1	no	—
49	C₂×C₃×C₃⋊C₈	—	2	1	no	—
50	C₂×C₃×C₃⋊C₈	—	2	1	no	—
51	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
55	#3710	—	2	2	no	—
56	#3710	C₈	1	4	yes	complemented
57	#1564	—	9	1	no	Sylow
58	C₂×D₃×C₈	—	3	1	no	—
59	C₂×D₃×C₈	—	3	1	no	—
60	C₂×D₃×C₈	—	3	1	no	—
61	C₂×D₃×C₈	—	3	1	no	—
62	C₄²×D₃	—	3	1	no	—
63	C₄²×D₃	—	3	1	no	—
64	C₄²×D₃	—	6	1	no	—
65	C₃×C₄⋊C₈	—	2	1	no	—
66	C₃×C₄⋊C₈	—	3	1	no	—
67	C₃×C₄⋊C₈	—	3	1	no	—
68	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
69	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
70	C₃×C₄×C₈	—	3	1	no	—
71	C₃×C₄×C₈	—	3	1	no	—
72	(C₂×D₃)⋊C₈	—	6	1	no	—
73	(C₂×D₃)⋊C₈	—	6	1	no	—
74	(C₂×D₃)⋊C₈	—	6	1	no	—
75	(C₂×D₃)⋊C₈	—	6	1	no	—
76	(C₂×D₃)⋊C₈	—	6	2	no	—
77	C₈×Dic₃	—	3	1	no	—
78	C₈×Dic₃	—	3	1	no	—
79	C₈×Dic₃	—	3	1	no	—
80	C₈×Dic₃	—	3	1	no	—
81	C₈×Dic₃	—	6	1	no	—
82	C₈×Dic₃	—	6	1	no	—
83	C₈×Dic₃	—	6	1	no	—
84	C₈×Dic₃	—	6	1	no	—
85	C₈×Dic₃	—	6	2	no	—
86	C₄⋊(C₃⋊C₈)	—	3	1	no	—
87	C₄⋊(C₃⋊C₈)	—	3	1	no	—
88	C₄⋊(C₃⋊C₈)	—	6	1	no	—
89	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
90	C₄⋊(C₃⋊C₈)	C₂×D₃	1	1	yes	characteristic
91	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
92	C₄×C₃⋊C₈	C₂×D₃	1	1	yes	characteristic
93	C₂²×C₃⋊D₃	C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
97	C₂×C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₃⋊Dic₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄×C₃⋊D₃	—	2	1	no	—
101	C₄×C₃⋊D₃	—	2	1	no	—
102	C₄×C₃⋊D₃	—	2	2	no	—
103	C₄×C₃⋊D₃	C₂×C₈	1	2	yes	—
104	C₄×C₃⋊D₃	M₁₆	1	2	yes	—
105	C₃²×C₈	—	4	1	no	—
106	C₃⋊(C₃⋊C₈)	—	4	1	no	—
107	C₃×C₃⋊C₈	—	2	1	no	—
108	C₃×C₃⋊C₈	—	2	1	no	—
109	C₃×C₃⋊C₈	—	4	1	no	—
110	C₃×C₃⋊C₈	—	4	1	no	—
111	C₃×C₃⋊C₈	C₂×C₈	1	2	yes	—
112	C₃×C₃⋊C₈	C₂×C₈	1	2	yes	—
113	#431	—	9	1	no	—
114	#431	—	9	1	no	—
115	C₂×C₄×C₈	—	9	1	no	—
116	#335	—	3	1	no	—
117	#335	—	9	1	no	—
118	#334	—	3	1	no	—
119	#334	—	9	1	no	—
120	C₂×C₄×D₃	—	3	1	no	—
121	C₂×C₄×D₃	—	3	1	no	—
122	C₂×C₄×D₃	—	3	1	no	—
123	C₂×C₄×D₃	—	3	1	no	—
124	C₂×C₄×D₃	—	3	1	no	—
125	C₂×C₄×D₃	—	3	1	no	—
126	C₂×C₄×D₃	—	6	1	no	—
127	C₂×C₄×D₃	—	6	1	no	—
128	C₂×C₄×D₃	—	6	1	no	—
129	C₂×C₃×C₈	—	2	2	no	—
130	C₂×C₃×C₈	—	2	1	no	—
131	C₂×C₃×C₈	—	2	1	no	—
132	C₂×C₃×C₈	—	3	1	no	—
133	C₂×C₃×C₈	—	3	1	no	—
134	C₂×C₃×C₈	—	3	1	no	—
135	C₂×C₃×C₈	—	3	1	no	—
136	C₂×C₃×C₈	—	6	1	no	—
137	C₂×C₃×C₈	—	6	1	no	—
138	C₃×C₄²	—	2	1	no	—
139	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
140	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
141	C₄×Dic₃	—	3	1	no	—
142	C₄×Dic₃	—	3	1	no	—
143	C₄×Dic₃	—	3	1	no	—
144	C₄×Dic₃	—	3	1	no	—
145	C₄×Dic₃	—	3	1	no	—
146	C₄×Dic₃	—	3	1	no	—
147	C₄×Dic₃	—	6	1	no	—
148	C₄×Dic₃	—	6	1	no	—
149	C₄×Dic₃	—	6	1	no	—
150	C₂×C₃⋊C₈	—	2	1	no	—
151	C₂×C₃⋊C₈	—	2	1	no	—
152	C₂×C₃⋊C₈	—	6	2	no	—
153	C₂×C₃⋊C₈	—	6	1	no	—
154	C₂×C₃⋊C₈	—	6	1	no	—
155	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
156	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
157	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
158	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
159	D₃×C₈	—	3	4	no	—
160	D₃×C₈	—	3	4	no	—
161	D₃×C₈	—	6	2	no	—
162	D₃×C₈	—	6	2	no	—
163	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₃⋊D₃	—	2	1	no	—
165	C₂×C₃⋊D₃	—	2	1	no	—
166	C₂×C₃⋊D₃	C₄⋊C₈	1	2	yes	—
167	C₃²×C₄	—	2	1	no	—
168	C₃²×C₄	C₂²×C₈	1	1	yes	characteristic
169	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
170	C₃²×C₄	#135	1	1	yes	characteristic
171	C₃²×C₄	#135	1	1	yes	characteristic
172	C₃⋊Dic₃	—	2	1	no	—
173	C₃⋊Dic₃	C₄⋊C₈	1	2	yes	—
174	C₃⋊Dic₃	#108	1	2	yes	—
175	C₂²×C₈	—	9	1	no	—
176	C₂²×C₈	—	9	1	no	—
177	C₂×C₄²	—	9	1	no	—
178	C₄⋊C₈	—	3	1	no	—
179	C₄⋊C₈	—	3	1	no	—
180	C₄⋊C₈	—	9	1	no	—
181	C₄⋊C₈	D₃²	1	1	yes	characteristic
182	C₂²⋊C₈	—	18	1	no	—
183	C₂²⋊C₈	—	18	1	no	—
184	C₄×C₈	—	3	1	no	—
185	C₄×C₈	—	3	1	no	—
186	C₄×C₈	—	9	1	no	—
187	C₄×C₈	—	9	1	no	—
188	C₄×C₈	—	18	1	no	—
189	C₄×C₈	—	18	1	no	—
190	C₂²×D₃	—	3	1	no	—
191	C₂²×D₃	—	3	1	no	—
192	C₂²×D₃	—	6	1	no	—
193	C₂×C₃×C₄	—	2	1	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
198	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
199	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
200	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
201	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
202	C₂×Dic₃	—	3	1	no	—
203	C₂×Dic₃	—	3	1	no	—
204	C₂×Dic₃	—	3	1	no	—
205	C₂×Dic₃	—	3	1	no	—
206	C₂×Dic₃	—	3	1	no	—
207	C₂×Dic₃	—	3	1	no	—
208	C₂×Dic₃	—	6	1	no	—
209	C₂×Dic₃	—	6	1	no	—
210	C₂×Dic₃	—	6	1	no	—
211	C₄×D₃	—	3	2	no	—
212	C₄×D₃	—	3	2	no	—
213	C₄×D₃	—	3	2	no	—
214	C₄×D₃	—	3	2	no	—
215	C₄×D₃	—	6	2	no	—
216	C₄×D₃	—	6	2	no	—
217	C₄×D₃	—	6	2	no	—
218	C₄×D₃	—	6	2	no	—
219	C₄×D₃	—	6	1	no	—
220	C₄×D₃	—	6	1	no	—
221	C₄×D₃	—	6	1	no	—
222	C₄×D₃	—	6	1	no	—
223	C₄×D₃	—	12	2	no	—
224	C₄×D₃	—	12	1	no	—
225	C₄×D₃	—	12	1	no	—
226	C₃×C₈	—	3	2	no	—
227	C₃×C₈	—	3	2	no	—
228	C₃×C₈	—	4	2	no	—
229	C₃×C₈	—	4	1	no	—
230	C₃×C₈	—	4	1	no	—
231	C₃×C₈	—	6	1	no	—
232	C₃×C₈	—	6	1	no	—
233	C₃×C₈	—	12	1	no	—
234	C₃×C₈	—	12	1	no	—
235	C₃⋊C₈	—	2	1	no	—
236	C₃⋊C₈	—	2	1	no	—
237	C₃⋊C₈	—	4	1	no	—
238	C₃⋊C₈	—	4	1	no	—
239	C₃⋊C₈	—	12	2	no	—
240	C₃⋊C₈	—	12	1	no	—
241	C₃⋊C₈	—	12	1	no	—
242	C₃⋊C₈	D₃×C₈	1	2	yes	—
243	C₃⋊C₈	D₃×C₈	1	2	yes	—
244	C₂×C₃²	C₂×C₄⋊C₈	1	1	yes	characteristic
245	C₂×C₃²	#427	1	1	yes	characteristic
246	C₂×C₃²	C₂×#108	1	1	yes	characteristic
247	C₃⋊D₃	—	2	2	no	—
248	C₂²×C₄	—	9	1	no	—
249	C₂²×C₄	—	9	1	no	—
250	C₂²×C₄	—	9	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	3	1	no	—
253	C₂×C₈	—	3	1	no	—
254	C₂×C₈	—	3	1	no	—
255	C₂×C₈	—	3	1	no	—
256	C₂×C₈	—	6	1	no	—
257	C₂×C₈	—	6	1	no	—
258	C₂×C₈	—	9	4	no	—
259	C₂×C₈	—	18	2	no	—
260	C₂×C₈	—	18	1	no	—
261	C₄²	—	9	1	no	—
262	C₄²	—	9	1	no	—
263	C₄²	—	9	1	no	—
264	C₄²	C₂×D₃²	1	1	yes	characteristic
265	C₂²×C₃	—	2	1	no	—
266	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
267	C₂²×C₃	D₃×C₄⋊C₄	1	1	yes	characteristic
268	C₂×D₃	—	3	2	no	—
269	C₂×D₃	—	3	2	no	—
270	C₂×D₃	—	6	2	no	—
271	C₂×D₃	—	6	1	no	—
272	C₂×D₃	—	6	1	no	—
273	C₂×D₃	—	6	1	no	—
274	C₂×D₃	—	6	1	no	—
275	C₂×D₃	—	12	1	no	—
276	C₂×D₃	—	12	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	4	1	no	—
284	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
285	C₃×C₄	C₂×D₃×C₈	1	1	yes	characteristic
286	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
287	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
288	C₃×C₄	#912	1	1	yes	characteristic
289	C₃×C₄	#916	1	1	yes	characteristic
290	C₃×C₄	#916	1	1	yes	characteristic
291	C₃×C₄	#919	1	1	yes	characteristic
292	Dic₃	—	3	2	no	—
293	Dic₃	—	3	2	no	—
294	Dic₃	—	3	2	no	—
295	Dic₃	—	3	2	no	—
296	Dic₃	—	6	2	no	—
297	Dic₃	—	6	2	no	—
298	Dic₃	—	6	1	no	—
299	Dic₃	—	6	1	no	—
300	Dic₃	—	12	1	no	—
301	C₃²	#1564	1	1	yes	characteristic, complemented, Sylow
302	C₂³	—	9	1	no	—
303	C₂×C₄	—	9	1	no	—
304	C₂×C₄	—	9	1	no	—
305	C₂×C₄	—	9	2	no	—
306	C₂×C₄	—	9	2	no	—
307	C₂×C₄	—	9	1	no	—
308	C₂×C₄	—	18	2	no	—
309	C₂×C₄	—	18	1	no	—
310	C₂×C₄	—	18	1	no	—
311	C₂×C₄	#3797	1	1	yes	characteristic
312	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
313	C₂×C₄	#3811	1	1	yes	characteristic
314	C₈	—	3	2	no	—
315	C₈	—	3	2	no	—
316	C₈	—	4	1	no	—
317	C₈	—	6	1	no	—
318	C₈	—	6	1	no	—
319	C₈	—	12	1	no	—
320	C₈	—	12	1	no	—
321	C₈	—	36	1	no	—
322	C₂×C₃	—	2	1	no	—
323	C₂×C₃	—	2	1	no	—
324	C₂×C₃	—	2	1	no	—
325	C₂×C₃	D₃×C₄⋊C₈	1	1	yes	characteristic
326	C₂×C₃	D₃×C₄⋊C₈	1	1	yes	characteristic
327	C₂×C₃	#4860	1	1	yes	characteristic
328	C₂×C₃	#4880	1	1	yes	characteristic
329	C₂×C₃	D₃×#108	1	1	yes	characteristic
330	C₂×C₃	D₃×#108	1	1	yes	characteristic
331	D₃	—	6	2	no	—
332	D₃	—	6	2	no	—
333	D₃	—	12	2	no	—
334	C₂²	—	9	2	no	—
335	C₂²	—	18	1	no	—
336	C₂²	—	18	1	no	—
337	C₂²	#63883	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	9	2	no	—
340	C₄	—	9	2	no	—
341	C₄	—	18	1	no	—
342	C₄	D₃²×C₈	1	1	yes	characteristic
343	C₄	#63791	1	1	yes	characteristic
344	C₄	#63936	1	1	yes	characteristic
345	C₄	#63950	1	1	yes	characteristic
346	C₃	—	2	1	no	—
347	C₃	#68527	1	1	yes	characteristic, complemented
348	C₃	#68567	1	1	yes	characteristic, complemented
349	C₂	—	18	2	no	—
350	C₂	#10590574	1	1	yes	characteristic
351	C₂	#10590658	1	1	yes	characteristic
352	C₂	#10590972	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#3811
commutator	C₃²×C₄	C₂²×C₈
Frattini	C₄²	C₂×D₃²
Fitting	C₃²×C₄⋊C₈	C₂²
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂²×C₃⋊D₃	C₄⋊C₄
cosocle	C₃²×C₄²	C₂³
hypercenter	C₄⋊C₈	D₃²
hypocenter	C₃²	#1564
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. C₃×#4480	C₂	semidirect
5. C₃×#4479	C₂	semidirect
6. #10589099	C₂	semidirect
7. #10589097	C₂	semidirect
35. #4480	D₃	semidirect
37. #4479	D₃	semidirect
56. #3710	C₈	semidirect
301. C₃²	#1564	semidirect
347. C₃	#68527	semidirect
348. C₃	#68567	semidirect

Maximal subgroups

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

subgroup	index
#10592532	2
C₄×#3710	2
#10590256	2
C₃×#4480	2
C₃×#4479	2
#10589099	2
#10589097	2
#68567	3
#68527	3