Subgroups — #10685556

Order 768.

562 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#2920	C₂	1	1	yes	maximal, characteristic
2	#82060	C₂	1	1	yes	maximal, characteristic
3	#82049	C₂	1	1	yes	maximal, characteristic
4	#67686	C₂	1	4	yes	maximal
5	#8113	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
7	C₃×#432	—	2	1	no	—
8	C₃×#432	C₄	1	2	yes	complemented
9	C₃×#431	—	2	1	no	—
10	C₃×#431	C₄	1	2	yes	complemented
11	C₃×#423	—	2	1	no	—
12	C₃×#407	C₂²	1	2	yes	—
13	C₃×C₄×M₁₆	—	2	1	no	—
14	#5212	C₄	1	2	yes	—
15	#5211	C₂²	1	2	yes	—
16	C₄×#217	C₄	1	2	yes	—
17	#5207	C₂²	1	2	yes	—
18	#4937	—	2	1	no	—
19	#4936	—	2	1	no	—
20	C₄×C₄⋊Dic₃	—	2	1	no	—
21	#4932	—	2	1	no	—
22	#4926	—	2	1	no	—
23	#4926	C₄	1	2	yes	complemented
24	#4925	—	2	1	no	—
25	#4925	C₄	1	2	yes	complemented
26	#4924	—	2	1	no	—
27	C₄×#208	—	2	1	no	—
28	#2920	—	3	1	no	—
29	#2920	D₃	1	1	yes	characteristic
30	#2283	—	3	1	no	—
31	#1306	—	3	4	no	—
32	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
35	C₂×C₃×M₁₆	—	2	2	no	—
36	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
41	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
42	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
43	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
44	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
45	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
46	C₂×C₃×C₄²	Q₈	1	1	yes	characteristic
47	C₂×C₃×C₄²	Q₈	1	1	yes	characteristic
48	C₂×#217	C₂×C₄	1	4	yes	—
49	C₂×C₄⋊Dic₃	—	2	2	no	—
50	C₂×C₄×Dic₃	—	2	2	no	—
51	C₂×#208	—	2	2	no	—
52	C₃×C₄⋊C₈	—	2	2	no	—
53	C₃×C₄⋊C₈	—	2	2	no	—
54	C₃×C₂²⋊C₈	—	2	2	no	—
55	C₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
56	C₃×#97	—	2	2	no	—
57	C₃×C₄×C₈	—	2	2	no	—
58	C₃⋊#95	—	2	2	no	—
59	C₃⋊#95	—	2	2	no	—
60	C₃⋊#95	—	2	2	no	—
61	C₃⋊#95	—	2	2	no	—
62	#830	—	2	2	no	—
63	#830	C₂×C₄	1	4	yes	—
64	C₄⋊(C₃⋊C₈)	—	2	2	no	—
65	C₄⋊(C₃⋊C₈)	—	2	2	no	—
66	#803	—	2	2	no	—
67	C₄×C₃⋊C₈	—	2	2	no	—
68	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic
69	#432	—	2	1	no	—
70	#432	—	3	2	no	—
71	#432	—	6	1	no	—
72	#432	Dic₃	1	2	yes	complemented
73	#431	—	2	1	no	—
74	#431	—	3	2	no	—
75	#431	—	6	1	no	—
76	#431	Dic₃	1	2	yes	complemented
77	#423	—	2	1	no	—
78	#423	—	6	1	no	—
79	#407	—	3	2	no	—
80	#407	C₂×D₃	1	2	yes	—
81	C₄×M₁₆	—	2	1	no	—
82	C₄×M₁₆	—	6	1	no	—
83	#383	—	6	1	no	—
84	#382	—	6	1	no	—
85	#381	—	3	2	no	—
86	#380	—	3	2	no	—
87	C₄×C₄⋊C₄	—	6	1	no	—
88	C₄×C₂²⋊C₄	—	3	2	no	—
89	#376	—	6	1	no	—
90	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₃×C₄	—	4	2	no	—
92	C₂²×C₃×C₄	—	4	2	no	—
93	C₂²×C₃×C₄	—	4	2	no	—
94	C₂²×C₃×C₄	C₄²	1	1	yes	characteristic
95	C₂²×C₃×C₄	C₄²	1	1	yes	characteristic
96	C₂²×C₃×C₄	C₄²	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₄²	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
112	C₂²×Dic₃	—	2	4	no	—
113	C₃×M₁₆	—	4	4	no	—
114	C₂×C₃×C₈	—	2	4	no	—
115	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
116	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
117	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
118	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
119	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
120	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
121	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
122	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
123	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
124	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
125	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
126	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
127	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
128	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
129	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
130	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
131	#217	—	4	4	no	—
132	C₄⋊Dic₃	—	4	4	no	—
133	C₄×Dic₃	—	4	4	no	—
134	#208	—	4	4	no	—
135	C₂×C₃⋊C₈	—	2	4	no	—
136	C₂³×C₄	C₄×D₃	1	1	yes	characteristic
137	C₂³×C₄	C₄×D₃	1	1	yes	characteristic
138	C₂³×C₄	C₂²×D₃	1	1	yes	characteristic
139	C₂×M₁₆	—	2	2	no	—
140	C₂×M₁₆	—	6	2	no	—
141	C₂×C₄⋊C₄	—	6	2	no	—
142	C₂×C₂²⋊C₄	—	3	4	no	—
143	C₂×C₄²	—	6	2	no	—
144	C₂×C₄²	Dic₆	1	1	yes	characteristic
145	C₂×C₄²	Dic₆	1	1	yes	characteristic
146	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
147	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
148	C₂×C₄²	D₁₂	1	1	yes	characteristic
149	C₂×C₄²	D₁₂	1	1	yes	characteristic
150	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
151	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
152	C₂×C₄²	#67	1	1	yes	characteristic
153	C₂×C₄²	#67	1	1	yes	characteristic
154	C₂×C₄²	#67	1	1	yes	characteristic
155	C₂×C₄²	#67	1	1	yes	characteristic
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	6	2	no	—
159	C₄⋊C₈	—	6	2	no	—
160	C₂²⋊C₈	—	2	2	no	—
161	C₂²⋊C₈	—	3	4	no	—
162	C₂²⋊C₈	—	6	2	no	—
163	C₂²⋊C₈	C₂×Dic₃	1	4	yes	—
164	#97	—	2	2	no	—
165	#97	—	6	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	—	6	2	no	—
168	#95	—	6	2	no	—
169	#95	—	6	2	no	—
170	#95	—	6	2	no	—
171	#95	—	6	2	no	—
172	C₂³×C₃	—	2	2	no	—
173	C₂³×C₃	—	4	1	no	—
174	C₂³×C₃	—	4	1	no	—
175	C₂³×C₃	C₂×C₄²	1	1	yes	characteristic
176	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
177	C₂³×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	4	1	no	—
187	C₂×C₃×C₄	—	4	1	no	—
188	C₂×C₃×C₄	—	4	1	no	—
189	C₂×C₃×C₄	—	4	1	no	—
190	C₂×C₃×C₄	—	4	1	no	—
191	C₂×C₃×C₄	—	4	1	no	—
192	C₂×C₃×C₄	—	4	1	no	—
193	C₂×C₃×C₄	—	4	1	no	—
194	C₂×C₃×C₄	—	4	1	no	—
195	C₂×C₃×C₄	—	4	1	no	—
196	C₂×C₃×C₄	#95	1	1	yes	characteristic
197	C₂×C₃×C₄	#95	1	1	yes	characteristic
198	C₂×C₃×C₄	#95	1	1	yes	characteristic
199	C₂×C₃×C₄	#95	1	1	yes	characteristic
200	C₂×C₃×C₄	#95	1	1	yes	characteristic
201	C₂×C₃×C₄	#95	1	1	yes	characteristic
202	C₂×C₃×C₄	#95	1	1	yes	characteristic
203	C₂×C₃×C₄	#95	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
208	C₂×Dic₃	—	4	4	no	—
209	C₂×Dic₃	—	4	4	no	—
210	C₂×Dic₃	—	4	4	no	—
211	C₃×C₈	—	4	4	no	—
212	C₃⋊C₈	—	4	4	no	—
213	C₂⁴	C₂×C₄×D₃	1	1	yes	characteristic
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	—	6	4	no	—
218	C₂²×C₄	C₄×Dic₃	1	1	yes	characteristic
219	C₂²×C₄	C₄×Dic₃	1	1	yes	characteristic
220	C₂²×C₄	C₄×Dic₃	1	1	yes	characteristic
221	C₂²×C₄	C₄×Dic₃	1	1	yes	characteristic
222	C₂²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
223	C₂²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
224	C₂²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
225	C₂²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
226	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
227	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
228	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
229	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
230	C₂²×C₄	C₂×Dic₆	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
233	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
234	C₂²×C₄	C₂×#67	1	1	yes	characteristic
235	C₂²×C₄	C₂×#67	1	1	yes	characteristic
236	M₁₆	—	4	4	no	—
237	M₁₆	—	12	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	6	4	no	—
240	C₄⋊C₄	—	12	4	no	—
241	C₂²⋊C₄	—	12	4	no	—
242	C₄²	—	12	4	no	—
243	C₄²	Dic₃⋊C₄	1	1	yes	characteristic
244	C₄²	Dic₃⋊C₄	1	1	yes	characteristic
245	C₄²	Dic₃⋊C₄	1	1	yes	characteristic
246	C₄²	Dic₃⋊C₄	1	1	yes	characteristic
247	C₄²	C₄⋊Dic₃	1	1	yes	characteristic
248	C₄²	C₄⋊Dic₃	1	1	yes	characteristic
249	C₄²	C₄⋊Dic₃	1	1	yes	characteristic
250	C₄²	C₄⋊Dic₃	1	1	yes	characteristic
251	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
252	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
253	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
254	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
255	C₄²	#217	1	1	yes	characteristic
256	C₄²	#217	1	1	yes	characteristic
257	C₄²	#217	1	1	yes	characteristic
258	C₄²	#217	1	1	yes	characteristic
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	—	2	1	no	—
262	C₂²×C₃	—	2	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	—	4	4	no	—
268	C₂²×C₃	C₂×#95	1	1	yes	characteristic
269	C₂²×C₃	#410	1	2	yes	—
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	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	#344	1	2	yes	—
281	C₃×C₄	#344	1	2	yes	—
282	Dic₃	—	8	4	no	—
283	C₂³	—	2	2	no	—
284	C₂³	—	4	1	no	—
285	C₂³	—	4	1	no	—
286	C₂³	C₂×C₄×Dic₃	1	1	yes	characteristic
287	C₂³	C₂×Dic₃⋊C₄	1	1	yes	characteristic
288	C₂³	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	12	4	no	—
308	C₂×C₄	—	12	4	no	—
309	C₂×C₄	—	12	4	no	—
310	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
311	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
312	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
313	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
314	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
315	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
316	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
317	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
318	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
319	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
320	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
321	C₂×C₄	C₂×#217	1	1	yes	characteristic
322	C₈	—	4	4	no	—
323	C₈	—	12	4	no	—
324	C₂×C₃	—	2	2	no	—
325	C₂×C₃	—	4	1	no	—
326	C₂×C₃	—	4	1	no	—
327	C₂×C₃	#1740	1	1	yes	characteristic
328	C₂×C₃	C₂×#344	1	1	yes	characteristic
329	C₂×C₃	C₂×#344	1	1	yes	characteristic
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	4	4	no	—
339	C₂²	#5006	1	2	yes	—
340	C₂²	C₂×C₃⋊#95	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	24	4	no	—
352	C₄	#4556	1	2	yes	—
353	C₄	#4559	1	2	yes	—
354	C₃	#8113	1	1	yes	characteristic, complemented, Sylow
355	C₂	—	2	2	no	—
356	C₂	—	4	1	no	—
357	C₂	—	4	1	no	—
358	C₂	#71593	1	1	yes	characteristic
359	C₂	C₂×#4556	1	1	yes	characteristic
360	C₂	C₂×#4559	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₃⋊#95
commutator	C₂³×C₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₃
Fitting	C₃×#2920	C₂
socle	C₂²×C₃	C₂×#95
prime radical	C₂⁴×C₃	C₂²×C₄
cosocle	C₂³×C₃×C₄	C₂³
hypercenter	#2920	D₃
hypocenter	C₃	#8113
LCS	G ▷ C₂³×C₃ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
8. C₃×#432	C₄	semidirect
10. C₃×#431	C₄	semidirect
23. #4926	C₄	semidirect
25. #4925	C₄	semidirect
72. #432	Dic₃	semidirect
76. #431	Dic₃	semidirect
354. C₃	#8113	semidirect

Maximal subgroups

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

subgroup	index
C₃×#2920	2
#82060	2
#82049	2
#67686	2
#8113	3