Subgroups — #10685133

Order 768.

542 conjugacy classes of subgroups.

The subgroup lattice has 352 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	#82049	C₂	1	1	yes	maximal, characteristic
3	C₂×#4925	C₂	1	1	yes	maximal, characteristic
4	#67711	C₂	1	4	yes	maximal, complemented
5	#8000	—	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	—
9	C₃×#431	—	2	1	no	—
10	C₃×#431	C₄	1	2	yes	—
11	C₃×#423	C₂²	1	2	yes	—
12	C₃×#407	—	2	1	no	—
13	C₃×C₄×M₁₆	—	2	1	no	—
14	#5207	—	2	1	no	—
15	C₂×#830	C₄	1	2	yes	—
16	#4926	—	2	1	no	—
17	#4926	C₄	1	2	yes	—
18	#4925	—	2	1	no	—
19	#4925	—	2	1	no	—
20	#4925	—	2	1	no	—
21	#4925	—	2	1	no	—
22	#4925	—	2	1	no	—
23	#4925	C₄	1	2	yes	—
24	#4924	C₂²	1	2	yes	—
25	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
26	C₄×#208	—	2	1	no	—
27	C₂×C₄×C₃⋊C₈	C₄	1	2	yes	—
28	#2920	—	3	1	no	—
29	#2920	D₃	1	1	yes	characteristic
30	C₂×#431	—	3	1	no	—
31	#1292	—	3	4	no	—
32	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
34	C₂³×C₃×C₄	Q₈	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	C₂×C₄	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	C₂³	1	1	yes	characteristic
48	C₂×#208	—	2	2	no	—
49	C₂²×C₃⋊C₈	C₂×C₄	1	4	yes	—
50	C₃×C₄⋊C₈	—	2	2	no	—
51	C₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
52	C₃×C₂²⋊C₈	—	2	2	no	—
53	C₃×C₂²⋊C₈	—	2	2	no	—
54	C₃×#97	—	2	2	no	—
55	C₃×C₄×C₈	—	2	2	no	—
56	#830	—	2	2	no	—
57	#830	—	2	2	no	—
58	#830	—	2	2	no	—
59	#830	—	2	2	no	—
60	C₄⋊(C₃⋊C₈)	—	2	2	no	—
61	C₄⋊(C₃⋊C₈)	—	2	2	no	—
62	C₄⋊(C₃⋊C₈)	—	2	2	no	—
63	C₄⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
64	#803	—	2	2	no	—
65	C₄×C₃⋊C₈	—	2	2	no	—
66	C₄×C₃⋊C₈	—	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	—
73	#431	—	2	1	no	—
74	#431	—	3	2	no	—
75	#431	—	6	1	no	—
76	#431	—	6	1	no	—
77	#431	—	6	1	no	—
78	#431	—	6	1	no	—
79	#431	—	6	1	no	—
80	#431	Dic₃	1	2	yes	—
81	#423	—	3	2	no	—
82	#423	C₂×D₃	1	2	yes	—
83	C₂×C₄⋊C₈	—	3	2	no	—
84	#407	—	2	1	no	—
85	#407	—	6	1	no	—
86	C₂×C₂²⋊C₈	—	3	2	no	—
87	C₄×M₁₆	—	2	1	no	—
88	C₄×M₁₆	—	6	1	no	—
89	C₂×C₄×C₈	—	3	2	no	—
90	C₂⁴×C₃	C₄⋊C₄	1	1	yes	characteristic
91	C₂²×C₃×C₄	—	2	4	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	C₂²⋊C₄	1	1	yes	characteristic
97	C₂²×C₃×C₄	C₂²⋊C₄	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	C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂²×C₄	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	C₃×M₁₆	—	4	4	no	—
113	C₂×C₃×C₈	—	2	4	no	—
114	C₃×C₄²	C₄²	1	1	yes	characteristic
115	C₃×C₄²	C₄²	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	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
127	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
128	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
129	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
130	#208	—	4	4	no	—
131	C₂×C₃⋊C₈	—	2	4	no	—
132	C₂×C₃⋊C₈	—	2	4	no	—
133	C₂×C₃⋊C₈	—	4	4	no	—
134	C₂³×C₄	Dic₆	1	1	yes	characteristic
135	C₂³×C₄	C₄×D₃	1	1	yes	characteristic
136	C₂³×C₄	#67	1	1	yes	characteristic
137	C₂×M₁₆	—	2	2	no	—
138	C₂×M₁₆	—	6	2	no	—
139	C₂²×C₈	—	3	4	no	—
140	C₂×C₄²	Dic₆	1	1	yes	characteristic
141	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
142	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
143	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
144	C₂×C₄²	D₁₂	1	1	yes	characteristic
145	C₂×C₄²	D₁₂	1	1	yes	characteristic
146	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
147	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
148	C₂×C₄²	#67	1	1	yes	characteristic
149	C₂×C₄²	#67	1	1	yes	characteristic
150	C₂×C₄²	#67	1	1	yes	characteristic
151	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	3	4	no	—
154	C₄⋊C₈	—	6	2	no	—
155	C₄⋊C₈	—	6	2	no	—
156	C₄⋊C₈	—	6	2	no	—
157	C₄⋊C₈	C₂×Dic₃	1	4	yes	—
158	C₂²⋊C₈	—	2	2	no	—
159	C₂²⋊C₈	—	2	2	no	—
160	C₂²⋊C₈	—	6	2	no	—
161	C₂²⋊C₈	—	6	2	no	—
162	C₂²⋊C₈	—	6	2	no	—
163	C₂²⋊C₈	—	6	2	no	—
164	#97	—	2	2	no	—
165	#97	—	6	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	—	6	2	no	—
168	C₄×C₈	—	6	2	no	—
169	C₄×C₈	—	6	2	no	—
170	C₂³×C₃	—	2	2	no	—
171	C₂³×C₃	—	4	1	no	—
172	C₂³×C₃	—	4	1	no	—
173	C₂³×C₃	#95	1	1	yes	characteristic
174	C₂³×C₃	#95	1	1	yes	characteristic
175	C₂³×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
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	—	2	2	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	#95	1	1	yes	characteristic
194	C₂×C₃×C₄	#95	1	1	yes	characteristic
195	C₂×C₃×C₄	#95	1	1	yes	characteristic
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	C₂×C₄²	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
205	C₂×C₃×C₄	#135	1	4	yes	—
206	C₃×C₈	—	4	4	no	—
207	C₃⋊C₈	—	4	4	no	—
208	C₃⋊C₈	—	4	4	no	—
209	C₂⁴	Dic₃⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	C₄×Dic₃	1	1	yes	characteristic
214	C₂²×C₄	C₄×Dic₃	1	1	yes	characteristic
215	C₂²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
216	C₂²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
217	C₂²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
218	C₂²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
219	C₂²×C₄	Dic₃⋊C₄	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	(C₂×D₃)⋊C₄	1	1	yes	characteristic
223	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
224	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
225	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
226	C₂²×C₄	#217	1	1	yes	characteristic
227	C₂²×C₄	#217	1	1	yes	characteristic
228	C₂²×C₄	C₂×Dic₆	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×#67	1	1	yes	characteristic
231	M₁₆	—	4	4	no	—
232	M₁₆	—	12	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	6	4	no	—
235	C₂×C₈	—	6	4	no	—
236	C₂×C₈	—	12	4	no	—
237	C₄²	C₄×Dic₃	1	1	yes	characteristic
238	C₄²	C₄×Dic₃	1	1	yes	characteristic
239	C₄²	Dic₃⋊C₄	1	1	yes	characteristic
240	C₄²	Dic₃⋊C₄	1	1	yes	characteristic
241	C₄²	C₄⋊Dic₃	1	1	yes	characteristic
242	C₄²	C₄⋊Dic₃	1	1	yes	characteristic
243	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
244	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
245	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
246	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
247	C₄²	#217	1	1	yes	characteristic
248	C₄²	#217	1	1	yes	characteristic
249	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
250	C₄²	C₂×D₁₂	1	1	yes	characteristic
251	C₄²	C₂²×Dic₃	1	1	yes	characteristic
252	C₄²	C₂×#67	1	1	yes	characteristic
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	—	4	4	no	—
262	C₂²×C₃	C₂×#95	1	1	yes	characteristic
263	C₂²×C₃	#427	1	2	yes	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	#338	1	2	yes	—
273	C₃×C₄	#413	1	2	yes	—
274	C₃×C₄	#416	1	2	yes	—
275	C₃×C₄	#416	1	2	yes	—
276	C₂³	—	2	2	no	—
277	C₂³	—	4	1	no	—
278	C₂³	—	4	1	no	—
279	C₂³	C₃⋊#95	1	1	yes	characteristic
280	C₂³	C₃⋊#95	1	1	yes	characteristic
281	C₂³	C₂×Dic₃⋊C₄	1	1	yes	characteristic
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
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	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
300	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
301	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
302	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
303	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
304	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
305	C₂×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
306	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
307	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
308	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
309	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
310	C₂×C₄	#950	1	4	yes	—
311	C₂×C₄	C₂×#217	1	1	yes	characteristic
312	C₈	—	4	4	no	—
313	C₈	—	12	4	no	—
314	C₈	—	12	4	no	—
315	C₂×C₃	—	2	2	no	—
316	C₂×C₃	—	4	1	no	—
317	C₂×C₃	—	4	1	no	—
318	C₂×C₃	C₂×#338	1	1	yes	characteristic
319	C₂×C₃	#1735	1	1	yes	characteristic
320	C₂×C₃	#1742	1	1	yes	characteristic
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	4	4	no	—
330	C₂²	#4998	1	2	yes	—
331	C₂²	C₂×C₃⋊#95	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	C₃⋊#338	1	2	yes	—
341	C₄	#5002	1	2	yes	—
342	C₄	#5132	1	2	yes	—
343	C₄	#5233	1	2	yes	—
344	C₃	#8000	1	1	yes	characteristic, complemented, Sylow
345	C₂	—	2	2	no	—
346	C₂	—	4	1	no	—
347	C₂	—	4	1	no	—
348	C₂	#71580	1	1	yes	characteristic
349	C₂	#71987	1	1	yes	characteristic
350	C₂	C₂×C₃⋊#338	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₃⋊#95
commutator	C₂×C₃×C₄	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×#2920	C₂
socle	C₂²×C₃	C₂×#95
prime radical	C₂⁴×C₃	C₄⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	#2920	D₃
hypocenter	C₃	#8000
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×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
4. #67711	C₂	semidirect
344. C₃	#8000	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
#82049	2
C₂×#4925	2
#67711	2
#8000	3