Subgroups — #10685447

Order 768.

628 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#82049	C₂	1	1	yes	maximal, characteristic
3	C₂×#4925	C₂	1	1	yes	maximal, characteristic
4	#67662	C₂	1	4	yes	maximal, complemented
5	#8084	—	3	1	no	maximal, Sylow
6	C₂²×C₃×M₁₆	—	2	1	no	—
7	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
8	C₃×C₄×M₁₆	—	2	1	no	—
9	C₃×C₄×M₁₆	—	2	1	no	—
10	C₃×C₄×M₁₆	C₄	1	2	yes	—
11	C₃×C₄×M₁₆	C₄	1	2	yes	—
12	C₂×C₃×#97	C₂²	1	2	yes	—
13	C₂×C₃×C₄×C₈	—	2	1	no	—
14	#5207	—	2	1	no	—
15	C₂×#830	C₄	1	2	yes	—
16	#4926	—	2	1	no	—
17	#4926	C₂²	1	2	yes	—
18	#4925	—	2	1	no	—
19	#4925	—	2	1	no	—
20	#4925	—	2	1	no	—
21	#4925	C₄	1	2	yes	—
22	#4925	C₄	1	2	yes	—
23	#4925	C₂²	1	2	yes	—
24	#4924	C₄	1	2	yes	—
25	C₂×C₄⋊(C₃⋊C₈)	—	2	1	no	—
26	C₄×#208	—	2	1	no	—
27	C₂×C₄×C₃⋊C₈	—	2	1	no	—
28	#2920	—	3	1	no	—
29	C₂×#431	—	3	1	no	—
30	C₂×C₄×M₁₆	D₃	1	1	yes	characteristic
31	#1302	—	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	8	no	—
36	C₂×C₃×M₁₆	—	2	2	no	—
37	C₂×C₃×M₁₆	—	2	2	no	—
38	C₂²×C₃×C₈	—	2	2	no	—
39	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃×C₄²	C₂×C₄	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	D₄	1	1	yes	characteristic
47	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
48	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
49	C₂×C₃×C₄²	Q₈	1	1	yes	characteristic
50	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
51	C₂×#208	—	2	2	no	—
52	C₂²×C₃⋊C₈	—	2	2	no	—
53	C₃×#97	—	2	2	no	—
54	C₃×#97	C₂×C₄	1	4	yes	—
55	C₃×C₄×C₈	—	2	2	no	—
56	C₃×C₄×C₈	—	2	2	no	—
57	#830	—	2	2	no	—
58	#830	—	2	2	no	—
59	#830	—	2	2	no	—
60	#830	C₂×C₄	1	4	yes	—
61	C₄⋊(C₃⋊C₈)	—	2	2	no	—
62	C₄⋊(C₃⋊C₈)	—	2	2	no	—
63	C₄⋊(C₃⋊C₈)	—	2	2	no	—
64	C₄⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
65	#803	—	2	2	no	—
66	C₄×C₃⋊C₈	—	2	2	no	—
67	C₄×C₃⋊C₈	—	2	2	no	—
68	C₄×C₃⋊C₈	—	2	2	no	—
69	C₂²×M₁₆	—	2	1	no	—
70	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic
71	#432	—	3	2	no	—
72	#432	—	6	1	no	—
73	#431	—	3	2	no	—
74	#431	—	3	2	no	—
75	#431	—	3	2	no	—
76	#431	—	6	1	no	—
77	#431	—	6	1	no	—
78	#431	—	6	1	no	—
79	#423	—	3	2	no	—
80	C₂×C₄⋊C₈	—	6	1	no	—
81	#407	—	6	1	no	—
82	C₂×C₂²⋊C₈	—	3	2	no	—
83	C₄×M₁₆	—	2	1	no	—
84	C₄×M₁₆	—	2	1	no	—
85	C₄×M₁₆	—	6	1	no	—
86	C₄×M₁₆	Dic₃	1	2	yes	—
87	C₄×M₁₆	Dic₃	1	2	yes	—
88	C₂×#97	C₂×D₃	1	2	yes	—
89	C₂×C₄×C₈	—	2	1	no	—
90	C₂×C₄×C₈	—	6	1	no	—
91	C₂⁴×C₃	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₃×C₄	—	2	4	no	—
93	C₂²×C₃×C₄	—	2	4	no	—
94	C₂²×C₃×C₄	—	4	2	no	—
95	C₂²×C₃×C₄	C₄²	1	1	yes	characteristic
96	C₂²×C₃×C₄	C₄²	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	C₂²⋊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	C₄⋊C₄	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
113	C₃×M₁₆	—	2	8	no	—
114	C₃×M₁₆	—	2	8	no	—
115	C₂×C₃×C₈	—	2	8	no	—
116	C₂×C₃×C₈	—	2	4	no	—
117	C₃×C₄²	C₄²	1	1	yes	characteristic
118	C₃×C₄²	C₄²	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	C₂²⋊C₄	1	1	yes	characteristic
124	C₃×C₄²	C₂²⋊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	C₂²×C₄	1	1	yes	characteristic
130	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
131	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
132	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
133	#208	—	4	4	no	—
134	C₂×C₃⋊C₈	—	2	4	no	—
135	C₂×C₃⋊C₈	—	2	4	no	—
136	C₂×C₃⋊C₈	—	4	4	no	—
137	C₂³×C₄	Dic₆	1	1	yes	characteristic
138	C₂³×C₄	C₄×D₃	1	1	yes	characteristic
139	C₂³×C₄	#67	1	1	yes	characteristic
140	C₂×M₁₆	—	2	8	no	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	2	2	no	—
143	C₂×M₁₆	—	6	2	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₂²×C₈	—	6	2	no	—
146	C₂×C₄²	Dic₆	1	1	yes	characteristic
147	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
148	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
149	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
150	C₂×C₄²	D₁₂	1	1	yes	characteristic
151	C₂×C₄²	D₁₂	1	1	yes	characteristic
152	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
153	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
154	C₂×C₄²	#67	1	1	yes	characteristic
155	C₂×C₄²	#67	1	1	yes	characteristic
156	C₂×C₄²	#67	1	1	yes	characteristic
157	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
158	C₄⋊C₈	—	3	4	no	—
159	C₄⋊C₈	—	6	2	no	—
160	C₄⋊C₈	—	6	2	no	—
161	C₄⋊C₈	—	6	2	no	—
162	C₂²⋊C₈	—	3	4	no	—
163	C₂²⋊C₈	—	6	2	no	—
164	C₂²⋊C₈	—	6	2	no	—
165	C₂²⋊C₈	—	6	2	no	—
166	#97	—	2	2	no	—
167	#97	—	6	2	no	—
168	#97	C₂×Dic₃	1	4	yes	—
169	C₄×C₈	—	2	2	no	—
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	—	6	2	no	—
172	C₄×C₈	—	6	2	no	—
173	C₄×C₈	—	6	2	no	—
174	C₂³×C₃	—	2	2	no	—
175	C₂³×C₃	—	2	2	no	—
176	C₂³×C₃	—	4	1	no	—
177	C₂³×C₃	#95	1	1	yes	characteristic
178	C₂³×C₃	#95	1	1	yes	characteristic
179	C₂³×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	2	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	#104	1	4	yes	—
203	C₂×C₃×C₄	#108	1	4	yes	—
204	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
210	C₃×C₈	—	2	8	no	—
211	C₃⋊C₈	—	4	4	no	—
212	C₃⋊C₈	—	4	4	no	—
213	C₂⁴	Dic₃⋊C₄	1	1	yes	characteristic
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	C₄×Dic₃	1	1	yes	characteristic
218	C₂²×C₄	C₄×Dic₃	1	1	yes	characteristic
219	C₂²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
220	C₂²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
221	C₂²×C₄	Dic₃⋊C₄	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	C₄⋊Dic₃	1	1	yes	characteristic
225	C₂²×C₄	C₄⋊Dic₃	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	#217	1	1	yes	characteristic
231	C₂²×C₄	#217	1	1	yes	characteristic
232	C₂²×C₄	C₂×Dic₆	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
234	C₂²×C₄	C₂×#67	1	1	yes	characteristic
235	M₁₆	—	2	8	no	—
236	M₁₆	—	2	8	no	—
237	M₁₆	—	12	4	no	—
238	C₂×C₈	—	2	8	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	6	4	no	—
241	C₂×C₈	—	6	4	no	—
242	C₂×C₈	—	12	4	no	—
243	C₄²	C₄×Dic₃	1	1	yes	characteristic
244	C₄²	C₄×Dic₃	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	(C₂×D₃)⋊C₄	1	1	yes	characteristic
250	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
251	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
252	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
253	C₄²	#217	1	1	yes	characteristic
254	C₄²	#217	1	1	yes	characteristic
255	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
256	C₄²	C₂×D₁₂	1	1	yes	characteristic
257	C₄²	C₂²×Dic₃	1	1	yes	characteristic
258	C₄²	C₂×#67	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	—	4	4	no	—
267	C₂²×C₃	#339	1	2	yes	—
268	C₂²×C₃	C₂×#95	1	1	yes	characteristic
269	C₂²×C₃	C₂×#108	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	#339	1	2	yes	—
278	C₃×C₄	#341	1	2	yes	—
279	C₃×C₄	#341	1	2	yes	—
280	C₃×C₄	#344	1	2	yes	—
281	C₃×C₄	C₂×#104	1	2	yes	—
282	C₂³	—	2	2	no	—
283	C₂³	—	2	2	no	—
284	C₂³	—	4	1	no	—
285	C₂³	C₃⋊#95	1	1	yes	characteristic
286	C₂³	C₃⋊#95	1	1	yes	characteristic
287	C₂³	C₂×Dic₃⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	#805	1	4	yes	—
305	C₂×C₄	#822	1	4	yes	—
306	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
307	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
308	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
309	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
310	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
311	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
313	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
315	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
316	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
317	C₂×C₄	C₂×#217	1	1	yes	characteristic
318	C₈	—	2	8	no	—
319	C₈	—	12	4	no	—
320	C₈	—	12	4	no	—
321	C₂×C₃	—	2	2	no	—
322	C₂×C₃	—	2	2	no	—
323	C₂×C₃	—	4	1	no	—
324	C₂×C₃	C₂×#339	1	1	yes	characteristic
325	C₂×C₃	C₂×#341	1	1	yes	characteristic
326	C₂×C₃	C₂×#344	1	1	yes	characteristic
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	4	4	no	—
335	C₂²	#4523	1	2	yes	—
336	C₂²	C₂×#822	1	2	yes	—
337	C₂²	C₂×C₃⋊#95	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	1	no	—
345	C₄	#4523	1	2	yes	—
346	C₄	#4558	1	2	yes	—
347	C₄	#4558	1	2	yes	—
348	C₄	#4559	1	2	yes	—
349	C₄	C₂×#805	1	2	yes	—
350	C₃	#8084	1	1	yes	characteristic, complemented, Sylow
351	C₂	—	2	2	no	—
352	C₂	—	2	2	no	—
353	C₂	—	4	1	no	—
354	C₂	C₂×#4523	1	1	yes	characteristic
355	C₂	C₂×#4558	1	1	yes	characteristic
356	C₂	C₂×#4559	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₂×C₃×C₄×M₁₆	C₂
socle	C₂²×C₃	C₂×#95
prime radical	C₂⁴×C₃	C₄⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₂×C₄×M₁₆	D₃
hypocenter	C₃	#8084
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #67662	C₂	semidirect
350. C₃	#8084	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×M₁₆	2
#82049	2
C₂×#4925	2
#67662	2
#8084	3