Subgroups — #10684635

Order 768.

684 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#117	C₂	1	1	yes	maximal, characteristic
2	#82049	C₂	1	1	yes	maximal, characteristic
3	C₂×#4925	C₂	1	1	yes	maximal, characteristic
4	#67611	C₂	1	4	yes	maximal, complemented
5	#7869	—	3	1	no	maximal, Sylow
6	C₂×C₃×#117	—	2	1	no	—
7	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
8	C₃×C₄×C₄⋊C₄	—	2	1	no	—
9	C₃×C₄×C₄⋊C₄	C₄	1	2	yes	—
10	C₃×C₄×C₂²⋊C₄	—	2	1	no	—
11	C₃×C₄×C₂²⋊C₄	C₄	1	2	yes	—
12	C₃×#376	—	2	1	no	—
13	C₃×C₄³	C₂²	1	2	yes	—
14	#5207	—	2	1	no	—
15	C₂×#830	—	2	1	no	—
16	#4926	—	2	1	no	—
17	#4926	C₄	1	2	yes	complemented
18	#4925	—	2	1	no	—
19	#4925	—	2	1	no	—
20	#4925	—	2	1	no	—
21	#4925	C₄	1	2	yes	complemented
22	#4925	C₂²	1	2	yes	—
23	#4925	C₂²	1	2	yes	—
24	#4924	C₄	1	2	yes	complemented
25	C₂×C₄⋊(C₃⋊C₈)	C₄	1	2	yes	complemented
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	C₄×#117	D₃	1	1	yes	characteristic
31	#1276	—	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	D₄	1	1	yes	characteristic
35	C₃×#117	—	2	8	no	—
36	C₂×C₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
38	C₂×C₃×C₄²	—	2	2	no	—
39	C₂×C₃×C₄²	—	2	2	no	—
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	C₂×C₄	1	4	yes	—
45	C₂×C₃×C₄²	C₂×C₄	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	D₄	1	1	yes	characteristic
50	C₂×C₃×C₄²	Q₈	1	1	yes	characteristic
51	C₂×C₃×C₄²	Q₈	1	1	yes	characteristic
52	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
53	C₂×#208	—	2	2	no	—
54	C₂²×C₃⋊C₈	—	2	2	no	—
55	C₃×#95	—	2	2	no	—
56	C₃×#95	—	2	2	no	—
57	#830	—	2	2	no	—
58	#830	—	2	2	no	—
59	#830	—	2	2	no	—
60	#830	—	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	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	C₂×#117	—	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	—	3	2	no	—
81	#407	—	6	1	no	—
82	C₂×C₂²⋊C₈	—	6	1	no	—
83	C₄×M₁₆	—	6	1	no	—
84	C₂×C₄×C₈	—	6	1	no	—
85	C₄×C₄⋊C₄	—	2	1	no	—
86	C₄×C₄⋊C₄	Dic₃	1	2	yes	—
87	C₄×C₂²⋊C₄	—	2	1	no	—
88	C₄×C₂²⋊C₄	Dic₃	1	2	yes	—
89	#376	—	2	1	no	—
90	C₄³	C₂×D₃	1	2	yes	—
91	C₂⁴×C₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₃×C₄	—	2	4	no	—
93	C₂²×C₃×C₄	—	2	4	no	—
94	C₂²×C₃×C₄	—	2	4	no	—
95	C₂²×C₃×C₄	—	4	2	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₂²⋊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	C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
114	C₃×C₄⋊C₄	—	2	8	no	—
115	C₃×C₂²⋊C₄	—	2	8	no	—
116	C₃×C₄²	—	2	8	no	—
117	C₃×C₄²	—	2	8	no	—
118	C₃×C₄²	—	4	4	no	—
119	C₃×C₄²	C₄²	1	1	yes	characteristic
120	C₃×C₄²	C₄²	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	C₄⋊C₄	1	1	yes	characteristic
130	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
131	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
132	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
133	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
134	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
135	#208	—	4	4	no	—
136	C₂×C₃⋊C₈	—	2	4	no	—
137	C₂×C₃⋊C₈	—	2	4	no	—
138	C₂×C₃⋊C₈	—	4	4	no	—
139	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
140	C₂³×C₄	#67	1	1	yes	characteristic
141	C₂³×C₄	#67	1	1	yes	characteristic
142	C₂×M₁₆	—	6	2	no	—
143	C₂²×C₈	—	6	2	no	—
144	#117	—	2	8	no	—
145	C₂×C₄⋊C₄	—	2	2	no	—
146	C₂×C₂²⋊C₄	—	2	2	no	—
147	C₂×C₄²	—	2	2	no	—
148	C₂×C₄²	—	2	2	no	—
149	C₂×C₄²	Dic₆	1	1	yes	characteristic
150	C₂×C₄²	Dic₆	1	1	yes	characteristic
151	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
152	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
153	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
154	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
155	C₂×C₄²	D₁₂	1	1	yes	characteristic
156	C₂×C₄²	D₁₂	1	1	yes	characteristic
157	C₂×C₄²	C₂×Dic₃	1	4	yes	—
158	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
159	C₂×C₄²	#67	1	1	yes	characteristic
160	C₂×C₄²	#67	1	1	yes	characteristic
161	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
162	C₄⋊C₈	—	3	4	no	—
163	C₄⋊C₈	—	3	4	no	—
164	C₄⋊C₈	—	6	2	no	—
165	C₄⋊C₈	—	6	2	no	—
166	C₂²⋊C₈	—	6	2	no	—
167	C₂²⋊C₈	—	6	2	no	—
168	C₂²⋊C₈	—	6	2	no	—
169	C₂²⋊C₈	—	6	2	no	—
170	#97	—	6	2	no	—
171	C₄×C₈	—	6	2	no	—
172	C₄×C₈	—	6	2	no	—
173	C₄×C₈	—	6	2	no	—
174	#95	—	2	2	no	—
175	#95	—	2	2	no	—
176	C₂³×C₃	—	2	2	no	—
177	C₂³×C₃	—	2	2	no	—
178	C₂³×C₃	—	4	1	no	—
179	C₂³×C₃	#95	1	1	yes	characteristic
180	C₂³×C₃	#95	1	1	yes	characteristic
181	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	2	8	no	—
196	C₂×C₃×C₄	—	4	1	no	—
197	C₂×C₃×C₄	—	4	1	no	—
198	C₂×C₃×C₄	—	4	1	no	—
199	C₂×C₃×C₄	—	4	4	no	—
200	C₂×C₃×C₄	—	4	4	no	—
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	#95	1	1	yes	characteristic
205	C₂×C₃×C₄	#95	1	1	yes	characteristic
206	C₂×C₃×C₄	#95	1	1	yes	characteristic
207	C₂×C₃×C₄	#104	1	4	yes	—
208	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
214	C₂×C₃×C₄	#131	1	4	yes	—
215	C₃⋊C₈	—	4	4	no	—
216	C₃⋊C₈	—	4	4	no	—
217	C₂⁴	#217	1	1	yes	characteristic
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	C₄×Dic₃	1	1	yes	characteristic
223	C₂²×C₄	C₄×Dic₃	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	Dic₃⋊C₄	1	1	yes	characteristic
227	C₂²×C₄	Dic₃⋊C₄	1	1	yes	characteristic
228	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
229	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
230	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
231	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
232	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
233	C₂²×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
234	C₂²×C₄	#217	1	1	yes	characteristic
235	C₂²×C₄	#217	1	1	yes	characteristic
236	C₂²×C₄	#217	1	1	yes	characteristic
237	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×#67	1	1	yes	characteristic
239	C₂²×C₄	C₂×#67	1	1	yes	characteristic
240	M₁₆	—	12	4	no	—
241	C₂×C₈	—	6	4	no	—
242	C₂×C₈	—	6	4	no	—
243	C₂×C₈	—	12	4	no	—
244	C₄⋊C₄	—	2	8	no	—
245	C₂²⋊C₄	—	2	8	no	—
246	C₄²	—	2	8	no	—
247	C₄²	—	2	8	no	—
248	C₄²	—	4	4	no	—
249	C₄²	C₄×Dic₃	1	1	yes	characteristic
250	C₄²	C₄×Dic₃	1	1	yes	characteristic
251	C₄²	Dic₃⋊C₄	1	1	yes	characteristic
252	C₄²	Dic₃⋊C₄	1	1	yes	characteristic
253	C₄²	Dic₃⋊C₄	1	1	yes	characteristic
254	C₄²	Dic₃⋊C₄	1	1	yes	characteristic
255	C₄²	C₄⋊Dic₃	1	1	yes	characteristic
256	C₄²	C₄⋊Dic₃	1	1	yes	characteristic
257	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
258	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
259	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
260	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
261	C₄²	C₂×Dic₆	1	1	yes	characteristic
262	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
263	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
264	C₄²	C₂×D₁₂	1	1	yes	characteristic
265	C₂²×C₃	—	2	1	no	—
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	—	2	1	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	—	2	1	no	—
271	C₂²×C₃	—	2	1	no	—
272	C₂²×C₃	—	4	4	no	—
273	C₂²×C₃	#339	1	2	yes	—
274	C₂²×C₃	C₂×#95	1	1	yes	characteristic
275	C₂²×C₃	#408	1	2	yes	—
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	—	2	1	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	4	8	no	—
284	C₃×C₄	#339	1	2	yes	—
285	C₃×C₄	#405	1	2	yes	—
286	C₃×C₄	C₂×#104	1	2	yes	—
287	C₃×C₄	#421	1	2	yes	—
288	C₃×C₄	#424	1	2	yes	—
289	C₂³	—	2	2	no	—
290	C₂³	—	2	2	no	—
291	C₂³	—	4	1	no	—
292	C₂³	C₃⋊#95	1	1	yes	characteristic
293	C₂³	C₃⋊#95	1	1	yes	characteristic
294	C₂³	C₂×#217	1	1	yes	characteristic
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	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	8	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	4	4	no	—
314	C₂×C₄	#805	1	4	yes	—
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	C₃⋊#95	1	1	yes	characteristic
319	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
320	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
321	C₂×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
322	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
323	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
324	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
325	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
326	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
327	C₂×C₄	#948	1	4	yes	—
328	C₈	—	12	4	no	—
329	C₈	—	12	4	no	—
330	C₂×C₃	—	2	2	no	—
331	C₂×C₃	—	2	2	no	—
332	C₂×C₃	—	4	1	no	—
333	C₂×C₃	#1729	1	1	yes	characteristic
334	C₂×C₃	C₂×#339	1	1	yes	characteristic
335	C₂×C₃	#1733	1	1	yes	characteristic
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	4	4	no	—
344	C₂²	#4523	1	2	yes	—
345	C₂²	C₂×C₃⋊#95	1	1	yes	characteristic
346	C₂²	#5234	1	2	yes	—
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	4	8	no	—
355	C₄	#4523	1	2	yes	—
356	C₄	C₂×#805	1	2	yes	—
357	C₄	#4997	1	2	yes	—
358	C₄	#4999	1	2	yes	—
359	C₄	#5005	1	2	yes	—
360	C₃	#7869	1	1	yes	characteristic, complemented, Sylow
361	C₂	—	2	2	no	—
362	C₂	—	2	2	no	—
363	C₂	—	4	1	no	—
364	C₂	C₂×#4523	1	1	yes	characteristic
365	C₂	#71578	1	1	yes	characteristic
366	C₂	#71579	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₄×#117	C₂
socle	C₂²×C₃	C₂×#95
prime radical	C₂⁴×C₃	C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₄×#117	D₃
hypocenter	C₃	#7869
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
4. #67611	C₂	semidirect
17. #4926	C₄	semidirect
21. #4925	C₄	semidirect
24. #4924	C₄	semidirect
25. C₂×C₄⋊(C₃⋊C₈)	C₄	semidirect
360. C₃	#7869	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×#117	2
#82049	2
C₂×#4925	2
#67611	2
#7869	3