Subgroups — #10764375

Order 768.

618 conjugacy classes of subgroups.

The subgroup lattice has 360 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×#4926	C₂	1	1	yes	maximal, characteristic
4	#70752	C₂	1	4	yes	maximal, complemented
5	#70751	C₂	1	4	yes	maximal, complemented
6	#69851	C₂	1	4	yes	maximal, complemented
7	#17821	—	3	1	no	maximal, Sylow
8	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
9	C₃×#432	C₄	1	2	yes	—
10	C₃×#432	C₄	1	2	yes	—
11	C₃×#431	C₂²	1	2	yes	—
12	C₃×#431	C₂²	1	2	yes	—
13	C₃×#423	C₄	1	2	yes	—
14	C₃×#407	C₄	1	2	yes	—
15	C₃×C₄×M₁₆	C₂²	1	2	yes	—
16	#5207	—	2	1	no	—
17	C₂×#830	—	2	1	no	—
18	#4926	—	2	1	no	—
19	#4926	—	2	1	no	—
20	#4926	—	2	1	no	—
21	#4926	C₂²	1	2	yes	—
22	#4926	C₂²	1	2	yes	—
23	#4926	C₂²	1	2	yes	—
24	#4925	—	2	1	no	—
25	#4925	C₂²	1	2	yes	—
26	#4924	C₂²	1	2	yes	—
27	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
28	C₄×#208	—	2	1	no	—
29	C₂×#803	—	2	1	no	—
30	#4458	C₂²	1	8	yes	complemented
31	#4457	C₂²	1	8	yes	complemented
32	#2920	—	3	1	no	—
33	#2920	D₃	1	1	yes	characteristic
34	C₂×#432	—	3	1	no	—
35	#1573	—	3	4	no	—
36	#1572	—	3	4	no	—
37	#1569	—	3	4	no	—
38	C₂³×C₃×C₄	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₃×C₄	D₄	1	1	yes	characteristic
40	C₂³×C₃×C₄	Q₈	1	1	yes	characteristic
41	C₂×C₃×M₁₆	C₂×C₄	1	4	yes	—
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	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	Q₈	1	1	yes	characteristic
49	C₂×C₃×C₄²	Q₈	1	1	yes	characteristic
50	C₂×C₃×C₄²	Q₈	1	1	yes	characteristic
51	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
53	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
54	C₂×#208	—	2	2	no	—
55	C₂²×C₃⋊C₈	—	2	2	no	—
56	C₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
57	C₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
58	C₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
59	C₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
60	C₃×#97	C₂×C₄	1	4	yes	—
61	C₃×C₄×C₈	C₂³	1	4	yes	—
62	#830	—	2	2	no	—
63	#830	—	2	2	no	—
64	#830	—	2	2	no	—
65	#830	—	2	2	no	—
66	C₄⋊(C₃⋊C₈)	—	2	2	no	—
67	C₄⋊(C₃⋊C₈)	—	2	2	no	—
68	C₄⋊(C₃⋊C₈)	C₂³	1	4	yes	—
69	C₄⋊(C₃⋊C₈)	C₂³	1	4	yes	—
70	#803	—	2	2	no	—
71	#803	—	2	2	no	—
72	#803	—	2	2	no	—
73	C₄×C₃⋊C₈	—	2	2	no	—
74	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic
75	#432	—	3	2	no	—
76	#432	—	3	2	no	—
77	#432	—	3	2	no	—
78	#432	—	6	1	no	—
79	#432	—	6	1	no	—
80	#432	—	6	1	no	—
81	#432	Dic₃	1	2	yes	—
82	#432	Dic₃	1	2	yes	—
83	#431	—	3	2	no	—
84	#431	—	6	1	no	—
85	#431	C₂×D₃	1	2	yes	—
86	#431	C₂×D₃	1	2	yes	—
87	#423	—	3	2	no	—
88	#423	Dic₃	1	2	yes	—
89	C₂×C₄⋊C₈	—	3	2	no	—
90	#407	—	6	1	no	—
91	#407	Dic₃	1	2	yes	—
92	C₂×C₂²⋊C₈	—	6	1	no	—
93	C₄×M₁₆	—	6	1	no	—
94	C₄×M₁₆	C₂×D₃	1	2	yes	—
95	C₂×#97	—	6	1	no	—
96	#335	—	3	8	no	—
97	#334	—	3	8	no	—
98	C₂⁴×C₃	C₄⋊C₄	1	1	yes	characteristic
99	C₂²×C₃×C₄	—	4	2	no	—
100	C₂²×C₃×C₄	—	4	2	no	—
101	C₂²×C₃×C₄	—	4	2	no	—
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	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	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
120	C₃×M₁₆	—	4	4	no	—
121	C₂×C₃×C₈	C₂²×C₄	1	8	yes	—
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	C₄⋊C₄	1	1	yes	characteristic
127	C₃×C₄²	C₄⋊C₄	1	1	yes	characteristic
128	C₃×C₄²	C₂²×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	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
134	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
135	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
136	C₃×C₄²	C₂×Q₈	1	1	yes	characteristic
137	C₃×C₄²	C₂⁴	1	1	yes	characteristic
138	#208	—	4	4	no	—
139	C₂×C₃⋊C₈	—	2	4	no	—
140	C₂×C₃⋊C₈	—	2	4	no	—
141	C₂×C₃⋊C₈	—	4	4	no	—
142	C₂³×C₄	Dic₆	1	1	yes	characteristic
143	C₂³×C₄	D₁₂	1	1	yes	characteristic
144	C₂³×C₄	C₂×Dic₃	1	1	yes	characteristic
145	C₂×M₁₆	—	6	2	no	—
146	C₂×M₁₆	C₂×Dic₃	1	4	yes	—
147	C₂²×C₈	—	6	2	no	—
148	C₂×C₄²	Dic₆	1	1	yes	characteristic
149	C₂×C₄²	Dic₆	1	1	yes	characteristic
150	C₂×C₄²	Dic₆	1	1	yes	characteristic
151	C₂×C₄²	D₁₂	1	1	yes	characteristic
152	C₂×C₄²	D₁₂	1	1	yes	characteristic
153	C₂×C₄²	D₁₂	1	1	yes	characteristic
154	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
155	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
156	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
157	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
158	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
159	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
160	C₄⋊C₈	—	3	4	no	—
161	C₄⋊C₈	—	3	4	no	—
162	C₄⋊C₈	—	6	2	no	—
163	C₄⋊C₈	—	6	2	no	—
164	C₄⋊C₈	C₂×Dic₃	1	4	yes	—
165	C₄⋊C₈	C₂×Dic₃	1	4	yes	—
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	C₂²⋊C₈	C₂×Dic₃	1	4	yes	—
171	C₂²⋊C₈	C₂×Dic₃	1	4	yes	—
172	#97	—	6	2	no	—
173	#97	—	6	2	no	—
174	#97	—	6	2	no	—
175	#97	C₂×Dic₃	1	4	yes	—
176	C₄×C₈	—	6	2	no	—
177	C₄×C₈	C₂²×D₃	1	4	yes	—
178	C₂³×C₃	—	2	2	no	—
179	C₂³×C₃	—	4	1	no	—
180	C₂³×C₃	—	4	1	no	—
181	C₂³×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂³×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
183	C₂³×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
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	—	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	—	4	1	no	—
197	C₂×C₃×C₄	—	4	1	no	—
198	C₂×C₃×C₄	—	4	1	no	—
199	C₂×C₃×C₄	—	4	1	no	—
200	C₂×C₃×C₄	—	4	1	no	—
201	C₂×C₃×C₄	—	4	1	no	—
202	C₂×C₃×C₄	C₂×C₄⋊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	C₂×C₄⋊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×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂²×Q₈	1	1	yes	characteristic
214	C₃×C₈	—	2	8	no	—
215	C₃⋊C₈	—	4	4	no	—
216	C₃⋊C₈	—	4	4	no	—
217	C₂⁴	C₄⋊Dic₃	1	1	yes	characteristic
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
222	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
223	C₂²×C₄	C₄⋊Dic₃	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	C₄⋊Dic₃	1	1	yes	characteristic
227	C₂²×C₄	C₄⋊Dic₃	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×Dic₆	1	1	yes	characteristic
231	C₂²×C₄	C₂×Dic₆	1	1	yes	characteristic
232	C₂²×C₄	C₂×Dic₆	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₁₂	1	1	yes	characteristic
236	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
237	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
238	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
239	M₁₆	—	4	4	no	—
240	M₁₆	—	12	4	no	—
241	C₂×C₈	—	6	4	no	—
242	C₂×C₈	—	6	4	no	—
243	C₂×C₈	—	12	4	no	—
244	C₂×C₈	C₂²×Dic₃	1	8	yes	—
245	C₄²	C₄⋊Dic₃	1	1	yes	characteristic
246	C₄²	C₄⋊Dic₃	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×Dic₆	1	1	yes	characteristic
252	C₄²	C₂×Dic₆	1	1	yes	characteristic
253	C₄²	C₂×Dic₆	1	1	yes	characteristic
254	C₄²	C₂×D₁₂	1	1	yes	characteristic
255	C₄²	C₂×D₁₂	1	1	yes	characteristic
256	C₄²	C₂×D₁₂	1	1	yes	characteristic
257	C₄²	C₂²×Dic₃	1	1	yes	characteristic
258	C₄²	C₂²×Dic₃	1	1	yes	characteristic
259	C₄²	C₂²×Dic₃	1	1	yes	characteristic
260	C₄²	C₂³×D₃	1	1	yes	characteristic
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	—	2	1	no	—
268	C₂²×C₃	—	2	1	no	—
269	C₂²×C₃	—	4	4	no	—
270	C₂²×C₃	#430	1	2	yes	—
271	C₂²×C₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
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	—	2	1	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	#568	1	2	yes	—
283	C₃×C₄	#575	1	2	yes	—
284	C₂³	—	2	2	no	—
285	C₂³	—	4	1	no	—
286	C₂³	—	4	1	no	—
287	C₂³	C₂×C₄⋊Dic₃	1	1	yes	characteristic
288	C₂³	C₂×C₄⋊Dic₃	1	1	yes	characteristic
289	C₂³	C₂×C₄⋊Dic₃	1	1	yes	characteristic
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	—	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	—	4	1	no	—
308	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
309	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
310	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
311	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
312	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
317	C₂×C₄	C₂²×Dic₆	1	1	yes	characteristic
318	C₂×C₄	C₂²×D₁₂	1	1	yes	characteristic
319	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
320	C₈	—	2	8	no	—
321	C₈	—	12	4	no	—
322	C₈	—	12	4	no	—
323	C₂×C₃	—	2	2	no	—
324	C₂×C₃	—	4	1	no	—
325	C₂×C₃	—	4	1	no	—
326	C₂×C₃	#2905	1	1	yes	characteristic
327	C₂×C₃	#2911	1	1	yes	characteristic
328	C₂×C₃	C₂×#430	1	1	yes	characteristic
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	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	4	4	no	—
338	C₂²	#5125	1	2	yes	—
339	C₂²	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
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	—	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	#5748	1	2	yes	—
351	C₄	#5819	1	2	yes	—
352	C₃	#17821	1	1	yes	characteristic, complemented, Sylow
353	C₂	—	2	2	no	—
354	C₂	—	4	1	no	—
355	C₂	—	4	1	no	—
356	C₂	#82204	1	1	yes	characteristic
357	C₂	#83752	1	1	yes	characteristic
358	C₂	C₂×#5125	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₂
commutator	C₂×C₃×C₄	C₂³×C₄
Frattini	C₄²	C₂³×D₃
Fitting	C₃×#2920	C₂
socle	C₂²×C₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₃	C₄⋊C₄
cosocle	C₃×C₄²	C₂⁴
hypercenter	#2920	D₃
hypocenter	C₃	#17821
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. #70752	C₂	semidirect
5. #70751	C₂	semidirect
6. #69851	C₂	semidirect
30. #4458	C₂²	semidirect
31. #4457	C₂²	semidirect
352. C₃	#17821	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×#4926	2
#70752	2
#70751	2
#69851	2
#17821	3