Subgroups — #10685139

Order 768.

598 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×#2920	C₂	1	1	yes	maximal, characteristic
2	#82049	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#67711	C₂	1	4	yes	maximal, complemented
5	#8002	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
7	C₂³×C₃⋊C₈	C₄	1	2	yes	—
8	C₃×#432	—	2	1	no	—
9	C₃×#432	C₄	1	2	yes	—
10	C₃×#431	—	2	1	no	—
11	C₃×#431	C₄	1	2	yes	—
12	C₃×#423	C₂²	1	2	yes	—
13	C₃×#407	—	2	1	no	—
14	C₃×C₄×M₁₆	—	2	1	no	—
15	#5207	—	2	1	no	—
16	#4926	—	2	1	no	—
17	#4926	C₄	1	2	yes	—
18	#4925	—	2	1	no	—
19	#4925	C₄	1	2	yes	—
20	#4924	C₂²	1	2	yes	—
21	C₂×C₄⋊(C₃⋊C₈)	—	2	1	no	—
22	C₂×C₄⋊(C₃⋊C₈)	—	2	1	no	—
23	C₂×C₄⋊(C₃⋊C₈)	—	2	1	no	—
24	C₂×C₄⋊(C₃⋊C₈)	—	2	1	no	—
25	C₂×C₄⋊(C₃⋊C₈)	C₄	1	2	yes	—
26	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
27	C₄×#208	—	2	1	no	—
28	#2920	—	3	1	no	—
29	#2920	D₃	1	1	yes	characteristic
30	C₂²×C₄⋊C₈	—	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	—	2	2	no	—
50	C₂²×C₃⋊C₈	—	2	2	no	—
51	C₂²×C₃⋊C₈	—	4	4	no	—
52	C₂²×C₃⋊C₈	C₂×C₄	1	4	yes	—
53	C₃×C₄⋊C₈	—	2	2	no	—
54	C₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
55	C₃×C₂²⋊C₈	—	2	2	no	—
56	C₃×C₂²⋊C₈	—	2	2	no	—
57	C₃×#97	—	2	2	no	—
58	C₃×C₄×C₈	—	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)	—	2	2	no	—
64	C₄⋊(C₃⋊C₈)	—	2	2	no	—
65	C₄⋊(C₃⋊C₈)	—	2	2	no	—
66	C₄⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
67	#803	—	2	2	no	—
68	C₄×C₃⋊C₈	—	2	2	no	—
69	C₂³×C₈	—	3	2	no	—
70	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic
71	#432	—	2	1	no	—
72	#432	—	3	2	no	—
73	#432	—	6	1	no	—
74	#432	Dic₃	1	2	yes	—
75	#431	—	2	1	no	—
76	#431	—	3	2	no	—
77	#431	—	6	1	no	—
78	#431	Dic₃	1	2	yes	—
79	#423	—	3	2	no	—
80	#423	C₂×D₃	1	2	yes	—
81	C₂×C₄⋊C₈	—	3	2	no	—
82	C₂×C₄⋊C₈	—	3	2	no	—
83	C₂×C₄⋊C₈	—	6	1	no	—
84	C₂×C₄⋊C₈	—	6	1	no	—
85	C₂×C₄⋊C₈	—	6	1	no	—
86	C₂×C₄⋊C₈	—	6	1	no	—
87	#407	—	2	1	no	—
88	#407	—	6	1	no	—
89	C₄×M₁₆	—	2	1	no	—
90	C₄×M₁₆	—	6	1	no	—
91	C₂⁴×C₃	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₃×C₄	—	2	2	no	—
93	C₂²×C₃×C₄	—	2	2	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	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	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
114	C₃×M₁₆	—	4	4	no	—
115	C₂×C₃×C₈	—	2	4	no	—
116	C₃×C₄²	C₄²	1	1	yes	characteristic
117	C₃×C₄²	C₄²	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	C₂²⋊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	C₂×D₄	1	1	yes	characteristic
131	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
132	#208	—	4	4	no	—
133	C₂×C₃⋊C₈	—	2	4	no	—
134	C₂×C₃⋊C₈	—	2	4	no	—
135	C₂×C₃⋊C₈	—	4	4	no	—
136	C₂×C₃⋊C₈	—	4	4	no	—
137	C₂×C₃⋊C₈	—	4	4	no	—
138	C₂³×C₄	Dic₆	1	1	yes	characteristic
139	C₂³×C₄	C₄×D₃	1	1	yes	characteristic
140	C₂³×C₄	#67	1	1	yes	characteristic
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	6	2	no	—
143	C₂²×C₈	—	3	4	no	—
144	C₂²×C₈	—	6	2	no	—
145	C₂²×C₈	—	6	2	no	—
146	C₂²×C₈	—	12	4	no	—
147	C₂×C₄²	Dic₆	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	C₄×D₃	1	1	yes	characteristic
151	C₂×C₄²	D₁₂	1	1	yes	characteristic
152	C₂×C₄²	D₁₂	1	1	yes	characteristic
153	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
154	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
155	C₂×C₄²	#67	1	1	yes	characteristic
156	C₂×C₄²	#67	1	1	yes	characteristic
157	C₂×C₄²	#67	1	1	yes	characteristic
158	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	3	4	no	—
161	C₄⋊C₈	—	6	2	no	—
162	C₄⋊C₈	—	6	2	no	—
163	C₄⋊C₈	—	6	2	no	—
164	C₄⋊C₈	—	6	2	no	—
165	C₄⋊C₈	—	6	2	no	—
166	C₄⋊C₈	C₂×Dic₃	1	4	yes	—
167	C₂²⋊C₈	—	2	2	no	—
168	C₂²⋊C₈	—	2	2	no	—
169	C₂²⋊C₈	—	6	2	no	—
170	C₂²⋊C₈	—	6	2	no	—
171	#97	—	2	2	no	—
172	#97	—	6	2	no	—
173	C₄×C₈	—	2	2	no	—
174	C₄×C₈	—	6	2	no	—
175	C₂³×C₃	—	2	2	no	—
176	C₂³×C₃	—	2	2	no	—
177	C₂³×C₃	—	2	2	no	—
178	C₂³×C₃	#95	1	1	yes	characteristic
179	C₂³×C₃	#95	1	1	yes	characteristic
180	C₂³×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
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	1	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	2	1	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	—	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	#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	#95	1	1	yes	characteristic
205	C₂×C₃×C₄	#100	1	2	yes	—
206	C₂×C₃×C₄	#101	1	2	yes	—
207	C₂×C₃×C₄	C₂×C₄²	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×C₂²⋊C₄	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	#135	1	4	yes	—
214	C₃×C₈	—	4	4	no	—
215	C₃⋊C₈	—	4	4	no	—
216	C₃⋊C₈	—	4	4	no	—
217	C₂⁴	Dic₃⋊C₄	1	1	yes	characteristic
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	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	Dic₃⋊C₄	1	1	yes	characteristic
229	C₂²×C₄	C₄⋊Dic₃	1	1	yes	characteristic
230	C₂²×C₄	C₄⋊Dic₃	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	(C₂×D₃)⋊C₄	1	1	yes	characteristic
235	C₂²×C₄	#217	1	1	yes	characteristic
236	C₂²×C₄	#217	1	1	yes	characteristic
237	C₂²×C₄	C₂×Dic₆	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×#67	1	1	yes	characteristic
240	M₁₆	—	4	4	no	—
241	M₁₆	—	12	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	6	4	no	—
244	C₂×C₈	—	6	4	no	—
245	C₂×C₈	—	12	4	no	—
246	C₂×C₈	—	12	4	no	—
247	C₂×C₈	—	12	4	no	—
248	C₄²	C₄×Dic₃	1	1	yes	characteristic
249	C₄²	C₄×Dic₃	1	1	yes	characteristic
250	C₄²	Dic₃⋊C₄	1	1	yes	characteristic
251	C₄²	Dic₃⋊C₄	1	1	yes	characteristic
252	C₄²	C₄⋊Dic₃	1	1	yes	characteristic
253	C₄²	C₄⋊Dic₃	1	1	yes	characteristic
254	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
255	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
256	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
257	C₄²	(C₂×D₃)⋊C₄	1	1	yes	characteristic
258	C₄²	#217	1	1	yes	characteristic
259	C₄²	#217	1	1	yes	characteristic
260	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
261	C₄²	C₂×D₁₂	1	1	yes	characteristic
262	C₄²	C₂²×Dic₃	1	1	yes	characteristic
263	C₄²	C₂×#67	1	1	yes	characteristic
264	C₂²×C₃	—	2	8	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	—	2	1	no	—
270	C₂²×C₃	—	2	1	no	—
271	C₂²×C₃	#338	1	2	yes	—
272	C₂²×C₃	#343	1	2	yes	—
273	C₂²×C₃	C₂×#95	1	1	yes	characteristic
274	C₂²×C₃	#427	1	2	yes	—
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	—	2	1	no	—
283	C₃×C₄	#343	1	2	yes	—
284	C₃×C₄	C₂×#100	1	1	yes	characteristic
285	C₃×C₄	C₂×#101	1	1	yes	characteristic
286	C₃×C₄	#416	1	2	yes	—
287	C₃×C₄	#416	1	2	yes	—
288	C₂³	—	2	2	no	—
289	C₂³	—	2	2	no	—
290	C₂³	—	2	2	no	—
291	C₂³	C₃⋊#95	1	1	yes	characteristic
292	C₂³	C₃⋊#95	1	1	yes	characteristic
293	C₂³	C₂×Dic₃⋊C₄	1	1	yes	characteristic
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	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	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	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	#823	1	2	yes	—
313	C₂×C₄	#824	1	2	yes	—
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	C₃⋊#95	1	1	yes	characteristic
319	C₂×C₄	C₃⋊#95	1	1	yes	characteristic
320	C₂×C₄	C₂×C₄×Dic₃	1	1	yes	characteristic
321	C₂×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
322	C₂×C₄	C₂×C₄⋊Dic₃	1	1	yes	characteristic
323	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
324	C₂×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
325	C₂×C₄	#950	1	4	yes	—
326	C₂×C₄	C₂×#217	1	1	yes	characteristic
327	C₈	—	4	4	no	—
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	—	2	2	no	—
333	C₂×C₃	C₂×#338	1	1	yes	characteristic
334	C₂×C₃	#1735	1	1	yes	characteristic
335	C₂×C₃	C₂×#343	1	1	yes	characteristic
336	C₂²	—	2	8	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	#4554	1	2	yes	—
344	C₂²	C₃⋊#338	1	2	yes	—
345	C₂²	#4998	1	2	yes	—
346	C₂²	C₂×C₃⋊#95	1	1	yes	characteristic
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	—	2	1	no	—
355	C₄	#4554	1	2	yes	—
356	C₄	#5002	1	2	yes	—
357	C₄	C₂×#823	1	1	yes	characteristic
358	C₄	C₂×#824	1	1	yes	characteristic
359	C₄	#5233	1	2	yes	—
360	C₃	#8002	1	1	yes	characteristic, complemented, Sylow
361	C₂	—	2	2	no	—
362	C₂	—	2	2	no	—
363	C₂	—	2	2	no	—
364	C₂	#71580	1	1	yes	characteristic
365	C₂	C₂×#4554	1	1	yes	characteristic
366	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	#8002
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
360. C₃	#8002	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
C₂²×C₄⋊(C₃⋊C₈)	2
#67711	2
#8002	3