Subgroups — #10712913

Order 768.

394 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×(C₂×D₃)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×(C₂×D₃)⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic
6	#67804	C₂	1	1	yes	maximal, characteristic, complemented
7	#67657	C₂	1	1	yes	maximal, characteristic, complemented
8	#13120	—	3	1	no	maximal, Sylow
9	C₂²×D₃×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×(C₂×D₃)⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₃	C₂²	1	1	yes	characteristic
15	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₄×(C₂×D₃)⋊C₄	C₄	1	1	yes	characteristic
18	C₃×C₄⋊C₁₆	—	2	2	no	—
19	#4550	C₄	1	1	yes	characteristic
20	#4524	C₄	1	1	yes	characteristic
21	(C₂×D₃)⋊C₁₆	—	2	2	no	—
22	(C₂×D₃)⋊C₁₆	—	2	2	no	—
23	Dic₃×C₁₆	—	2	2	no	—
24	C₂×C₄⋊C₁₆	D₃	1	1	yes	characteristic, complemented
25	C₂×C₂²⋊C₁₆	—	3	1	no	—
26	C₂×C₂²⋊C₁₆	—	3	1	no	—
27	C₂×C₄×C₁₆	—	3	1	no	—
28	C₈×C₂²⋊C₄	—	3	1	no	—
29	#1374	—	3	1	no	—
30	#1374	—	3	1	no	—
31	C₂²×C₄×D₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
35	C₂×(C₂×D₃)⋊C₄	C₈	1	1	yes	characteristic
36	C₂×(C₂×D₃)⋊C₄	C₈	1	1	yes	characteristic
37	C₂×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×D₃×C₈	—	2	1	no	—
41	C₂×D₃×C₈	—	2	1	no	—
42	C₂×D₃×C₈	—	2	1	no	—
43	C₂×D₃×C₈	—	2	1	no	—
44	C₂×D₃×C₈	—	2	1	no	—
45	C₂×D₃×C₈	—	2	1	no	—
46	C₂×C₃×C₁₆	—	2	1	no	—
47	C₂×C₃×C₁₆	—	2	1	no	—
48	C₂×C₃×C₁₆	—	2	1	no	—
49	C₂×C₃×C₁₆	—	2	1	no	—
50	C₂×C₃×C₁₆	—	2	1	no	—
51	C₂×C₃×C₁₆	D₄	1	2	yes	—
52	C₃×C₄×C₈	—	2	1	no	—
53	C₃×C₄×C₈	—	2	1	no	—
54	C₃⋊#95	C₈	1	1	yes	characteristic
55	C₃⋊#95	C₈	1	1	yes	characteristic
56	(C₂×D₃)⋊C₈	—	2	2	no	—
57	C₈×Dic₃	—	2	1	no	—
58	C₈×Dic₃	—	2	1	no	—
59	C₂×C₃⋊C₁₆	—	2	1	no	—
60	C₂×C₃⋊C₁₆	—	2	1	no	—
61	C₂×C₃⋊C₁₆	—	2	1	no	—
62	C₂×C₃⋊C₁₆	—	2	1	no	—
63	C₂×C₃⋊C₁₆	—	2	1	no	—
64	C₂×C₃⋊C₁₆	D₄	1	2	yes	—
65	C₂³×C₈	—	3	1	no	—
66	C₂²×C₁₆	—	3	1	no	—
67	C₂²×C₁₆	—	3	1	no	—
68	C₂²×C₁₆	C₂×D₃	1	1	yes	characteristic
69	C₂²×C₁₆	C₂×D₃	1	1	yes	characteristic
70	C₂×C₂²⋊C₈	—	3	1	no	—
71	C₂×C₄×C₈	—	3	1	no	—
72	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
73	C₄×C₂²⋊C₄	—	3	1	no	—
74	C₄⋊C₁₆	—	2	2	no	—
75	C₂²⋊C₁₆	—	6	2	no	—
76	C₂²⋊C₁₆	—	6	2	no	—
77	C₄×C₁₆	—	6	2	no	—
78	#336	—	3	1	no	—
79	#336	—	3	1	no	—
80	C₂³×D₃	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
85	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
86	C₂²×Dic₃	C₂×C₈	1	1	yes	characteristic
87	C₂×C₄×D₃	—	2	1	no	—
88	C₂×C₄×D₃	—	2	1	no	—
89	C₂×C₄×D₃	—	2	1	no	—
90	C₂×C₄×D₃	—	2	1	no	—
91	C₂×C₄×D₃	—	2	1	no	—
92	C₂×C₄×D₃	—	2	1	no	—
93	C₂×C₃×C₈	—	2	1	no	—
94	C₂×C₃×C₈	—	2	1	no	—
95	C₂×C₃×C₈	—	2	1	no	—
96	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂×C₃×C₈	#41	1	1	yes	characteristic
99	C₂×C₃×C₈	#41	1	1	yes	characteristic
100	C₂×C₃×C₈	#41	1	1	yes	characteristic
101	C₂×C₃×C₈	#41	1	1	yes	characteristic
102	C₃×C₄²	—	2	1	no	—
103	C₃×C₄²	—	2	1	no	—
104	(C₂×D₃)⋊C₄	—	2	2	no	—
105	(C₂×D₃)⋊C₄	C₁₆	1	4	yes	complemented
106	C₄×Dic₃	—	2	1	no	—
107	C₄×Dic₃	—	2	1	no	—
108	C₂×C₃⋊C₈	—	2	1	no	—
109	C₂×C₃⋊C₈	—	2	1	no	—
110	C₂×C₃⋊C₈	—	2	1	no	—
111	C₂×C₃⋊C₈	—	2	1	no	—
112	C₂×C₃⋊C₈	—	2	1	no	—
113	C₂×C₃⋊C₈	M₁₆	1	2	yes	—
114	D₃×C₈	—	4	1	no	—
115	D₃×C₈	—	4	1	no	—
116	D₃×C₈	—	4	1	no	—
117	D₃×C₈	—	4	1	no	—
118	C₃×C₁₆	—	2	2	no	—
119	C₃×C₁₆	—	4	1	no	—
120	C₃⋊C₁₆	—	2	2	no	—
121	C₃⋊C₁₆	—	4	1	no	—
122	C₂³×C₄	—	3	1	no	—
123	C₂²×C₈	—	3	1	no	—
124	C₂²×C₈	—	3	1	no	—
125	C₂²×C₈	—	6	1	no	—
126	C₂²×C₈	—	6	1	no	—
127	C₂²×C₈	—	6	1	no	—
128	C₂²×C₈	—	6	1	no	—
129	C₂²×C₈	—	6	1	no	—
130	C₂²×C₈	—	6	1	no	—
131	C₂²×C₈	C₄×D₃	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
133	C₂×C₂²⋊C₄	—	3	1	no	—
134	C₂×C₂²⋊C₄	—	3	1	no	—
135	C₂×C₄²	—	3	1	no	—
136	C₂×C₄²	C₄×D₃	1	1	yes	characteristic
137	C₂×C₁₆	—	2	1	no	—
138	C₂×C₁₆	—	2	1	no	—
139	C₂×C₁₆	—	2	1	no	—
140	C₂×C₁₆	—	2	1	no	—
141	C₂×C₁₆	—	2	1	no	—
142	C₂×C₁₆	—	3	2	no	—
143	C₂×C₁₆	—	6	1	no	—
144	C₂×C₁₆	—	6	1	no	—
145	C₂×C₁₆	—	6	1	no	—
146	C₂×C₁₆	—	6	1	no	—
147	C₂×C₁₆	—	6	1	no	—
148	C₂×C₁₆	#67	1	2	yes	—
149	C₂²⋊C₈	—	6	2	no	—
150	C₄×C₈	—	2	1	no	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	—	6	1	no	—
153	C₄×C₈	—	6	1	no	—
154	#95	—	3	1	no	—
155	#95	—	3	1	no	—
156	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
157	C₂²×D₃	—	2	1	no	—
158	C₂²×D₃	—	2	1	no	—
159	C₂²×D₃	—	2	1	no	—
160	C₂²×D₃	—	2	1	no	—
161	C₂²×D₃	—	2	1	no	—
162	C₂²×D₃	—	2	1	no	—
163	C₂²×D₃	C₂×C₁₆	1	2	yes	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	C₂×C₁₆	1	2	yes	—
170	C₂×C₃×C₄	#117	1	1	yes	characteristic
171	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
172	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
174	C₂×C₃×C₄	#131	1	1	yes	characteristic
175	C₂×C₃×C₄	#131	1	1	yes	characteristic
176	C₂×Dic₃	—	2	1	no	—
177	C₂×Dic₃	—	2	1	no	—
178	C₂×Dic₃	—	2	1	no	—
179	C₂×Dic₃	—	2	1	no	—
180	C₂×Dic₃	—	2	1	no	—
181	C₂×Dic₃	—	2	1	no	—
182	C₂×Dic₃	—	2	1	no	—
183	C₂×Dic₃	C₂×C₁₆	1	2	yes	—
184	C₂×Dic₃	M₃₂	1	2	yes	—
185	C₄×D₃	—	4	1	no	—
186	C₄×D₃	—	4	1	no	—
187	C₄×D₃	—	4	1	no	—
188	C₄×D₃	—	4	1	no	—
189	C₃×C₈	—	4	1	no	—
190	C₃×C₈	#121	1	1	yes	characteristic
191	C₃×C₈	#123	1	1	yes	characteristic
192	C₃×C₈	#124	1	1	yes	characteristic
193	C₃×C₈	#126	1	1	yes	characteristic
194	C₃⋊C₈	—	2	2	no	—
195	C₃⋊C₈	—	4	1	no	—
196	C₂⁴	—	3	1	no	—
197	C₂²×C₄	—	3	1	no	—
198	C₂²×C₄	—	3	1	no	—
199	C₂²×C₄	—	3	1	no	—
200	C₂²×C₄	—	6	1	no	—
201	C₂²×C₄	—	6	1	no	—
202	C₂²×C₄	—	6	1	no	—
203	C₂²×C₄	—	6	1	no	—
204	C₂²×C₄	—	6	1	no	—
205	C₂²×C₄	—	6	1	no	—
206	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
207	C₂²×C₄	D₃×C₈	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	3	2	no	—
213	C₂×C₈	—	6	1	no	—
214	C₂×C₈	—	6	1	no	—
215	C₂×C₈	—	6	1	no	—
216	C₂×C₈	—	6	1	no	—
217	C₂×C₈	—	6	1	no	—
218	C₂×C₈	—	12	1	no	—
219	C₂×C₈	—	12	1	no	—
220	C₂×C₈	—	12	1	no	—
221	C₂×C₈	—	12	1	no	—
222	C₂×C₈	#235	1	1	yes	characteristic
223	C₂×C₈	D₃×D₄	1	1	yes	characteristic
224	C₂×C₈	#237	1	1	yes	characteristic
225	C₂×C₈	#239	1	1	yes	characteristic
226	C₂×C₈	#239	1	1	yes	characteristic
227	C₂×C₈	C₂×#67	1	1	yes	characteristic
228	C₂²⋊C₄	—	3	4	no	—
229	C₂²⋊C₄	—	6	2	no	—
230	C₄²	—	2	1	no	—
231	C₄²	—	2	1	no	—
232	C₄²	—	6	1	no	—
233	C₄²	—	6	1	no	—
234	C₁₆	—	2	2	no	—
235	C₁₆	—	4	1	no	—
236	C₁₆	—	6	2	no	—
237	C₁₆	—	12	1	no	—
238	C₂²×C₃	#431	1	1	yes	characteristic
239	C₂²×C₃	C₈×D₄	1	1	yes	characteristic
240	C₂²×C₃	C₈×D₄	1	1	yes	characteristic
241	C₂²×C₃	C₂²×C₁₆	1	1	yes	characteristic
242	C₂²×C₃	C₂×M₃₂	1	1	yes	characteristic
243	C₂²×C₃	#504	1	1	yes	characteristic
244	C₂²×C₃	#504	1	1	yes	characteristic
245	C₂×D₃	—	2	2	no	—
246	C₂×D₃	—	2	2	no	—
247	C₂×D₃	—	2	2	no	—
248	C₂×D₃	—	4	1	no	—
249	C₂×D₃	—	4	1	no	—
250	C₂×D₃	—	4	1	no	—
251	C₂×D₃	—	4	1	no	—
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	—	4	1	no	—
254	C₃×C₄	#388	1	1	yes	characteristic
255	C₃×C₄	#433	1	1	yes	characteristic
256	C₃×C₄	#435	1	1	yes	characteristic
257	C₃×C₄	#436	1	1	yes	characteristic
258	Dic₃	—	2	2	no	—
259	Dic₃	—	2	2	no	—
260	Dic₃	—	4	1	no	—
261	C₂³	—	3	2	no	—
262	C₂³	—	6	1	no	—
263	C₂³	—	6	1	no	—
264	C₂³	—	6	1	no	—
265	C₂³	—	6	1	no	—
266	C₂³	—	6	1	no	—
267	C₂³	—	6	1	no	—
268	C₂³	C₂×D₃×C₈	1	1	yes	characteristic
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	3	2	no	—
275	C₂×C₄	—	3	2	no	—
276	C₂×C₄	—	6	1	no	—
277	C₂×C₄	—	6	1	no	—
278	C₂×C₄	—	6	1	no	—
279	C₂×C₄	—	6	1	no	—
280	C₂×C₄	—	6	1	no	—
281	C₂×C₄	—	6	1	no	—
282	C₂×C₄	—	6	1	no	—
283	C₂×C₄	—	12	1	no	—
284	C₂×C₄	—	12	1	no	—
285	C₂×C₄	—	12	1	no	—
286	C₂×C₄	—	12	1	no	—
287	C₂×C₄	D₃×C₁₆	1	2	yes	—
288	C₂×C₄	#892	1	1	yes	characteristic
289	C₂×C₄	#893	1	1	yes	characteristic
290	C₂×C₄	#901	1	1	yes	characteristic
291	C₂×C₄	D₃×M₁₆	1	1	yes	characteristic
292	C₂×C₄	#907	1	1	yes	characteristic
293	C₂×C₄	C₄×#67	1	1	yes	characteristic
294	C₈	—	4	1	no	—
295	C₈	—	6	2	no	—
296	C₈	—	12	1	no	—
297	C₈	#894	1	1	yes	characteristic
298	C₈	#898	1	1	yes	characteristic
299	C₈	#939	1	1	yes	characteristic
300	C₈	#947	1	1	yes	characteristic
301	C₂×C₃	#1923	1	1	yes	characteristic
302	C₂×C₃	#2162	1	1	yes	characteristic
303	C₂×C₃	#2164	1	1	yes	characteristic
304	C₂×C₃	D₄×C₁₆	1	1	yes	characteristic
305	C₂×C₃	D₄×C₁₆	1	1	yes	characteristic
306	C₂×C₃	#2168	1	1	yes	characteristic
307	C₂×C₃	#2169	1	1	yes	characteristic
308	D₃	—	4	2	no	—
309	C₂²	—	6	2	no	—
310	C₂²	—	6	2	no	—
311	C₂²	—	6	2	no	—
312	C₂²	—	12	1	no	—
313	C₂²	—	12	1	no	—
314	C₂²	—	12	1	no	—
315	C₂²	—	12	1	no	—
316	C₂²	#4833	1	1	yes	characteristic
317	C₂²	#4834	1	1	yes	characteristic
318	C₂²	C₂×D₃×C₁₆	1	1	yes	characteristic
319	C₂²	#4901	1	1	yes	characteristic
320	C₂²	D₃×M₃₂	1	1	yes	characteristic
321	C₂²	#4907	1	1	yes	characteristic
322	C₂²	C₈×#67	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	4	1	no	—
325	C₄	—	6	2	no	—
326	C₄	—	6	2	no	—
327	C₄	—	12	1	no	—
328	C₄	#4836	1	1	yes	characteristic
329	C₄	#4840	1	1	yes	characteristic
330	C₄	#4990	1	1	yes	characteristic
331	C₄	#5128	1	1	yes	characteristic
332	C₃	#13120	1	1	yes	characteristic, complemented, Sylow
333	C₂	—	12	2	no	—
334	C₂	#69274	1	1	yes	characteristic
335	C₂	#69276	1	1	yes	characteristic
336	C₂	#69277	1	1	yes	characteristic
337	C₂	#69278	1	1	yes	characteristic
338	C₂	#70449	1	1	yes	characteristic
339	C₂	C₁₆×#67	1	1	yes	characteristic
340	C₂	#71430	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₁₆
Fitting	C₂×C₃×C₄⋊C₁₆	C₂
socle	C₂³×C₃	C₂²×C₈
prime radical	C₂³×D₃	C₂×C₈
cosocle	C₂²×C₃×C₈	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₃
hypocenter	C₃	#13120
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄⋊C₁₆	C₂	semidirect
4. C₂×Dic₃×C₁₆	C₂	semidirect
6. #67804	C₂	semidirect
7. #67657	C₂	semidirect
24. C₂×C₄⋊C₁₆	D₃	semidirect
105. (C₂×D₃)⋊C₄	C₁₆	semidirect
332. C₃	#13120	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⋊C₁₆	2
C₂×(C₂×D₃)⋊C₁₆	2
C₂×Dic₃×C₁₆	2
C₈×(C₂×D₃)⋊C₄	2
#67804	2
#67657	2
#13120	3