Subgroups — #10948854

Order 768.

570 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#237	C₂	1	1	yes	maximal, characteristic
2	C₃×D₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₆×#67	C₂	1	2	yes	maximal, complemented
4	C₂×Dic₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	D₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	D₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#69280	C₂	1	1	yes	maximal, characteristic, complemented
8	#67⋊C₁₆	C₂	1	2	yes	maximal, complemented
9	#69092	C₂	1	2	yes	maximal, complemented
10	#69052	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₆×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#33502	—	3	1	no	maximal, Sylow
13	C₄×#237	C₄	1	1	yes	characteristic
14	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
15	C₃×C₈×D₄	C₂²	1	1	yes	characteristic
16	C₈×#67	C₂²	1	2	yes	—
17	C₂×C₈×Dic₃	C₂²	1	2	yes	—
18	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
19	D₄×C₃⋊C₈	C₄	1	1	yes	characteristic
20	C₂×D₃×C₁₆	C₂²	1	2	yes	—
21	#4833	C₄	1	1	yes	characteristic
22	#4830	C₄	1	1	yes	characteristic
23	#67⋊C₈	C₄	1	2	yes	—
24	#4718	C₄	1	2	yes	—
25	C₄×D₃×C₈	C₂²	1	1	yes	characteristic
26	C₈×Dic₆	C₂²	1	1	yes	characteristic
27	C₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
29	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
30	#4549	C₂²	1	2	yes	—
31	(C₂×D₃)⋊C₁₆	C₂²	1	1	yes	characteristic
32	(C₂×D₃)⋊C₁₆	C₂²	1	1	yes	characteristic
33	Dic₃⋊C₁₆	C₂²	1	2	yes	—
34	Dic₃⋊C₁₆	C₂²	1	2	yes	—
35	Dic₃⋊C₁₆	C₂²	1	2	yes	—
36	Dic₃⋊C₁₆	C₂²	1	1	yes	characteristic
37	Dic₃⋊C₁₆	C₂²	1	1	yes	characteristic
38	Dic₃×C₁₆	C₂²	1	2	yes	—
39	Dic₃×C₁₆	C₂²	1	2	yes	—
40	C₄⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
41	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
42	C₈×#41	—	3	1	no	—
43	Q₈×C₁₆	—	3	1	no	—
44	Q₈×C₁₆	—	3	1	no	—
45	D₄×C₁₆	—	3	2	no	—
46	D₄×C₁₆	—	3	1	no	—
47	D₄×C₁₆	—	3	2	no	—
48	D₄×C₁₆	D₃	1	1	yes	characteristic, complemented
49	#2162	—	3	2	no	—
50	#2162	—	3	1	no	—
51	C₂×C₄⋊C₁₆	—	3	2	no	—
52	C₂×C₄⋊C₁₆	—	3	1	no	—
53	C₂×#237	C₈	1	1	yes	characteristic
54	C₂²×C₃×C₈	C₂³	1	2	yes	—
55	C₃×C₄×D₄	C₂×C₄	1	1	yes	characteristic
56	D₄×Dic₃	C₈	1	1	yes	characteristic
57	C₄×#67	C₂×C₄	1	2	yes	—
58	C₂×C₄×Dic₃	C₂×C₄	1	2	yes	—
59	C₂²×C₃⋊C₈	C₂×C₄	1	2	yes	—
60	C₂×D₃×C₈	C₂×C₄	1	1	yes	characteristic
61	C₂×D₃×C₈	C₂³	1	1	yes	characteristic
62	#892	C₈	1	1	yes	characteristic
63	#887	C₈	1	1	yes	characteristic
64	#67⋊C₄	C₈	1	2	yes	—
65	#877	C₈	1	2	yes	—
66	C₄²×D₃	C₂×C₄	1	1	yes	characteristic
67	C₄×Dic₆	C₂×C₄	1	1	yes	characteristic
68	C₂×C₃×C₁₆	—	4	2	no	—
69	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
70	C₂×C₃×C₁₆	C₂³	1	2	yes	—
71	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
72	C₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
73	C₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
74	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
75	#830	C₂×C₄	1	2	yes	—
76	(C₂×D₃)⋊C₈	C₂×C₄	1	2	yes	—
77	Dic₃⋊C₈	C₂×C₄	1	2	yes	—
78	Dic₃⋊C₈	C₂×C₄	1	2	yes	—
79	C₈×Dic₃	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₃	C₂×C₄	1	2	yes	—
81	C₈×Dic₃	C₂×C₄	1	2	yes	—
82	C₈×Dic₃	C₂³	1	2	yes	—
83	C₈×Dic₃	C₂³	1	1	yes	characteristic
84	C₂×C₃⋊C₁₆	—	4	2	no	—
85	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
86	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
87	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
88	C₄⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
89	C₄×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
90	D₃×C₁₆	—	4	2	no	—
91	C₄×#41	—	3	1	no	—
92	C₂²×C₁₆	—	3	2	no	—
93	C₂²×C₁₆	—	3	2	no	—
94	C₂²×C₁₆	C₂×D₃	1	2	yes	—
95	C₈×Q₈	—	3	1	no	—
96	C₈×Q₈	—	3	1	no	—
97	C₈×D₄	—	3	2	no	—
98	C₈×D₄	—	3	1	no	—
99	C₈×D₄	—	3	2	no	—
100	C₈×D₄	C₂×D₃	1	1	yes	characteristic
101	#431	—	3	1	no	—
102	#431	—	3	2	no	—
103	C₂×C₄×C₈	—	3	2	no	—
104	C₂×C₄×C₈	—	3	1	no	—
105	C₄⋊C₁₆	—	3	2	no	—
106	C₄⋊C₁₆	—	3	1	no	—
107	C₄⋊C₁₆	—	3	2	no	—
108	C₄⋊C₁₆	—	3	2	no	—
109	C₄⋊C₁₆	—	3	1	no	—
110	C₄⋊C₁₆	—	3	1	no	—
111	C₄⋊C₁₆	C₂×D₃	1	1	yes	characteristic
112	C₂²⋊C₁₆	—	3	2	no	—
113	C₂²⋊C₁₆	—	3	1	no	—
114	C₂²⋊C₁₆	—	3	1	no	—
115	C₂²⋊C₁₆	C₂×D₃	1	2	yes	—
116	C₄×C₁₆	—	3	2	no	—
117	C₄×C₁₆	—	3	2	no	—
118	C₄×C₁₆	—	3	1	no	—
119	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
120	C₂×C₃×D₄	C₂×C₈	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
122	C₂×#67	C₂×C₈	1	2	yes	—
123	C₂²×Dic₃	C₂×C₈	1	2	yes	—
124	C₂²×Dic₃	C₂×C₈	1	2	yes	—
125	#237	C₁₆	1	8	yes	complemented
126	C₂×C₄×D₃	C₂×C₈	1	1	yes	characteristic
127	C₂×C₄×D₃	C₂×C₈	1	1	yes	characteristic
128	C₂×C₄×D₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₆	C₂×C₈	1	1	yes	characteristic
130	C₂×C₃×C₈	—	2	2	no	—
131	C₂×C₃×C₈	—	2	2	no	—
132	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
133	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
135	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
136	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
137	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
138	#217	C₂×C₈	1	2	yes	—
139	(C₂×D₃)⋊C₄	C₂×C₈	1	2	yes	—
140	C₄⋊Dic₃	C₂×C₈	1	1	yes	characteristic
141	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
142	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
143	C₄×Dic₃	C₂×C₈	1	1	yes	characteristic
144	C₄×Dic₃	C₂×C₈	1	1	yes	characteristic
145	C₄×Dic₃	C₂×C₈	1	2	yes	—
146	C₄×Dic₃	C₂×C₈	1	2	yes	—
147	C₄×Dic₃	C₂²×C₄	1	1	yes	characteristic
148	C₄×Dic₃	C₂²×C₄	1	2	yes	—
149	C₂×C₃⋊C₈	—	4	2	no	—
150	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
151	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
152	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
153	D₃×C₈	—	2	1	no	—
154	D₃×C₈	—	2	1	no	—
155	D₃×C₈	—	4	1	no	—
156	C₃×C₁₆	—	2	2	no	—
157	C₃×C₁₆	—	2	1	no	—
158	C₃×C₁₆	—	2	1	no	—
159	C₃⋊C₁₆	—	2	2	no	—
160	C₃⋊C₁₆	—	2	1	no	—
161	C₃⋊C₁₆	—	2	1	no	—
162	C₂×#41	—	3	1	no	—
163	C₂²×C₈	—	3	2	no	—
164	C₂²×C₈	—	3	1	no	—
165	C₂²×C₈	—	3	1	no	—
166	C₂²×C₈	C₂²×D₃	1	2	yes	—
167	C₄×Q₈	—	3	1	no	—
168	C₄×Q₈	—	3	1	no	—
169	C₄×D₄	—	3	1	no	—
170	C₄×D₄	—	3	2	no	—
171	C₄×D₄	—	3	2	no	—
172	C₄×D₄	C₄×D₃	1	1	yes	characteristic
173	#117	—	3	1	no	—
174	#117	—	3	2	no	—
175	C₂×C₄²	—	3	2	no	—
176	C₂×C₄²	—	3	1	no	—
177	C₂×C₁₆	—	3	1	no	—
178	C₂×C₁₆	—	3	2	no	—
179	C₂×C₁₆	—	3	1	no	—
180	C₂×C₁₆	—	4	2	no	—
181	C₂×C₁₆	—	12	2	no	—
182	C₂×C₁₆	—	12	2	no	—
183	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
184	C₂×C₁₆	C₂²×D₃	1	2	yes	—
185	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
186	C₄⋊C₈	—	3	2	no	—
187	C₄⋊C₈	—	3	2	no	—
188	C₄⋊C₈	—	3	1	no	—
189	C₄⋊C₈	C₄×D₃	1	1	yes	characteristic
190	C₂²⋊C₈	—	3	2	no	—
191	C₂²⋊C₈	—	3	2	no	—
192	C₂²⋊C₈	C₄×D₃	1	2	yes	—
193	C₄×C₈	—	3	2	no	—
194	C₄×C₈	—	3	1	no	—
195	C₄×C₈	—	3	2	no	—
196	C₄×C₈	—	3	2	no	—
197	C₄×C₈	—	3	1	no	—
198	C₄×C₈	—	3	1	no	—
199	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
200	C₂³×C₃	C₂²×C₈	1	2	yes	—
201	C₂²×D₃	C₂²×C₈	1	1	yes	characteristic
202	C₃×D₄	C₂×C₁₆	1	4	yes	complemented
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
206	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
209	#67	C₂×C₁₆	1	8	yes	complemented
210	C₂×Dic₃	—	4	2	no	—
211	C₂×Dic₃	C₂×C₁₆	1	8	yes	complemented
212	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
213	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
214	C₂×Dic₃	C₂²×C₈	1	2	yes	—
215	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
216	C₂×Dic₃	C₂²×C₈	1	2	yes	—
217	C₄×D₃	—	2	1	no	—
218	C₄×D₃	—	2	1	no	—
219	C₄×D₃	—	4	1	no	—
220	C₄×D₃	C₂×C₁₆	1	4	yes	complemented
221	Dic₆	C₂×C₁₆	1	4	yes	complemented
222	C₃×C₈	—	2	2	no	—
223	C₃×C₈	—	2	1	no	—
224	C₃×C₈	H(5,𝔽₂)	1	1	yes	characteristic
225	C₃×C₈	2₋¹⁺⁴	1	1	yes	characteristic
226	C₃⋊C₈	—	2	1	no	—
227	C₃⋊C₈	—	2	1	no	—
228	C₃⋊C₈	—	2	2	no	—
229	#41	—	3	8	no	—
230	C₂×Q₈	—	3	1	no	—
231	C₂×D₄	—	3	2	no	—
232	C₂×D₄	D₃×C₈	1	1	yes	characteristic
233	C₂²×C₄	—	3	2	no	—
234	C₂²×C₄	—	3	1	no	—
235	C₂²×C₄	—	3	2	no	—
236	C₂²×C₄	—	3	1	no	—
237	C₂²×C₄	—	3	1	no	—
238	C₂²×C₄	C₂×C₄×D₃	1	2	yes	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	3	1	no	—
242	C₂×C₈	—	3	2	no	—
243	C₂×C₈	—	3	1	no	—
244	C₂×C₈	—	6	1	no	—
245	C₂×C₈	—	6	1	no	—
246	C₂×C₈	—	12	1	no	—
247	C₂×C₈	—	12	2	no	—
248	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
249	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
250	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
251	C₄⋊C₄	—	3	2	no	—
252	C₄⋊C₄	—	3	2	no	—
253	C₄⋊C₄	—	3	1	no	—
254	C₄⋊C₄	D₃×C₈	1	1	yes	characteristic
255	C₂²⋊C₄	—	3	2	no	—
256	C₂²⋊C₄	—	3	2	no	—
257	C₂²⋊C₄	D₃×C₈	1	2	yes	—
258	C₄²	—	3	2	no	—
259	C₄²	—	3	1	no	—
260	C₄²	—	3	2	no	—
261	C₄²	—	3	1	no	—
262	C₄²	—	3	1	no	—
263	C₄²	—	3	2	no	—
264	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
265	C₁₆	—	2	1	no	—
266	C₁₆	—	2	2	no	—
267	C₁₆	—	2	1	no	—
268	C₁₆	—	6	2	no	—
269	C₁₆	—	6	1	no	—
270	C₁₆	—	6	1	no	—
271	C₂²×C₃	—	2	2	no	—
272	C₂²×C₃	—	2	2	no	—
273	C₂²×C₃	C₂²×C₁₆	1	4	yes	—
274	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
275	C₂×D₃	—	2	1	no	—
276	C₂×D₃	—	2	1	no	—
277	C₂×D₃	C₂²×C₁₆	1	2	yes	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	2	no	—
280	C₃×C₄	C₂²×C₁₆	1	2	yes	—
281	C₃×C₄	#520	1	1	yes	characteristic
282	C₃×C₄	#568	1	1	yes	characteristic
283	Dic₃	—	2	1	no	—
284	Dic₃	—	2	2	no	—
285	Dic₃	—	2	1	no	—
286	Dic₃	C₂²×C₁₆	1	4	yes	—
287	Dic₃	C₂²×C₁₆	1	2	yes	—
288	C₂³	—	3	1	no	—
289	C₂³	C₂×D₃×C₈	1	2	yes	—
290	Q₈	—	3	4	no	—
291	D₄	—	3	8	no	—
292	D₄	D₃×C₁₆	1	4	yes	complemented
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	3	8	no	—
296	C₂×C₄	—	3	1	no	—
297	C₂×C₄	—	3	2	no	—
298	C₂×C₄	—	3	4	no	—
299	C₂×C₄	—	3	1	no	—
300	C₂×C₄	—	3	1	no	—
301	C₂×C₄	—	3	2	no	—
302	C₂×C₄	—	6	1	no	—
303	C₂×C₄	—	6	1	no	—
304	C₂×C₄	—	12	1	no	—
305	C₂×C₄	—	12	2	no	—
306	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
307	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
308	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	2	no	—
312	C₈	—	6	1	no	—
313	C₈	—	6	2	no	—
314	C₈	—	6	1	no	—
315	C₈	#1004	1	1	yes	characteristic
316	C₈	#1010	1	1	yes	characteristic
317	C₂×C₃	—	2	4	no	—
318	C₂×C₃	#2988	1	1	yes	characteristic
319	C₂×C₃	C₂³×C₁₆	1	1	yes	characteristic
320	C₂×C₃	#3407	1	1	yes	characteristic
321	D₃	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	3	2	no	—
325	C₂²	—	6	1	no	—
326	C₂²	—	6	1	no	—
327	C₂²	C₂×D₃×C₁₆	1	4	yes	—
328	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	2	1	no	—
331	C₄	—	3	2	no	—
332	C₄	—	3	4	no	—
333	C₄	—	6	1	no	—
334	C₄	—	6	1	no	—
335	C₄	—	6	2	no	—
336	C₄	C₂×D₃×C₁₆	1	2	yes	—
337	C₄	#5546	1	1	yes	characteristic
338	C₄	#5750	1	1	yes	characteristic
339	C₃	#33502	1	1	yes	characteristic, complemented, Sylow
340	C₂	—	2	4	no	—
341	C₂	—	6	2	no	—
342	C₂	#79405	1	1	yes	characteristic
343	C₂	C₂²×D₃×C₁₆	1	1	yes	characteristic
344	C₂	#82012	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator	C₂×C₃	C₂³×C₁₆
Fitting	C₃×D₄×C₁₆	C₂
socle	C₂²×C₃	C₂³×C₈
prime radical	C₂×#237	C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	D₄×C₁₆	D₃
hypocenter	C₃	#33502
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×D₄×C₁₆	C₂	semidirect
3. C₁₆×#67	C₂	semidirect
4. C₂×Dic₃⋊C₁₆	C₂	semidirect
5. D₄×C₃⋊C₁₆	C₂	semidirect
6. D₃×C₄⋊C₁₆	C₂	semidirect
7. #69280	C₂	semidirect
8. #67⋊C₁₆	C₂	semidirect
9. #69092	C₂	semidirect
10. #69052	C₂	semidirect
11. C₁₆×Dic₆	C₂	semidirect
48. D₄×C₁₆	D₃	semidirect
125. #237	C₁₆	semidirect
202. C₃×D₄	C₂×C₁₆	semidirect
209. #67	C₂×C₁₆	semidirect
211. C₂×Dic₃	C₂×C₁₆	semidirect
220. C₄×D₃	C₂×C₁₆	semidirect
221. Dic₆	C₂×C₁₆	semidirect
292. D₄	D₃×C₁₆	semidirect
339. C₃	#33502	semidirect

Maximal subgroups

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

subgroup	index
C₈×#237	2
C₃×D₄×C₁₆	2
C₁₆×#67	2
C₂×Dic₃⋊C₁₆	2
D₄×C₃⋊C₁₆	2
D₃×C₄⋊C₁₆	2
#69280	2
#67⋊C₁₆	2
#69092	2
#69052	2
C₁₆×Dic₆	2
#33502	3