Subgroups — #10948755

Order 768.

552 conjugacy classes of subgroups.

The subgroup lattice has 348 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	#71407	C₂	1	2	yes	maximal, complemented
4	C₂×#4502	C₂	1	2	yes	maximal, complemented
5	D₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#69281	C₂	1	1	yes	maximal, characteristic, complemented
7	#69275	C₂	1	1	yes	maximal, characteristic, complemented
8	#69102	C₂	1	2	yes	maximal, complemented
9	#69093	C₂	1	2	yes	maximal, complemented
10	C₄×#798	C₂	1	1	yes	maximal, characteristic, complemented
11	#69048	C₂	1	1	yes	maximal, characteristic, complemented
12	#33484	—	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×#798	C₂²	1	1	yes	characteristic
21	C₂×#798	C₂²	1	1	yes	characteristic
22	#4833	C₄	1	1	yes	characteristic
23	#4830	C₄	1	1	yes	characteristic
24	#67⋊C₈	C₄	1	2	yes	—
25	#4718	C₄	1	2	yes	—
26	C₄×D₃×C₈	C₂²	1	1	yes	characteristic
27	C₈×Dic₆	C₂²	1	1	yes	characteristic
28	C₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
29	C₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
30	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
31	#4549	C₂²	1	2	yes	—
32	(C₂×D₃)⋊C₁₆	C₂²	1	2	yes	—
33	#4502	C₂²	1	2	yes	—
34	#4502	C₂²	1	2	yes	—
35	#4502	C₂²	1	2	yes	—
36	#4502	C₂²	1	1	yes	characteristic
37	#4502	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	#2183	—	3	1	no	—
44	#2183	—	3	1	no	—
45	#2168	—	3	2	no	—
46	#2168	—	3	2	no	—
47	D₄×C₁₆	—	3	1	no	—
48	D₄×C₁₆	D₃	1	1	yes	characteristic, complemented
49	#2163	—	3	1	no	—
50	#2163	—	3	2	no	—
51	C₄×M₃₂	—	3	1	no	—
52	C₂×#346	—	3	2	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	—	2	4	no	—
69	C₂×C₃×C₁₆	C₂³	1	2	yes	—
70	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
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	—	2	4	no	—
85	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
86	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
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	#798	—	2	2	no	—
91	#798	—	2	2	no	—
92	C₄×#41	—	3	1	no	—
93	C₂×M₃₂	—	3	1	no	—
94	C₂×M₃₂	—	3	1	no	—
95	C₂²×C₁₆	—	3	2	no	—
96	C₂²×C₁₆	C₂×D₃	1	2	yes	—
97	C₈×Q₈	—	3	1	no	—
98	C₈×Q₈	—	3	1	no	—
99	C₈×D₄	—	3	2	no	—
100	C₈×D₄	—	3	1	no	—
101	C₈×D₄	—	3	2	no	—
102	C₈×D₄	C₂×D₃	1	1	yes	characteristic
103	#431	—	3	1	no	—
104	#431	—	3	2	no	—
105	C₂×C₄×C₈	—	3	2	no	—
106	C₂×C₄×C₈	—	3	1	no	—
107	C₄⋊C₁₆	—	3	2	no	—
108	C₄⋊C₁₆	—	3	2	no	—
109	C₄⋊C₁₆	—	3	1	no	—
110	C₄⋊C₁₆	C₂×D₃	1	1	yes	characteristic
111	C₂²⋊C₁₆	—	3	2	no	—
112	C₂²⋊C₁₆	—	3	2	no	—
113	C₂²⋊C₁₆	C₂×D₃	1	2	yes	—
114	#346	—	3	1	no	—
115	#346	—	3	2	no	—
116	#346	—	3	2	no	—
117	#346	—	3	2	no	—
118	#346	—	3	1	no	—
119	C₄×C₁₆	—	3	1	no	—
120	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
121	C₂×C₃×D₄	C₂×C₈	1	1	yes	characteristic
122	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
123	C₂×#67	C₂×C₈	1	2	yes	—
124	C₂²×Dic₃	C₂×C₈	1	2	yes	—
125	C₂²×Dic₃	C₂×C₈	1	2	yes	—
126	#237	—	2	4	no	—
127	C₂×C₄×D₃	C₂×C₈	1	1	yes	characteristic
128	C₂×C₄×D₃	C₂×C₈	1	1	yes	characteristic
129	C₂×C₄×D₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×Dic₆	C₂×C₈	1	1	yes	characteristic
131	C₂×C₃×C₈	—	2	2	no	—
132	C₂×C₃×C₈	—	2	2	no	—
133	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
134	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
136	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
137	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
138	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
139	#217	C₂×C₈	1	2	yes	—
140	(C₂×D₃)⋊C₄	C₂×C₈	1	2	yes	—
141	C₄⋊Dic₃	C₂×C₈	1	1	yes	characteristic
142	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
143	Dic₃⋊C₄	C₂×C₈	1	2	yes	—
144	C₄×Dic₃	C₂×C₈	1	1	yes	characteristic
145	C₄×Dic₃	C₂×C₈	1	1	yes	characteristic
146	C₄×Dic₃	C₂×C₈	1	2	yes	—
147	C₄×Dic₃	C₂×C₈	1	2	yes	—
148	C₄×Dic₃	C₂²×C₄	1	1	yes	characteristic
149	C₄×Dic₃	C₂²×C₄	1	2	yes	—
150	C₂×C₃⋊C₈	—	2	4	no	—
151	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
152	C₂×C₃⋊C₈	C₂²×C₄	1	2	yes	—
153	C₂×C₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
154	D₃×C₈	—	2	1	no	—
155	D₃×C₈	—	2	1	no	—
156	D₃×C₈	—	2	2	no	—
157	C₃×C₁₆	—	2	1	no	—
158	C₃×C₁₆	—	2	2	no	—
159	C₃×C₁₆	#41	1	2	yes	complemented
160	C₃⋊C₁₆	—	2	1	no	—
161	C₃⋊C₁₆	—	2	2	no	—
162	C₃⋊C₁₆	#41	1	2	yes	complemented
163	C₂×#41	—	3	1	no	—
164	C₂²×C₈	—	3	2	no	—
165	C₂²×C₈	—	3	1	no	—
166	C₂²×C₈	—	3	1	no	—
167	C₂²×C₈	C₂²×D₃	1	2	yes	—
168	C₄×Q₈	—	3	1	no	—
169	C₄×Q₈	—	3	1	no	—
170	C₄×D₄	—	3	1	no	—
171	C₄×D₄	—	3	2	no	—
172	C₄×D₄	—	3	2	no	—
173	C₄×D₄	C₄×D₃	1	1	yes	characteristic
174	#117	—	3	1	no	—
175	#117	—	3	2	no	—
176	C₂×C₄²	—	3	2	no	—
177	C₂×C₄²	—	3	1	no	—
178	M₃₂	—	6	2	no	—
179	M₃₂	—	6	2	no	—
180	C₂×C₁₆	—	2	4	no	—
181	C₂×C₁₆	—	3	1	no	—
182	C₂×C₁₆	—	3	1	no	—
183	C₂×C₁₆	—	3	2	no	—
184	C₂×C₁₆	—	6	4	no	—
185	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
186	C₂×C₁₆	C₂²×D₃	1	2	yes	—
187	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
188	C₄⋊C₈	—	3	2	no	—
189	C₄⋊C₈	—	3	2	no	—
190	C₄⋊C₈	—	3	1	no	—
191	C₄⋊C₈	C₄×D₃	1	1	yes	characteristic
192	C₂²⋊C₈	—	3	2	no	—
193	C₂²⋊C₈	—	3	2	no	—
194	C₂²⋊C₈	C₄×D₃	1	2	yes	—
195	C₄×C₈	—	3	2	no	—
196	C₄×C₈	—	3	1	no	—
197	C₄×C₈	—	3	2	no	—
198	C₄×C₈	—	3	1	no	—
199	C₄×C₈	—	3	2	no	—
200	C₄×C₈	—	3	1	no	—
201	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
202	C₂³×C₃	C₂²×C₈	1	2	yes	—
203	C₂²×D₃	C₂²×C₈	1	1	yes	characteristic
204	C₃×D₄	M₃₂	1	4	yes	complemented
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
208	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
211	#67	—	2	4	no	—
212	C₂×Dic₃	—	2	4	no	—
213	C₂×Dic₃	—	4	2	no	—
214	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
215	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
216	C₂×Dic₃	C₂²×C₈	1	1	yes	characteristic
217	C₂×Dic₃	C₂²×C₈	1	2	yes	—
218	C₂×Dic₃	C₂²×C₈	1	2	yes	—
219	C₄×D₃	—	2	1	no	—
220	C₄×D₃	—	2	1	no	—
221	C₄×D₃	—	2	2	no	—
222	C₄×D₃	—	4	1	no	—
223	Dic₆	—	2	2	no	—
224	C₃×C₈	—	2	2	no	—
225	C₃×C₈	—	2	1	no	—
226	C₃×C₈	C₂×#41	1	1	yes	characteristic
227	C₃×C₈	C₂×#41	1	1	yes	characteristic
228	C₃⋊C₈	—	2	1	no	—
229	C₃⋊C₈	—	2	2	no	—
230	C₃⋊C₈	#131	1	2	yes	—
231	#41	—	6	4	no	—
232	C₂×Q₈	—	3	1	no	—
233	C₂×D₄	—	3	2	no	—
234	C₂×D₄	D₃×C₈	1	1	yes	characteristic
235	C₂²×C₄	—	3	2	no	—
236	C₂²×C₄	—	3	2	no	—
237	C₂²×C₄	—	3	1	no	—
238	C₂²×C₄	—	3	1	no	—
239	C₂²×C₄	—	3	1	no	—
240	C₂²×C₄	C₂×C₄×D₃	1	2	yes	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	3	1	no	—
244	C₂×C₈	—	3	2	no	—
245	C₂×C₈	—	3	1	no	—
246	C₂×C₈	—	6	2	no	—
247	C₂×C₈	—	6	4	no	—
248	C₂×C₈	—	6	1	no	—
249	C₂×C₈	—	6	1	no	—
250	C₂×C₈	C₂×C₄×D₃	1	2	yes	—
251	C₂×C₈	C₂×C₄×D₃	1	1	yes	characteristic
252	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
253	C₄⋊C₄	—	3	2	no	—
254	C₄⋊C₄	—	3	2	no	—
255	C₄⋊C₄	—	3	1	no	—
256	C₄⋊C₄	D₃×C₈	1	1	yes	characteristic
257	C₂²⋊C₄	—	3	2	no	—
258	C₂²⋊C₄	—	3	2	no	—
259	C₂²⋊C₄	D₃×C₈	1	2	yes	—
260	C₄²	—	3	2	no	—
261	C₄²	—	3	1	no	—
262	C₄²	—	3	1	no	—
263	C₄²	—	3	2	no	—
264	C₄²	—	3	1	no	—
265	C₄²	—	3	2	no	—
266	C₄²	C₂×C₄×D₃	1	1	yes	characteristic
267	C₁₆	—	2	2	no	—
268	C₁₆	—	2	1	no	—
269	C₁₆	—	3	2	no	—
270	C₁₆	—	6	1	no	—
271	C₁₆	—	6	2	no	—
272	C₁₆	#237	1	2	yes	complemented
273	C₂²×C₃	—	2	2	no	—
274	C₂²×C₃	—	2	2	no	—
275	C₂²×C₃	C₂×M₃₂	1	4	yes	—
276	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
277	C₂×D₃	—	2	1	no	—
278	C₂×D₃	—	2	1	no	—
279	C₂×D₃	—	2	1	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	2	2	no	—
282	C₃×C₄	C₂×M₃₂	1	2	yes	—
283	C₃×C₄	C₄×#41	1	1	yes	characteristic
284	C₃×C₄	C₂×#131	1	1	yes	characteristic
285	Dic₃	—	2	2	no	—
286	Dic₃	—	2	1	no	—
287	Dic₃	—	2	1	no	—
288	Dic₃	—	2	2	no	—
289	Dic₃	—	2	1	no	—
290	C₂³	—	3	1	no	—
291	C₂³	C₂×D₃×C₈	1	2	yes	—
292	Q₈	—	6	2	no	—
293	D₄	—	6	4	no	—
294	D₄	#798	1	4	yes	complemented
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	3	1	no	—
298	C₂×C₄	—	3	1	no	—
299	C₂×C₄	—	3	1	no	—
300	C₂×C₄	—	3	2	no	—
301	C₂×C₄	—	3	2	no	—
302	C₂×C₄	—	6	4	no	—
303	C₂×C₄	—	6	2	no	—
304	C₂×C₄	—	6	1	no	—
305	C₂×C₄	—	6	1	no	—
306	C₂×C₄	—	12	1	no	—
307	C₂×C₄	—	12	2	no	—
308	C₂×C₄	C₂×D₃×C₈	1	2	yes	—
309	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
310	C₂×C₄	C₂×D₃×C₈	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	2	no	—
314	C₈	—	3	2	no	—
315	C₈	—	6	2	no	—
316	C₈	—	6	1	no	—
317	C₈	C₂×#237	1	1	yes	characteristic
318	C₈	D₃×#41	1	1	yes	characteristic
319	C₂×C₃	—	2	4	no	—
320	C₂×C₃	C₈×#41	1	1	yes	characteristic
321	C₂×C₃	C₂²×M₃₂	1	1	yes	characteristic
322	C₂×C₃	#3407	1	1	yes	characteristic
323	D₃	—	4	1	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	6	1	no	—
327	C₂²	—	6	1	no	—
328	C₂²	—	6	1	no	—
329	C₂²	C₂×#798	1	4	yes	—
330	C₂²	C₂²×D₃×C₈	1	1	yes	characteristic
331	C₄	—	2	2	no	—
332	C₄	—	2	1	no	—
333	C₄	—	6	1	no	—
334	C₄	—	6	1	no	—
335	C₄	—	6	2	no	—
336	C₄	—	6	1	no	—
337	C₄	—	6	2	no	—
338	C₄	C₂×#798	1	2	yes	—
339	C₄	C₄×#237	1	1	yes	characteristic
340	C₄	D₃×#131	1	1	yes	characteristic
341	C₃	#33484	1	1	yes	characteristic, complemented, Sylow
342	C₂	—	2	4	no	—
343	C₂	—	12	1	no	—
344	C₂	C₈×#237	1	1	yes	characteristic
345	C₂	C₂²×#798	1	1	yes	characteristic
346	C₂	#82012	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×D₄×C₁₆	C₂
prime radical	C₂×#237	C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	D₄×C₁₆	D₃
hypocenter	C₃	#33484
LCS	G ▷ C₂²×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. #71407	C₂	semidirect
4. C₂×#4502	C₂	semidirect
5. D₄×C₃⋊C₁₆	C₂	semidirect
6. #69281	C₂	semidirect
7. #69275	C₂	semidirect
8. #69102	C₂	semidirect
9. #69093	C₂	semidirect
10. C₄×#798	C₂	semidirect
11. #69048	C₂	semidirect
48. D₄×C₁₆	D₃	semidirect
159. C₃×C₁₆	#41	semidirect
162. C₃⋊C₁₆	#41	semidirect
204. C₃×D₄	M₃₂	semidirect
272. C₁₆	#237	semidirect
294. D₄	#798	semidirect
341. C₃	#33484	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
#71407	2
C₂×#4502	2
D₄×C₃⋊C₁₆	2
#69281	2
#69275	2
#69102	2
#69093	2
C₄×#798	2
#69048	2
#33484	3