Subgroups — #10948752

Order 768.

516 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	C₃×#2966	C₂	1	1	yes	maximal, characteristic
2	#70949	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#70858	C₂	1	1	yes	maximal, characteristic, complemented
5	#70858	C₂	1	2	yes	maximal, complemented
6	#70857	C₂	1	2	yes	maximal, complemented
7	D₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#69883	C₂	1	2	yes	maximal, complemented
9	#69881	C₂	1	2	yes	maximal, complemented
10	#69744	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#812	C₂	1	1	yes	maximal, characteristic, complemented
12	#33483	—	3	1	no	maximal, Sylow
13	C₃×#529	C₄	1	1	yes	characteristic
14	C₃×#446	C₄	1	1	yes	characteristic
15	C₃×C₈×Q₈	C₂²	1	1	yes	characteristic
16	C₃×#436	C₄	1	1	yes	characteristic
17	C₃×#435	C₄	1	2	yes	—
18	C₃×C₈×D₄	C₂²	1	1	yes	characteristic
19	C₃×C₈×D₄	C₂²	1	2	yes	—
20	C₃×#433	C₄	1	2	yes	—
21	C₃×#432	C₄	1	1	yes	characteristic
22	C₃×#405	C₂²	1	2	yes	—
23	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
24	C₂×#812	C₂²	1	1	yes	characteristic
25	C₂×#812	C₂²	1	1	yes	characteristic
26	C₂×#812	C₂²	1	2	yes	—
27	C₂²×C₃⋊C₁₆	C₂²	1	2	yes	—
28	#4549	C₂²	1	2	yes	—
29	#4549	C₂²	1	2	yes	—
30	#4549	C₂²	1	2	yes	—
31	C₄⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
32	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
33	C₄⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
34	C₄⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
35	#4461	C₂²	1	2	yes	—
36	#4461	C₂²	1	1	yes	characteristic
37	#4461	C₂²	1	1	yes	characteristic
38	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
39	C₄×C₃⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
41	C₄×C₃⋊C₁₆	C₂²	1	2	yes	—
42	#2966	D₃	1	1	yes	characteristic
43	#2183	—	3	1	no	—
44	Q₈×C₁₆	—	3	1	no	—
45	#2169	—	3	1	no	—
46	#2169	—	3	2	no	—
47	#2168	—	3	2	no	—
48	D₄×C₁₆	—	3	1	no	—
49	#2164	—	3	2	no	—
50	#2162	—	3	1	no	—
51	#2108	—	3	2	no	—
52	C₄×M₃₂	—	3	1	no	—
53	C₂×C₃×M₁₆	C₂×C₄	1	2	yes	—
54	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
56	C₂²×C₃×C₈	C₂³	1	2	yes	—
57	C₃×#126	C₈	1	2	yes	—
58	C₃×#125	C₈	1	1	yes	characteristic
59	C₃×#124	C₈	1	1	yes	characteristic
60	C₃×#123	C₈	1	2	yes	—
61	C₃×C₂²⋊Q₈	C₈	1	1	yes	characteristic
62	C₃×#121	C₈	1	1	yes	characteristic
63	C₃×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
64	C₃×C₄×D₄	C₂×C₄	1	2	yes	—
65	C₃×C₄×D₄	C₂×C₄	1	1	yes	characteristic
66	C₃×#117	C₂×C₄	1	2	yes	—
67	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
68	C₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
69	C₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
70	C₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
71	C₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
72	C₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
73	C₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
74	C₃×C₂²⋊C₈	C₂×C₄	1	2	yes	—
75	C₃×#97	C₂×C₄	1	2	yes	—
76	C₃×#97	C₂×C₄	1	2	yes	—
77	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
78	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
79	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
80	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
81	C₃×C₄×C₈	C₂³	1	2	yes	—
82	#812	—	2	2	no	—
83	#812	—	2	2	no	—
84	#812	—	2	4	no	—
85	C₂×C₃⋊C₁₆	—	2	4	no	—
86	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
87	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
88	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
89	C₂×C₃⋊C₁₆	C₂³	1	2	yes	—
90	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
91	C₂×C₃⋊C₁₆	C₂³	1	1	yes	characteristic
92	#529	Dic₃	1	1	yes	characteristic
93	C₂×M₃₂	—	3	1	no	—
94	C₂×M₃₂	—	3	1	no	—
95	C₂×M₃₂	—	3	2	no	—
96	C₂²×C₁₆	—	3	2	no	—
97	#446	Dic₃	1	1	yes	characteristic
98	C₈×Q₈	C₂×D₃	1	1	yes	characteristic
99	#436	Dic₃	1	1	yes	characteristic
100	#435	Dic₃	1	2	yes	—
101	C₈×D₄	C₂×D₃	1	1	yes	characteristic
102	C₈×D₄	C₂×D₃	1	2	yes	—
103	#433	Dic₃	1	2	yes	—
104	#432	Dic₃	1	1	yes	characteristic
105	#405	C₂×D₃	1	2	yes	—
106	C₂×C₄×C₈	C₂×D₃	1	1	yes	characteristic
107	C₄⋊C₁₆	—	3	1	no	—
108	C₄⋊C₁₆	—	3	1	no	—
109	C₄⋊C₁₆	—	3	2	no	—
110	C₄⋊C₁₆	—	3	2	no	—
111	C₂²⋊C₁₆	—	3	2	no	—
112	C₂²⋊C₁₆	—	3	2	no	—
113	C₂²⋊C₁₆	—	3	2	no	—
114	#346	—	3	1	no	—
115	#346	—	3	1	no	—
116	#346	—	3	2	no	—
117	C₄×C₁₆	—	3	1	no	—
118	C₄×C₁₆	—	3	1	no	—
119	C₄×C₁₆	—	3	2	no	—
120	C₄×C₁₆	—	3	2	no	—
121	C₂×C₃×Q₈	C₂×C₈	1	1	yes	characteristic
122	C₂×C₃×D₄	C₂×C₈	1	1	yes	characteristic
123	C₂×C₃×D₄	C₂×C₈	1	2	yes	—
124	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₂×C₈	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
127	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
128	C₃×M₁₆	—	2	4	no	—
129	C₂×C₃×C₈	—	2	2	no	—
130	C₂×C₃×C₈	—	2	2	no	—
131	C₂×C₃×C₈	—	2	1	no	—
132	C₂×C₃×C₈	—	2	1	no	—
133	C₂×C₃×C₈	—	2	2	no	—
134	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
136	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
137	C₂×C₃×C₈	C₂²×C₄	1	2	yes	—
138	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
140	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
141	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
142	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
143	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
144	C₃×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
145	C₃×C₄⋊C₄	C₂×C₈	1	2	yes	—
146	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
147	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
148	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
149	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
150	C₃×C₂²⋊C₄	C₂×C₈	1	2	yes	—
151	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
152	C₃×C₄²	C₂×C₈	1	1	yes	characteristic
153	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
154	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
155	C₃×C₄²	C₂²×C₄	1	2	yes	—
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	#41	1	2	yes	—
161	C₃⋊C₁₆	#41	1	2	yes	—
162	C₂×M₁₆	C₂×Dic₃	1	2	yes	—
163	C₂²×C₈	C₂×Dic₃	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²×D₃	1	2	yes	—
166	#126	C₃⋊C₈	1	2	yes	—
167	#125	C₃⋊C₈	1	1	yes	characteristic
168	#124	C₃⋊C₈	1	1	yes	characteristic
169	#123	C₃⋊C₈	1	2	yes	—
170	C₂²⋊Q₈	C₃⋊C₈	1	1	yes	characteristic
171	#121	C₃⋊C₈	1	1	yes	characteristic
172	C₄×Q₈	C₂×Dic₃	1	1	yes	characteristic
173	C₄×D₄	C₂×Dic₃	1	2	yes	—
174	C₄×D₄	C₂×Dic₃	1	1	yes	characteristic
175	#117	C₂×Dic₃	1	2	yes	—
176	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
177	M₃₂	—	6	2	no	—
178	M₃₂	—	6	2	no	—
179	M₃₂	—	6	4	no	—
180	C₂×C₁₆	—	3	1	no	—
181	C₂×C₁₆	—	3	1	no	—
182	C₂×C₁₆	—	3	2	no	—
183	C₂×C₁₆	—	3	1	no	—
184	C₂×C₁₆	—	3	1	no	—
185	C₂×C₁₆	—	3	2	no	—
186	C₂×C₁₆	—	6	4	no	—
187	C₄⋊C₈	C₂×Dic₃	1	1	yes	characteristic
188	C₄⋊C₈	C₂×Dic₃	1	1	yes	characteristic
189	C₄⋊C₈	C₂×Dic₃	1	2	yes	—
190	C₄⋊C₈	C₂×Dic₃	1	2	yes	—
191	C₂²⋊C₈	C₂×Dic₃	1	2	yes	—
192	C₂²⋊C₈	C₂×Dic₃	1	2	yes	—
193	C₂²⋊C₈	C₂×Dic₃	1	2	yes	—
194	#97	C₂×Dic₃	1	2	yes	—
195	#97	C₂×Dic₃	1	2	yes	—
196	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
197	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
198	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
199	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
200	C₄×C₈	C₂²×D₃	1	2	yes	—
201	C₂³×C₃	C₂²×C₈	1	2	yes	—
202	C₂³×C₃	C₂²×C₈	1	1	yes	characteristic
203	C₃×Q₈	—	2	2	no	—
204	C₃×D₄	—	2	2	no	—
205	C₃×D₄	—	4	2	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	4	1	no	—
212	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
217	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
218	C₂×C₃×C₄	C₂²×C₈	1	2	yes	—
219	C₂×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
221	C₃×C₈	—	2	2	no	—
222	C₃×C₈	—	2	1	no	—
223	C₃×C₈	—	2	1	no	—
224	C₃×C₈	—	2	2	no	—
225	C₃×C₈	#131	1	2	yes	—
226	C₃×C₈	C₂×#41	1	1	yes	characteristic
227	C₃×C₈	C₂×#41	1	1	yes	characteristic
228	C₂×Q₈	C₂×C₃⋊C₈	1	1	yes	characteristic
229	C₂×D₄	C₂×C₃⋊C₈	1	1	yes	characteristic
230	C₂×D₄	C₂×C₃⋊C₈	1	2	yes	—
231	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
233	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
234	C₂²×C₄	C₂²×Dic₃	1	2	yes	—
235	M₁₆	—	2	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
242	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
243	C₂×C₈	C₂²×Dic₃	1	2	yes	—
244	C₂×C₈	C₂²×Dic₃	1	2	yes	—
245	C₂×C₈	C₂²×Dic₃	1	1	yes	characteristic
246	C₂×C₈	C₂³×D₃	1	1	yes	characteristic
247	C₄⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
248	C₄⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
249	C₄⋊C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
250	C₄⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
251	C₄⋊C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
252	C₄⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
253	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
254	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
255	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
256	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
257	C₂²⋊C₄	C₂×C₃⋊C₈	1	2	yes	—
258	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
259	C₄²	C₂×C₃⋊C₈	1	1	yes	characteristic
260	C₄²	C₂²×Dic₃	1	1	yes	characteristic
261	C₄²	C₂²×Dic₃	1	1	yes	characteristic
262	C₄²	C₂²×Dic₃	1	2	yes	—
263	C₁₆	—	3	2	no	—
264	C₁₆	—	3	2	no	—
265	C₁₆	—	6	2	no	—
266	C₁₆	—	6	1	no	—
267	C₁₆	—	6	1	no	—
268	C₁₆	—	6	2	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	—	2	2	no	—
271	C₂²×C₃	—	2	1	no	—
272	C₂²×C₃	—	2	2	no	—
273	C₂²×C₃	—	2	1	no	—
274	C₂²×C₃	—	2	2	no	—
275	C₂²×C₃	C₂³×C₈	1	1	yes	characteristic
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	2	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	2	2	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	2	no	—
283	C₃×C₄	#504	1	2	yes	—
284	C₃×C₄	C₄×#41	1	1	yes	characteristic
285	C₃×C₄	C₂×#131	1	1	yes	characteristic
286	C₂³	C₂²×C₃⋊C₈	1	2	yes	—
287	C₂³	C₂²×C₃⋊C₈	1	1	yes	characteristic
288	Q₈	—	2	2	no	—
289	D₄	—	2	2	no	—
290	D₄	—	4	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
298	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
302	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
303	C₂×C₄	C₂²×C₃⋊C₈	1	2	yes	—
304	C₂×C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
305	C₂×C₄	C₂³×Dic₃	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	2	2	no	—
308	C₈	—	2	2	no	—
309	C₈	—	2	1	no	—
310	C₈	#948	1	2	yes	—
311	C₈	D₃×#41	1	1	yes	characteristic
312	C₈	D₃×#41	1	1	yes	characteristic
313	C₂×C₃	—	4	1	no	—
314	C₂×C₃	—	4	2	no	—
315	C₂×C₃	C₈×#41	1	1	yes	characteristic
316	C₂×C₃	C₂×#504	1	1	yes	characteristic
317	C₂×C₃	#3407	1	1	yes	characteristic
318	C₂²	—	2	1	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	2	no	—
324	C₂²	C₂³×C₃⋊C₈	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	2	no	—
332	C₄	#5140	1	2	yes	—
333	C₄	C₂×#948	1	1	yes	characteristic
334	C₄	Dic₃×#41	1	1	yes	characteristic
335	C₃	#33483	1	1	yes	characteristic, complemented, Sylow
336	C₂	—	4	1	no	—
337	C₂	—	4	2	no	—
338	C₂	#41×C₃⋊C₈	1	1	yes	characteristic
339	C₂	C₂×#5140	1	1	yes	characteristic
340	C₂	#83780	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₈
Fitting	C₃×#2966	C₂
prime radical	C₃×#121	C₈
cosocle	C₂×C₃×C₈	C₂⁴
hypercenter	#2966	D₃
hypocenter	C₃	#33483
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #2966

Complements

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

N	H ≅ G/N	type
2. #70949	C₂	semidirect
3. Q₈×C₃⋊C₁₆	C₂	semidirect
4. #70858	C₂	semidirect
5. #70858	C₂	semidirect
6. #70857	C₂	semidirect
7. D₄×C₃⋊C₁₆	C₂	semidirect
8. #69883	C₂	semidirect
9. #69881	C₂	semidirect
10. #69744	C₂	semidirect
11. C₄×#812	C₂	semidirect
335. C₃	#33483	semidirect

Maximal subgroups

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

subgroup	index
C₃×#2966	2
#70949	2
Q₈×C₃⋊C₁₆	2
#70858	2
#70857	2
D₄×C₃⋊C₁₆	2
#69883	2
#69881	2
#69744	2
C₄×#812	2
#33483	3