Subgroups — #147155

Order 512.

556 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2150	C₂	1	1	yes	maximal, characteristic
2	C₂²×#347	C₂	1	1	yes	maximal, characteristic
3	C₂×#1375	C₂	1	1	yes	maximal, characteristic
4	C₂×#1375	C₂	1	1	yes	maximal, characteristic
5	C₂×#1375	C₂	1	1	yes	maximal, characteristic
6	#12544	C₂	1	1	yes	maximal, characteristic
7	#10696	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×M₃₂	C₂²	1	1	yes	characteristic
12	C₂²×#97	C₂²	1	1	yes	characteristic
13	C₂²×#97	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
15	#2150	—	2	2	no	—
16	#2150	—	2	2	no	—
17	C₂×C₄⋊C₁₆	C₄	1	4	yes	complemented
18	C₂×#347	—	2	2	no	—
19	C₂×#347	—	2	2	no	—
20	C₂×#347	—	2	2	no	—
21	C₂×#336	C₄	1	1	yes	characteristic
22	C₂×#336	C₄	1	1	yes	characteristic
23	C₂×#336	C₄	1	1	yes	characteristic
24	C₄×#95	C₄	1	1	yes	characteristic
25	#1375	—	2	2	no	—
26	#1375	—	2	2	no	—
27	#1375	—	2	2	no	—
28	#1374	C₄	1	4	yes	complemented
29	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₈	C₂³	1	1	yes	characteristic
33	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₂×M₃₂	—	2	1	no	—
37	C₂×M₃₂	—	2	1	no	—
38	C₂×M₃₂	—	2	1	no	—
39	C₂×M₃₂	—	2	1	no	—
40	C₂×M₃₂	—	2	1	no	—
41	C₂×M₃₂	—	2	1	no	—
42	C₂×M₃₂	—	2	1	no	—
43	C₂×M₃₂	—	2	1	no	—
44	C₂×M₃₂	—	2	1	no	—
45	C₂×M₃₂	—	2	1	no	—
46	C₂×M₃₂	—	2	1	no	—
47	C₂×M₃₂	—	2	1	no	—
48	C₂×M₃₂	—	2	1	no	—
49	C₂×M₃₂	—	2	1	no	—
50	C₂×M₃₂	—	2	1	no	—
51	C₂×M₃₂	—	2	1	no	—
52	C₂×M₃₂	—	2	1	no	—
53	C₂×M₃₂	—	2	1	no	—
54	C₂×M₃₂	—	2	1	no	—
55	C₂×M₃₂	—	2	1	no	—
56	C₂×M₃₂	D₄	1	2	yes	—
57	C₂×M₃₂	D₄	1	2	yes	—
58	C₂×M₃₂	D₄	1	2	yes	—
59	C₂×M₃₂	Q₈	1	2	yes	—
60	C₂²×C₁₆	C₂×C₄	1	2	yes	—
61	C₂²×C₁₆	C₂×C₄	1	2	yes	—
62	C₂²×C₁₆	C₂×C₄	1	2	yes	—
63	C₂²×C₁₆	C₂×C₄	1	2	yes	—
64	C₂×#97	—	2	1	no	—
65	C₂×#97	—	2	1	no	—
66	C₂×#97	—	2	1	no	—
67	C₂×#97	—	2	1	no	—
68	C₂×#97	—	2	1	no	—
69	C₂×#97	—	2	1	no	—
70	C₂×#97	—	2	1	no	—
71	C₂×#97	—	2	1	no	—
72	C₂×#97	—	2	1	no	—
73	C₂×#97	—	2	1	no	—
74	C₂×#97	D₄	1	2	yes	—
75	C₂×#97	Q₈	1	2	yes	—
76	C₂×C₄×C₈	—	2	1	no	—
77	C₂×C₄×C₈	—	2	1	no	—
78	C₂×C₄×C₈	—	2	1	no	—
79	C₂×C₄×C₈	—	2	1	no	—
80	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
81	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
82	C₂×#95	—	2	2	no	—
83	C₄⋊C₁₆	—	2	8	no	—
84	#347	—	2	8	no	—
85	#336	—	2	2	no	—
86	#336	—	2	2	no	—
87	#336	—	2	1	no	—
88	#336	—	2	1	no	—
89	C₂³×C₄	—	2	1	no	—
90	C₂³×C₄	—	2	1	no	—
91	C₂³×C₄	M₁₆	1	2	yes	—
92	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₈	—	2	1	no	—
94	C₂²×C₈	—	2	1	no	—
95	C₂²×C₈	—	2	1	no	—
96	C₂²×C₈	—	2	1	no	—
97	C₂²×C₈	—	2	1	no	—
98	C₂²×C₈	—	2	1	no	—
99	C₂²×C₈	—	2	1	no	—
100	C₂²×C₈	—	2	1	no	—
101	C₂²×C₈	—	2	1	no	—
102	C₂²×C₈	—	2	1	no	—
103	C₂²×C₈	—	2	1	no	—
104	C₂²×C₈	—	2	1	no	—
105	C₂²×C₈	—	2	1	no	—
106	C₂²×C₈	—	2	1	no	—
107	C₂²×C₈	—	2	1	no	—
108	C₂²×C₈	—	2	1	no	—
109	C₂²×C₈	—	2	1	no	—
110	C₂²×C₈	—	2	1	no	—
111	C₂²×C₈	C₄²	1	2	yes	—
112	C₂²×C₈	C₄⋊C₄	1	2	yes	—
113	C₂²×C₈	M₁₆	1	2	yes	—
114	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
121	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
122	C₂²×C₈	#41	1	1	yes	characteristic
123	C₂²×C₈	#41	1	1	yes	characteristic
124	C₂²×C₈	#41	1	1	yes	characteristic
125	C₂²×C₈	#41	1	1	yes	characteristic
126	C₂²×C₈	#41	1	1	yes	characteristic
127	C₂²×C₈	#41	1	1	yes	characteristic
128	C₂×C₄²	—	2	1	no	—
129	C₂×C₄²	—	2	1	no	—
130	C₂×C₄²	—	2	1	no	—
131	C₂×C₄²	—	2	1	no	—
132	C₂×C₄²	—	2	1	no	—
133	C₂×C₄²	—	2	1	no	—
134	C₂×C₄²	—	2	1	no	—
135	C₂×C₄²	—	2	1	no	—
136	C₂×C₄²	—	2	1	no	—
137	C₂×C₄²	—	2	1	no	—
138	C₂×C₄²	—	2	1	no	—
139	C₂×C₄²	—	2	1	no	—
140	C₂×C₄²	—	2	1	no	—
141	C₂×C₄²	—	2	1	no	—
142	C₂×C₄²	C₄²	1	2	yes	—
143	C₂×C₄²	C₄⋊C₄	1	2	yes	—
144	C₂×C₄²	M₁₆	1	2	yes	—
145	C₂×C₄²	M₁₆	1	1	yes	characteristic
146	C₂×C₄²	M₁₆	1	1	yes	characteristic
147	M₃₂	—	2	2	no	—
148	M₃₂	—	2	2	no	—
149	M₃₂	—	2	2	no	—
150	M₃₂	—	2	2	no	—
151	M₃₂	—	2	2	no	—
152	M₃₂	—	2	2	no	—
153	M₃₂	—	2	2	no	—
154	M₃₂	—	2	2	no	—
155	M₃₂	—	4	1	no	—
156	M₃₂	—	4	1	no	—
157	M₃₂	—	4	1	no	—
158	M₃₂	—	4	1	no	—
159	M₃₂	—	4	1	no	—
160	M₃₂	—	4	1	no	—
161	M₃₂	—	4	1	no	—
162	M₃₂	—	4	1	no	—
163	C₂×C₁₆	—	2	2	no	—
164	C₂×C₁₆	—	2	2	no	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	2	2	no	—
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	—	2	2	no	—
170	C₂×C₁₆	—	2	2	no	—
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	C₂²⋊C₄	1	4	yes	—
174	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
175	#97	—	2	2	no	—
176	#97	—	2	2	no	—
177	#97	—	2	2	no	—
178	#97	—	2	2	no	—
179	#97	—	4	1	no	—
180	#97	—	4	1	no	—
181	#97	—	4	1	no	—
182	#97	—	4	1	no	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	—	2	2	no	—
187	#95	—	4	4	no	—
188	C₂⁴	C₂×M₁₆	1	1	yes	characteristic
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	—	4	1	no	—
201	C₂²×C₄	—	4	1	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	#97	1	2	yes	—
204	C₂²×C₄	#97	1	2	yes	—
205	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	#117	1	1	yes	characteristic
208	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
209	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
210	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
211	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
212	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
213	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
214	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
215	C₂²×C₄	#131	1	1	yes	characteristic
216	C₂²×C₄	#131	1	1	yes	characteristic
217	C₂²×C₄	#131	1	1	yes	characteristic
218	C₂²×C₄	#131	1	1	yes	characteristic
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	4	1	no	—
240	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
241	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
242	C₂×C₈	#117	1	1	yes	characteristic
243	C₂×C₈	#117	1	1	yes	characteristic
244	C₂×C₈	C₄×D₄	1	1	yes	characteristic
245	C₂×C₈	C₄×D₄	1	1	yes	characteristic
246	C₂×C₈	C₄×D₄	1	1	yes	characteristic
247	C₂×C₈	C₄×D₄	1	1	yes	characteristic
248	C₂×C₈	C₄×D₄	1	1	yes	characteristic
249	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
250	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
251	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
252	C₂×C₈	#120	1	1	yes	characteristic
253	C₂×C₈	#121	1	1	yes	characteristic
254	C₂×C₈	#121	1	1	yes	characteristic
255	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
256	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
257	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
258	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
259	C₂×C₈	#123	1	1	yes	characteristic
260	C₂×C₈	#123	1	1	yes	characteristic
261	C₂×C₈	#123	1	1	yes	characteristic
262	C₂×C₈	#124	1	1	yes	characteristic
263	C₂×C₈	#125	1	1	yes	characteristic
264	C₂×C₈	#125	1	1	yes	characteristic
265	C₂×C₈	#126	1	1	yes	characteristic
266	C₂×C₈	#126	1	1	yes	characteristic
267	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	4	1	no	—
280	C₄²	—	4	1	no	—
281	C₄²	—	4	1	no	—
282	C₁₆	—	2	4	no	—
283	C₁₆	—	2	4	no	—
284	C₁₆	—	4	2	no	—
285	C₁₆	—	4	2	no	—
286	C₂³	—	2	2	no	—
287	C₂³	—	2	2	no	—
288	C₂³	C₂×#97	1	1	yes	characteristic
289	C₂³	#423	1	1	yes	characteristic
290	C₂³	#432	1	1	yes	characteristic
291	C₂³	#435	1	1	yes	characteristic
292	C₂³	#435	1	1	yes	characteristic
293	C₂³	#436	1	1	yes	characteristic
294	C₂³	#446	1	1	yes	characteristic
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	8	2	no	—
309	C₂×C₄	—	8	2	no	—
310	C₂×C₄	—	8	2	no	—
311	C₂×C₄	—	8	2	no	—
312	C₂×C₄	#347	1	4	yes	complemented
313	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
314	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
315	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
316	C₂×C₄	#385	1	1	yes	characteristic
317	C₂×C₄	#387	1	1	yes	characteristic
318	C₂×C₄	#388	1	1	yes	characteristic
319	C₂×C₄	#389	1	1	yes	characteristic
320	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
321	C₂×C₄	#405	1	1	yes	characteristic
322	C₂×C₄	#405	1	1	yes	characteristic
323	C₂×C₄	#423	1	1	yes	characteristic
324	C₂×C₄	#424	1	1	yes	characteristic
325	C₂×C₄	#424	1	1	yes	characteristic
326	C₂×C₄	#432	1	1	yes	characteristic
327	C₂×C₄	#433	1	1	yes	characteristic
328	C₂×C₄	#433	1	1	yes	characteristic
329	C₂×C₄	#435	1	1	yes	characteristic
330	C₂×C₄	#435	1	1	yes	characteristic
331	C₂×C₄	#435	1	1	yes	characteristic
332	C₂×C₄	#435	1	1	yes	characteristic
333	C₂×C₄	#435	1	1	yes	characteristic
334	C₂×C₄	#435	1	1	yes	characteristic
335	C₂×C₄	#436	1	1	yes	characteristic
336	C₂×C₄	#436	1	1	yes	characteristic
337	C₂×C₄	#436	1	1	yes	characteristic
338	C₂×C₄	#446	1	1	yes	characteristic
339	C₂×C₄	#446	1	1	yes	characteristic
340	C₂×C₄	#446	1	1	yes	characteristic
341	C₈	—	4	2	no	—
342	C₈	—	4	2	no	—
343	C₈	—	4	2	no	—
344	C₈	#385	1	1	yes	characteristic
345	C₈	#386	1	1	yes	characteristic
346	C₈	#387	1	1	yes	characteristic
347	C₈	#387	1	1	yes	characteristic
348	C₈	#387	1	1	yes	characteristic
349	C₈	#388	1	1	yes	characteristic
350	C₈	#389	1	1	yes	characteristic
351	C₈	#391	1	1	yes	characteristic
352	C₂²	—	2	2	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	2	no	—
359	C₂²	—	2	2	no	—
360	C₂²	#1752	1	1	yes	characteristic
361	C₂²	#1767	1	1	yes	characteristic
362	C₂²	#1770	1	1	yes	characteristic
363	C₂²	#1851	1	1	yes	characteristic
364	C₂²	#1917	1	1	yes	characteristic
365	C₂²	#1924	1	1	yes	characteristic
366	C₂²	#1940	1	1	yes	characteristic
367	C₂²	C₂×#347	1	2	yes	—
368	C₂²	#2110	1	2	yes	—
369	C₄	—	2	4	no	—
370	C₄	—	4	2	no	—
371	C₄	—	8	2	no	—
372	C₄	—	8	2	no	—
373	C₄	#1436	1	1	yes	characteristic
374	C₄	#1753	1	1	yes	characteristic
375	C₄	#1768	1	1	yes	characteristic
376	C₄	#1771	1	1	yes	characteristic
377	C₄	#1851	1	1	yes	characteristic
378	C₄	#1917	1	1	yes	characteristic
379	C₄	#1924	1	1	yes	characteristic
380	C₄	#1940	1	1	yes	characteristic
381	C₂	—	2	1	no	—
382	C₂	—	2	1	no	—
383	C₂	—	2	1	no	—
384	C₂	—	2	1	no	—
385	C₂	#10877	1	1	yes	characteristic
386	C₂	#12616	1	2	yes	—
387	C₂	#12657	1	1	yes	characteristic
388	C₂	#12658	1	1	yes	characteristic
389	C₂	#12666	1	2	yes	—

	subgroup	G/H
center, commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×#97
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
17. C₂×C₄⋊C₁₆	C₄	semidirect
28. #1374	C₄	semidirect
312. C₂×C₄	#347	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2150	2
C₂²×#347	2
C₂×#1375	2
#12544	2
#10696	2