Subgroups — #147139

Order 512.

604 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#364	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×#1376	C₂	1	1	yes	maximal, characteristic
4	C₂×#1376	C₂	1	1	yes	maximal, characteristic
5	C₂×#1376	C₂	1	1	yes	maximal, characteristic
6	#12544	C₂	1	1	yes	maximal, characteristic
7	C₈×#95	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×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×#364	—	2	2	no	—
16	C₂×#364	—	2	2	no	—
17	C₂×#364	C₄	1	4	yes	complemented
18	C₄×M₃₂	—	2	2	no	—
19	C₄×M₃₂	—	2	2	no	—
20	C₂×#346	—	2	1	no	—
21	C₂×C₄×C₁₆	—	2	1	no	—
22	C₂×#336	C₄	1	1	yes	characteristic
23	C₂×#336	C₄	1	1	yes	characteristic
24	C₂×#336	C₄	1	1	yes	characteristic
25	C₄×#95	C₄	1	1	yes	characteristic
26	#1376	—	2	2	no	—
27	#1376	—	2	2	no	—
28	#1376	C₄	1	4	yes	complemented
29	#1374	—	2	1	no	—
30	#1374	—	2	1	no	—
31	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
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	C₂×C₄	1	2	yes	—
55	C₂×M₃₂	C₂×C₄	1	2	yes	—
56	C₂×M₃₂	C₂×C₄	1	2	yes	—
57	C₂×M₃₂	C₂×C₄	1	2	yes	—
58	C₂×M₃₂	D₄	1	2	yes	—
59	C₂×M₃₂	D₄	1	2	yes	—
60	C₂×M₃₂	D₄	1	2	yes	—
61	C₂×M₃₂	Q₈	1	2	yes	—
62	C₂²×C₁₆	—	2	1	no	—
63	C₂²×C₁₆	—	2	1	no	—
64	C₂²×C₁₆	—	2	1	no	—
65	C₂²×C₁₆	—	2	1	no	—
66	C₂×C₄×C₈	—	2	1	no	—
67	C₂×C₄×C₈	—	2	1	no	—
68	C₂×C₄×C₈	—	2	1	no	—
69	C₂×C₄×C₈	—	2	1	no	—
70	C₂×C₄×C₈	—	2	1	no	—
71	C₂×C₄×C₈	—	2	1	no	—
72	C₂×C₄×C₈	—	2	1	no	—
73	C₂×C₄×C₈	—	2	1	no	—
74	C₂×C₄×C₈	—	2	1	no	—
75	C₂×C₄×C₈	—	2	1	no	—
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×C₄×C₈	D₄	1	1	yes	characteristic
83	C₂×C₄×C₈	D₄	1	1	yes	characteristic
84	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
85	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
86	C₂×#95	—	2	2	no	—
87	#364	—	2	8	no	—
88	#346	—	2	4	no	—
89	C₄×C₁₆	—	2	4	no	—
90	#336	—	2	2	no	—
91	#336	C₈	1	4	yes	—
92	#336	C₈	1	4	yes	—
93	C₂³×C₄	—	2	1	no	—
94	C₂³×C₄	—	2	1	no	—
95	C₂³×C₄	M₁₆	1	2	yes	—
96	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
113	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
114	C₂²×C₈	C₂×C₈	1	2	yes	—
115	C₂²×C₈	C₂×C₈	1	2	yes	—
116	C₂²×C₈	C₂×C₈	1	2	yes	—
117	C₂²×C₈	C₂×C₈	1	2	yes	—
118	C₂²×C₈	M₁₆	1	2	yes	—
119	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
126	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
127	C₂²×C₈	#41	1	1	yes	characteristic
128	C₂²×C₈	#41	1	1	yes	characteristic
129	C₂²×C₈	#41	1	1	yes	characteristic
130	C₂²×C₈	#41	1	1	yes	characteristic
131	C₂²×C₈	#41	1	1	yes	characteristic
132	C₂²×C₈	#41	1	1	yes	characteristic
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	—	2	1	no	—
143	C₂×C₄²	—	2	1	no	—
144	C₂×C₄²	—	2	1	no	—
145	C₂×C₄²	—	2	1	no	—
146	C₂×C₄²	—	2	1	no	—
147	C₂×C₄²	C₄²	1	2	yes	—
148	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄²	C₂×C₈	1	2	yes	—
151	C₂×C₄²	C₂×C₈	1	2	yes	—
152	M₃₂	—	2	2	no	—
153	M₃₂	—	2	2	no	—
154	M₃₂	—	2	2	no	—
155	M₃₂	—	2	2	no	—
156	M₃₂	—	2	2	no	—
157	M₃₂	—	2	2	no	—
158	M₃₂	—	2	2	no	—
159	M₃₂	—	2	2	no	—
160	M₃₂	—	2	2	no	—
161	M₃₂	—	2	2	no	—
162	M₃₂	—	4	1	no	—
163	M₃₂	—	4	1	no	—
164	M₃₂	—	4	1	no	—
165	M₃₂	—	4	1	no	—
166	M₃₂	C₂²⋊C₄	1	4	yes	—
167	M₃₂	C₄⋊C₄	1	4	yes	—
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	—	2	2	no	—
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	4	1	no	—
177	C₂×C₁₆	—	4	1	no	—
178	C₂×C₁₆	—	4	1	no	—
179	C₂×C₁₆	—	4	1	no	—
180	C₄×C₈	—	2	2	no	—
181	C₄×C₈	—	2	2	no	—
182	C₄×C₈	—	2	2	no	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	—	2	1	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	—	2	1	no	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	4	1	no	—
193	C₄×C₈	—	4	1	no	—
194	C₄×C₈	—	4	1	no	—
195	C₄×C₈	—	4	1	no	—
196	#95	—	2	8	no	—
197	C₂⁴	C₂×M₁₆	1	1	yes	characteristic
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	4	1	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	—	4	1	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	C₄⋊C₈	1	2	yes	—
213	C₂²×C₄	C₄⋊C₈	1	2	yes	—
214	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
216	C₂²×C₄	#117	1	1	yes	characteristic
217	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
218	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
219	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
220	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
221	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
222	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
223	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
224	C₂²×C₄	#131	1	1	yes	characteristic
225	C₂²×C₄	#131	1	1	yes	characteristic
226	C₂²×C₄	#131	1	1	yes	characteristic
227	C₂²×C₄	#131	1	1	yes	characteristic
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	1	no	—
249	C₂×C₈	C₄×C₈	1	4	yes	—
250	C₂×C₈	C₄⋊C₈	1	2	yes	—
251	C₂×C₈	C₄⋊C₈	1	2	yes	—
252	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
253	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
254	C₂×C₈	#117	1	1	yes	characteristic
255	C₂×C₈	#117	1	1	yes	characteristic
256	C₂×C₈	C₄×D₄	1	1	yes	characteristic
257	C₂×C₈	C₄×D₄	1	1	yes	characteristic
258	C₂×C₈	C₄×D₄	1	1	yes	characteristic
259	C₂×C₈	C₄×D₄	1	1	yes	characteristic
260	C₂×C₈	C₄×D₄	1	1	yes	characteristic
261	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
262	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
263	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
264	C₂×C₈	#120	1	1	yes	characteristic
265	C₂×C₈	#121	1	1	yes	characteristic
266	C₂×C₈	#121	1	1	yes	characteristic
267	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
268	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
269	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
270	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
271	C₂×C₈	#123	1	1	yes	characteristic
272	C₂×C₈	#123	1	1	yes	characteristic
273	C₂×C₈	#123	1	1	yes	characteristic
274	C₂×C₈	#124	1	1	yes	characteristic
275	C₂×C₈	#125	1	1	yes	characteristic
276	C₂×C₈	#125	1	1	yes	characteristic
277	C₂×C₈	#126	1	1	yes	characteristic
278	C₂×C₈	#126	1	1	yes	characteristic
279	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
280	C₄²	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	4	1	no	—
291	C₄²	—	4	1	no	—
292	C₄²	—	4	1	no	—
293	C₄²	—	4	1	no	—
294	C₁₆	—	2	4	no	—
295	C₁₆	—	2	4	no	—
296	C₁₆	—	4	2	no	—
297	C₁₆	—	4	2	no	—
298	C₂³	—	2	2	no	—
299	C₂³	—	2	2	no	—
300	C₂³	C₄×M₁₆	1	1	yes	characteristic
301	C₂³	C₂×C₄⋊C₈	1	1	yes	characteristic
302	C₂³	#431	1	1	yes	characteristic
303	C₂³	#435	1	1	yes	characteristic
304	C₂³	#435	1	1	yes	characteristic
305	C₂³	#436	1	1	yes	characteristic
306	C₂³	#446	1	1	yes	characteristic
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	4	no	—
321	C₂×C₄	—	4	4	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	#364	1	4	yes	complemented
325	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
326	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
327	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
328	C₂×C₄	#385	1	1	yes	characteristic
329	C₂×C₄	#387	1	1	yes	characteristic
330	C₂×C₄	#388	1	1	yes	characteristic
331	C₂×C₄	#389	1	1	yes	characteristic
332	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
333	C₂×C₄	#405	1	1	yes	characteristic
334	C₂×C₄	#405	1	1	yes	characteristic
335	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
336	C₂×C₄	#424	1	1	yes	characteristic
337	C₂×C₄	#424	1	1	yes	characteristic
338	C₂×C₄	#431	1	1	yes	characteristic
339	C₂×C₄	#433	1	1	yes	characteristic
340	C₂×C₄	#433	1	1	yes	characteristic
341	C₂×C₄	C₈×D₄	1	1	yes	characteristic
342	C₂×C₄	C₈×D₄	1	1	yes	characteristic
343	C₂×C₄	C₈×D₄	1	1	yes	characteristic
344	C₂×C₄	C₈×D₄	1	1	yes	characteristic
345	C₂×C₄	C₈×D₄	1	1	yes	characteristic
346	C₂×C₄	C₈×D₄	1	1	yes	characteristic
347	C₂×C₄	#435	1	1	yes	characteristic
348	C₂×C₄	#435	1	1	yes	characteristic
349	C₂×C₄	#436	1	1	yes	characteristic
350	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
351	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
352	C₂×C₄	#446	1	1	yes	characteristic
353	C₈	—	2	2	no	—
354	C₈	—	2	2	no	—
355	C₈	—	2	4	no	—
356	C₈	—	4	2	no	—
357	C₈	#385	1	1	yes	characteristic
358	C₈	#386	1	1	yes	characteristic
359	C₈	#387	1	1	yes	characteristic
360	C₈	#387	1	1	yes	characteristic
361	C₈	#387	1	1	yes	characteristic
362	C₈	#388	1	1	yes	characteristic
363	C₈	#389	1	1	yes	characteristic
364	C₈	#391	1	1	yes	characteristic
365	C₂²	—	2	2	no	—
366	C₂²	—	2	2	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	2	no	—
372	C₂²	—	2	2	no	—
373	C₂²	#1753	1	1	yes	characteristic
374	C₂²	C₄×C₄⋊C₈	1	1	yes	characteristic
375	C₂²	#1771	1	1	yes	characteristic
376	C₂²	#1850	1	1	yes	characteristic
377	C₂²	#1916	1	1	yes	characteristic
378	C₂²	#1923	1	1	yes	characteristic
379	C₂²	#1939	1	1	yes	characteristic
380	C₂²	C₂×#364	1	2	yes	—
381	C₂²	#2153	1	2	yes	—
382	C₄	—	2	4	no	—
383	C₄	—	4	4	no	—
384	C₄	—	4	4	no	—
385	C₄	—	4	2	no	—
386	C₄	#1436	1	1	yes	characteristic
387	C₄	C₈×C₂²⋊C₄	1	1	yes	characteristic
388	C₄	#1768	1	1	yes	characteristic
389	C₄	C₈×C₄⋊C₄	1	1	yes	characteristic
390	C₄	#1850	1	1	yes	characteristic
391	C₄	#1916	1	1	yes	characteristic
392	C₄	#1923	1	1	yes	characteristic
393	C₄	#1939	1	1	yes	characteristic
394	C₂	—	2	1	no	—
395	C₂	—	2	1	no	—
396	C₂	—	2	1	no	—
397	C₂	—	2	1	no	—
398	C₂	#10876	1	1	yes	characteristic
399	C₂	C₄×#364	1	1	yes	characteristic
400	C₂	#12656	1	1	yes	characteristic
401	C₂	#12914	1	2	yes	—
402	C₂	#13163	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×C₄⋊C₈
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈ ◁ 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×#364	C₄	semidirect
28. #1376	C₄	semidirect
324. C₂×C₄	#364	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#364	2
C₂×C₄×M₃₂	2
C₂×#1376	2
#12544	2
C₈×#95	2