Subgroups — #147146

Order 512.

572 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2151	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#1379	C₂	1	1	yes	maximal, characteristic
4	C₂×#1379	C₂	1	1	yes	maximal, characteristic
5	C₂×#1376	C₂	1	1	yes	maximal, characteristic
6	#12545	C₂	1	1	yes	maximal, characteristic
7	#10706	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×M₃₂	C₂²	1	1	yes	characteristic
10	C₂³×C₁₆	C₂²	1	1	yes	characteristic
11	C₂³×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×#405	C₂²	1	1	yes	characteristic
13	C₂×#405	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
15	#2151	—	2	2	no	—
16	#2151	C₄	1	4	yes	complemented
17	C₂×C₄⋊C₁₆	—	2	2	no	—
18	C₂×C₄×C₁₆	—	2	2	no	—
19	C₂×C₄×C₁₆	—	2	2	no	—
20	C₂×C₄×C₁₆	—	2	2	no	—
21	C₂×#341	C₄	1	1	yes	characteristic
22	#1735	C₄	1	1	yes	characteristic
23	C₂×#339	C₄	1	1	yes	characteristic
24	C₂×#339	C₄	1	1	yes	characteristic
25	#1379	—	2	2	no	—
26	#1379	C₄	1	4	yes	complemented
27	#1376	—	2	2	no	—
28	#1374	—	2	2	no	—
29	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₈	C₂³	1	1	yes	characteristic
33	C₂×#117	C₂×C₄	1	1	yes	characteristic
34	C₂×#117	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	C₂×C₄	1	2	yes	—
47	C₂×M₃₂	C₂×C₄	1	2	yes	—
48	C₂²×C₁₆	—	2	1	no	—
49	C₂²×C₁₆	—	2	1	no	—
50	C₂²×C₁₆	—	2	1	no	—
51	C₂²×C₁₆	—	2	1	no	—
52	C₂²×C₁₆	—	2	1	no	—
53	C₂²×C₁₆	—	2	1	no	—
54	C₂²×C₁₆	—	2	1	no	—
55	C₂²×C₁₆	—	2	1	no	—
56	C₂²×C₁₆	—	2	1	no	—
57	C₂²×C₁₆	—	2	1	no	—
58	C₂²×C₁₆	C₂×C₄	1	2	yes	—
59	C₂²×C₁₆	C₂×C₄	1	2	yes	—
60	C₂²×C₁₆	D₄	1	2	yes	—
61	C₂²×C₁₆	D₄	1	2	yes	—
62	C₂²×C₁₆	D₄	1	2	yes	—
63	C₂²×C₁₆	Q₈	1	2	yes	—
64	#405	—	2	1	no	—
65	#405	—	2	1	no	—
66	#405	—	2	1	no	—
67	#405	—	2	1	no	—
68	#405	—	2	1	no	—
69	#405	—	2	1	no	—
70	#405	C₂×C₄	1	2	yes	—
71	#405	C₂×C₄	1	2	yes	—
72	C₂×#97	—	2	1	no	—
73	C₂×#97	—	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	D₄	1	2	yes	—
81	C₂×C₄×C₈	Q₈	1	2	yes	—
82	C₄⋊C₁₆	—	2	8	no	—
83	C₄×C₁₆	—	2	8	no	—
84	#341	—	2	2	no	—
85	#340	C₈	1	4	yes	—
86	#339	—	2	2	no	—
87	#339	C₈	1	4	yes	—
88	C₂³×C₄	—	2	1	no	—
89	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
90	C₂×M₁₆	—	2	1	no	—
91	C₂×M₁₆	—	2	1	no	—
92	C₂×M₁₆	—	2	1	no	—
93	C₂×M₁₆	—	2	1	no	—
94	C₂×M₁₆	—	2	1	no	—
95	C₂×M₁₆	—	2	1	no	—
96	C₂×M₁₆	—	2	1	no	—
97	C₂×M₁₆	—	2	1	no	—
98	C₂×M₁₆	—	2	1	no	—
99	C₂×M₁₆	—	2	1	no	—
100	C₂×M₁₆	C₄²	1	2	yes	—
101	C₂×M₁₆	C₂×C₈	1	2	yes	—
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	C₄⋊C₄	1	2	yes	—
109	C₂²×C₈	C₂×C₈	1	2	yes	—
110	C₂²×C₈	M₁₆	1	2	yes	—
111	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
118	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
119	C₂²×C₈	#41	1	1	yes	characteristic
120	C₂²×C₈	#41	1	1	yes	characteristic
121	C₂²×C₈	#41	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	#117	—	2	1	no	—
126	#117	—	2	1	no	—
127	#117	—	2	1	no	—
128	#117	—	2	1	no	—
129	#117	—	2	1	no	—
130	#117	—	2	1	no	—
131	#117	C₄²	1	2	yes	—
132	#117	C₂×C₈	1	2	yes	—
133	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
134	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
135	C₂×C₂²⋊C₄	—	2	1	no	—
136	C₂×C₂²⋊C₄	—	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	C₄⋊C₄	1	2	yes	—
143	C₂×C₄²	C₂×C₈	1	2	yes	—
144	C₂×C₄²	M₁₆	1	2	yes	—
145	M₃₂	—	2	2	no	—
146	M₃₂	—	2	2	no	—
147	M₃₂	—	2	2	no	—
148	M₃₂	—	2	2	no	—
149	M₃₂	—	4	1	no	—
150	M₃₂	—	4	1	no	—
151	M₃₂	—	4	1	no	—
152	M₃₂	—	4	1	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	—	2	2	no	—
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	—	2	2	no	—
158	C₂×C₁₆	—	2	2	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	—	2	2	no	—
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	—	2	2	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	—	4	1	no	—
168	C₂×C₁₆	—	4	1	no	—
169	C₂×C₁₆	—	4	1	no	—
170	C₂×C₁₆	—	4	1	no	—
171	C₂×C₁₆	C₂²⋊C₄	1	4	yes	—
172	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
173	#97	—	2	2	no	—
174	#97	—	2	2	no	—
175	#97	—	4	1	no	—
176	#97	—	4	1	no	—
177	C₄×C₈	—	2	2	no	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	—	2	2	no	—
180	C₄×C₈	—	2	2	no	—
181	C₄×C₈	—	2	2	no	—
182	C₄×C₈	—	2	2	no	—
183	C₄×C₈	—	4	1	no	—
184	C₄×C₈	—	4	1	no	—
185	C₂⁴	#131	1	1	yes	characteristic
186	C₂²×C₄	—	2	8	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	C₂×C₄²	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
194	C₂²×C₄	#117	1	1	yes	characteristic
195	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
196	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
197	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
198	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
199	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
201	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
202	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
203	C₂²×C₄	#131	1	1	yes	characteristic
204	C₂²×C₄	#131	1	1	yes	characteristic
205	C₂²×C₄	#131	1	1	yes	characteristic
206	M₁₆	—	2	2	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	—	2	2	no	—
209	M₁₆	—	2	2	no	—
210	M₁₆	—	4	1	no	—
211	M₁₆	—	4	1	no	—
212	M₁₆	—	4	1	no	—
213	M₁₆	—	4	1	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	C₄⋊C₈	1	4	yes	—
227	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
229	C₂×C₈	#117	1	1	yes	characteristic
230	C₂×C₈	#117	1	1	yes	characteristic
231	C₂×C₈	C₄×D₄	1	1	yes	characteristic
232	C₂×C₈	C₄×D₄	1	1	yes	characteristic
233	C₂×C₈	C₄×D₄	1	1	yes	characteristic
234	C₂×C₈	C₄×D₄	1	1	yes	characteristic
235	C₂×C₈	C₄×D₄	1	1	yes	characteristic
236	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
237	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
238	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
239	C₂×C₈	#120	1	1	yes	characteristic
240	C₂×C₈	#121	1	1	yes	characteristic
241	C₂×C₈	#121	1	1	yes	characteristic
242	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
243	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
244	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
245	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
246	C₂×C₈	#123	1	1	yes	characteristic
247	C₂×C₈	#123	1	1	yes	characteristic
248	C₂×C₈	#123	1	1	yes	characteristic
249	C₂×C₈	#124	1	1	yes	characteristic
250	C₂×C₈	#125	1	1	yes	characteristic
251	C₂×C₈	#125	1	1	yes	characteristic
252	C₂×C₈	#126	1	1	yes	characteristic
253	C₂×C₈	#126	1	1	yes	characteristic
254	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	C₄×C₈	1	4	yes	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	2	2	no	—
261	C₂²⋊C₄	—	4	1	no	—
262	C₂²⋊C₄	—	4	1	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	4	1	no	—
269	C₄²	—	4	1	no	—
270	C₄²	C₄⋊C₈	1	4	yes	—
271	C₁₆	—	2	4	no	—
272	C₁₆	—	2	4	no	—
273	C₁₆	—	4	2	no	—
274	C₁₆	—	4	2	no	—
275	C₂³	—	2	4	no	—
276	C₂³	#405	1	1	yes	characteristic
277	C₂³	#424	1	1	yes	characteristic
278	C₂³	#433	1	1	yes	characteristic
279	C₂³	C₈×D₄	1	1	yes	characteristic
280	C₂³	C₈×D₄	1	1	yes	characteristic
281	C₂³	#435	1	1	yes	characteristic
282	C₂³	#446	1	1	yes	characteristic
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	2	8	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
292	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
293	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
294	C₂×C₄	#385	1	1	yes	characteristic
295	C₂×C₄	#387	1	1	yes	characteristic
296	C₂×C₄	#388	1	1	yes	characteristic
297	C₂×C₄	#389	1	1	yes	characteristic
298	C₂×C₄	C₂×C₄×C₈	1	1	yes	characteristic
299	C₂×C₄	C₄×M₁₆	1	1	yes	characteristic
300	C₂×C₄	#405	1	1	yes	characteristic
301	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
302	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
303	C₂×C₄	#424	1	1	yes	characteristic
304	C₂×C₄	#431	1	1	yes	characteristic
305	C₂×C₄	#431	1	1	yes	characteristic
306	C₂×C₄	#433	1	1	yes	characteristic
307	C₂×C₄	C₈×D₄	1	1	yes	characteristic
308	C₂×C₄	C₈×D₄	1	1	yes	characteristic
309	C₂×C₄	C₈×D₄	1	1	yes	characteristic
310	C₂×C₄	C₈×D₄	1	1	yes	characteristic
311	C₂×C₄	#435	1	1	yes	characteristic
312	C₂×C₄	#435	1	1	yes	characteristic
313	C₂×C₄	#435	1	1	yes	characteristic
314	C₂×C₄	#436	1	1	yes	characteristic
315	C₂×C₄	#436	1	1	yes	characteristic
316	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
317	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
318	C₂×C₄	#446	1	1	yes	characteristic
319	C₈	—	2	4	no	—
320	C₈	—	4	2	no	—
321	C₈	—	4	2	no	—
322	C₈	#385	1	1	yes	characteristic
323	C₈	#386	1	1	yes	characteristic
324	C₈	#387	1	1	yes	characteristic
325	C₈	#387	1	1	yes	characteristic
326	C₈	#387	1	1	yes	characteristic
327	C₈	#388	1	1	yes	characteristic
328	C₈	#389	1	1	yes	characteristic
329	C₈	#391	1	1	yes	characteristic
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	C₈×C₂²⋊C₄	1	1	yes	characteristic
337	C₂²	#1768	1	1	yes	characteristic
338	C₂²	#1771	1	1	yes	characteristic
339	C₂²	#1850	1	1	yes	characteristic
340	C₂²	#1916	1	1	yes	characteristic
341	C₂²	#1923	1	1	yes	characteristic
342	C₂²	#1939	1	1	yes	characteristic
343	C₂²	#2170	1	4	yes	—
344	C₄	—	2	8	no	—
345	C₄	—	4	4	no	—
346	C₄	—	8	2	no	—
347	C₄	#1436	1	1	yes	characteristic
348	C₄	#1753	1	1	yes	characteristic
349	C₄	C₄×C₄⋊C₈	1	1	yes	characteristic
350	C₄	C₈×C₄⋊C₄	1	1	yes	characteristic
351	C₄	#1850	1	1	yes	characteristic
352	C₄	#1916	1	1	yes	characteristic
353	C₄	#1923	1	1	yes	characteristic
354	C₄	#1939	1	1	yes	characteristic
355	C₂	—	2	1	no	—
356	C₂	—	2	1	no	—
357	C₂	—	2	1	no	—
358	C₂	—	2	1	no	—
359	C₂	#10876	1	1	yes	characteristic
360	C₂	#12614	1	2	yes	—
361	C₂	#13124	1	2	yes	—
362	C₂	#13161	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₈×D₄
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
16. #2151	C₄	semidirect
26. #1379	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2151	2
C₂²×C₄×C₁₆	2
C₂×#1379	2
C₂×#1376	2
#12545	2
#10706	2