Subgroups — #149108

Order 512.

462 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#1379	C₂	1	1	yes	maximal, characteristic
2	C₂×#1379	C₂	1	1	yes	maximal, characteristic
3	C₂×#1376	C₂	1	1	yes	maximal, characteristic
4	#12545	C₂	1	1	yes	maximal, characteristic
5	#12545	C₂	1	1	yes	maximal, characteristic
6	C₂×#1374	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	C₂×#341	C₄	1	1	yes	characteristic
16	#1735	C₄	1	1	yes	characteristic
17	C₂×#339	C₄	1	1	yes	characteristic
18	C₂×#339	C₄	1	1	yes	characteristic
19	#1379	—	2	2	no	—
20	#1379	—	2	2	no	—
21	#1376	—	2	2	no	—
22	#1374	—	2	2	no	—
23	#1374	—	2	2	no	—
24	#1374	—	2	2	no	—
25	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂²×M₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₈	C₂³	1	1	yes	characteristic
29	C₂×#117	C₂×C₄	1	1	yes	characteristic
30	C₂×#117	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₂×M₃₂	—	2	1	no	—
33	C₂×M₃₂	—	2	1	no	—
34	C₂×M₃₂	—	2	1	no	—
35	C₂×M₃₂	—	2	1	no	—
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	D₄	1	2	yes	—
43	C₂×M₃₂	Q₈	1	2	yes	—
44	C₂²×C₁₆	—	2	1	no	—
45	C₂²×C₁₆	—	2	1	no	—
46	C₂²×C₁₆	—	2	1	no	—
47	C₂²×C₁₆	—	2	1	no	—
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	D₄	1	2	yes	—
59	C₂²×C₁₆	D₄	1	2	yes	—
60	#405	—	2	1	no	—
61	#405	—	2	1	no	—
62	#405	—	2	1	no	—
63	#405	—	2	1	no	—
64	#405	—	2	1	no	—
65	#405	—	2	1	no	—
66	#405	—	2	1	no	—
67	#405	—	2	1	no	—
68	C₂×#97	—	2	1	no	—
69	C₂×#97	—	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	#341	—	2	2	no	—
79	#340	C₈	1	2	yes	—
80	#340	C₈	1	2	yes	—
81	#339	—	2	2	no	—
82	#339	—	2	2	no	—
83	C₂³×C₄	—	2	1	no	—
84	C₂³×C₄	C₂²×C₄	1	1	yes	characteristic
85	C₂×M₁₆	—	2	1	no	—
86	C₂×M₁₆	—	2	1	no	—
87	C₂×M₁₆	—	2	1	no	—
88	C₂×M₁₆	—	2	1	no	—
89	C₂×M₁₆	—	2	1	no	—
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	M₁₆	1	2	yes	—
96	C₂×M₁₆	M₁₆	1	2	yes	—
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	C₂×C₈	1	2	yes	—
106	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
110	C₂²×C₈	#41	1	1	yes	characteristic
111	C₂²×C₈	#41	1	1	yes	characteristic
112	C₂²×C₈	#41	1	1	yes	characteristic
113	C₂²×C₈	#41	1	1	yes	characteristic
114	C₂²×C₈	#41	1	1	yes	characteristic
115	C₂²×C₈	#41	1	1	yes	characteristic
116	C₂²×C₈	#41	1	1	yes	characteristic
117	C₂²×C₈	#41	1	1	yes	characteristic
118	C₂²×C₈	#41	1	1	yes	characteristic
119	C₂²×C₈	#41	1	1	yes	characteristic
120	#117	—	2	1	no	—
121	#117	—	2	1	no	—
122	#117	—	2	1	no	—
123	#117	—	2	1	no	—
124	#117	—	2	1	no	—
125	#117	—	2	1	no	—
126	#117	—	2	1	no	—
127	#117	—	2	1	no	—
128	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
129	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
130	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
131	C₂×C₂²⋊C₄	—	2	1	no	—
132	C₂×C₂²⋊C₄	—	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	M₁₆	1	2	yes	—
141	M₃₂	—	2	2	no	—
142	M₃₂	—	2	2	no	—
143	M₃₂	—	2	2	no	—
144	M₃₂	—	2	2	no	—
145	M₃₂	—	4	1	no	—
146	M₃₂	—	4	1	no	—
147	M₃₂	—	4	1	no	—
148	M₃₂	—	4	1	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	—	2	2	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	—	4	1	no	—
158	C₂×C₁₆	—	4	1	no	—
159	C₂×C₁₆	—	4	1	no	—
160	C₂×C₁₆	—	4	1	no	—
161	C₂×C₁₆	—	4	1	no	—
162	C₂×C₁₆	—	4	1	no	—
163	C₂×C₁₆	—	4	1	no	—
164	C₂×C₁₆	—	4	1	no	—
165	C₂×C₁₆	—	4	1	no	—
166	C₂×C₁₆	—	4	1	no	—
167	C₂×C₁₆	—	4	1	no	—
168	C₂×C₁₆	—	4	1	no	—
169	#97	—	4	1	no	—
170	#97	—	4	1	no	—
171	#97	—	4	1	no	—
172	#97	—	4	1	no	—
173	C₄×C₈	—	4	1	no	—
174	C₄×C₈	—	4	1	no	—
175	C₄×C₈	—	4	1	no	—
176	C₄×C₈	—	4	1	no	—
177	C₄×C₈	—	4	1	no	—
178	C₄×C₈	—	4	1	no	—
179	C₄×C₈	—	4	1	no	—
180	C₄×C₈	—	4	1	no	—
181	C₂⁴	#131	1	1	yes	characteristic
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	4	1	no	—
189	C₂²×C₄	—	4	1	no	—
190	C₂²×C₄	#117	1	1	yes	characteristic
191	C₂²×C₄	#117	1	1	yes	characteristic
192	C₂²×C₄	#117	1	1	yes	characteristic
193	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
194	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
195	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
196	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
197	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
198	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
199	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
200	C₂²×C₄	C₂×M₁₆	1	1	yes	characteristic
201	C₂²×C₄	#131	1	1	yes	characteristic
202	C₂²×C₄	#131	1	1	yes	characteristic
203	C₂²×C₄	#131	1	1	yes	characteristic
204	M₁₆	—	2	2	no	—
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	2	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	—	4	1	no	—
209	M₁₆	—	4	1	no	—
210	M₁₆	—	4	1	no	—
211	M₁₆	—	4	1	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	1	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	#121	1	1	yes	characteristic
226	C₂×C₈	#121	1	1	yes	characteristic
227	C₂×C₈	#121	1	1	yes	characteristic
228	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
229	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
231	C₂×C₈	#123	1	1	yes	characteristic
232	C₂×C₈	#123	1	1	yes	characteristic
233	C₂×C₈	#123	1	1	yes	characteristic
234	C₂×C₈	#123	1	1	yes	characteristic
235	C₂×C₈	#123	1	1	yes	characteristic
236	C₂×C₈	#123	1	1	yes	characteristic
237	C₂×C₈	#124	1	1	yes	characteristic
238	C₂×C₈	#124	1	1	yes	characteristic
239	C₂×C₈	#124	1	1	yes	characteristic
240	C₂×C₈	#125	1	1	yes	characteristic
241	C₂×C₈	#125	1	1	yes	characteristic
242	C₂×C₈	#125	1	1	yes	characteristic
243	C₂×C₈	#126	1	1	yes	characteristic
244	C₂×C₈	#126	1	1	yes	characteristic
245	C₂×C₈	#126	1	1	yes	characteristic
246	C₂×C₈	#126	1	1	yes	characteristic
247	C₂×C₈	#126	1	1	yes	characteristic
248	C₂×C₈	#126	1	1	yes	characteristic
249	C₂×C₈	#126	1	1	yes	characteristic
250	C₂×C₈	#126	1	1	yes	characteristic
251	C₂×C₈	#126	1	1	yes	characteristic
252	C₂×C₈	#126	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	4	1	no	—
260	C₂²⋊C₄	—	4	1	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	—	4	1	no	—
266	C₄²	—	4	1	no	—
267	C₄²	—	4	1	no	—
268	C₄²	—	4	1	no	—
269	C₄²	—	4	1	no	—
270	C₄²	—	4	1	no	—
271	C₁₆	—	4	2	no	—
272	C₁₆	—	4	2	no	—
273	C₁₆	—	4	2	no	—
274	C₁₆	—	4	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	#433	1	1	yes	characteristic
278	C₂³	#433	1	1	yes	characteristic
279	C₂³	#433	1	1	yes	characteristic
280	C₂³	C₈×D₄	1	1	yes	characteristic
281	C₂³	#435	1	1	yes	characteristic
282	C₂³	#436	1	1	yes	characteristic
283	C₂³	#446	1	1	yes	characteristic
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	#383	1	1	yes	characteristic
297	C₂×C₄	#387	1	1	yes	characteristic
298	C₂×C₄	#387	1	1	yes	characteristic
299	C₂×C₄	#387	1	1	yes	characteristic
300	C₂×C₄	#388	1	1	yes	characteristic
301	C₂×C₄	#388	1	1	yes	characteristic
302	C₂×C₄	#388	1	1	yes	characteristic
303	C₂×C₄	#431	1	1	yes	characteristic
304	C₂×C₄	#431	1	1	yes	characteristic
305	C₂×C₄	#431	1	1	yes	characteristic
306	C₂×C₄	#432	1	1	yes	characteristic
307	C₂×C₄	#432	1	1	yes	characteristic
308	C₂×C₄	#432	1	1	yes	characteristic
309	C₂×C₄	#433	1	1	yes	characteristic
310	C₂×C₄	#433	1	1	yes	characteristic
311	C₂×C₄	#433	1	1	yes	characteristic
312	C₂×C₄	C₈×D₄	1	1	yes	characteristic
313	C₂×C₄	C₈×D₄	1	1	yes	characteristic
314	C₂×C₄	#435	1	1	yes	characteristic
315	C₂×C₄	#435	1	1	yes	characteristic
316	C₂×C₄	#435	1	1	yes	characteristic
317	C₂×C₄	#435	1	1	yes	characteristic
318	C₂×C₄	#435	1	1	yes	characteristic
319	C₂×C₄	#436	1	1	yes	characteristic
320	C₂×C₄	#436	1	1	yes	characteristic
321	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
322	C₂×C₄	#446	1	1	yes	characteristic
323	C₂×C₄	#446	1	1	yes	characteristic
324	C₈	—	4	2	no	—
325	C₈	—	4	2	no	—
326	C₈	—	4	2	no	—
327	C₈	#396	1	1	yes	characteristic
328	C₈	#397	1	1	yes	characteristic
329	C₈	#397	1	1	yes	characteristic
330	C₈	#397	1	1	yes	characteristic
331	C₈	#400	1	1	yes	characteristic
332	C₈	#400	1	1	yes	characteristic
333	C₈	#400	1	1	yes	characteristic
334	C₈	#401	1	1	yes	characteristic
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	#1840	1	1	yes	characteristic
344	C₂²	#1916	1	1	yes	characteristic
345	C₂²	#1917	1	1	yes	characteristic
346	C₂²	#1917	1	1	yes	characteristic
347	C₂²	#1923	1	1	yes	characteristic
348	C₂²	#1923	1	1	yes	characteristic
349	C₂²	#1924	1	1	yes	characteristic
350	C₂²	#2170	1	2	yes	—
351	C₂²	#2171	1	2	yes	—
352	C₄	—	4	4	no	—
353	C₄	—	8	2	no	—
354	C₄	—	8	1	no	—
355	C₄	—	8	1	no	—
356	C₄	#1443	1	1	yes	characteristic
357	C₄	#1839	1	1	yes	characteristic
358	C₄	#1916	1	1	yes	characteristic
359	C₄	#1916	1	1	yes	characteristic
360	C₄	#1917	1	1	yes	characteristic
361	C₄	#1923	1	1	yes	characteristic
362	C₄	#1924	1	1	yes	characteristic
363	C₄	#1924	1	1	yes	characteristic
364	C₂	—	2	1	no	—
365	C₂	—	2	1	no	—
366	C₂	—	2	1	no	—
367	C₂	—	2	1	no	—
368	C₂	#11298	1	1	yes	characteristic
369	C₂	#13106	1	2	yes	—
370	C₂	#13122	1	2	yes	—
371	C₂	#13124	1	1	yes	characteristic
372	C₂	#13125	1	1	yes	characteristic

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

Maximal subgroups

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

subgroup	index
C₂×#1379	2
C₂×#1376	2
#12545	2
C₂×#1374	2
#10706	2