Subgroups — #149405

Order 512.

554 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#367	C₂	1	1	yes	maximal, characteristic
2	C₂×#1380	C₂	1	1	yes	maximal, characteristic
3	C₂×#1380	C₂	1	1	yes	maximal, characteristic
4	C₂×#1380	C₂	1	1	yes	maximal, characteristic
5	C₂×#1374	C₂	1	1	yes	maximal, characteristic
6	#10862	C₂	1	1	yes	maximal, characteristic
7	#10862	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₁₆	C₂²	1	1	yes	characteristic
9	C₂³×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×#107	C₂²	1	1	yes	characteristic
11	C₂²×#107	C₂²	1	1	yes	characteristic
12	C₂²×#107	C₂²	1	1	yes	characteristic
13	C₂²×#107	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×#367	—	2	2	no	—
16	C₂×#367	—	2	2	no	—
17	C₂×#367	C₄	1	4	yes	complemented
18	C₂×#340	—	2	1	no	—
19	C₂×#340	—	2	1	no	—
20	C₄⋊#95	—	2	1	no	—
21	C₄⋊#95	—	2	1	no	—
22	#1380	—	2	2	no	—
23	#1380	—	2	2	no	—
24	#1380	—	2	2	no	—
25	#1374	—	2	2	no	—
26	C₂³×C₈	D₄	1	1	yes	characteristic
27	C₂³×C₈	C₂³	1	1	yes	characteristic
28	C₂²×C₄⋊C₄	—	2	1	no	—
29	C₂²×C₄⋊C₄	—	2	1	no	—
30	C₂²×C₄⋊C₄	—	2	1	no	—
31	C₂²×C₄⋊C₄	—	2	1	no	—
32	C₂²×C₄²	D₄	1	1	yes	characteristic
33	C₂²×C₁₆	—	2	1	no	—
34	C₂²×C₁₆	—	2	1	no	—
35	C₂²×C₁₆	—	2	1	no	—
36	C₂²×C₁₆	—	2	1	no	—
37	C₂²×C₁₆	—	2	1	no	—
38	C₂²×C₁₆	—	2	1	no	—
39	C₂²×C₁₆	—	2	1	no	—
40	C₂²×C₁₆	—	2	1	no	—
41	C₂²×C₁₆	—	2	1	no	—
42	C₂²×C₁₆	—	2	1	no	—
43	C₂²×C₁₆	C₂×C₄	1	2	yes	—
44	C₂²×C₁₆	D₄	1	2	yes	—
45	C₂²×C₁₆	D₄	1	2	yes	—
46	C₂²×C₁₆	D₄	1	2	yes	—
47	C₂×#107	—	2	1	no	—
48	C₂×#107	—	2	1	no	—
49	C₂×#107	—	2	1	no	—
50	C₂×#107	—	2	1	no	—
51	C₂×#107	—	2	1	no	—
52	C₂×#107	—	2	1	no	—
53	C₂×#107	—	2	1	no	—
54	C₂×#107	—	2	1	no	—
55	C₂×#107	—	2	1	no	—
56	C₂×#107	—	2	1	no	—
57	C₂×#107	—	2	1	no	—
58	C₂×#107	—	2	1	no	—
59	C₂×#107	—	2	1	no	—
60	C₂×#107	—	2	1	no	—
61	C₂×#107	—	2	1	no	—
62	C₂×#107	—	2	1	no	—
63	C₂×#107	—	2	1	no	—
64	C₂×#107	—	2	1	no	—
65	C₂×#107	—	2	1	no	—
66	C₂×#107	—	2	1	no	—
67	C₂×#107	C₂×C₄	1	2	yes	—
68	C₂×#107	C₂×C₄	1	2	yes	—
69	C₂×#107	Q₈	1	2	yes	—
70	C₂×#107	Q₈	1	2	yes	—
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×#95	—	4	2	no	—
78	C₂×#95	—	4	2	no	—
79	#367	—	2	8	no	—
80	#340	—	4	2	no	—
81	#340	—	4	2	no	—
82	C₂³×C₄	—	4	1	no	—
83	C₂³×C₄	—	4	1	no	—
84	C₂³×C₄	—	4	1	no	—
85	C₂³×C₄	—	4	1	no	—
86	C₂³×C₄	D₈	1	2	yes	—
87	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₈	—	2	1	no	—
89	C₂²×C₈	—	2	1	no	—
90	C₂²×C₈	—	2	1	no	—
91	C₂²×C₈	—	2	1	no	—
92	C₂²×C₈	—	2	1	no	—
93	C₂²×C₈	—	2	1	no	—
94	C₂²×C₈	D₈	1	2	yes	—
95	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
100	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
101	C₂²×C₈	#41	1	1	yes	characteristic
102	C₂²×C₈	#41	1	1	yes	characteristic
103	C₂²×C₈	#41	1	1	yes	characteristic
104	C₂²×C₈	#41	1	1	yes	characteristic
105	C₂²×C₈	#41	1	1	yes	characteristic
106	C₂²×C₈	#41	1	1	yes	characteristic
107	C₂²×C₈	#41	1	1	yes	characteristic
108	C₂²×C₈	#41	1	1	yes	characteristic
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	4	1	no	—
114	C₂×C₄⋊C₄	—	4	1	no	—
115	C₂×C₄⋊C₄	—	4	1	no	—
116	C₂×C₄⋊C₄	—	4	1	no	—
117	C₂×C₄⋊C₄	—	4	1	no	—
118	C₂×C₄⋊C₄	—	4	1	no	—
119	C₂×C₄⋊C₄	—	4	1	no	—
120	C₂×C₄⋊C₄	—	4	1	no	—
121	C₂×C₄⋊C₄	—	4	1	no	—
122	C₂×C₄⋊C₄	—	4	1	no	—
123	C₂×C₄⋊C₄	—	4	1	no	—
124	C₂×C₄⋊C₄	—	4	1	no	—
125	C₂×C₄⋊C₄	—	4	1	no	—
126	C₂×C₄⋊C₄	—	4	1	no	—
127	C₂×C₄⋊C₄	—	4	1	no	—
128	C₂×C₄⋊C₄	—	4	1	no	—
129	C₂×C₄⋊C₄	—	4	1	no	—
130	C₂×C₄⋊C₄	—	4	1	no	—
131	C₂×C₄⋊C₄	—	4	1	no	—
132	C₂×C₄⋊C₄	—	4	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×C₁₆	—	2	2	no	—
140	C₂×C₁₆	—	2	2	no	—
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	—	2	2	no	—
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	2	2	no	—
146	C₂×C₁₆	—	2	2	no	—
147	C₂×C₁₆	—	2	2	no	—
148	C₂×C₁₆	—	4	1	no	—
149	C₂×C₁₆	—	4	1	no	—
150	C₂×C₁₆	—	4	1	no	—
151	C₂×C₁₆	—	4	1	no	—
152	C₂×C₁₆	C₂²⋊C₄	1	4	yes	—
153	#107	—	2	2	no	—
154	#107	—	2	2	no	—
155	#107	—	2	2	no	—
156	#107	—	2	2	no	—
157	#107	—	2	2	no	—
158	#107	—	2	2	no	—
159	#107	—	2	2	no	—
160	#107	—	2	2	no	—
161	#107	—	4	1	no	—
162	#107	—	4	1	no	—
163	#107	—	4	1	no	—
164	#107	—	4	1	no	—
165	#107	—	4	1	no	—
166	#107	—	4	1	no	—
167	#107	—	4	1	no	—
168	#107	—	4	1	no	—
169	C₄×C₈	—	4	1	no	—
170	C₄×C₈	—	4	1	no	—
171	C₄×C₈	—	4	1	no	—
172	C₄×C₈	—	4	1	no	—
173	#95	—	8	4	no	—
174	#95	—	8	4	no	—
175	C₂⁴	C₂×D₈	1	1	yes	characteristic
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	—	8	1	no	—
187	C₂²×C₄	—	8	1	no	—
188	C₂²×C₄	—	8	1	no	—
189	C₂²×C₄	—	8	1	no	—
190	C₂²×C₄	—	8	1	no	—
191	C₂²×C₄	—	8	1	no	—
192	C₂²×C₄	—	8	1	no	—
193	C₂²×C₄	—	8	1	no	—
194	C₂²×C₄	—	8	2	no	—
195	C₂²×C₄	—	8	2	no	—
196	C₂²×C₄	—	8	2	no	—
197	C₂²×C₄	—	8	2	no	—
198	C₂²×C₄	—	8	2	no	—
199	C₂²×C₄	—	8	2	no	—
200	C₂²×C₄	—	8	2	no	—
201	C₂²×C₄	—	8	2	no	—
202	C₂²×C₄	SD₃₂	1	4	yes	—
203	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
204	C₂²×C₄	#121	1	1	yes	characteristic
205	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
206	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
207	C₂²×C₄	#123	1	1	yes	characteristic
208	C₂²×C₄	#123	1	1	yes	characteristic
209	C₂²×C₄	#127	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
211	C₂²×C₄	#135	1	1	yes	characteristic
212	C₂²×C₄	#135	1	1	yes	characteristic
213	C₂²×C₄	#137	1	1	yes	characteristic
214	C₂²×C₄	#137	1	1	yes	characteristic
215	C₂²×C₄	#137	1	1	yes	characteristic
216	C₂²×C₄	#137	1	1	yes	characteristic
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
225	C₂×C₈	#117	1	1	yes	characteristic
226	C₂×C₈	#117	1	1	yes	characteristic
227	C₂×C₈	C₄×D₄	1	1	yes	characteristic
228	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
229	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
230	C₂×C₈	#121	1	1	yes	characteristic
231	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
233	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
234	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
235	C₂×C₈	#123	1	1	yes	characteristic
236	C₂×C₈	#123	1	1	yes	characteristic
237	C₂×C₈	#123	1	1	yes	characteristic
238	C₂×C₈	#123	1	1	yes	characteristic
239	C₂×C₈	#123	1	1	yes	characteristic
240	C₂×C₈	#123	1	1	yes	characteristic
241	C₂×C₈	#124	1	1	yes	characteristic
242	C₂×C₈	#124	1	1	yes	characteristic
243	C₂×C₈	#125	1	1	yes	characteristic
244	C₂×C₈	#125	1	1	yes	characteristic
245	C₂×C₈	#125	1	1	yes	characteristic
246	C₂×C₈	#125	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	C₄⋊Q₈	1	1	yes	characteristic
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	8	1	no	—
261	C₄⋊C₄	—	8	1	no	—
262	C₄⋊C₄	—	8	1	no	—
263	C₄⋊C₄	—	8	1	no	—
264	C₄⋊C₄	—	8	1	no	—
265	C₄⋊C₄	—	8	1	no	—
266	C₄⋊C₄	—	8	1	no	—
267	C₄⋊C₄	—	8	1	no	—
268	C₄²	—	4	1	no	—
269	C₄²	—	4	1	no	—
270	C₄²	—	4	1	no	—
271	C₄²	—	4	1	no	—
272	C₁₆	—	2	4	no	—
273	C₁₆	—	4	2	no	—
274	C₂³	C₄×D₈	1	1	yes	characteristic
275	C₂³	#466	1	1	yes	characteristic
276	C₂³	#474	1	1	yes	characteristic
277	C₂³	#474	1	1	yes	characteristic
278	C₂³	#480	1	1	yes	characteristic
279	C₂³	#480	1	1	yes	characteristic
280	C₂³	#493	1	1	yes	characteristic
281	C₂³	C₂×SD₃₂	1	2	yes	—
282	C₂³	#508	1	2	yes	—
283	C₂³	#509	1	2	yes	—
284	C₂³	#510	1	2	yes	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	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	—	8	2	no	—
293	C₂×C₄	—	8	2	no	—
294	C₂×C₄	—	8	4	no	—
295	C₂×C₄	—	8	4	no	—
296	C₂×C₄	—	8	2	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	—	8	4	no	—
299	C₂×C₄	—	8	4	no	—
300	C₂×C₄	—	16	2	no	—
301	C₂×C₄	—	16	2	no	—
302	C₂×C₄	—	16	2	no	—
303	C₂×C₄	—	16	2	no	—
304	C₂×C₄	—	16	1	no	—
305	C₂×C₄	—	16	1	no	—
306	C₂×C₄	—	16	1	no	—
307	C₂×C₄	—	16	1	no	—
308	C₂×C₄	—	16	1	no	—
309	C₂×C₄	—	16	1	no	—
310	C₂×C₄	—	16	1	no	—
311	C₂×C₄	—	16	1	no	—
312	C₂×C₄	#387	1	1	yes	characteristic
313	C₂×C₄	#387	1	1	yes	characteristic
314	C₂×C₄	#390	1	1	yes	characteristic
315	C₂×C₄	#397	1	1	yes	characteristic
316	C₂×C₄	#398	1	1	yes	characteristic
317	C₂×C₄	#399	1	1	yes	characteristic
318	C₂×C₄	#399	1	1	yes	characteristic
319	C₂×C₄	C₄×D₈	1	1	yes	characteristic
320	C₂×C₄	#441	1	1	yes	characteristic
321	C₂×C₄	#441	1	1	yes	characteristic
322	C₂×C₄	#466	1	1	yes	characteristic
323	C₂×C₄	#470	1	1	yes	characteristic
324	C₂×C₄	#470	1	1	yes	characteristic
325	C₂×C₄	#474	1	1	yes	characteristic
326	C₂×C₄	#474	1	1	yes	characteristic
327	C₂×C₄	#479	1	1	yes	characteristic
328	C₂×C₄	#479	1	1	yes	characteristic
329	C₂×C₄	#479	1	1	yes	characteristic
330	C₂×C₄	#479	1	1	yes	characteristic
331	C₂×C₄	#480	1	1	yes	characteristic
332	C₂×C₄	#480	1	1	yes	characteristic
333	C₂×C₄	#485	1	1	yes	characteristic
334	C₂×C₄	#485	1	1	yes	characteristic
335	C₂×C₄	#485	1	1	yes	characteristic
336	C₂×C₄	#485	1	1	yes	characteristic
337	C₂×C₄	#495	1	1	yes	characteristic
338	C₂×C₄	#497	1	1	yes	characteristic
339	C₂×C₄	#497	1	1	yes	characteristic
340	C₈	—	4	2	no	—
341	C₈	#381	1	1	yes	characteristic
342	C₈	#387	1	1	yes	characteristic
343	C₈	#387	1	1	yes	characteristic
344	C₈	#391	1	1	yes	characteristic
345	C₈	#397	1	1	yes	characteristic
346	C₈	#398	1	1	yes	characteristic
347	C₈	#400	1	1	yes	characteristic
348	C₈	#400	1	1	yes	characteristic
349	C₂²	#1918	1	1	yes	characteristic
350	C₂²	#1918	1	1	yes	characteristic
351	C₂²	#1970	1	1	yes	characteristic
352	C₂²	#2075	1	1	yes	characteristic
353	C₂²	#2086	1	1	yes	characteristic
354	C₂²	#2093	1	1	yes	characteristic
355	C₂²	#2093	1	1	yes	characteristic
356	C₂²	C₄×SD₃₂	1	2	yes	—
357	C₂²	#2175	1	2	yes	—
358	C₂²	#2217	1	2	yes	—
359	C₂²	#2220	1	2	yes	—
360	C₂²	#2225	1	2	yes	—
361	C₂²	#2226	1	2	yes	—
362	C₂²	#2228	1	2	yes	—
363	C₂²	#2229	1	2	yes	—
364	C₂²	#2233	1	2	yes	—
365	C₂²	#2234	1	2	yes	—
366	C₂²	#2235	1	2	yes	—
367	C₂²	#2237	1	2	yes	—
368	C₂²	#2248	1	1	yes	characteristic
369	C₂²	#2249	1	1	yes	characteristic
370	C₂²	#2250	1	1	yes	characteristic
371	C₂²	#2251	1	1	yes	characteristic
372	C₄	—	4	4	no	—
373	C₄	—	16	2	no	—
374	C₄	—	16	2	no	—
375	C₄	—	16	2	no	—
376	C₄	—	16	2	no	—
377	C₄	#1445	1	1	yes	characteristic
378	C₄	#1918	1	1	yes	characteristic
379	C₄	#1918	1	1	yes	characteristic
380	C₄	#1971	1	1	yes	characteristic
381	C₄	#2075	1	1	yes	characteristic
382	C₄	#2086	1	1	yes	characteristic
383	C₄	#2097	1	1	yes	characteristic
384	C₄	#2097	1	1	yes	characteristic
385	C₂	#11379	1	1	yes	characteristic
386	C₂	#13110	1	2	yes	—
387	C₂	#13111	1	2	yes	—
388	C₂	#13237	1	1	yes	characteristic
389	C₂	#13238	1	1	yes	characteristic
390	C₂	#13494	1	2	yes	—
391	C₂	#13516	1	2	yes	—
392	C₂	#13527	1	1	yes	characteristic
393	C₂	#13528	1	1	yes	characteristic
394	C₂	#13529	1	1	yes	characteristic
395	C₂	#13531	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×D₈
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ 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×#367	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#367	2
C₂×#1380	2
C₂×#1374	2
#10862	2