Subgroups — #149204

Order 512.

626 conjugacy classes of subgroups.

The subgroup lattice has 407 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	C₂²×#366	C₂	1	1	yes	maximal, characteristic
3	C₂×#1846	C₂	1	1	yes	maximal, characteristic
4	C₂×#1380	C₂	1	1	yes	maximal, characteristic
5	C₂×#1380	C₂	1	1	yes	maximal, characteristic
6	C₂×#1374	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×#106	—	2	1	no	—
15	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×#382	—	2	1	no	—
17	C₂×#367	—	2	2	no	—
18	C₂×#367	—	2	2	no	—
19	C₂×#367	—	2	2	no	—
20	C₂×#366	—	2	2	no	—
21	C₂×#366	—	2	2	no	—
22	C₂×#366	—	2	2	no	—
23	#1846	—	2	2	no	—
24	#1846	—	2	2	no	—
25	C₂×#340	—	2	1	no	—
26	C₄⋊#95	—	2	1	no	—
27	#1380	—	2	2	no	—
28	#1380	—	2	2	no	—
29	#1374	C₄	1	4	yes	complemented
30	C₂³×C₈	D₄	1	1	yes	characteristic
31	C₂³×C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₄⋊C₄	—	2	1	no	—
33	C₂²×C₄⋊C₄	—	2	1	no	—
34	C₂²×C₄⋊C₄	—	2	1	no	—
35	C₂²×C₄⋊C₄	—	2	1	no	—
36	C₂²×C₄²	D₄	1	1	yes	characteristic
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	—	2	1	no	—
44	C₂²×C₁₆	—	2	1	no	—
45	C₂²×C₁₆	C₂×C₄	1	2	yes	—
46	C₂²×C₁₆	C₂×C₄	1	2	yes	—
47	C₂²×C₁₆	D₄	1	2	yes	—
48	C₂²×C₁₆	D₄	1	2	yes	—
49	C₂²×C₁₆	Q₈	1	2	yes	—
50	C₂²×C₁₆	Q₈	1	2	yes	—
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	—	2	1	no	—
68	C₂×#107	—	2	1	no	—
69	C₂×#107	—	2	1	no	—
70	C₂×#107	—	2	1	no	—
71	C₂×#107	D₄	1	2	yes	—
72	C₂×#107	D₄	1	2	yes	—
73	C₂×#107	D₄	1	2	yes	—
74	C₂×#107	Q₈	1	2	yes	—
75	C₂×#106	—	2	4	no	—
76	C₂×#106	—	4	2	no	—
77	C₂×#106	—	4	2	no	—
78	C₂×C₄×C₈	—	2	1	no	—
79	C₂×C₄×C₈	—	2	1	no	—
80	C₂×C₄×C₈	—	2	1	no	—
81	C₂×C₄×C₈	—	2	1	no	—
82	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
83	C₂×C₄×C₈	Q₈	1	2	yes	—
84	#382	—	2	4	no	—
85	#382	—	4	2	no	—
86	C₂×#95	—	4	2	no	—
87	C₂×#95	—	4	2	no	—
88	#367	—	2	8	no	—
89	#366	—	2	8	no	—
90	#340	—	4	2	no	—
91	C₂³×C₄	—	4	1	no	—
92	C₂³×C₄	—	4	1	no	—
93	C₂³×C₄	—	4	1	no	—
94	C₂³×C₄	—	4	1	no	—
95	C₂³×C₄	D₈	1	2	yes	—
96	C₂³×C₄	C₂×D₄	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	C₄⋊C₄	1	2	yes	—
102	C₂²×C₈	D₈	1	2	yes	—
103	C₂²×C₈	Q₁₆	1	2	yes	—
104	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
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×D₄	1	1	yes	characteristic
110	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
111	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
112	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
113	C₂²×C₈	C₂×Q₈	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	C₂×C₄⋊C₄	—	2	2	no	—
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	2	2	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×C₄⋊C₄	—	4	1	no	—
134	C₂×C₄⋊C₄	—	4	1	no	—
135	C₂×C₄⋊C₄	—	4	1	no	—
136	C₂×C₄⋊C₄	—	4	1	no	—
137	C₂×C₄⋊C₄	—	4	1	no	—
138	C₂×C₄⋊C₄	—	4	1	no	—
139	C₂×C₄⋊C₄	—	4	1	no	—
140	C₂×C₄⋊C₄	—	4	1	no	—
141	C₂×C₄⋊C₄	—	4	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	C₄⋊C₄	1	2	yes	—
147	C₂×C₄²	Q₁₆	1	2	yes	—
148	C₂×C₁₆	—	2	2	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	—	2	2	no	—
158	C₂×C₁₆	—	2	2	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
161	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
162	#107	—	2	2	no	—
163	#107	—	2	2	no	—
164	#107	—	2	2	no	—
165	#107	—	2	2	no	—
166	#107	—	2	2	no	—
167	#107	—	2	2	no	—
168	#107	—	2	2	no	—
169	#107	—	2	2	no	—
170	#107	—	4	1	no	—
171	#107	—	4	1	no	—
172	#107	—	4	1	no	—
173	#107	—	4	1	no	—
174	#107	—	4	1	no	—
175	#107	—	4	1	no	—
176	#107	—	4	1	no	—
177	#107	—	4	1	no	—
178	#106	—	4	8	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	#95	—	4	8	no	—
184	#95	—	8	4	no	—
185	C₂⁴	C₂×D₈	1	1	yes	characteristic
186	C₂²×C₄	—	2	2	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	—	4	2	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	8	1	no	—
197	C₂²×C₄	—	8	1	no	—
198	C₂²×C₄	—	8	1	no	—
199	C₂²×C₄	—	8	1	no	—
200	C₂²×C₄	—	8	1	no	—
201	C₂²×C₄	—	8	1	no	—
202	C₂²×C₄	—	8	1	no	—
203	C₂²×C₄	—	8	1	no	—
204	C₂²×C₄	—	8	2	no	—
205	C₂²×C₄	—	8	2	no	—
206	C₂²×C₄	—	8	2	no	—
207	C₂²×C₄	—	8	2	no	—
208	C₂²×C₄	—	8	2	no	—
209	C₂²×C₄	—	8	2	no	—
210	C₂²×C₄	—	8	2	no	—
211	C₂²×C₄	—	8	2	no	—
212	C₂²×C₄	#107	1	4	yes	—
213	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
214	C₂²×C₄	#120	1	1	yes	characteristic
215	C₂²×C₄	#121	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
217	C₂²×C₄	#123	1	1	yes	characteristic
218	C₂²×C₄	#127	1	1	yes	characteristic
219	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
221	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
222	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
223	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
224	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
225	C₂²×C₄	#137	1	1	yes	characteristic
226	C₂²×C₄	#137	1	1	yes	characteristic
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	#107	1	4	yes	—
234	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
235	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
236	C₂×C₈	C₄×D₄	1	1	yes	characteristic
237	C₂×C₈	C₄×D₄	1	1	yes	characteristic
238	C₂×C₈	C₄×D₄	1	1	yes	characteristic
239	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
240	C₂×C₈	#120	1	1	yes	characteristic
241	C₂×C₈	#120	1	1	yes	characteristic
242	C₂×C₈	#121	1	1	yes	characteristic
243	C₂×C₈	#121	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	C₂²⋊Q₈	1	1	yes	characteristic
247	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
248	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
250	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
251	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
252	C₂×C₈	#123	1	1	yes	characteristic
253	C₂×C₈	#123	1	1	yes	characteristic
254	C₂×C₈	#125	1	1	yes	characteristic
255	C₂×C₈	#125	1	1	yes	characteristic
256	C₂×C₈	#125	1	1	yes	characteristic
257	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
258	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
259	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
260	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
261	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	8	1	no	—
271	C₄⋊C₄	—	8	1	no	—
272	C₄⋊C₄	—	8	1	no	—
273	C₄⋊C₄	—	8	1	no	—
274	C₄⋊C₄	—	8	1	no	—
275	C₄⋊C₄	—	8	1	no	—
276	C₄⋊C₄	—	8	1	no	—
277	C₄⋊C₄	—	8	1	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₁₆	—	2	4	no	—
283	C₁₆	—	2	4	no	—
284	C₂³	C₂×#107	1	1	yes	characteristic
285	C₂³	#447	1	1	yes	characteristic
286	C₂³	#459	1	1	yes	characteristic
287	C₂³	#474	1	1	yes	characteristic
288	C₂³	#480	1	1	yes	characteristic
289	C₂³	#493	1	1	yes	characteristic
290	C₂³	#500	1	1	yes	characteristic
291	C₂³	#508	1	2	yes	—
292	C₂³	#508	1	2	yes	—
293	C₂³	#509	1	2	yes	—
294	C₂³	#510	1	2	yes	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	8	2	no	—
303	C₂×C₄	—	8	4	no	—
304	C₂×C₄	—	8	4	no	—
305	C₂×C₄	—	8	4	no	—
306	C₂×C₄	—	8	4	no	—
307	C₂×C₄	—	8	2	no	—
308	C₂×C₄	—	8	2	no	—
309	C₂×C₄	—	8	2	no	—
310	C₂×C₄	—	16	2	no	—
311	C₂×C₄	—	16	2	no	—
312	C₂×C₄	—	16	2	no	—
313	C₂×C₄	—	16	2	no	—
314	C₂×C₄	—	16	1	no	—
315	C₂×C₄	—	16	1	no	—
316	C₂×C₄	—	16	1	no	—
317	C₂×C₄	—	16	1	no	—
318	C₂×C₄	—	16	1	no	—
319	C₂×C₄	—	16	1	no	—
320	C₂×C₄	—	16	1	no	—
321	C₂×C₄	—	16	1	no	—
322	C₂×C₄	#384	1	1	yes	characteristic
323	C₂×C₄	#385	1	1	yes	characteristic
324	C₂×C₄	#389	1	1	yes	characteristic
325	C₂×C₄	#392	1	1	yes	characteristic
326	C₂×C₄	#395	1	1	yes	characteristic
327	C₂×C₄	#398	1	1	yes	characteristic
328	C₂×C₄	#399	1	1	yes	characteristic
329	C₂×C₄	C₂×#107	1	1	yes	characteristic
330	C₂×C₄	#428	1	1	yes	characteristic
331	C₂×C₄	#428	1	1	yes	characteristic
332	C₂×C₄	#447	1	1	yes	characteristic
333	C₂×C₄	#451	1	1	yes	characteristic
334	C₂×C₄	#451	1	1	yes	characteristic
335	C₂×C₄	#459	1	1	yes	characteristic
336	C₂×C₄	#462	1	1	yes	characteristic
337	C₂×C₄	#462	1	1	yes	characteristic
338	C₂×C₄	#474	1	1	yes	characteristic
339	C₂×C₄	#477	1	1	yes	characteristic
340	C₂×C₄	#477	1	1	yes	characteristic
341	C₂×C₄	#480	1	1	yes	characteristic
342	C₂×C₄	#484	1	1	yes	characteristic
343	C₂×C₄	#484	1	1	yes	characteristic
344	C₂×C₄	#494	1	1	yes	characteristic
345	C₂×C₄	#496	1	1	yes	characteristic
346	C₂×C₄	#497	1	1	yes	characteristic
347	C₂×C₄	#500	1	1	yes	characteristic
348	C₂×C₄	#501	1	1	yes	characteristic
349	C₂×C₄	#501	1	1	yes	characteristic
350	C₈	—	2	4	no	—
351	C₈	#385	1	1	yes	characteristic
352	C₈	#385	1	1	yes	characteristic
353	C₈	#389	1	1	yes	characteristic
354	C₈	#389	1	1	yes	characteristic
355	C₈	#395	1	1	yes	characteristic
356	C₈	#395	1	1	yes	characteristic
357	C₈	#398	1	1	yes	characteristic
358	C₈	#398	1	1	yes	characteristic
359	C₂²	#1845	1	1	yes	characteristic
360	C₂²	#1852	1	1	yes	characteristic
361	C₂²	#1947	1	1	yes	characteristic
362	C₂²	#2011	1	1	yes	characteristic
363	C₂²	#2057	1	1	yes	characteristic
364	C₂²	#2083	1	1	yes	characteristic
365	C₂²	#2093	1	1	yes	characteristic
366	C₂²	#2158	1	2	yes	—
367	C₂²	#2159	1	2	yes	—
368	C₂²	#2186	1	2	yes	—
369	C₂²	#2189	1	2	yes	—
370	C₂²	#2210	1	2	yes	—
371	C₂²	#2211	1	2	yes	—
372	C₂²	#2228	1	2	yes	—
373	C₂²	#2229	1	2	yes	—
374	C₂²	#2234	1	2	yes	—
375	C₂²	#2237	1	2	yes	—
376	C₂²	#2249	1	1	yes	characteristic
377	C₂²	#2249	1	1	yes	characteristic
378	C₂²	#2250	1	1	yes	characteristic
379	C₂²	#2251	1	1	yes	characteristic
380	C₂²	#2253	1	2	yes	—
381	C₂²	#2255	1	2	yes	—
382	C₄	—	4	4	no	—
383	C₄	—	16	2	no	—
384	C₄	—	16	2	no	—
385	C₄	—	16	2	no	—
386	C₄	—	16	2	no	—
387	C₄	#1444	1	1	yes	characteristic
388	C₄	#1847	1	1	yes	characteristic
389	C₄	#1852	1	1	yes	characteristic
390	C₄	#1947	1	1	yes	characteristic
391	C₄	#2016	1	1	yes	characteristic
392	C₄	#2057	1	1	yes	characteristic
393	C₄	#2083	1	1	yes	characteristic
394	C₄	#2095	1	1	yes	characteristic
395	C₂	#11324	1	1	yes	characteristic
396	C₂	#12907	1	1	yes	characteristic
397	C₂	#12908	1	1	yes	characteristic
398	C₂	#12921	1	2	yes	—
399	C₂	#13178	1	2	yes	—
400	C₂	#13326	1	1	yes	characteristic
401	C₂	#13329	1	1	yes	characteristic
402	C₂	#13453	1	2	yes	—
403	C₂	#13507	1	2	yes	—
404	C₂	#13528	1	1	yes	characteristic
405	C₂	#13529	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
29. #1374	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
C₂²×#366	2
C₂×#1846	2
C₂×#1380	2
C₂×#1374	2
#10862	2