Subgroups — #149262

Order 512.

534 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	C₂×#2159	C₂	1	1	yes	maximal, characteristic
2	#21181	C₂	1	1	yes	maximal, characteristic
3	#21156	C₂	1	1	yes	maximal, characteristic
4	C₂×#1389	C₂	1	1	yes	maximal, characteristic
5	C₂×#1387	C₂	1	1	yes	maximal, characteristic
6	#12567	C₂	1	1	yes	maximal, characteristic
7	#12554	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₃₂	C₂²	1	1	yes	characteristic
9	C₂²×M₃₂	C₂²	1	1	yes	characteristic
10	C₂²×#108	C₂²	1	1	yes	characteristic
11	#2911	—	2	1	no	—
12	C₂×#427	C₂²	1	1	yes	characteristic
13	C₂×#427	C₂²	1	1	yes	characteristic
14	C₂²×#107	C₂²	1	1	yes	characteristic
15	C₂×#405	C₂²	1	1	yes	characteristic
16	#2317	—	2	1	no	—
17	#2159	—	2	2	no	—
18	#2159	—	2	2	no	—
19	C₂×#367	—	2	1	no	—
20	C₂×#366	—	2	1	no	—
21	#1952	—	2	1	no	—
22	#1952	—	2	1	no	—
23	#1951	—	2	2	no	—
24	#1944	—	2	2	no	—
25	#1942	—	2	2	no	—
26	#1742	—	2	1	no	—
27	#1736	—	2	1	no	—
28	#1389	C₄	1	4	yes	complemented
29	#1387	—	2	2	no	—
30	#1386	—	2	2	no	—
31	#1379	—	2	2	no	—
32	C₂²×M₁₆	—	2	1	no	—
33	C₂²×M₁₆	D₄	1	1	yes	characteristic
34	C₂³×C₈	C₂³	1	1	yes	characteristic
35	C₂×#117	—	2	1	no	—
36	C₂×#117	—	2	1	no	—
37	C₂×#117	D₄	1	1	yes	characteristic
38	C₂²×C₄⋊C₄	—	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	C₂×C₄	1	2	yes	—
48	C₂×M₃₂	D₄	1	2	yes	—
49	C₂×M₃₂	Q₈	1	2	yes	—
50	C₂×M₃₂	Q₈	1	2	yes	—
51	C₂²×C₁₆	—	2	1	no	—
52	C₂²×C₁₆	—	2	1	no	—
53	C₂×#108	—	2	1	no	—
54	C₂×#108	—	2	1	no	—
55	C₂×#108	—	2	1	no	—
56	C₂×#108	C₂×C₄	1	2	yes	—
57	C₂×#108	D₄	1	2	yes	—
58	C₂×#108	Q₈	1	2	yes	—
59	#428	—	2	2	no	—
60	#428	—	2	2	no	—
61	#428	—	4	1	no	—
62	#428	—	4	1	no	—
63	#427	—	2	1	no	—
64	#427	—	2	1	no	—
65	#427	—	2	1	no	—
66	#427	—	2	1	no	—
67	#427	—	2	1	no	—
68	#427	—	2	1	no	—
69	#427	—	4	2	no	—
70	#427	C₂×C₄	1	2	yes	—
71	#427	Q₈	1	2	yes	—
72	C₂×#107	—	2	1	no	—
73	C₂×#107	—	2	1	no	—
74	C₂×#107	—	2	1	no	—
75	C₂×#107	—	2	1	no	—
76	C₂×#107	—	2	1	no	—
77	C₂×#107	—	2	1	no	—
78	C₂×#107	D₄	1	2	yes	—
79	C₂×#107	D₄	1	1	yes	characteristic
80	C₂×#107	D₄	1	1	yes	characteristic
81	C₂×#106	—	2	1	no	—
82	C₂×#106	D₄	1	2	yes	—
83	#405	—	2	1	no	—
84	#405	—	2	1	no	—
85	#405	—	2	1	no	—
86	#405	—	2	1	no	—
87	C₂×#97	—	2	1	no	—
88	C₂×C₄×C₈	—	2	1	no	—
89	#389	—	2	4	no	—
90	#385	—	4	2	no	—
91	#384	—	4	1	no	—
92	#382	—	4	1	no	—
93	#367	—	2	4	no	—
94	#366	—	2	4	no	—
95	#343	—	2	2	no	—
96	#343	—	2	2	no	—
97	#340	—	2	4	no	—
98	C₂³×C₄	—	4	1	no	—
99	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
100	C₂×M₁₆	—	2	1	no	—
101	C₂×M₁₆	—	2	2	no	—
102	C₂×M₁₆	—	2	2	no	—
103	C₂×M₁₆	—	2	1	no	—
104	C₂×M₁₆	—	2	1	no	—
105	C₂×M₁₆	—	2	2	no	—
106	C₂×M₁₆	—	2	1	no	—
107	C₂×M₁₆	—	2	1	no	—
108	C₂×M₁₆	—	4	1	no	—
109	C₂×M₁₆	—	4	1	no	—
110	C₂×M₁₆	—	4	1	no	—
111	C₂×M₁₆	Q₁₆	1	2	yes	—
112	C₂²×C₈	—	2	1	no	—
113	C₂²×C₈	—	4	1	no	—
114	C₂²×C₈	C₂²×C₄	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×D₄	1	1	yes	characteristic
118	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
121	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
122	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
123	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
124	C₂²×C₈	#41	1	1	yes	characteristic
125	C₂²×C₈	#41	1	1	yes	characteristic
126	C₂²×C₈	#41	1	1	yes	characteristic
127	C₂²×C₈	#41	1	1	yes	characteristic
128	#117	—	2	2	no	—
129	#117	—	2	1	no	—
130	#117	—	2	1	no	—
131	#117	—	2	2	no	—
132	#117	—	2	1	no	—
133	#117	—	2	1	no	—
134	#117	—	4	1	no	—
135	#117	—	4	1	no	—
136	#117	—	4	1	no	—
137	#117	—	4	1	no	—
138	#117	—	4	1	no	—
139	#117	—	4	1	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	2	1	no	—
143	C₂×C₄⋊C₄	—	2	1	no	—
144	C₂×C₄⋊C₄	—	4	1	no	—
145	C₂×C₄⋊C₄	—	4	1	no	—
146	C₂×C₄⋊C₄	—	4	1	no	—
147	C₂×C₄⋊C₄	—	4	1	no	—
148	C₂×C₄⋊C₄	—	4	1	no	—
149	C₂×C₄⋊C₄	—	4	4	no	—
150	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
151	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
152	C₂×C₂²⋊C₄	—	2	1	no	—
153	C₂×C₂²⋊C₄	—	4	1	no	—
154	C₂×C₂²⋊C₄	—	4	1	no	—
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	—	4	1	no	—
157	C₂×C₄²	—	4	1	no	—
158	M₃₂	—	2	2	no	—
159	M₃₂	—	2	2	no	—
160	M₃₂	—	2	2	no	—
161	M₃₂	—	2	2	no	—
162	M₃₂	—	2	2	no	—
163	M₃₂	—	4	1	no	—
164	M₃₂	—	4	1	no	—
165	M₃₂	C₄⋊C₄	1	4	yes	complemented
166	C₂×C₁₆	—	2	2	no	—
167	C₂×C₁₆	—	2	2	no	—
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	—	2	2	no	—
170	C₂×C₁₆	—	4	1	no	—
171	C₂×C₁₆	—	4	1	no	—
172	#108	—	2	2	no	—
173	#108	—	2	2	no	—
174	#108	—	2	2	no	—
175	#108	C₄⋊C₄	1	4	yes	complemented
176	#107	—	2	2	no	—
177	#107	—	2	2	no	—
178	#107	—	2	1	no	—
179	#107	—	2	2	no	—
180	#107	—	2	1	no	—
181	#107	—	2	2	no	—
182	#107	—	2	1	no	—
183	#107	—	2	1	no	—
184	#107	—	4	2	no	—
185	#107	—	4	2	no	—
186	#107	—	4	1	no	—
187	#107	—	4	1	no	—
188	#106	—	2	2	no	—
189	#106	—	4	2	no	—
190	#106	—	4	2	no	—
191	#106	—	4	1	no	—
192	#106	—	4	1	no	—
193	#106	C₄⋊C₄	1	4	yes	—
194	#97	—	4	1	no	—
195	#97	—	4	1	no	—
196	C₄×C₈	—	4	1	no	—
197	C₄×C₈	—	4	1	no	—
198	#95	—	4	4	no	—
199	C₂⁴	AGL(1,ℤ/8)	1	1	yes	characteristic
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	4	1	no	—
205	C₂²×C₄	—	4	1	no	—
206	C₂²×C₄	—	8	1	no	—
207	C₂²×C₄	—	8	1	no	—
208	C₂²×C₄	—	16	2	no	—
209	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
210	C₂²×C₄	#120	1	1	yes	characteristic
211	C₂²×C₄	#120	1	1	yes	characteristic
212	C₂²×C₄	#121	1	1	yes	characteristic
213	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
214	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
215	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
217	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
218	C₂²×C₄	#135	1	1	yes	characteristic
219	C₂²×C₄	#135	1	1	yes	characteristic
220	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
221	C₂²×C₄	#137	1	1	yes	characteristic
222	C₂²×C₄	#137	1	1	yes	characteristic
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	2	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	1	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	4	1	no	—
230	M₁₆	—	4	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
239	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
240	C₂×C₈	C₄×D₄	1	1	yes	characteristic
241	C₂×C₈	C₄×D₄	1	1	yes	characteristic
242	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
243	C₂×C₈	#120	1	1	yes	characteristic
244	C₂×C₈	#121	1	1	yes	characteristic
245	C₂×C₈	#121	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	C₂²⋊Q₈	1	1	yes	characteristic
253	C₂×C₈	#123	1	1	yes	characteristic
254	C₂×C₈	#123	1	1	yes	characteristic
255	C₂×C₈	#123	1	1	yes	characteristic
256	C₂×C₈	#125	1	1	yes	characteristic
257	C₂×C₈	#125	1	1	yes	characteristic
258	C₂×C₈	#125	1	1	yes	characteristic
259	C₂×C₈	#127	1	1	yes	characteristic
260	C₂×C₈	C₄⋊Q₈	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	C₄⋊Q₈	1	1	yes	characteristic
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	8	1	no	—
278	C₄⋊C₄	—	8	1	no	—
279	C₄⋊C₄	—	16	4	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	4	1	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	8	1	no	—
285	C₂²⋊C₄	—	8	1	no	—
286	C₄²	—	4	1	no	—
287	C₄²	—	4	2	no	—
288	C₄²	—	4	1	no	—
289	C₄²	—	4	2	no	—
290	C₄²	—	8	1	no	—
291	C₄²	—	8	1	no	—
292	C₁₆	—	2	4	no	—
293	C₁₆	—	4	2	no	—
294	C₂³	—	4	1	no	—
295	C₂³	—	4	1	no	—
296	C₂³	#440	1	1	yes	characteristic
297	C₂³	#447	1	1	yes	characteristic
298	C₂³	#449	1	1	yes	characteristic
299	C₂³	#469	1	1	yes	characteristic
300	C₂³	#477	1	1	yes	characteristic
301	C₂³	#478	1	1	yes	characteristic
302	C₂³	#501	1	1	yes	characteristic
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	8	2	no	—
309	C₂×C₄	—	8	2	no	—
310	C₂×C₄	—	8	4	no	—
311	C₂×C₄	—	8	1	no	—
312	C₂×C₄	—	8	1	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	—	16	2	no	—
315	C₂×C₄	—	16	2	no	—
316	C₂×C₄	—	16	1	no	—
317	C₂×C₄	—	16	1	no	—
318	C₂×C₄	#385	1	1	yes	characteristic
319	C₂×C₄	#385	1	1	yes	characteristic
320	C₂×C₄	#389	1	1	yes	characteristic
321	C₂×C₄	#392	1	1	yes	characteristic
322	C₂×C₄	#395	1	1	yes	characteristic
323	C₂×C₄	#395	1	1	yes	characteristic
324	C₂×C₄	#398	1	1	yes	characteristic
325	C₂×C₄	C₄×D₈	1	1	yes	characteristic
326	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
327	C₂×C₄	#440	1	1	yes	characteristic
328	C₂×C₄	#447	1	1	yes	characteristic
329	C₂×C₄	#449	1	1	yes	characteristic
330	C₂×C₄	#451	1	1	yes	characteristic
331	C₂×C₄	#451	1	1	yes	characteristic
332	C₂×C₄	#452	1	1	yes	characteristic
333	C₂×C₄	#452	1	1	yes	characteristic
334	C₂×C₄	#466	1	1	yes	characteristic
335	C₂×C₄	#467	1	1	yes	characteristic
336	C₂×C₄	#470	1	1	yes	characteristic
337	C₂×C₄	#474	1	1	yes	characteristic
338	C₂×C₄	#474	1	1	yes	characteristic
339	C₂×C₄	#477	1	1	yes	characteristic
340	C₂×C₄	#478	1	1	yes	characteristic
341	C₂×C₄	#479	1	1	yes	characteristic
342	C₂×C₄	#479	1	1	yes	characteristic
343	C₂×C₄	#499	1	1	yes	characteristic
344	C₂×C₄	#500	1	1	yes	characteristic
345	C₂×C₄	#501	1	1	yes	characteristic
346	C₈	—	4	4	no	—
347	C₈	—	4	2	no	—
348	C₈	#384	1	1	yes	characteristic
349	C₈	#385	1	1	yes	characteristic
350	C₈	#389	1	1	yes	characteristic
351	C₈	#389	1	1	yes	characteristic
352	C₈	#395	1	1	yes	characteristic
353	C₈	#398	1	1	yes	characteristic
354	C₈	#398	1	1	yes	characteristic
355	C₈	#399	1	1	yes	characteristic
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	4	1	no	—
363	C₂²	—	4	1	no	—
364	C₂²	—	4	1	no	—
365	C₂²	—	4	1	no	—
366	C₂²	#1862	1	1	yes	characteristic
367	C₂²	#1863	1	1	yes	characteristic
368	C₂²	#1944	1	1	yes	characteristic
369	C₂²	#1999	1	1	yes	characteristic
370	C₂²	#2057	1	1	yes	characteristic
371	C₂²	#2059	1	1	yes	characteristic
372	C₂²	#2087	1	1	yes	characteristic
373	C₄	—	8	4	no	—
374	C₄	—	8	1	no	—
375	C₄	—	8	1	no	—
376	C₄	—	16	2	no	—
377	C₄	—	16	1	no	—
378	C₄	—	16	1	no	—
379	C₄	#1444	1	1	yes	characteristic
380	C₄	#1862	1	1	yes	characteristic
381	C₄	#1863	1	1	yes	characteristic
382	C₄	#1942	1	1	yes	characteristic
383	C₄	#2001	1	1	yes	characteristic
384	C₄	#2057	1	1	yes	characteristic
385	C₄	#2059	1	1	yes	characteristic
386	C₄	#2086	1	1	yes	characteristic
387	C₂	—	2	1	no	—
388	C₂	—	2	1	no	—
389	C₂	—	2	1	no	—
390	C₂	—	2	1	no	—
391	C₂	—	2	1	no	—
392	C₂	—	2	1	no	—
393	C₂	#11328	1	1	yes	characteristic
394	C₂	#13310	1	1	yes	characteristic
395	C₂	#13311	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#1999
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
28. #1389	C₄	semidirect
165. M₃₂	C₄⋊C₄	semidirect
175. #108	C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#2159	2
#21181	2
#21156	2
C₂×#1389	2
C₂×#1387	2
#12567	2
#12554	2