Subgroups — #147261

Order 512.

554 conjugacy classes of subgroups.

The subgroup lattice has 412 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	#20455	C₂	1	1	yes	maximal, characteristic
3	C₂×#1387	C₂	1	1	yes	maximal, characteristic
4	C₂×#1382	C₂	1	1	yes	maximal, characteristic
5	C₂×#1380	C₂	1	1	yes	maximal, characteristic
6	C₂×#1379	C₂	1	1	yes	maximal, characteristic
7	#10870	C₂	1	1	yes	maximal, characteristic
8	C₂²×M₃₂	C₂²	1	1	yes	characteristic
9	C₂³×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×#108	C₂²	1	1	yes	characteristic
11	C₂×#427	C₂²	1	1	yes	characteristic
12	C₂²×#107	C₂²	1	1	yes	characteristic
13	C₂²×#107	C₂²	1	1	yes	characteristic
14	C₂×#405	C₂²	1	1	yes	characteristic
15	#2274	—	2	1	no	—
16	#2159	—	2	2	no	—
17	#2159	C₄	1	4	yes	complemented
18	C₂×#367	—	2	1	no	—
19	C₂×#366	—	2	1	no	—
20	#1776	—	2	2	no	—
21	C₄×#107	—	2	1	no	—
22	C₄×#107	—	2	1	no	—
23	C₂×#343	—	2	1	no	—
24	#1736	—	2	1	no	—
25	#1736	—	2	1	no	—
26	#1387	C₄	1	4	yes	complemented
27	#1382	—	2	2	no	—
28	#1380	—	2	2	no	—
29	#1379	C₄	1	4	yes	complemented
30	C₂²×M₁₆	—	2	1	no	—
31	C₂²×M₁₆	D₄	1	1	yes	characteristic
32	C₂³×C₈	C₂³	1	1	yes	characteristic
33	C₂×#117	—	2	1	no	—
34	C₂×#117	D₄	1	1	yes	characteristic
35	C₂²×C₄⋊C₄	—	2	1	no	—
36	C₂²×C₄⋊C₄	—	2	1	no	—
37	C₂×M₃₂	—	2	1	no	—
38	C₂×M₃₂	—	2	1	no	—
39	C₂×M₃₂	—	2	1	no	—
40	C₂×M₃₂	C₂×C₄	1	2	yes	—
41	C₂×M₃₂	C₂×C₄	1	2	yes	—
42	C₂×M₃₂	C₂×C₄	1	2	yes	—
43	C₂²×C₁₆	—	2	1	no	—
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	C₂×C₄	1	2	yes	—
51	C₂×#108	—	2	1	no	—
52	C₂×#108	—	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	#427	—	2	1	no	—
58	#427	—	2	1	no	—
59	#427	C₂×C₄	1	2	yes	—
60	#427	D₄	1	2	yes	—
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	C₂×C₄	1	2	yes	—
70	C₂×#107	C₂×C₄	1	2	yes	—
71	C₂×#107	D₄	1	1	yes	characteristic
72	C₂×#107	D₄	1	1	yes	characteristic
73	C₂×#107	D₄	1	2	yes	—
74	C₂×#107	Q₈	1	2	yes	—
75	C₂×#106	—	2	1	no	—
76	#405	—	2	1	no	—
77	#405	—	2	1	no	—
78	#405	C₂×C₄	1	2	yes	—
79	#405	Q₈	1	2	yes	—
80	C₂×#97	—	2	1	no	—
81	C₂×C₄×C₈	—	2	1	no	—
82	C₄×C₄⋊C₄	—	2	2	no	—
83	C₄×C₄⋊C₄	—	2	2	no	—
84	C₄×C₄⋊C₄	—	4	1	no	—
85	C₄×C₄⋊C₄	—	4	1	no	—
86	C₄×C₂²⋊C₄	—	4	2	no	—
87	#367	—	2	4	no	—
88	#366	—	2	4	no	—
89	#343	—	4	2	no	—
90	#340	—	2	2	no	—
91	#340	—	2	2	no	—
92	#340	—	4	1	no	—
93	#340	—	4	1	no	—
94	C₂³×C₄	—	4	1	no	—
95	C₂³×C₄	—	4	1	no	—
96	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂×M₁₆	—	2	1	no	—
98	C₂×M₁₆	—	2	1	no	—
99	C₂×M₁₆	—	2	1	no	—
100	C₂×M₁₆	—	2	1	no	—
101	C₂×M₁₆	—	2	1	no	—
102	C₂×M₁₆	—	2	2	no	—
103	C₂×M₁₆	—	4	1	no	—
104	C₂×M₁₆	—	4	1	no	—
105	C₂×M₁₆	—	4	1	no	—
106	C₂×M₁₆	—	4	1	no	—
107	C₂×M₁₆	—	4	1	no	—
108	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
109	C₂²×C₈	—	2	1	no	—
110	C₂²×C₈	—	4	1	no	—
111	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₈	C₂²×C₄	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	2	no	—
128	#117	—	2	2	no	—
129	#117	—	4	1	no	—
130	#117	—	4	1	no	—
131	#117	C₄⋊C₄	1	2	yes	—
132	#117	Q₁₆	1	2	yes	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	1	no	—
136	C₂×C₄⋊C₄	—	2	1	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	4	1	no	—
140	C₂×C₄⋊C₄	—	4	1	no	—
141	C₂×C₄⋊C₄	—	4	1	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄⋊C₄	—	4	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	D₈	1	1	yes	characteristic
148	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
149	C₂×C₂²⋊C₄	—	2	1	no	—
150	C₂×C₂²⋊C₄	—	4	1	no	—
151	C₂×C₄²	—	2	1	no	—
152	C₂×C₄²	—	4	1	no	—
153	C₂×C₄²	—	4	2	no	—
154	C₂×C₄²	—	4	2	no	—
155	M₃₂	—	2	2	no	—
156	M₃₂	—	2	2	no	—
157	M₃₂	—	2	2	no	—
158	M₃₂	C₄²	1	4	yes	complemented
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	—	4	1	no	—
166	C₂×C₁₆	—	4	1	no	—
167	C₂×C₁₆	—	4	1	no	—
168	C₂×C₁₆	—	4	1	no	—
169	#108	—	2	2	no	—
170	#108	—	2	2	no	—
171	#108	—	4	1	no	—
172	#108	—	4	1	no	—
173	#107	—	2	2	no	—
174	#107	—	2	2	no	—
175	#107	—	2	2	no	—
176	#107	—	2	1	no	—
177	#107	—	2	1	no	—
178	#107	—	2	2	no	—
179	#107	—	2	2	no	—
180	#107	—	4	1	no	—
181	#107	—	4	1	no	—
182	#107	C₂²⋊C₄	1	2	yes	—
183	#107	C₂²⋊C₄	1	2	yes	—
184	#107	C₄⋊C₄	1	4	yes	—
185	#106	—	2	2	no	—
186	#106	—	2	2	no	—
187	#97	—	2	2	no	—
188	#97	—	2	2	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	—	2	2	no	—
191	#95	—	4	2	no	—
192	#95	—	4	2	no	—
193	C₂⁴	AGL(1,ℤ/8)	1	1	yes	characteristic
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	8	1	no	—
203	C₂²×C₄	—	8	1	no	—
204	C₂²×C₄	—	16	1	no	—
205	C₂²×C₄	—	16	1	no	—
206	C₂²×C₄	—	16	1	no	—
207	C₂²×C₄	—	16	1	no	—
208	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
209	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
210	C₂²×C₄	C₄×D₄	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	#123	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
216	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
217	C₂²×C₄	#135	1	1	yes	characteristic
218	C₂²×C₄	#135	1	1	yes	characteristic
219	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
220	C₂²×C₄	#137	1	1	yes	characteristic
221	C₂²×C₄	#137	1	1	yes	characteristic
222	M₁₆	—	2	2	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	4	1	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	8	1	no	—
229	M₁₆	—	8	1	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	8	1	no	—
235	C₂×C₈	—	8	1	no	—
236	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
237	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
238	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
239	C₂×C₈	#117	1	1	yes	characteristic
240	C₂×C₈	#117	1	1	yes	characteristic
241	C₂×C₈	#117	1	1	yes	characteristic
242	C₂×C₈	C₄×D₄	1	1	yes	characteristic
243	C₂×C₈	C₄×D₄	1	1	yes	characteristic
244	C₂×C₈	C₄×D₄	1	1	yes	characteristic
245	C₂×C₈	C₄×D₄	1	1	yes	characteristic
246	C₂×C₈	C₄×D₄	1	1	yes	characteristic
247	C₂×C₈	C₄×D₄	1	1	yes	characteristic
248	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
249	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
250	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
251	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
252	C₂×C₈	#121	1	1	yes	characteristic
253	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
254	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
255	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
256	C₂×C₈	#123	1	1	yes	characteristic
257	C₂×C₈	#123	1	1	yes	characteristic
258	C₂×C₈	#124	1	1	yes	characteristic
259	C₂×C₈	#125	1	1	yes	characteristic
260	C₂×C₈	#125	1	1	yes	characteristic
261	C₂×C₈	#126	1	1	yes	characteristic
262	C₂×C₈	#126	1	1	yes	characteristic
263	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	8	1	no	—
277	C₄⋊C₄	—	8	1	no	—
278	C₄⋊C₄	#107	1	2	yes	—
279	C₄⋊C₄	#107	1	2	yes	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	4	2	no	—
287	C₄²	—	4	2	no	—
288	C₄²	—	16	2	no	—
289	C₄²	—	16	2	no	—
290	C₁₆	—	2	4	no	—
291	C₁₆	—	4	2	no	—
292	C₂³	—	2	1	no	—
293	C₂³	—	2	1	no	—
294	C₂³	—	2	1	no	—
295	C₂³	—	2	1	no	—
296	C₂³	#428	1	1	yes	characteristic
297	C₂³	#440	1	1	yes	characteristic
298	C₂³	#442	1	1	yes	characteristic
299	C₂³	#447	1	1	yes	characteristic
300	C₂³	#463	1	1	yes	characteristic
301	C₂³	#478	1	1	yes	characteristic
302	C₂³	#484	1	1	yes	characteristic
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	8	1	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	—	8	1	no	—
315	C₂×C₄	—	8	1	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	—	8	1	no	—
318	C₂×C₄	—	8	2	no	—
319	C₂×C₄	—	16	2	no	—
320	C₂×C₄	—	16	2	no	—
321	C₂×C₄	—	16	2	no	—
322	C₂×C₄	—	16	1	no	—
323	C₂×C₄	—	16	1	no	—
324	C₂×C₄	—	16	2	no	—
325	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
326	C₂×C₄	#385	1	1	yes	characteristic
327	C₂×C₄	#385	1	1	yes	characteristic
328	C₂×C₄	#386	1	1	yes	characteristic
329	C₂×C₄	#387	1	1	yes	characteristic
330	C₂×C₄	#388	1	1	yes	characteristic
331	C₂×C₄	#389	1	1	yes	characteristic
332	C₂×C₄	C₂×#107	1	1	yes	characteristic
333	C₂×C₄	#427	1	1	yes	characteristic
334	C₂×C₄	#428	1	1	yes	characteristic
335	C₂×C₄	C₄×D₈	1	1	yes	characteristic
336	C₂×C₄	C₄×D₈	1	1	yes	characteristic
337	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
338	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
339	C₂×C₄	#440	1	1	yes	characteristic
340	C₂×C₄	#441	1	1	yes	characteristic
341	C₂×C₄	#449	1	1	yes	characteristic
342	C₂×C₄	#451	1	1	yes	characteristic
343	C₂×C₄	#452	1	1	yes	characteristic
344	C₂×C₄	#459	1	1	yes	characteristic
345	C₂×C₄	#462	1	1	yes	characteristic
346	C₂×C₄	#464	1	1	yes	characteristic
347	C₂×C₄	#474	1	1	yes	characteristic
348	C₂×C₄	#477	1	1	yes	characteristic
349	C₂×C₄	#479	1	1	yes	characteristic
350	C₂×C₄	#480	1	1	yes	characteristic
351	C₂×C₄	#482	1	1	yes	characteristic
352	C₂×C₄	#485	1	1	yes	characteristic
353	C₈	—	4	2	no	—
354	C₈	—	8	2	no	—
355	C₈	C₄×C₂²⋊C₄	1	1	yes	characteristic
356	C₈	C₄×C₄⋊C₄	1	1	yes	characteristic
357	C₈	#387	1	1	yes	characteristic
358	C₈	#387	1	1	yes	characteristic
359	C₈	#387	1	1	yes	characteristic
360	C₈	#388	1	1	yes	characteristic
361	C₈	#389	1	1	yes	characteristic
362	C₈	#391	1	1	yes	characteristic
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	2	1	no	—
374	C₂²	—	2	1	no	—
375	C₂²	#1776	1	1	yes	characteristic
376	C₂²	#1852	1	1	yes	characteristic
377	C₂²	#1862	1	1	yes	characteristic
378	C₂²	#1879	1	1	yes	characteristic
379	C₂²	#1921	1	1	yes	characteristic
380	C₂²	#1928	1	1	yes	characteristic
381	C₂²	#1949	1	1	yes	characteristic
382	C₂²	#2192	1	1	yes	characteristic
383	C₂²	#2193	1	1	yes	characteristic
384	C₂²	#2194	1	1	yes	characteristic
385	C₂²	#2195	1	1	yes	characteristic
386	C₄	—	4	2	no	—
387	C₄	—	4	2	no	—
388	C₄	—	8	4	no	—
389	C₄	—	8	2	no	—
390	C₄	—	8	2	no	—
391	C₄	—	16	2	no	—
392	C₄	#1436	1	1	yes	characteristic
393	C₄	C₄×#107	1	1	yes	characteristic
394	C₄	#1854	1	1	yes	characteristic
395	C₄	#1863	1	1	yes	characteristic
396	C₄	#1878	1	1	yes	characteristic
397	C₄	#1918	1	1	yes	characteristic
398	C₄	#1925	1	1	yes	characteristic
399	C₄	#1947	1	1	yes	characteristic
400	C₂	—	2	1	no	—
401	C₂	—	2	1	no	—
402	C₂	—	2	1	no	—
403	C₂	—	2	1	no	—
404	C₂	#10907	1	1	yes	characteristic
405	C₂	#12925	1	1	yes	characteristic
406	C₂	#12926	1	1	yes	characteristic
407	C₂	#12940	1	1	yes	characteristic
408	C₂	#12941	1	1	yes	characteristic
409	C₂	#13022	1	1	yes	characteristic
410	C₂	#13023	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ 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
17. #2159	C₄	semidirect
26. #1387	C₄	semidirect
29. #1379	C₄	semidirect
158. M₃₂	C₄²	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
#20455	2
C₂×#1387	2
C₂×#1382	2
C₂×#1380	2
C₂×#1379	2
#10870	2