Subgroups — #109855

Order 512.

634 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#1571	C₂	1	1	yes	maximal, characteristic
3	#10844	C₂	1	1	yes	maximal, characteristic
4	C₂×#1303	C₂	1	1	yes	maximal, characteristic
5	C₂×#1303	C₂	1	1	yes	maximal, characteristic
6	#7877	C₂	1	1	yes	maximal, characteristic
7	#7872	C₂	1	1	yes	maximal, characteristic
8	C₂×#431	C₂²	1	1	yes	characteristic
9	C₂×#431	C₂²	1	1	yes	characteristic
10	C₂×#423	C₂²	1	1	yes	characteristic
11	C₂×#423	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₈	—	2	1	no	—
14	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₈×C₄⋊C₄	—	2	2	no	—
17	C₈×C₄⋊C₄	—	2	2	no	—
18	C₂×#341	—	2	1	no	—
19	C₂×#341	—	2	1	no	—
20	C₂×#336	—	2	1	no	—
21	#1571	—	2	2	no	—
22	#1571	C₄	1	4	yes	complemented
23	C₂×#335	—	2	2	no	—
24	#1303	—	2	2	no	—
25	#1303	C₄	1	4	yes	—
26	#1277	—	2	2	no	—
27	#1277	C₄	1	4	yes	complemented
28	C₂²×M₁₆	—	2	1	no	—
29	C₂²×M₁₆	—	2	1	no	—
30	C₂³×C₈	—	2	1	no	—
31	C₂³×C₈	D₄	1	1	yes	characteristic
32	C₂³×C₈	D₄	1	1	yes	characteristic
33	C₂²×C₄⋊C₄	—	2	1	no	—
34	C₂²×C₄²	—	2	1	no	—
35	C₂²×C₄²	C₂³	1	1	yes	characteristic
36	#431	—	2	1	no	—
37	#431	—	2	1	no	—
38	#431	—	2	1	no	—
39	#431	—	2	1	no	—
40	#431	—	2	1	no	—
41	#431	—	2	1	no	—
42	#431	C₂×C₄	1	2	yes	—
43	#431	C₂×C₄	1	2	yes	—
44	#423	—	2	1	no	—
45	#423	—	2	1	no	—
46	#423	—	2	1	no	—
47	#423	—	2	1	no	—
48	#423	—	2	1	no	—
49	#423	—	2	1	no	—
50	#423	C₂×C₄	1	2	yes	—
51	#423	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₈	—	2	1	no	—
53	C₂×C₄⋊C₈	—	2	1	no	—
54	C₂×C₄⋊C₈	—	2	1	no	—
55	C₂×C₄⋊C₈	—	2	1	no	—
56	C₂×C₄⋊C₈	—	2	1	no	—
57	C₂×C₄⋊C₈	—	2	1	no	—
58	C₂×C₄⋊C₈	—	2	1	no	—
59	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
61	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
62	C₂×C₄⋊C₈	D₄	1	2	yes	—
63	C₂×C₄⋊C₈	D₄	1	2	yes	—
64	C₂×C₂²⋊C₈	—	2	1	no	—
65	C₂×C₂²⋊C₈	—	2	1	no	—
66	C₂×C₄×C₈	—	2	1	no	—
67	C₂×C₄×C₈	—	2	1	no	—
68	C₂×C₄×C₈	—	2	1	no	—
69	C₂×C₄×C₈	—	2	1	no	—
70	C₂×C₄×C₈	—	2	1	no	—
71	C₂×C₄×C₈	—	2	4	no	—
72	C₂×C₄×C₈	—	2	1	no	—
73	C₂×C₄×C₈	—	4	2	no	—
74	C₂×C₄×C₈	—	4	2	no	—
75	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
76	C₂×C₄×C₈	Q₈	1	2	yes	—
77	C₄×C₄⋊C₄	—	2	1	no	—
78	C₄×C₄⋊C₄	—	2	1	no	—
79	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
80	C₄×C₄⋊C₄	D₄	1	2	yes	—
81	C₂×#95	Q₈	1	2	yes	—
82	#341	—	4	2	no	—
83	#341	—	4	2	no	—
84	#336	—	2	4	no	—
85	#335	—	2	8	no	—
86	C₂³×C₄	—	2	2	no	—
87	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
88	C₂³×C₄	C₂×Q₈	1	1	yes	characteristic
89	C₂³×C₄	#41	1	1	yes	characteristic
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	—	2	2	no	—
92	C₂×M₁₆	—	4	1	no	—
93	C₂×M₁₆	—	4	1	no	—
94	C₂×M₁₆	—	4	1	no	—
95	C₂×M₁₆	—	4	1	no	—
96	C₂×M₁₆	—	4	1	no	—
97	C₂×M₁₆	—	4	1	no	—
98	C₂×M₁₆	—	4	1	no	—
99	C₂×M₁₆	—	4	1	no	—
100	C₂×M₁₆	—	4	1	no	—
101	C₂×M₁₆	—	4	1	no	—
102	C₂²×C₈	—	2	1	no	—
103	C₂²×C₈	—	2	1	no	—
104	C₂²×C₈	—	2	1	no	—
105	C₂²×C₈	—	2	1	no	—
106	C₂²×C₈	—	2	2	no	—
107	C₂²×C₈	—	2	2	no	—
108	C₂²×C₈	—	2	2	no	—
109	C₂²×C₈	—	2	1	no	—
110	C₂²×C₈	—	2	2	no	—
111	C₂²×C₈	—	2	1	no	—
112	C₂²×C₈	—	2	1	no	—
113	C₂²×C₈	—	2	1	no	—
114	C₂²×C₈	—	2	1	no	—
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	—	2	2	no	—
117	C₂²×C₈	—	4	1	no	—
118	C₂²×C₈	—	4	1	no	—
119	C₂²×C₈	—	4	1	no	—
120	C₂²×C₈	—	4	1	no	—
121	C₂²×C₈	—	4	1	no	—
122	C₂²×C₈	—	4	1	no	—
123	C₂²×C₈	C₄⋊C₄	1	2	yes	—
124	C₂²×C₈	C₄⋊C₄	1	2	yes	—
125	C₂²×C₈	D₈	1	2	yes	—
126	C₂²×C₈	Q₁₆	1	2	yes	—
127	C₂×C₄⋊C₄	—	2	8	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄²	—	2	2	no	—
131	C₂×C₄²	—	2	2	no	—
132	C₂×C₄²	—	4	1	no	—
133	C₂×C₄²	—	4	1	no	—
134	C₂×C₄²	—	4	1	no	—
135	C₂×C₄²	—	4	1	no	—
136	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
137	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
138	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
140	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
142	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
143	C₂×C₄²	#41	1	1	yes	characteristic
144	C₂×C₄²	#41	1	1	yes	characteristic
145	C₂×C₄²	#41	1	1	yes	characteristic
146	C₂×C₄²	#41	1	1	yes	characteristic
147	C₂×C₄²	#41	1	1	yes	characteristic
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	2	2	no	—
151	C₄⋊C₈	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	4	1	no	—
159	C₄⋊C₈	—	4	1	no	—
160	C₄⋊C₈	—	4	1	no	—
161	C₄⋊C₈	—	4	1	no	—
162	C₄⋊C₈	C₄²	1	4	yes	—
163	C₄⋊C₈	C₂²⋊C₄	1	4	yes	—
164	C₂²⋊C₈	—	2	2	no	—
165	C₂²⋊C₈	—	2	2	no	—
166	C₂²⋊C₈	—	4	1	no	—
167	C₂²⋊C₈	—	4	1	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	—	2	2	no	—
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	—	2	2	no	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	—	2	2	no	—
174	C₄×C₈	—	4	8	no	—
175	C₄×C₈	—	4	1	no	—
176	C₄×C₈	—	4	1	no	—
177	#95	—	2	2	no	—
178	#95	C₄⋊C₄	1	4	yes	—
179	C₂⁴	C₂²⋊Q₈	1	1	yes	characteristic
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	4	4	no	—
185	C₂²×C₄	—	4	2	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	#117	1	1	yes	characteristic
190	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
191	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
192	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
193	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
194	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
195	C₂²×C₄	C₄×Q₈	1	1	yes	characteristic
196	C₂²×C₄	#120	1	1	yes	characteristic
197	C₂²×C₄	#121	1	1	yes	characteristic
198	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
199	C₂²×C₄	#123	1	1	yes	characteristic
200	C₂²×C₄	#123	1	1	yes	characteristic
201	C₂²×C₄	#125	1	1	yes	characteristic
202	C₂²×C₄	#125	1	1	yes	characteristic
203	C₂²×C₄	#126	1	1	yes	characteristic
204	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
206	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
207	C₂²×C₄	#135	1	1	yes	characteristic
208	C₂²×C₄	#135	1	1	yes	characteristic
209	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
210	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
211	C₂²×C₄	#137	1	1	yes	characteristic
212	C₂²×C₄	#137	1	1	yes	characteristic
213	M₁₆	—	4	2	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	8	1	no	—
218	M₁₆	—	8	1	no	—
219	M₁₆	—	8	1	no	—
220	M₁₆	—	8	1	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	4	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	1	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	4	no	—
247	C₂×C₈	—	8	1	no	—
248	C₂×C₈	—	8	1	no	—
249	C₂×C₈	—	8	1	no	—
250	C₂×C₈	—	8	1	no	—
251	C₂×C₈	#107	1	4	yes	—
252	C₄⋊C₄	—	2	8	no	—
253	C₄⋊C₄	—	2	8	no	—
254	C₄²	—	4	2	no	—
255	C₄²	—	4	2	no	—
256	C₄²	—	4	2	no	—
257	C₄²	—	4	2	no	—
258	C₄²	C₂×C₄²	1	1	yes	characteristic
259	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
260	C₄²	#117	1	1	yes	characteristic
261	C₄²	#117	1	1	yes	characteristic
262	C₄²	C₄×D₄	1	1	yes	characteristic
263	C₄²	C₄×D₄	1	1	yes	characteristic
264	C₄²	C₄×D₄	1	1	yes	characteristic
265	C₄²	C₄×D₄	1	1	yes	characteristic
266	C₄²	C₄×Q₈	1	1	yes	characteristic
267	C₄²	C₄×Q₈	1	1	yes	characteristic
268	C₄²	#121	1	1	yes	characteristic
269	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
270	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
271	C₄²	#123	1	1	yes	characteristic
272	C₄²	#124	1	1	yes	characteristic
273	C₄²	#126	1	1	yes	characteristic
274	C₂³	—	2	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	#385	1	1	yes	characteristic
277	C₂³	#387	1	1	yes	characteristic
278	C₂³	#389	1	1	yes	characteristic
279	C₂³	#474	1	1	yes	characteristic
280	C₂³	#477	1	1	yes	characteristic
281	C₂³	#478	1	1	yes	characteristic
282	C₂³	#479	1	1	yes	characteristic
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	8	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	8	no	—
297	C₂×C₄	—	4	8	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
300	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
301	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
302	C₂×C₄	#385	1	1	yes	characteristic
303	C₂×C₄	#386	1	1	yes	characteristic
304	C₂×C₄	#387	1	1	yes	characteristic
305	C₂×C₄	#387	1	1	yes	characteristic
306	C₂×C₄	#387	1	1	yes	characteristic
307	C₂×C₄	#388	1	1	yes	characteristic
308	C₂×C₄	#388	1	1	yes	characteristic
309	C₂×C₄	#389	1	1	yes	characteristic
310	C₂×C₄	#391	1	1	yes	characteristic
311	C₂×C₄	C₂×#107	1	1	yes	characteristic
312	C₂×C₄	#427	1	1	yes	characteristic
313	C₂×C₄	#428	1	1	yes	characteristic
314	C₂×C₄	#428	1	1	yes	characteristic
315	C₂×C₄	C₄×D₈	1	1	yes	characteristic
316	C₂×C₄	C₄×D₈	1	1	yes	characteristic
317	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
318	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
319	C₂×C₄	#440	1	1	yes	characteristic
320	C₂×C₄	#440	1	1	yes	characteristic
321	C₂×C₄	#441	1	1	yes	characteristic
322	C₂×C₄	#442	1	1	yes	characteristic
323	C₂×C₄	#443	1	4	yes	—
324	C₂×C₄	#447	1	1	yes	characteristic
325	C₂×C₄	#449	1	1	yes	characteristic
326	C₂×C₄	#451	1	1	yes	characteristic
327	C₂×C₄	#452	1	1	yes	characteristic
328	C₂×C₄	#459	1	1	yes	characteristic
329	C₂×C₄	#462	1	1	yes	characteristic
330	C₂×C₄	#463	1	1	yes	characteristic
331	C₂×C₄	#464	1	1	yes	characteristic
332	C₂×C₄	#480	1	1	yes	characteristic
333	C₂×C₄	#482	1	1	yes	characteristic
334	C₂×C₄	#484	1	1	yes	characteristic
335	C₂×C₄	#485	1	1	yes	characteristic
336	C₈	—	2	4	no	—
337	C₈	—	4	2	no	—
338	C₈	—	4	4	no	—
339	C₈	—	4	4	no	—
340	C₈	—	8	2	no	—
341	C₈	—	8	2	no	—
342	C₈	—	8	2	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	2	2	no	—
350	C₂²	—	2	2	no	—
351	C₂²	#1436	1	1	yes	characteristic
352	C₂²	#1862	1	1	yes	characteristic
353	C₂²	#1863	1	1	yes	characteristic
354	C₂²	#1918	1	1	yes	characteristic
355	C₂²	#1921	1	1	yes	characteristic
356	C₂²	#1945	1	2	yes	—
357	C₂²	#1947	1	1	yes	characteristic
358	C₂²	#1949	1	1	yes	characteristic
359	C₂²	#1953	1	2	yes	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	4	8	no	—
367	C₄	#1764	1	2	yes	—
368	C₄	C₄×#107	1	1	yes	characteristic
369	C₄	#1776	1	1	yes	characteristic
370	C₄	#1852	1	1	yes	characteristic
371	C₄	#1854	1	1	yes	characteristic
372	C₄	#1878	1	1	yes	characteristic
373	C₄	#1879	1	1	yes	characteristic
374	C₄	#1925	1	1	yes	characteristic
375	C₄	#1928	1	1	yes	characteristic
376	C₄	#1937	1	2	yes	—
377	C₂	—	2	1	no	—
378	C₂	—	2	1	no	—
379	C₂	—	2	1	no	—
380	C₂	—	2	1	no	—
381	C₂	#9627	1	2	yes	—
382	C₂	#9629	1	2	yes	—
383	C₂	#10882	1	1	yes	characteristic
384	C₂	#10898	1	1	yes	characteristic
385	C₂	#10907	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#389
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
22. #1571	C₄	semidirect
27. #1277	C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×C₄⋊C₄	2
C₂×#1571	2
#10844	2
C₂×#1303	2
#7877	2
#7872	2