Subgroups — #108710

Order 512.

600 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
2	#10853	C₂	1	1	yes	maximal, characteristic
3	#10851	C₂	1	1	yes	maximal, characteristic
4	C₂×#1303	C₂	1	1	yes	maximal, characteristic
5	#8039	C₂	1	1	yes	maximal, characteristic
6	#7897	C₂	1	1	yes	maximal, characteristic
7	C₂×#1279	C₂	1	1	yes	maximal, characteristic
8	C₂×#431	C₂²	1	1	yes	characteristic
9	C₂×#423	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₈	C₂²	1	1	yes	characteristic
13	#2285	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×#341	—	2	1	no	—
16	C₂×#341	—	2	1	no	—
17	C₂×#339	—	2	1	no	—
18	C₂×#339	—	2	1	no	—
19	C₈×M₁₆	—	2	2	no	—
20	C₈×M₁₆	C₄	1	4	yes	complemented
21	C₂×#322	—	2	1	no	—
22	C₂×C₈²	—	2	1	no	—
23	#1303	C₄	1	4	yes	complemented
24	#1296	C₄	1	2	yes	complemented
25	#1296	C₄	1	2	yes	complemented
26	#1279	—	2	2	no	—
27	#1279	—	2	1	no	—
28	#1279	—	2	1	no	—
29	C₂²×M₁₆	—	2	1	no	—
30	C₂²×M₁₆	D₄	1	1	yes	characteristic
31	C₂²×M₁₆	D₄	1	1	yes	characteristic
32	C₂³×C₈	—	2	1	no	—
33	C₂³×C₈	—	2	1	no	—
34	C₂×#117	—	2	1	no	—
35	C₂²×C₄⋊C₄	—	2	1	no	—
36	C₂²×C₄²	—	2	1	no	—
37	C₂²×C₄²	C₂³	1	1	yes	characteristic
38	#431	—	2	1	no	—
39	#431	—	2	1	no	—
40	#431	C₂×C₄	1	2	yes	—
41	#431	C₂×C₄	1	2	yes	—
42	#423	—	2	1	no	—
43	#423	—	2	1	no	—
44	#423	—	2	1	no	—
45	#423	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₈	—	2	1	no	—
47	C₂×C₄⋊C₈	—	2	1	no	—
48	C₂×C₄⋊C₈	—	2	1	no	—
49	C₂×C₄⋊C₈	—	2	1	no	—
50	C₂×C₄⋊C₈	—	2	1	no	—
51	C₂×C₄⋊C₈	—	2	1	no	—
52	C₂×C₄⋊C₈	—	2	1	no	—
53	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₂×C₂²⋊C₈	—	2	1	no	—
57	C₄×M₁₆	—	2	1	no	—
58	C₄×M₁₆	—	2	1	no	—
59	C₄×M₁₆	C₂×C₄	1	2	yes	—
60	C₄×M₁₆	Q₈	1	2	yes	—
61	C₂×#97	—	2	1	no	—
62	C₂×C₄×C₈	—	2	1	no	—
63	C₂×C₄×C₈	—	2	1	no	—
64	C₂×C₄×C₈	—	2	1	no	—
65	C₂×C₄×C₈	—	2	1	no	—
66	C₂×C₄×C₈	—	2	1	no	—
67	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
68	C₂×C₄×C₈	D₄	1	1	yes	characteristic
69	C₂×C₄×C₈	D₄	1	1	yes	characteristic
70	C₂×C₄×C₈	D₄	1	2	yes	—
71	#384	C₂×C₄	1	2	yes	—
72	#382	—	2	1	no	—
73	#380	—	2	1	no	—
74	#380	—	2	1	no	—
75	C₄×C₄⋊C₄	—	2	1	no	—
76	C₄×C₄⋊C₄	—	2	1	no	—
77	C₄×C₄⋊C₄	—	2	1	no	—
78	C₄×C₄⋊C₄	—	2	1	no	—
79	C₄×C₄⋊C₄	—	2	1	no	—
80	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
81	C₂×#95	—	2	1	no	—
82	#341	—	4	2	no	—
83	#341	—	4	2	no	—
84	#339	—	4	2	no	—
85	#339	—	4	2	no	—
86	#322	—	2	4	no	—
87	C₈²	—	2	4	no	—
88	C₂³×C₄	—	4	1	no	—
89	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
90	C₂³×C₄	#41	1	1	yes	characteristic
91	C₂³×C₄	#41	1	1	yes	characteristic
92	C₂×M₁₆	—	2	1	no	—
93	C₂×M₁₆	—	2	1	no	—
94	C₂×M₁₆	—	2	1	no	—
95	C₂×M₁₆	—	2	1	no	—
96	C₂×M₁₆	—	2	1	no	—
97	C₂×M₁₆	—	2	1	no	—
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	4	1	no	—
100	C₂×M₁₆	—	4	1	no	—
101	C₂×M₁₆	—	4	1	no	—
102	C₂×M₁₆	—	4	1	no	—
103	C₂×M₁₆	—	4	1	no	—
104	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
105	C₂×M₁₆	C₄⋊C₄	1	2	yes	—
106	C₂×M₁₆	D₈	1	2	yes	—
107	C₂×M₁₆	SD₁₆	1	2	yes	—
108	C₂×M₁₆	SD₁₆	1	2	yes	—
109	C₂×M₁₆	Q₁₆	1	2	yes	—
110	C₂²×C₈	—	2	1	no	—
111	C₂²×C₈	—	2	1	no	—
112	C₂²×C₈	—	2	1	no	—
113	C₂²×C₈	—	2	2	no	—
114	C₂²×C₈	—	2	1	no	—
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	—	2	2	no	—
117	C₂²×C₈	—	2	1	no	—
118	C₂²×C₈	—	2	2	no	—
119	C₂²×C₈	—	4	4	no	—
120	C₂²×C₈	—	4	1	no	—
121	C₂²×C₈	—	4	1	no	—
122	C₂²×C₈	—	4	1	no	—
123	C₂²×C₈	—	4	1	no	—
124	C₂²×C₈	—	4	1	no	—
125	C₂²×C₈	—	4	1	no	—
126	C₂²×C₈	—	4	1	no	—
127	#117	—	4	1	no	—
128	#117	—	4	1	no	—
129	#117	—	4	1	no	—
130	#117	—	4	1	no	—
131	C₂×C₄⋊C₄	—	2	4	no	—
132	C₂×C₄⋊C₄	—	2	2	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	2	no	—
136	C₂×C₄⋊C₄	—	4	2	no	—
137	C₂×C₄⋊C₄	—	4	1	no	—
138	C₂×C₂²⋊C₄	—	4	1	no	—
139	C₂×C₄²	—	2	2	no	—
140	C₂×C₄²	—	4	1	no	—
141	C₂×C₄²	—	4	1	no	—
142	C₂×C₄²	—	4	1	no	—
143	C₂×C₄²	—	4	1	no	—
144	C₂×C₄²	—	4	1	no	—
145	C₂×C₄²	—	4	1	no	—
146	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
147	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
149	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
150	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
151	C₂×C₄²	C₂×Q₈	1	1	yes	characteristic
152	C₂×C₄²	#41	1	1	yes	characteristic
153	C₂×C₄²	#41	1	1	yes	characteristic
154	C₂×C₄²	#41	1	1	yes	characteristic
155	C₂×C₄²	#41	1	1	yes	characteristic
156	C₂×C₄²	#41	1	1	yes	characteristic
157	C₂×C₄²	#41	1	1	yes	characteristic
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	2	1	no	—
161	C₄⋊C₈	—	2	1	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	4	1	no	—
165	C₄⋊C₈	—	4	1	no	—
166	C₄⋊C₈	—	4	1	no	—
167	C₄⋊C₈	—	4	1	no	—
168	C₄⋊C₈	C₄²	1	2	yes	—
169	C₄⋊C₈	C₄²	1	2	yes	—
170	C₂²⋊C₈	—	2	2	no	—
171	C₂²⋊C₈	—	2	2	no	—
172	#97	—	2	2	no	—
173	#97	—	2	2	no	—
174	C₄×C₈	—	2	2	no	—
175	C₄×C₈	—	2	2	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	2	2	no	—
178	C₄×C₈	—	2	2	no	—
179	C₄×C₈	—	2	1	no	—
180	C₄×C₈	—	2	2	no	—
181	C₄×C₈	—	2	1	no	—
182	C₄×C₈	C₂²⋊C₄	1	2	yes	—
183	C₄×C₈	C₂²⋊C₄	1	2	yes	—
184	#95	—	2	2	no	—
185	#95	—	4	1	no	—
186	#95	—	4	1	no	—
187	#95	—	4	1	no	—
188	C₂⁴	#124	1	1	yes	characteristic
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	4	4	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	8	2	no	—
196	C₂²×C₄	—	8	1	no	—
197	C₂²×C₄	—	8	1	no	—
198	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
199	C₂²×C₄	#117	1	1	yes	characteristic
200	C₂²×C₄	#117	1	1	yes	characteristic
201	C₂²×C₄	#117	1	1	yes	characteristic
202	C₂²×C₄	#117	1	1	yes	characteristic
203	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
204	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
205	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
206	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
207	C₂²×C₄	#121	1	1	yes	characteristic
208	C₂²×C₄	#121	1	1	yes	characteristic
209	C₂²×C₄	#123	1	1	yes	characteristic
210	C₂²×C₄	#123	1	1	yes	characteristic
211	C₂²×C₄	#124	1	1	yes	characteristic
212	C₂²×C₄	#125	1	1	yes	characteristic
213	C₂²×C₄	#125	1	1	yes	characteristic
214	C₂²×C₄	#126	1	1	yes	characteristic
215	C₂²×C₄	#126	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
217	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
218	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
219	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
220	C₂²×C₄	#135	1	1	yes	characteristic
221	C₂²×C₄	#135	1	1	yes	characteristic
222	C₂²×C₄	#135	1	1	yes	characteristic
223	C₂²×C₄	#135	1	1	yes	characteristic
224	M₁₆	—	2	2	no	—
225	M₁₆	—	2	2	no	—
226	M₁₆	—	2	2	no	—
227	M₁₆	—	2	2	no	—
228	M₁₆	—	2	2	no	—
229	M₁₆	—	2	2	no	—
230	M₁₆	—	4	2	no	—
231	M₁₆	—	4	2	no	—
232	M₁₆	—	8	1	no	—
233	M₁₆	—	8	1	no	—
234	M₁₆	#106	1	4	yes	—
235	M₁₆	#107	1	4	yes	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	4	no	—
248	C₂×C₈	—	4	4	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	8	1	no	—
257	C₂×C₈	—	8	1	no	—
258	C₂×C₈	—	8	1	no	—
259	C₂×C₈	—	8	1	no	—
260	C₂×C₈	—	8	1	no	—
261	C₂×C₈	—	8	1	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	4	4	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	—	8	2	no	—
270	C₂²⋊C₄	—	8	1	no	—
271	C₂²⋊C₄	—	8	1	no	—
272	C₄²	—	4	2	no	—
273	C₄²	—	4	2	no	—
274	C₄²	—	8	1	no	—
275	C₄²	—	8	1	no	—
276	C₄²	—	8	1	no	—
277	C₄²	—	8	1	no	—
278	C₄²	C₂×C₄²	1	1	yes	characteristic
279	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
280	C₄²	#117	1	1	yes	characteristic
281	C₄²	#117	1	1	yes	characteristic
282	C₄²	#117	1	1	yes	characteristic
283	C₄²	#117	1	1	yes	characteristic
284	C₄²	C₄×D₄	1	1	yes	characteristic
285	C₄²	C₄×D₄	1	1	yes	characteristic
286	C₄²	C₄×Q₈	1	1	yes	characteristic
287	C₄²	C₄×Q₈	1	1	yes	characteristic
288	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
289	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
290	C₄²	#123	1	1	yes	characteristic
291	C₄²	#123	1	1	yes	characteristic
292	C₄²	#126	1	1	yes	characteristic
293	C₄²	#126	1	1	yes	characteristic
294	C₂³	—	2	2	no	—
295	C₂³	—	2	2	no	—
296	C₂³	#382	1	1	yes	characteristic
297	C₂³	#388	1	1	yes	characteristic
298	C₂³	#388	1	1	yes	characteristic
299	C₂³	#486	1	1	yes	characteristic
300	C₂³	#487	1	1	yes	characteristic
301	C₂³	#488	1	1	yes	characteristic
302	C₂³	#488	1	1	yes	characteristic
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	8	2	no	—
318	C₂×C₄	—	8	4	no	—
319	C₂×C₄	—	8	4	no	—
320	C₂×C₄	—	8	2	no	—
321	C₂×C₄	—	8	2	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	—	8	2	no	—
324	C₂×C₄	#376	1	1	yes	characteristic
325	C₂×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
326	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
327	C₂×C₄	#380	1	1	yes	characteristic
328	C₂×C₄	#381	1	1	yes	characteristic
329	C₂×C₄	#383	1	1	yes	characteristic
330	C₂×C₄	#387	1	1	yes	characteristic
331	C₂×C₄	#387	1	1	yes	characteristic
332	C₂×C₄	#387	1	1	yes	characteristic
333	C₂×C₄	#387	1	1	yes	characteristic
334	C₂×C₄	#388	1	1	yes	characteristic
335	C₂×C₄	#388	1	1	yes	characteristic
336	C₂×C₄	C₂×#106	1	1	yes	characteristic
337	C₂×C₄	C₂×#107	1	1	yes	characteristic
338	C₂×C₄	#427	1	1	yes	characteristic
339	C₂×C₄	#427	1	1	yes	characteristic
340	C₂×C₄	C₄×D₈	1	1	yes	characteristic
341	C₂×C₄	C₄×D₈	1	1	yes	characteristic
342	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
343	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
344	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
345	C₂×C₄	C₄×SD₁₆	1	1	yes	characteristic
346	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
347	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
348	C₂×C₄	#443	1	2	yes	—
349	C₂×C₄	#444	1	2	yes	—
350	C₂×C₄	#465	1	1	yes	characteristic
351	C₂×C₄	#465	1	1	yes	characteristic
352	C₂×C₄	#465	1	1	yes	characteristic
353	C₂×C₄	#465	1	1	yes	characteristic
354	C₂×C₄	#466	1	1	yes	characteristic
355	C₂×C₄	#466	1	1	yes	characteristic
356	C₂×C₄	#467	1	1	yes	characteristic
357	C₂×C₄	#467	1	1	yes	characteristic
358	C₂×C₄	#486	1	1	yes	characteristic
359	C₂×C₄	#487	1	1	yes	characteristic
360	C₂×C₄	#488	1	1	yes	characteristic
361	C₂×C₄	#488	1	1	yes	characteristic
362	C₈	—	2	4	no	—
363	C₈	—	2	4	no	—
364	C₈	—	2	4	no	—
365	C₈	—	4	2	no	—
366	C₈	—	4	2	no	—
367	C₈	—	4	2	no	—
368	C₈	—	8	2	no	—
369	C₈	—	8	2	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	2	1	no	—
374	C₂²	—	2	2	no	—
375	C₂²	—	2	2	no	—
376	C₂²	—	2	2	no	—
377	C₂²	—	2	2	no	—
378	C₂²	#1434	1	1	yes	characteristic
379	C₂²	#1834	1	1	yes	characteristic
380	C₂²	#1835	1	1	yes	characteristic
381	C₂²	#1929	1	2	yes	—
382	C₂²	#1932	1	2	yes	—
383	C₂²	#1933	1	1	yes	characteristic
384	C₂²	#1933	1	1	yes	characteristic
385	C₂²	#1934	1	1	yes	characteristic
386	C₂²	#1934	1	1	yes	characteristic
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	1	no	—
390	C₄	—	2	1	no	—
391	C₄	—	2	1	no	—
392	C₄	—	2	1	no	—
393	C₄	—	8	4	no	—
394	C₄	—	16	2	no	—
395	C₄	#1764	1	1	yes	characteristic
396	C₄	#1765	1	1	yes	characteristic
397	C₄	C₄×#106	1	1	yes	characteristic
398	C₄	C₄×#107	1	1	yes	characteristic
399	C₄	#1807	1	1	yes	characteristic
400	C₄	#1808	1	1	yes	characteristic
401	C₄	#1922	1	2	yes	—
402	C₄	#1933	1	1	yes	characteristic
403	C₄	#1933	1	1	yes	characteristic
404	C₄	#1934	1	1	yes	characteristic
405	C₄	#1934	1	1	yes	characteristic
406	C₂	—	2	1	no	—
407	C₂	—	2	1	no	—
408	C₂	—	2	1	no	—
409	C₂	—	2	1	no	—
410	C₂	#9589	1	2	yes	—
411	C₂	#9590	1	2	yes	—
412	C₂	#10809	1	1	yes	characteristic
413	C₂	#10820	1	1	yes	characteristic
414	C₂	#10826	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#388
prime radical	C₂⁴	#124
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
20. C₈×M₁₆	C₄	semidirect
23. #1303	C₄	semidirect
24. #1296	C₄	semidirect
25. #1296	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×M₁₆	2
#10853	2
#10851	2
C₂×#1303	2
#8039	2
#7897	2
C₂×#1279	2