Subgroups — #72227000

Order 2048.

580 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72226997	C₂	1	4	yes	maximal, complemented
2	#72225400	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×D₁₆	C₂	1	1	yes	maximal, characteristic
4	#72216220	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×D₁₆	C₄	1	1	yes	characteristic
6	C₂×C₁₆×D₈	—	2	1	no	—
7	#151042	—	2	1	no	—
8	C₈×M₆₄	—	2	1	no	—
9	C₁₆×D₁₆	—	2	1	no	—
10	C₁₆×D₁₆	—	2	1	no	—
11	C₁₆×D₁₆	C₄	1	2	yes	—
12	C₁₆×D₁₆	C₂²	1	2	yes	—
13	C₂×#7100	C₄	1	1	yes	characteristic
14	C₂×#7068	—	2	1	no	—
15	C₂×C₁₆²	C₂²	1	1	yes	characteristic
16	#93883	C₂²	1	2	yes	complemented
17	#93879	—	2	1	no	—
18	#93665	—	2	1	no	—
19	#93665	C₂²	1	2	yes	complemented
20	C₂×C₄×D₁₆	C₈	1	1	yes	characteristic
21	C₂×D₄×C₁₆	—	4	1	no	—
22	C₂×C₈×D₈	—	2	1	no	—
23	#13640	—	4	1	no	—
24	C₄×M₆₄	—	2	2	no	—
25	C₈×D₁₆	—	2	1	no	—
26	C₈×D₁₆	—	2	1	no	—
27	C₈×D₁₆	C₈	1	2	yes	—
28	C₈×D₁₆	C₂×C₄	1	2	yes	—
29	C₁₆×D₈	—	2	2	no	—
30	C₁₆×D₈	—	2	2	no	—
31	C₁₆×D₈	—	4	1	no	—
32	C₁₆×D₈	—	4	1	no	—
33	C₂×#1371	C₈	1	1	yes	characteristic
34	C₂×#1370	—	2	1	no	—
35	C₂×#1331	—	2	1	no	—
36	C₂×D₄⋊C₁₆	—	4	1	no	—
37	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₆	D₄	1	1	yes	characteristic
39	C₂×C₈×C₁₆	D₄	1	1	yes	characteristic
40	#7458	—	2	2	no	—
41	#7455	—	4	1	no	—
42	#7329	—	2	2	no	—
43	C₈×C₃₂	—	2	2	no	—
44	#7100	—	2	1	no	—
45	#7100	C₂×C₄	1	2	yes	—
46	#7068	—	2	2	no	—
47	#7068	—	2	2	no	—
48	C₁₆²	C₂×C₄	1	1	yes	characteristic
49	C₁₆²	D₄	1	1	yes	characteristic
50	C₁₆²	D₄	1	1	yes	characteristic
51	C₁₆²	C₂³	1	1	yes	characteristic
52	C₂²×D₁₆	C₁₆	1	1	yes	characteristic
53	C₂³×C₁₆	—	8	1	no	—
54	C₂×C₄×D₈	—	2	1	no	—
55	C₂×C₈×D₄	—	4	1	no	—
56	C₂×M₆₄	—	2	4	no	—
57	C₄×D₁₆	—	2	1	no	—
58	C₄×D₁₆	—	2	1	no	—
59	C₄×D₁₆	C₁₆	1	2	yes	—
60	C₄×D₁₆	C₂×C₈	1	2	yes	—
61	D₄×C₁₆	—	4	2	no	—
62	D₄×C₁₆	—	4	2	no	—
63	D₄×C₁₆	—	8	1	no	—
64	D₄×C₁₆	—	8	1	no	—
65	C₂×#366	C₁₆	1	1	yes	characteristic
66	C₂×C₄⋊C₁₆	—	4	1	no	—
67	C₂×#357	—	2	1	no	—
68	C₂×C₂²⋊C₁₆	—	8	1	no	—
69	C₂×C₄×C₁₆	—	2	1	no	—
70	C₂×C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×C₁₆	M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₁₆	D₈	1	1	yes	characteristic
73	C₂×C₄×C₁₆	SD₁₆	1	1	yes	characteristic
74	C₈×D₈	—	2	2	no	—
75	C₈×D₈	—	2	2	no	—
76	C₈×D₈	—	4	1	no	—
77	C₈×D₈	—	4	1	no	—
78	C₂×#335	—	2	1	no	—
79	C₂×D₄⋊C₈	—	4	1	no	—
80	C₂×C₈²	C₂²⋊C₄	1	1	yes	characteristic
81	C₄⋊C₃₂	—	4	2	no	—
82	C₂²⋊C₃₂	—	8	1	no	—
83	#1397	—	2	4	no	—
84	#1397	—	8	1	no	—
85	C₄×C₃₂	—	2	4	no	—
86	#1371	—	2	1	no	—
87	#1371	C₂×C₈	1	2	yes	—
88	#1370	—	2	2	no	—
89	#1370	—	4	1	no	—
90	#1331	—	2	2	no	—
91	#1331	—	2	2	no	—
92	D₄⋊C₁₆	—	4	2	no	—
93	D₄⋊C₁₆	—	4	2	no	—
94	C₈×C₁₆	—	2	1	no	—
95	C₈×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
96	C₈×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
97	C₈×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
98	C₈×C₁₆	C₂×C₈	1	1	yes	characteristic
99	C₈×C₁₆	M₁₆	1	1	yes	characteristic
100	C₈×C₁₆	D₈	1	1	yes	characteristic
101	C₈×C₁₆	SD₁₆	1	1	yes	characteristic
102	C₈×C₁₆	C₂²×C₄	1	1	yes	characteristic
103	C₈×C₁₆	C₂×D₄	1	1	yes	characteristic
104	C₈×C₁₆	C₂×D₄	1	1	yes	characteristic
105	C₂²×D₈	—	2	1	no	—
106	C₂³×C₈	—	8	1	no	—
107	C₂×C₄×D₄	—	4	1	no	—
108	C₂×D₁₆	—	2	4	no	—
109	C₂×D₁₆	—	2	1	no	—
110	C₂×D₁₆	C₂×C₁₆	1	2	yes	—
111	C₂²×C₁₆	—	2	2	no	—
112	C₂²×C₁₆	—	2	1	no	—
113	C₂²×C₁₆	—	2	1	no	—
114	C₂²×C₁₆	—	8	1	no	—
115	C₂²×C₁₆	—	8	2	no	—
116	C₂²×C₁₆	—	8	2	no	—
117	C₂²×C₁₆	—	16	4	no	—
118	C₂²×C₁₆	C₂×C₁₆	1	1	yes	characteristic
119	C₂²×C₁₆	M₃₂	1	1	yes	characteristic
120	C₂²×C₁₆	D₁₆	1	1	yes	characteristic
121	C₂²×C₁₆	SD₃₂	1	1	yes	characteristic
122	C₄×D₈	—	2	2	no	—
123	C₄×D₈	—	2	2	no	—
124	C₄×D₈	—	4	1	no	—
125	C₄×D₈	—	4	1	no	—
126	C₈×D₄	—	4	2	no	—
127	C₈×D₄	—	4	2	no	—
128	C₈×D₄	—	8	1	no	—
129	C₈×D₄	—	8	1	no	—
130	C₂×#107	—	2	1	no	—
131	C₂×C₄⋊C₈	—	4	1	no	—
132	C₂×#102	—	4	1	no	—
133	C₂×C₂²⋊C₈	—	8	1	no	—
134	C₂×C₄×C₈	C₂²⋊C₈	1	1	yes	characteristic
135	C₂×C₄×C₈	#102	1	1	yes	characteristic
136	C₂×C₄×C₈	#104	1	1	yes	characteristic
137	M₆₄	—	4	8	no	—
138	C₂×C₃₂	—	2	8	no	—
139	C₂×C₃₂	—	8	2	no	—
140	#366	—	2	1	no	—
141	#366	C₂×C₁₆	1	2	yes	—
142	C₄⋊C₁₆	—	4	2	no	—
143	C₄⋊C₁₆	—	8	1	no	—
144	#357	—	2	2	no	—
145	#357	—	2	2	no	—
146	C₂²⋊C₁₆	—	8	2	no	—
147	C₂²⋊C₁₆	—	8	2	no	—
148	C₄×C₁₆	—	2	2	no	—
149	C₄×C₁₆	—	2	1	no	—
150	C₄×C₁₆	—	2	1	no	—
151	C₄×C₁₆	—	2	1	no	—
152	C₄×C₁₆	—	2	1	no	—
153	C₄×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
154	C₄×C₁₆	C₂²⋊C₈	1	1	yes	characteristic
155	C₄×C₁₆	#102	1	1	yes	characteristic
156	C₄×C₁₆	#102	1	1	yes	characteristic
157	C₄×C₁₆	C₂×C₁₆	1	1	yes	characteristic
158	C₄×C₁₆	M₃₂	1	1	yes	characteristic
159	C₄×C₁₆	D₁₆	1	1	yes	characteristic
160	C₄×C₁₆	SD₃₂	1	1	yes	characteristic
161	C₄×C₁₆	C₂²×C₈	1	1	yes	characteristic
162	C₄×C₁₆	C₂×M₁₆	1	1	yes	characteristic
163	C₄×C₁₆	C₂×D₈	1	1	yes	characteristic
164	C₄×C₁₆	C₂×SD₁₆	1	1	yes	characteristic
165	#335	—	2	2	no	—
166	#335	—	4	1	no	—
167	D₄⋊C₈	—	4	2	no	—
168	D₄⋊C₈	—	4	2	no	—
169	C₈²	C₂²⋊C₈	1	1	yes	characteristic
170	C₈²	#102	1	1	yes	characteristic
171	C₈²	#104	1	1	yes	characteristic
172	C₈²	C₂×C₂²⋊C₄	1	1	yes	characteristic
173	C₂²×D₄	—	4	1	no	—
174	C₂³×C₄	—	8	1	no	—
175	C₂×D₈	—	2	4	no	—
176	C₂×D₈	—	2	2	no	—
177	C₂×D₈	—	4	2	no	—
178	C₂×D₈	—	4	1	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	8	1	no	—
181	C₂²×C₈	—	8	2	no	—
182	C₂²×C₈	—	8	2	no	—
183	C₂²×C₈	—	16	2	no	—
184	C₂²×C₈	—	16	2	no	—
185	C₂²×C₈	C₂²⋊C₁₆	1	1	yes	characteristic
186	C₂²×C₈	#350	1	1	yes	characteristic
187	C₂²×C₈	#357	1	1	yes	characteristic
188	C₂²×C₈	#359	1	1	yes	characteristic
189	C₄×D₄	—	4	2	no	—
190	C₄×D₄	—	4	2	no	—
191	C₄×D₄	—	8	1	no	—
192	C₄×D₄	—	8	1	no	—
193	C₂×C₄⋊C₄	—	4	1	no	—
194	C₂×C₂²⋊C₄	—	8	1	no	—
195	C₂×C₄²	D₄⋊C₈	1	1	yes	characteristic
196	D₁₆	—	2	4	no	—
197	D₁₆	—	2	2	no	—
198	D₁₆	M₆₄	1	4	yes	complemented
199	C₂×C₁₆	—	2	2	no	—
200	C₂×C₁₆	—	2	2	no	—
201	C₂×C₁₆	—	2	8	no	—
202	C₂×C₁₆	—	2	1	no	—
203	C₂×C₁₆	—	2	1	no	—
204	C₂×C₁₆	—	2	2	no	—
205	C₂×C₁₆	—	2	2	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	—	4	1	no	—
209	C₂×C₁₆	—	4	1	no	—
210	C₂×C₁₆	—	4	2	no	—
211	C₂×C₁₆	—	4	2	no	—
212	C₂×C₁₆	—	8	2	no	—
213	C₂×C₁₆	—	16	4	no	—
214	C₂×C₁₆	—	16	2	no	—
215	C₂×C₁₆	—	16	4	no	—
216	C₂×C₁₆	C₂²⋊C₁₆	1	1	yes	characteristic
217	C₂×C₁₆	C₂²⋊C₁₆	1	1	yes	characteristic
218	C₂×C₁₆	#357	1	1	yes	characteristic
219	C₂×C₁₆	#357	1	1	yes	characteristic
220	C₂×C₁₆	C₂²×C₁₆	1	1	yes	characteristic
221	C₂×C₁₆	C₂×M₃₂	1	1	yes	characteristic
222	C₂×C₁₆	C₂×D₁₆	1	1	yes	characteristic
223	C₂×C₁₆	C₂×SD₃₂	1	1	yes	characteristic
224	#107	—	2	2	no	—
225	#107	—	4	1	no	—
226	C₄⋊C₈	—	4	2	no	—
227	C₄⋊C₈	—	8	1	no	—
228	#102	—	4	2	no	—
229	#102	—	4	2	no	—
230	C₂²⋊C₈	—	8	2	no	—
231	C₂²⋊C₈	—	8	2	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	D₄⋊C₈	1	1	yes	characteristic
234	C₄×C₈	D₄⋊C₈	1	1	yes	characteristic
235	C₄×C₈	D₄⋊C₈	1	1	yes	characteristic
236	C₄×C₈	C₂²⋊C₁₆	1	1	yes	characteristic
237	C₄×C₈	#350	1	1	yes	characteristic
238	C₄×C₈	#357	1	1	yes	characteristic
239	C₄×C₈	#359	1	1	yes	characteristic
240	C₄×C₈	C₂×C₂²⋊C₈	1	1	yes	characteristic
241	C₄×C₈	C₂×#102	1	1	yes	characteristic
242	C₄×C₈	C₂×#104	1	1	yes	characteristic
243	C₃₂	—	4	8	no	—
244	C₃₂	—	16	2	no	—
245	C₂⁴	—	8	1	no	—
246	C₂×D₄	—	4	4	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	8	2	no	—
249	C₂×D₄	—	8	1	no	—
250	C₂²×C₄	—	8	1	no	—
251	C₂²×C₄	—	8	2	no	—
252	C₂²×C₄	—	8	2	no	—
253	C₂²×C₄	—	16	2	no	—
254	C₂²×C₄	—	16	2	no	—
255	C₂²×C₄	D₄⋊C₁₆	1	1	yes	characteristic
256	C₂²×C₄	#1331	1	1	yes	characteristic
257	C₂²×C₄	#1335	1	1	yes	characteristic
258	D₈	—	2	4	no	—
259	D₈	—	4	4	no	—
260	D₈	—	4	2	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	8	2	no	—
271	C₂×C₈	—	16	4	no	—
272	C₂×C₈	—	16	2	no	—
273	C₂×C₈	—	16	2	no	—
274	C₂×C₈	—	16	2	no	—
275	C₂×C₈	D₄⋊C₁₆	1	1	yes	characteristic
276	C₂×C₈	D₄⋊C₁₆	1	1	yes	characteristic
277	C₂×C₈	#1331	1	1	yes	characteristic
278	C₂×C₈	#1331	1	1	yes	characteristic
279	C₂×C₈	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
280	C₂×C₈	C₂×#350	1	1	yes	characteristic
281	C₂×C₈	C₂×#357	1	1	yes	characteristic
282	C₂×C₈	C₂×#359	1	1	yes	characteristic
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	8	1	no	—
285	C₂²⋊C₄	—	8	2	no	—
286	C₂²⋊C₄	—	8	2	no	—
287	C₄²	D₄⋊C₁₆	1	1	yes	characteristic
288	C₄²	#1331	1	1	yes	characteristic
289	C₄²	#1335	1	1	yes	characteristic
290	C₄²	C₂×D₄⋊C₈	1	1	yes	characteristic
291	C₁₆	—	2	4	no	—
292	C₁₆	—	2	4	no	—
293	C₁₆	—	2	2	no	—
294	C₁₆	—	2	2	no	—
295	C₁₆	—	2	1	no	—
296	C₁₆	—	2	1	no	—
297	C₁₆	—	2	1	no	—
298	C₁₆	—	2	1	no	—
299	C₁₆	—	2	1	no	—
300	C₁₆	—	2	1	no	—
301	C₁₆	—	4	1	no	—
302	C₁₆	—	4	1	no	—
303	C₁₆	—	16	2	no	—
304	C₁₆	#1419	1	2	yes	—
305	C₁₆	C₂×M₆₄	1	2	yes	—
306	C₂³	—	8	4	no	—
307	C₂³	—	8	2	no	—
308	C₂³	—	16	2	no	—
309	C₂³	—	16	2	no	—
310	C₂³	#7068	1	1	yes	characteristic
311	D₄	—	4	4	no	—
312	D₄	—	8	4	no	—
313	D₄	—	8	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	8	2	no	—
320	C₂×C₄	—	16	4	no	—
321	C₂×C₄	—	16	2	no	—
322	C₂×C₄	—	16	2	no	—
323	C₂×C₄	—	16	2	no	—
324	C₂×C₄	#7068	1	1	yes	characteristic
325	C₂×C₄	#7068	1	1	yes	characteristic
326	C₂×C₄	#7068	1	1	yes	characteristic
327	C₂×C₄	C₂×D₄⋊C₁₆	1	1	yes	characteristic
328	C₂×C₄	C₂×#1331	1	1	yes	characteristic
329	C₂×C₄	C₂×#1335	1	1	yes	characteristic
330	C₈	—	2	2	no	—
331	C₈	—	2	2	no	—
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	16	2	no	—
339	C₈	#13553	1	1	yes	characteristic
340	C₈	#13558	1	1	yes	characteristic
341	C₈	#13622	1	1	yes	characteristic
342	C₈	C₂×#1419	1	1	yes	characteristic
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	8	4	no	—
346	C₂²	—	16	8	no	—
347	C₂²	—	16	2	no	—
348	C₂²	—	16	2	no	—
349	C₂²	#93752	1	2	yes	—
350	C₂²	C₂×#7068	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	16	2	no	—
356	C₄	#150044	1	1	yes	characteristic
357	C₄	#150051	1	1	yes	characteristic
358	C₄	#150064	1	1	yes	characteristic
359	C₄	#150069	1	1	yes	characteristic
360	C₂	—	2	2	no	—
361	C₂	—	16	4	no	—
362	C₂	C₂×#93752	1	1	yes	characteristic
363	C₂	#72226992	1	1	yes	characteristic
364	C₂	#72226999	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×#357
commutator	C₂×C₁₆	C₂²×C₁₆
Frattini, cosocle	C₁₆²	C₂³
socle	C₂²	C₂×#7068
prime radical	C₂²×D₁₆	C₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₁₆ ◁ C₂×C₈×C₁₆ ◁ C₂×C₁₆² ◁ G
LECS	G ▷ C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72226997	C₂	semidirect
2. #72225400	C₂	semidirect
4. #72216220	C₂	semidirect
16. #93883	C₂²	semidirect
19. #93665	C₂²	semidirect
198. D₁₆	M₆₄	semidirect

Maximal subgroups

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

subgroup	index
#72226997	2
#72225400	2
C₂×C₁₆×D₁₆	2
#72216220	2