Subgroups — #72322666

Order 2048.

632 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72321739	C₂	1	1	yes	maximal, characteristic
2	#72321145	C₂	1	2	yes	maximal, complemented
3	#72320480	C₂	1	1	yes	maximal, characteristic
4	#72286519	C₂	1	2	yes	maximal, complemented
5	#72232012	C₂	1	1	yes	maximal, characteristic
6	#512545	C₂²	1	1	yes	characteristic
7	#494211	—	2	1	no	—
8	#483341	—	2	1	no	—
9	#153583	C₄	1	4	yes	complemented
10	#153582	—	2	2	no	—
11	#153570	—	2	1	no	—
12	#153430	—	2	1	no	—
13	C₂×#7511	C₂²	1	2	yes	—
14	#152409	C₄	1	2	yes	—
15	#152391	—	2	1	no	—
16	#152300	—	2	1	no	—
17	#152295	C₄	1	1	yes	characteristic
18	#152294	C₄	1	1	yes	characteristic
19	#151998	—	2	1	no	—
20	#151996	C₂²	1	2	yes	—
21	#151038	—	2	1	no	—
22	#151038	—	2	2	no	—
23	#151037	C₄	1	2	yes	complemented
24	#151037	C₂²	1	1	yes	characteristic
25	C₂×#7463	C₄	1	2	yes	complemented
26	C₂×#7463	C₂²	1	1	yes	characteristic
27	C₂×#2258	C₂×C₄	1	1	yes	characteristic
28	C₂×#2258	D₄	1	1	yes	characteristic
29	C₄×#504	D₄	1	1	yes	characteristic
30	#20787	—	4	1	no	—
31	#17789	—	4	1	no	—
32	C₂×M₁₂₈	—	2	2	no	—
33	#13638	—	2	2	no	—
34	C₂×#1425	C₂×C₄	1	2	yes	—
35	#13633	C₂×C₄	1	2	yes	—
36	#13549	C₂×C₄	1	1	yes	characteristic
37	#13549	C₂×C₄	1	2	yes	—
38	#13549	D₄	1	1	yes	characteristic
39	#13549	Q₈	1	2	yes	—
40	C₄×M₆₄	D₄	1	2	yes	—
41	C₄×M₆₄	D₄	1	1	yes	characteristic
42	C₂×#1397	C₂×C₄	1	2	yes	—
43	C₂×#1397	C₂³	1	1	yes	characteristic
44	#12911	—	2	2	no	—
45	#12893	—	4	1	no	—
46	#12840	—	2	1	no	—
47	#12839	—	2	1	no	—
48	#12832	—	4	1	no	—
49	#12670	—	2	2	no	—
50	#12668	—	4	1	no	—
51	#12009	—	4	1	no	—
52	#12009	—	4	2	no	—
53	#12008	—	2	2	no	—
54	#12008	—	2	1	no	—
55	C₂×#1372	—	2	2	no	—
56	C₂×#1372	—	2	1	no	—
57	#7544	—	2	4	no	—
58	#7513	—	2	4	no	—
59	#7511	—	2	2	no	—
60	#7511	C₂×C₄	1	4	yes	—
61	#7476	—	2	1	no	—
62	#7475	—	2	1	no	—
63	#7473	—	2	2	no	—
64	#7463	—	2	2	no	—
65	#7463	—	2	2	no	—
66	#7463	—	2	1	no	—
67	#7463	—	2	1	no	—
68	#7463	C₂×C₄	1	1	yes	characteristic
69	#7463	C₂×C₄	1	2	yes	—
70	#7463	C₂×C₄	1	1	yes	characteristic
71	C₂×#504	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×#504	SD₁₆	1	1	yes	characteristic
73	#2988	—	8	1	no	—
74	C₄×#131	D₈	1	1	yes	characteristic
75	#2258	—	2	2	no	—
76	#2258	—	2	2	no	—
77	#2258	—	4	2	no	—
78	C₂×M₆₄	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×M₆₄	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×M₆₄	C₂²⋊C₄	1	2	yes	—
81	C₂×M₆₄	C₄⋊C₄	1	2	yes	—
82	C₂²×C₃₂	C₄²	1	2	yes	—
83	C₂²×C₃₂	C₄⋊C₄	1	2	yes	—
84	C₂²×C₃₂	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₃₂	C₂×D₄	1	1	yes	characteristic
86	C₂×#368	—	2	2	no	—
87	#2158	—	2	2	no	—
88	#2155	—	4	2	no	—
89	#2108	C₂²⋊C₄	1	1	yes	characteristic
90	#2108	C₄⋊C₄	1	2	yes	—
91	#2108	SD₁₆	1	1	yes	characteristic
92	#2108	Q₁₆	1	2	yes	—
93	C₄×M₃₂	D₈	1	1	yes	characteristic
94	C₄×M₃₂	SD₁₆	1	2	yes	—
95	C₂×C₄×C₁₆	C₂²⋊C₄	1	2	yes	—
96	C₂×C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
97	#1841	—	8	1	no	—
98	#1833	—	4	2	no	—
99	#1827	—	8	1	no	—
100	#1813	—	4	1	no	—
101	#1812	—	4	1	no	—
102	#1772	—	4	2	no	—
103	#1757	—	8	1	no	—
104	#1747	—	8	1	no	—
105	#1565	—	8	1	no	—
106	#1565	—	8	2	no	—
107	#1564	—	4	2	no	—
108	#1564	—	4	1	no	—
109	C₂×#335	—	4	2	no	—
110	C₂×#335	—	4	1	no	—
111	M₁₂₈	—	2	4	no	—
112	M₁₂₈	—	4	2	no	—
113	C₂×C₆₄	—	4	2	no	—
114	#1426	—	4	4	no	—
115	#1425	—	2	2	no	—
116	#1425	—	2	2	no	—
117	#1424	—	2	2	no	—
118	#1423	—	2	2	no	—
119	#1397	C₄²	1	2	yes	—
120	#1397	C₂²⋊C₄	1	2	yes	—
121	#1397	C₄⋊C₄	1	2	yes	—
122	#1397	C₂²×C₄	1	1	yes	characteristic
123	#1397	C₂²×C₄	1	1	yes	characteristic
124	#1397	C₂×D₄	1	1	yes	characteristic
125	#1397	C₂×Q₈	1	1	yes	characteristic
126	C₄×C₃₂	C₂²⋊C₄	1	2	yes	—
127	C₄×C₃₂	C₄⋊C₄	1	2	yes	—
128	C₄×C₃₂	C₄⋊C₄	1	2	yes	—
129	#1385	—	4	1	no	—
130	#1384	—	4	1	no	—
131	#1382	—	4	2	no	—
132	#1372	—	2	1	no	—
133	#1372	—	2	2	no	—
134	#1372	—	2	1	no	—
135	#1372	—	4	2	no	—
136	#1372	—	4	1	no	—
137	#1371	—	4	2	no	—
138	#1371	—	4	1	no	—
139	C₂×#131	#102	1	1	yes	characteristic
140	C₂×#131	SD₃₂	1	1	yes	characteristic
141	C₄×#41	D₁₆	1	1	yes	characteristic
142	#504	—	2	2	no	—
143	#504	—	2	2	no	—
144	#504	—	4	2	no	—
145	C₂×M₃₂	#102	1	1	yes	characteristic
146	C₂×M₃₂	#102	1	1	yes	characteristic
147	C₂×M₃₂	#103	1	2	yes	—
148	C₂×M₃₂	#106	1	2	yes	—
149	C₂²×C₁₆	#95	1	2	yes	—
150	C₂²×C₁₆	#103	1	2	yes	—
151	C₂²×C₁₆	C₂×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₁₆	C₂×SD₁₆	1	1	yes	characteristic
153	C₈×Q₈	—	16	1	no	—
154	C₈×D₄	—	16	2	no	—
155	C₈×D₄	—	16	1	no	—
156	#431	—	8	2	no	—
157	#431	—	8	1	no	—
158	C₂×#108	—	4	2	no	—
159	#427	—	4	2	no	—
160	C₂×C₄⋊C₈	—	8	2	no	—
161	C₂×C₄⋊C₈	—	8	1	no	—
162	#413	—	8	1	no	—
163	#410	—	8	1	no	—
164	#405	—	8	2	no	—
165	#405	#102	1	1	yes	characteristic
166	#405	#107	1	2	yes	—
167	#405	SD₃₂	1	1	yes	characteristic
168	#405	Q₃₂	1	2	yes	—
169	C₄×M₁₆	D₁₆	1	1	yes	characteristic
170	C₄×M₁₆	SD₃₂	1	2	yes	—
171	C₂×C₄×C₈	#102	1	2	yes	—
172	C₂×C₄×C₈	C₂×D₈	1	1	yes	characteristic
173	M₆₄	—	2	2	no	—
174	M₆₄	—	2	2	no	—
175	M₆₄	—	2	4	no	—
176	M₆₄	—	4	2	no	—
177	C₂×C₃₂	—	2	2	no	—
178	C₂×C₃₂	—	2	2	no	—
179	C₂×C₃₂	—	2	2	no	—
180	C₂×C₃₂	—	4	2	no	—
181	C₂×C₃₂	#95	1	2	yes	—
182	C₂×C₃₂	#95	1	2	yes	—
183	C₂×C₃₂	C₂×C₄²	1	1	yes	characteristic
184	C₂×C₃₂	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	C₂×C₃₂	C₂×C₄⋊C₄	1	1	yes	characteristic
186	C₂×C₃₂	C₂×C₄⋊C₄	1	1	yes	characteristic
187	C₂×C₃₂	#131	1	2	yes	—
188	C₂×C₃₂	#131	1	2	yes	—
189	#368	—	4	2	no	—
190	#368	—	4	2	no	—
191	#367	—	4	2	no	—
192	#366	—	4	2	no	—
193	#365	—	8	4	no	—
194	#346	#103	1	2	yes	—
195	#346	#106	1	2	yes	—
196	#346	#107	1	2	yes	—
197	C₄×C₁₆	#95	1	2	yes	—
198	C₄×C₁₆	#102	1	2	yes	—
199	C₄×C₁₆	#103	1	2	yes	—
200	C₄×C₁₆	C₂×C₂²⋊C₄	1	1	yes	characteristic
201	C₄×C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
202	C₄×C₁₆	C₂×SD₁₆	1	1	yes	characteristic
203	C₄×C₁₆	C₂×Q₁₆	1	1	yes	characteristic
204	#344	—	8	2	no	—
205	#339	—	8	2	no	—
206	#338	—	8	1	no	—
207	H(3,ℤ/4)	—	8	1	no	—
208	H(3,ℤ/4)	—	8	2	no	—
209	#335	—	4	1	no	—
210	#335	—	4	2	no	—
211	#335	—	4	1	no	—
212	#335	—	8	2	no	—
213	#335	—	8	1	no	—
214	#334	—	8	2	no	—
215	#334	—	8	1	no	—
216	C₆₄	—	4	4	no	—
217	C₂×#41	#357	1	1	yes	characteristic
218	#131	—	2	2	no	—
219	#131	—	2	2	no	—
220	#131	—	4	2	no	—
221	C₂×M₁₆	—	8	2	no	—
222	C₂×M₁₆	#357	1	1	yes	characteristic
223	C₂×M₁₆	#357	1	1	yes	characteristic
224	C₂×M₁₆	#358	1	2	yes	—
225	C₂×M₁₆	#367	1	2	yes	—
226	C₂²×C₈	—	16	2	no	—
227	C₂²×C₈	—	16	1	no	—
228	C₂²×C₈	#340	1	2	yes	—
229	C₂²×C₈	#358	1	2	yes	—
230	C₂²×C₈	C₂×#102	1	1	yes	characteristic
231	C₂²×C₈	C₂×SD₃₂	1	1	yes	characteristic
232	C₄×Q₈	—	2	1	no	—
233	C₄×D₄	—	2	2	no	—
234	C₄×D₄	—	2	1	no	—
235	#117	—	8	2	no	—
236	#117	#357	1	1	yes	characteristic
237	#117	#366	1	2	yes	—
238	C₂×C₄²	#357	1	2	yes	—
239	C₂×C₄²	C₂×D₁₆	1	1	yes	characteristic
240	M₃₂	—	2	2	no	—
241	M₃₂	—	2	2	no	—
242	M₃₂	—	2	4	no	—
243	M₃₂	—	4	2	no	—
244	C₂×C₁₆	—	2	2	no	—
245	C₂×C₁₆	—	2	2	no	—
246	C₂×C₁₆	—	2	2	no	—
247	C₂×C₁₆	—	4	2	no	—
248	C₂×C₁₆	#340	1	2	yes	—
249	C₂×C₁₆	#340	1	2	yes	—
250	C₂×C₁₆	C₂×#95	1	1	yes	characteristic
251	C₂×C₁₆	#408	1	2	yes	—
252	C₂×C₁₆	C₂×#103	1	1	yes	characteristic
253	C₂×C₁₆	C₂×#103	1	1	yes	characteristic
254	C₂×C₁₆	#421	1	2	yes	—
255	C₂×C₁₆	C₂×#106	1	1	yes	characteristic
256	#108	—	8	2	no	—
257	#108	—	8	2	no	—
258	#107	—	8	2	no	—
259	#106	—	8	2	no	—
260	C₄⋊C₈	—	8	1	no	—
261	C₄⋊C₈	—	8	2	no	—
262	C₄⋊C₈	—	8	1	no	—
263	C₄⋊C₈	—	16	2	no	—
264	C₄⋊C₈	—	16	1	no	—
265	#101	—	16	1	no	—
266	#100	—	16	1	no	—
267	#99	—	16	2	no	—
268	C₂²⋊C₈	—	16	2	no	—
269	C₂²⋊C₈	—	16	1	no	—
270	#97	—	16	2	no	—
271	#97	#358	1	2	yes	—
272	#97	#366	1	2	yes	—
273	#97	#367	1	2	yes	—
274	C₄×C₈	—	16	2	no	—
275	C₄×C₈	—	16	1	no	—
276	C₄×C₈	—	16	2	no	—
277	C₄×C₈	#340	1	2	yes	—
278	C₄×C₈	#357	1	2	yes	—
279	C₄×C₈	#358	1	2	yes	—
280	C₄×C₈	C₂×#102	1	1	yes	characteristic
281	C₄×C₈	C₂×#107	1	1	yes	characteristic
282	C₄×C₈	C₂×SD₃₂	1	1	yes	characteristic
283	C₄×C₈	C₂×Q₃₂	1	1	yes	characteristic
284	C₃₂	—	2	2	no	—
285	C₃₂	—	2	1	no	—
286	C₃₂	—	2	1	no	—
287	C₃₂	—	2	2	no	—
288	C₃₂	#405	1	1	yes	characteristic
289	C₃₂	#405	1	1	yes	characteristic
290	C₃₂	#424	1	2	yes	—
291	#41	—	2	2	no	—
292	#41	—	2	2	no	—
293	C₂×Q₈	#1418	1	1	yes	characteristic
294	C₂×D₄	#1418	1	2	yes	—
295	C₂×D₄	#1418	1	1	yes	characteristic
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	1	no	—
298	C₂²×C₄	#1380	1	2	yes	—
299	C₂²×C₄	C₂×#357	1	1	yes	characteristic
300	M₁₆	—	2	2	no	—
301	M₁₆	—	2	2	no	—
302	M₁₆	—	2	4	no	—
303	M₁₆	—	4	2	no	—
304	M₁₆	—	16	2	no	—
305	M₁₆	—	16	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	2	2	no	—
309	C₂×C₈	—	4	2	no	—
310	C₂×C₈	—	16	1	no	—
311	C₂×C₈	—	16	2	no	—
312	C₂×C₈	—	16	1	no	—
313	C₂×C₈	—	16	2	no	—
314	C₂×C₈	—	64	1	no	—
315	C₂×C₈	—	64	2	no	—
316	C₂×C₈	#1380	1	2	yes	—
317	C₂×C₈	#1380	1	2	yes	—
318	C₂×C₈	#1480	1	2	yes	—
319	C₂×C₈	C₂×#340	1	1	yes	characteristic
320	C₂×C₈	C₂×#358	1	1	yes	characteristic
321	C₂×C₈	C₂×#358	1	1	yes	characteristic
322	C₂×C₈	#2143	1	2	yes	—
323	C₂×C₈	C₂×#367	1	1	yes	characteristic
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	16	2	no	—
327	C₂²⋊C₄	—	2	2	no	—
328	C₂²⋊C₄	—	2	1	no	—
329	C₂²⋊C₄	—	16	2	no	—
330	C₄²	—	2	2	no	—
331	C₄²	—	2	1	no	—
332	C₄²	—	16	2	no	—
333	C₄²	#1380	1	2	yes	—
334	C₄²	C₂×#357	1	1	yes	characteristic
335	C₄²	C₂×#366	1	1	yes	characteristic
336	C₁₆	—	2	1	no	—
337	C₁₆	—	2	1	no	—
338	C₁₆	—	2	2	no	—
339	C₁₆	—	2	2	no	—
340	C₁₆	#1481	1	2	yes	—
341	C₁₆	#1729	1	1	yes	characteristic
342	C₁₆	#1733	1	1	yes	characteristic
343	C₂³	#7477	1	2	yes	—
344	C₂³	C₂×#1418	1	1	yes	characteristic
345	Q₈	—	2	2	no	—
346	D₄	—	2	4	no	—
347	D₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	8	1	no	—
354	C₂×C₄	—	8	2	no	—
355	C₂×C₄	—	32	2	no	—
356	C₂×C₄	#7477	1	2	yes	—
357	C₂×C₄	#11645	1	2	yes	—
358	C₂×C₄	C₂×#1380	1	1	yes	characteristic
359	C₂×C₄	C₂×#1418	1	1	yes	characteristic
360	C₂×C₄	#13625	1	2	yes	—
361	C₈	—	2	2	no	—
362	C₈	—	2	1	no	—
363	C₈	—	2	1	no	—
364	C₈	—	2	2	no	—
365	C₈	—	32	1	no	—
366	C₈	—	32	2	no	—
367	C₈	—	32	1	no	—
368	C₈	—	32	2	no	—
369	C₈	#7893	1	1	yes	characteristic
370	C₈	#11646	1	2	yes	—
371	C₈	#12561	1	1	yes	characteristic
372	C₂²	—	2	2	no	—
373	C₂²	—	4	1	no	—
374	C₂²	—	4	2	no	—
375	C₂²	#150123	1	2	yes	—
376	C₂²	C₂×#7477	1	1	yes	characteristic
377	C₄	—	2	2	no	—
378	C₄	—	4	1	no	—
379	C₄	—	4	2	no	—
380	C₄	—	4	1	no	—
381	C₄	—	64	2	no	—
382	C₄	#126830	1	1	yes	characteristic
383	C₄	#150123	1	2	yes	—
384	C₄	#151778	1	1	yes	characteristic
385	C₂	—	2	1	no	—
386	C₂	—	2	2	no	—
387	C₂	—	4	2	no	—
388	C₂	#72322665	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72322665
commutator	C₂×C₃₂	C₂×C₄²
Frattini, cosocle	C₂×#1397	C₂³
prime radical	C₂×#41	#357
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ C₄×#504 ◁ #512545 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #72321145	C₂	semidirect
4. #72286519	C₂	semidirect
9. #153583	C₄	semidirect
23. #151037	C₄	semidirect
25. C₂×#7463	C₄	semidirect

Maximal subgroups

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

subgroup	index
#72321739	2
#72321145	2
#72320480	2
#72286519	2
#72232012	2