Finite Group Databasebeta

← C2×#23782247 #35227871

Subgroups — C2×#23782247 #35227871

Order 20864.

1336 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782247C2116yesmaximal, direct product factor
2C2×#19104730C211yesmaximal, characteristic, complemented
3C2×#19104728C211yesmaximal, characteristic, complemented
4C2×#19104722C211yesmaximal, characteristic, complemented
5C2×#19104721C211yesmaximal, characteristic, complemented
6C2×#19104718C211yesmaximal, characteristic, complemented
7C2×#19104716C211yesmaximal, characteristic, complemented
8C2×#19104713C211yesmaximal, characteristic, complemented
9C2×#19104707C211yesmaximal, characteristic, complemented
10C2×#19104706C211yesmaximal, characteristic, complemented
11C2×#19104703C211yesmaximal, characteristic, complemented
12C2×#19104703C211yesmaximal, characteristic, complemented
13C2×C4×C22⋊D163C211yesmaximal, characteristic, complemented
14C2×(C22⋊D163)⋊C4C211yesmaximal, characteristic, complemented
15C2×C22⋊C4×D163C211yesmaximal, characteristic, complemented
16C2×#121×C163C211yesmaximal, characteristic, complemented
17C24×D163C2211yescharacteristic
18C22×Dic326C2211yescharacteristic, complemented
19C22×C22⋊D163C2211yescharacteristic
20C22×C22⋊D163C2211yescharacteristic
21C22×C22⋊D163C2211yescharacteristic
22C22×C22⋊D163C2211yescharacteristic
23C23×Dic163C2211yescharacteristic, complemented
24C22×C4×D163C2211yescharacteristic
25C22×C4×D163C2211yescharacteristic
26#19104730C2218yescomplemented
27#19104728C2218yescomplemented
28#19104722C2218yescomplemented
29#19104721C2218yescomplemented
30#19104718C2218yescomplemented
31#19104716C2218yescomplemented
32#19104713C2218yescomplemented
33#19104707C2218yescomplemented
34#19104706C2218yescomplemented
35#19104703C2218yescomplemented
36#19104703C2218yescomplemented
37C4×C22⋊D163C2218yescomplemented
38(C22⋊D163)⋊C4C2218yescomplemented
39C2×C4⋊Dic163C2211yescharacteristic, complemented
40C2×Dic163⋊C4C2211yescharacteristic, complemented
41C2×Dic163⋊C4C2211yescharacteristic, complemented
42C2×Dic163⋊C4C2211yescharacteristic, complemented
43C2×Dic163⋊C4C2211yescharacteristic, complemented
44C2×C22⋊Dic163C2211yescharacteristic
45C2×C22⋊Dic163C2211yescharacteristic
46C2×C22⋊Dic163C2211yescharacteristic
47C2×C22⋊Dic163C2211yescharacteristic, complemented
48C2×C22⋊Dic163C2211yescharacteristic, complemented
49C2×C22⋊Dic163C2211yescharacteristic, complemented
50C2×(C2×D163)⋊C4C2211yescharacteristic
51C2×(C2×D163)⋊C4C2211yescharacteristic
52C2×(C2×D163)⋊C4C2211yescharacteristic
53C2×(C2×D163)⋊C4C2211yescharacteristic
54C2×(C2×D163)⋊C4C2211yescharacteristic
55C2×(C2×D163)⋊C4C2211yescharacteristic
56C22⋊C4×D163C2218yescomplemented
57C2×C4×Dic163C2211yescharacteristic, complemented
58C2×C4×Dic163C2211yescharacteristic, complemented
59C22×D4×C163C2211yescharacteristic
60C22×D4×C163C2211yescharacteristic
61C22×D4×C163C2211yescharacteristic, complemented
62C23×C4×C163C2211yescharacteristic, complemented
63#121×C163C2218yescomplemented
64C2×C4⋊C4×C163C2211yescharacteristic, complemented
65C2×C22⋊C4×C163C2211yescharacteristic
66C2×C22⋊C4×C163C2211yescharacteristic, complemented
67C23×D163—24no—
68C23×D163—42no—
69C23×D163—42no—
70C23×D163C2311yescharacteristic
71C23×D163C2311yescharacteristic
72C23×D163C2314yes—
73C2×Dic326—42no—
74C2×Dic326C2314yescomplemented
75C2×C22⋊D163—24no—
76C2×C22⋊D163—24no—
77C2×C22⋊D163—42no—
78C2×C22⋊D163—42no—
79C2×C22⋊D163C2314yes—
80C2×C22⋊D163C2314yes—
81C2×C22⋊D163C2314yes—
82C2×C22⋊D163C2314yes—
83C22×Dic163—42no—
84C22×Dic163C2311yescharacteristic
85C22×Dic163C2311yescharacteristic
86C22×Dic163C2311yescharacteristic
87C22×Dic163C2311yescharacteristic
88C22×Dic163C2311yescharacteristic
89C22×Dic163C2311yescharacteristic
90C22×Dic163C2314yescomplemented
91C2×C4×D163—42no—
92C2×C4×D163—42no—
93C2×C4×D163C2314yes—
94C2×C4×D163C2314yes—
95C4⋊Dic163C2314yescomplemented
96Dic163⋊C4C2314yescomplemented
97Dic163⋊C4C2314yescomplemented
98Dic163⋊C4C2314yescomplemented
99Dic163⋊C4C2314yescomplemented
100C22⋊Dic163C2314yes—
101C22⋊Dic163C2314yes—
102C22⋊Dic163C2314yes—
103C22⋊Dic163C2314yescomplemented
104C22⋊Dic163C2314yescomplemented
105C22⋊Dic163C2314yescomplemented
106(C2×D163)⋊C4C2314yes—
107(C2×D163)⋊C4C2314yes—
108(C2×D163)⋊C4C2314yes—
109(C2×D163)⋊C4C2314yes—
110(C2×D163)⋊C4C2314yes—
111(C2×D163)⋊C4C2314yes—
112C4×Dic163C2314yescomplemented
113C4×Dic163C2314yescomplemented
114C24×C163C2311yescharacteristic
115C24×C163C2311yescharacteristic
116C24×C163C2311yescharacteristic
117C2×D4×C163—42no—
118C2×D4×C163—42no—
119C2×D4×C163—42no—
120C2×D4×C163C2314yes—
121C2×D4×C163C2314yes—
122C2×D4×C163C2314yescomplemented
123C22×C4×C163—42no—
124C22×C4×C163C2311yescharacteristic
125C22×C4×C163C2311yescharacteristic
126C22×C4×C163C2311yescharacteristic
127C22×C4×C163C2311yescharacteristic
128C22×C4×C163C2314yescomplemented
129C4⋊C4×C163C2314yescomplemented
130C22⋊C4×C163C2314yes—
131C22⋊C4×C163C2314yescomplemented
132C22×D163—22no—
133C22×D163—22no—
134C22×D163—22no—
135C22×D163—22no—
136C22×D163—22no—
137C22×D163—28no—
138C22×D163—44no—
139C22×D163—44no—
140C22×D163—416no—
141C22×D163#4114yes—
142C22×D163C2412yes—
143C22×D163C2412yes—
144Dic326—44no—
145C22⋊D163—28no—
146C22⋊D163—28no—
147C22⋊D163—44no—
148C22⋊D163—44no—
149C2×Dic163—22no—
150C2×Dic163—22no—
151C2×Dic163—22no—
152C2×Dic163—22no—
153C2×Dic163—22no—
154C2×Dic163—22no—
155C2×Dic163—44no—
156C2×Dic163C2412yes—
157C2×Dic163C2412yes—
158C2×Dic163C2412yes—
159C2×Dic163C2412yes—
160C2×Dic163C2412yes—
161C2×Dic163C2412yes—
162C4×D163—44no—
163C4×D163—44no—
164C23×C163—22no—
165C23×C163—22no—
166C23×C163—22no—
167C23×C163—22no—
168C23×C163—22no—
169C23×C163—22no—
170C23×C163—22no—
171C23×C163—22no—
172C23×C163—22no—
173C23×C163C2412yes—
174C23×C163C2412yes—
175C23×C163C2412yes—
176C23×C163C2411yescharacteristic
177D4×C163—44no—
178D4×C163—44no—
179D4×C163—44no—
180C2×C4×C163—22no—
181C2×C4×C163—22no—
182C2×C4×C163—22no—
183C2×C4×C163—22no—
184C2×C4×C163—44no—
185C2×C4×C163C2412yes—
186C2×C4×C163C2412yes—
187C2×C4×C163C2412yes—
188C2×C4×C163C2412yes—
189C22×C163—22no—
190C22×C163—22no—
191C22×C163—22no—
192C22×C163—22no—
193C22×C163—22no—
194C22×C163—22no—
195C22×C163—22no—
196C22×C163—22no—
197C22×C163—22no—
198C22×C163—44no—
199C22×C163—44no—
200C22×C163—44no—
201C22×C163C2×#4112yes—
202C22×C163H(5,𝔽2)12yes—
203C22×C163H(5,𝔽2)12yes—
204C22×C163C2511yescharacteristic
205C2×D163—28no—
206C2×D163—22no—
207C2×D163—22no—
208C2×D163—22no—
209C2×D163—22no—
210C2×D163—22no—
211C2×D163—416no—
212C2×D163—44no—
213C2×D163C2×#4114yes—
214C4×C163—22no—
215C4×C163—22no—
216C4×C163—22no—
217C4×C163—22no—
218Dic163—22no—
219Dic163—22no—
220Dic163—22no—
221Dic163—22no—
222Dic163—22no—
223Dic163—22no—
224C2×C163—42no—
225C2×C163—42no—
226C2×C163—42no—
227C2×C163#53814yescomplemented
228C2×C163C22×#4111yescharacteristic
229C2×C163C2×H(5,𝔽2)11yescharacteristic
230C2×C163C2×H(5,𝔽2)11yescharacteristic
231D163—24no—
232D163—42no—
233C163C2×#53811yescharacteristic, complemented, Sylow
234C2×#538—1631nomaximal, Sylow
235#538—16316no—
236C2×#126—1631no—
237C2×#126—1631no—
238C2×#124—1631no—
239C2×#124—1631no—
240C2×#123—1631no—
241C2×#123—1631no—
242C2×C22⋊Q8—1631no—
243C2×#121—1631no—
244C2×#121—1631no—
245C2×#121D16311yescharacteristic, complemented
246C2×#120—1631no—
247C2×#120—1631no—
248C2×C4×D4—1631no—
249C2×C4×D4—1631no—
250C22×C22⋊C4—1631no—
251C25—1631no—
252C22×Q8—1631no—
253C22×D4—1631no—
254C22×D4—1631no—
255C22×D4—1631no—
256C22×D4—1631no—
257C22×D4C2×D16311yescharacteristic
258C22×D4C2×D16311yescharacteristic
259C22×D4C2×D16311yescharacteristic, complemented
260C23×C4—1631no—
261C23×C4—1631no—
262C23×C4—1631no—
263C23×C4C2×D16311yescharacteristic, complemented
264#126—1638no—
265#126—1638no—
266#124—1638no—
267#124—1638no—
268#123—1638no—
269#123—1638no—
270C22⋊Q8—1638no—
271#121—1638no—
272#121—1638no—
273#121C2×D16318yescomplemented
274#120—1638no—
275#120—1638no—
276C4×D4—1638no—
277C4×D4—1638no—
278C2×C4⋊C4—1631no—
279C2×C4⋊C4—1631no—
280C2×C4⋊C4—1631no—
281C2×C4⋊C4—1631no—
282C2×C4⋊C4—1631no—
283C2×C4⋊C4C2×D16311yescharacteristic, complemented
284C2×C22⋊C4—1631no—
285C2×C22⋊C4—1631no—
286C2×C22⋊C4—1631no—
287C2×C22⋊C4—1631no—
288C2×C22⋊C4—1631no—
289C2×C22⋊C4—1631no—
290C2×C22⋊C4—1631no—
291C2×C22⋊C4—1631no—
292C2×C22⋊C4—1631no—
293C2×C22⋊C4—1631no—
294C2×C22⋊C4—1631no—
295C2×C22⋊C4—1631no—
296C2×C22⋊C4—1638no—
297C2×C22⋊C4C2×D16311yescharacteristic
298C2×C22⋊C4C2×D16311yescharacteristic, complemented
299C2×C42—1631no—
300C2×C42—1631no—
301C24—3264no—
302C24—6522no—
303C24—6522no—
304C24—1631no—
305C24—1631no—
306C24—1634no—
307C24C22×D16311yescharacteristic
308C24C22×D16311yescharacteristic
309C24C22×D16311yescharacteristic
310C2×Q8—6522no—
311C2×Q8—1634no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—3264no—
316C2×D4—3264no—
317C2×D4—6522no—
318C2×D4—6522no—
319C2×D4—1634no—
320C2×D4—1634no—
321C2×D4—1634no—
322C2×D4—1634no—
323C2×D4C22×D16314yes—
324C2×D4C22×D16314yes—
325C2×D4C22×D16314yescomplemented
326C22×C4—42no—
327C22×C4—6522no—
328C22×C4—6522no—
329C22×C4—6522no—
330C22×C4—1631no—
331C22×C4—1631no—
332C22×C4—1631no—
333C22×C4—1631no—
334C22×C4—1631no—
335C22×C4—1631no—
336C22×C4—1634no—
337C22×C4—1634no—
338C22×C4—1634no—
339C22×C4C22×D16311yescharacteristic
340C22×C4C22×D16311yescharacteristic
341C22×C4C22×D16311yescharacteristic
342C22×C4C22×D16311yescharacteristic
343C22×C4C22×D16314yescomplemented
344C4⋊C4—1634no—
345C4⋊C4—1634no—
346C4⋊C4—1634no—
347C4⋊C4—1634no—
348C4⋊C4—1634no—
349C4⋊C4C22×D16314yescomplemented
350C22⋊C4—1634no—
351C22⋊C4—1634no—
352C22⋊C4—1634no—
353C22⋊C4—1634no—
354C22⋊C4—1634no—
355C22⋊C4—1634no—
356C22⋊C4—1634no—
357C22⋊C4—1634no—
358C22⋊C4—1634no—
359C22⋊C4—1634no—
360C22⋊C4—1634no—
361C22⋊C4—1634no—
362C22⋊C4C22×D16314yes—
363C22⋊C4C22×D16314yescomplemented
364C42—1634no—
365C42—1634no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—3268no—
376C23—3262no—
377C23—3262no—
378C23—3262no—
379C23—3262no—
380C23—3262no—
381C23—6524no—
382C23—6524no—
383C23—65216no—
384C23—1632no—
385C23—1634no—
386C23—1632no—
387C23C23×D16312yes—
388C23C23×D16312yes—
389C23C23×D16312yes—
390C23C23×D16311yescharacteristic
391Q8—6524no—
392D4—44no—
393D4—44no—
394D4—44no—
395D4—3268no—
396D4—3268no—
397D4—6524no—
398D4—6524no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—44no—
404C2×C4—3262no—
405C2×C4—3262no—
406C2×C4—3262no—
407C2×C4—3262no—
408C2×C4—3262no—
409C2×C4—3262no—
410C2×C4—6524no—
411C2×C4—6524no—
412C2×C4—6524no—
413C2×C4—1632no—
414C2×C4—1632no—
415C2×C4—1632no—
416C2×C4—1632no—
417C2×C4—1632no—
418C2×C4—1632no—
419C2×C4C23×D16312yes—
420C2×C4C23×D16312yes—
421C2×C4C23×D16312yes—
422C2×C4C23×D16312yes—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—44no—
433C22—44no—
434C22—44no—
435C22—3262no—
436C22—3262no—
437C22—3268no—
438C22—3262no—
439C22—3262no—
440C22—3262no—
441C22—65216no—
442C22—6524no—
443C22—1634no—
444C22#41×D16312yes—
445C22#1910478012yes—
446C22#1910478012yes—
447C22C24×D16311yescharacteristic
448C4—22no—
449C4—22no—
450C4—22no—
451C4—22no—
452C4—3262no—
453C4—3262no—
454C4—3262no—
455C4—3262no—
456C4—3262no—
457C4—3262no—
458C2—42no—
459C2—42no—
460C2—42no—
461C2—3264no—
462C2—6522no—
463C2#2378224714yesdirect product factor
464C2C2×#41×D16311yescharacteristic
465C2C2×#1910478011yescharacteristic
466C2C2×#1910478011yescharacteristic
subgroupG/H
centerC23C23×D163
commutator, cosocleC22×C163C25
FrattiniC22C24×D163
FittingC2×#121×C163C2
socleC23×C163C24
hypercenterC2×#121D163
hypocenterC163C2×#538
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #23782247C2direct (N × H)
2. C2×#19104730C2semidirect
3. C2×#19104728C2semidirect
4. C2×#19104722C2semidirect
5. C2×#19104721C2semidirect
6. C2×#19104718C2semidirect
7. C2×#19104716C2semidirect
8. C2×#19104713C2semidirect
9. C2×#19104707C2semidirect
10. C2×#19104706C2semidirect
11. C2×#19104703C2semidirect
12. C2×#19104703C2semidirect
13. C2×C4×C22⋊D163C2semidirect
14. C2×(C22⋊D163)⋊C4C2semidirect
15. C2×C22⋊C4×D163C2semidirect
16. C2×#121×C163C2semidirect
18. C22×Dic326C22semidirect
23. C23×Dic163C22semidirect
26. #19104730C22semidirect
27. #19104728C22semidirect
28. #19104722C22semidirect
29. #19104721C22semidirect
30. #19104718C22semidirect
31. #19104716C22semidirect
32. #19104713C22semidirect
33. #19104707C22semidirect
34. #19104706C22semidirect
35. #19104703C22semidirect
36. #19104703C22semidirect
37. C4×C22⋊D163C22semidirect
38. (C22⋊D163)⋊C4C22semidirect
39. C2×C4⋊Dic163C22semidirect
40. C2×Dic163⋊C4C22semidirect
41. C2×Dic163⋊C4C22semidirect
42. C2×Dic163⋊C4C22semidirect
43. C2×Dic163⋊C4C22semidirect
47. C2×C22⋊Dic163C22semidirect
48. C2×C22⋊Dic163C22semidirect
49. C2×C22⋊Dic163C22semidirect
56. C22⋊C4×D163C22semidirect
57. C2×C4×Dic163C22semidirect
58. C2×C4×Dic163C22semidirect
61. C22×D4×C163C22semidirect
62. C23×C4×C163C22semidirect
63. #121×C163C22semidirect
64. C2×C4⋊C4×C163C22semidirect
66. C2×C22⋊C4×C163C22semidirect
74. C2×Dic326C23semidirect
90. C22×Dic163C23semidirect
95. C4⋊Dic163C23semidirect
96. Dic163⋊C4C23semidirect
97. Dic163⋊C4C23semidirect
98. Dic163⋊C4C23semidirect
99. Dic163⋊C4C23semidirect
103. C22⋊Dic163C23semidirect
104. C22⋊Dic163C23semidirect
105. C22⋊Dic163C23semidirect
112. C4×Dic163C23semidirect
113. C4×Dic163C23semidirect
122. C2×D4×C163C23semidirect
128. C22×C4×C163C23semidirect
129. C4⋊C4×C163C23semidirect
131. C22⋊C4×C163C23semidirect
227. C2×C163#538semidirect
233. C163C2×#538semidirect
245. C2×#121D163semidirect
259. C22×D4C2×D163semidirect
263. C23×C4C2×D163semidirect
273. #121C2×D163semidirect
283. C2×C4⋊C4C2×D163semidirect
298. C2×C22⋊C4C2×D163semidirect
325. C2×D4C22×D163semidirect
343. C22×C4C22×D163semidirect
349. C4⋊C4C22×D163semidirect
363. C22⋊C4C22×D163semidirect
463. C2#23782247direct (N × H)

Maximal subgroups

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

subgroupindex
#237822472
C2×#191047302
C2×#191047282
C2×#191047222
C2×#191047212
C2×#191047182
C2×#191047162
C2×#191047132
C2×#191047072
C2×#191047062
C2×#191047032
C2×C4×C22⋊D1632
C2×(C22⋊D163)⋊C42
C2×C22⋊C4×D1632
C2×#121×C1632
C2×#538163