Finite Group Databasebeta

← #35217998

Subgroups — #35217998

Order 20864.

800 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104717C211yesmaximal, characteristic, complemented
2C2×#19104704C211yesmaximal, characteristic
3C2×C4×C22⋊D163C211yesmaximal, characteristic, complemented
4#23781601C211yesmaximal, characteristic, complemented
5#23781593C211yesmaximal, characteristic, complemented
6#23781581C211yesmaximal, characteristic, complemented
7#23781580C211yesmaximal, characteristic, complemented
8#23781559C211yesmaximal, characteristic, complemented
9#23781558C211yesmaximal, characteristic, complemented
10#23781544C211yesmaximal, characteristic, complemented
11#23781541C211yesmaximal, characteristic, complemented
12#23781538C211yesmaximal, characteristic, complemented
13#23781525C211yesmaximal, characteristic, complemented
14#23781521C211yesmaximal, characteristic, complemented
15C22×C22⋊C4×C163C211yesmaximal, characteristic, complemented
16C22×C22⋊D163C2211yescharacteristic
17C22×C22⋊D163C2211yescharacteristic
18C22×C22⋊D163C2211yescharacteristic
19C22×C4×D163C2211yescharacteristic, complemented
20#19104717—24no—
21#19104704—24no—
22C4×C22⋊D163—24no—
23C2×C4⋊Dic163C2211yescharacteristic, complemented
24C2×C4⋊Dic163C2211yescharacteristic, complemented
25C2×Dic163⋊C4C2211yescharacteristic, complemented
26C2×Dic163⋊C4C2211yescharacteristic, complemented
27C2×Dic163⋊C4C2211yescharacteristic, complemented
28C2×C22⋊Dic163C2211yescharacteristic
29C2×C22⋊Dic163C2211yescharacteristic
30C2×C22⋊Dic163C2211yescharacteristic
31C2×C22⋊Dic163C2211yescharacteristic
32C2×C22⋊Dic163C2211yescharacteristic
33C2×C22⋊Dic163C2211yescharacteristic
34C2×C22⋊Dic163C2211yescharacteristic
35C2×C22⋊Dic163C2211yescharacteristic
36C2×C22⋊Dic163C2211yescharacteristic
37C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
38C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
39C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
40C2×C4×Dic163C2211yescharacteristic, complemented
41C163⋊#95C2211yescharacteristic, complemented
42C163⋊#95C2211yescharacteristic, complemented
43C163⋊#95C2211yescharacteristic, complemented
44C163⋊#95C2211yescharacteristic, complemented
45C163⋊#95C2211yescharacteristic, complemented
46C163⋊#95C2211yescharacteristic, complemented
47C25×C163C2211yescharacteristic
48C23×C4×C163C2211yescharacteristic
49C23×C4×C163C2211yescharacteristic
50C2×C22⋊C4×C163—24no—
51C2×C22⋊C4×C163C2211yescharacteristic
52C2×C22⋊C4×C163C2211yescharacteristic
53C2×C22⋊C4×C163C2211yescharacteristic
54C2×C22⋊C4×C163C2211yescharacteristic
55C23×D163C2311yescharacteristic
56C2×C22⋊D163—22no—
57C2×C22⋊D163—22no—
58C2×C22⋊D163—22no—
59C2×C22⋊D163—22no—
60C2×C22⋊D163—22no—
61C2×C22⋊D163—22no—
62C2×C22⋊D163—41no—
63C2×C22⋊D163—41no—
64C2×C22⋊D163D414yes—
65C22×Dic163C2311yescharacteristic
66C22×Dic163C2311yescharacteristic
67C22×Dic163C2311yescharacteristic
68C22×Dic163C2311yescharacteristic
69C22×Dic163C2311yescharacteristic
70C22×Dic163C2311yescharacteristic
71C22×Dic163C2311yescharacteristic
72C2×C4×D163—22no—
73C2×C4×D163—22no—
74C2×C4×D163—41no—
75C4⋊Dic163—22no—
76C4⋊Dic163—41no—
77Dic163⋊C4—22no—
78Dic163⋊C4—22no—
79Dic163⋊C4—41no—
80C22⋊Dic163—22no—
81C22⋊Dic163—22no—
82C22⋊Dic163—22no—
83C22⋊Dic163—22no—
84C22⋊Dic163—22no—
85C22⋊Dic163—22no—
86C22⋊Dic163—22no—
87C22⋊Dic163—41no—
88C22⋊Dic163—41no—
89(C2×D163)⋊C4—22no—
90(C2×D163)⋊C4—22no—
91(C2×D163)⋊C4—41no—
92C4×Dic163—22no—
93C24×C163—22no—
94C24×C163—22no—
95C24×C163—22no—
96C24×C163—22no—
97C24×C163—22no—
98C24×C163—41no—
99C24×C163—41no—
100C24×C163C2311yescharacteristic
101C24×C163C2311yescharacteristic
102C24×C163C2311yescharacteristic
103C22×C4×C163—22no—
104C22×C4×C163—22no—
105C22×C4×C163—22no—
106C22×C4×C163—22no—
107C22×C4×C163—41no—
108C22×C4×C163—41no—
109C22×C4×C163C2311yescharacteristic
110C22×C4×C163C2311yescharacteristic
111C22×C4×C163C2311yescharacteristic
112C22×C4×C163C2311yescharacteristic
113C22⋊C4×C163—22no—
114C22⋊C4×C163—22no—
115C22⋊C4×C163—22no—
116C22⋊C4×C163D414yescomplemented
117C22×D163—21no—
118C22×D163—21no—
119C22×D163—21no—
120C22×D163—21no—
121C22×D163—21no—
122C22×D163—21no—
123C22×D163C2×D412yes—
124C22⋊D163—28no—
125C22⋊D163—44no—
126C22⋊D163—44no—
127C2×Dic163—21no—
128C2×Dic163—21no—
129C2×Dic163—21no—
130C2×Dic163—21no—
131C2×Dic163—21no—
132C2×Dic163—21no—
133C2×Dic163—21no—
134C2×Dic163—21no—
135C2×Dic163—21no—
136C2×Dic163—21no—
137C2×Dic163—21no—
138C2×Dic163—21no—
139C2×Dic163—21no—
140C2×Dic163—21no—
141C2×Dic163—21no—
142C2×Dic163—21no—
143C2×Dic163—21no—
144C2×Dic163—21no—
145C2×Dic163—21no—
146C2×Dic163—21no—
147C2×Dic163C2×D412yes—
148C4×D163—44no—
149C23×C163—21no—
150C23×C163—21no—
151C23×C163—21no—
152C23×C163—21no—
153C23×C163—28no—
154C23×C163—21no—
155C23×C163—21no—
156C23×C163—21no—
157C23×C163—21no—
158C23×C163—21no—
159C23×C163—21no—
160C23×C163—21no—
161C23×C163—21no—
162C23×C163—21no—
163C23×C163—21no—
164C23×C163—21no—
165C23×C163—21no—
166C23×C163—44no—
167C23×C163—44no—
168C23×C163—44no—
169C23×C163—44no—
170C23×C163—44no—
171C23×C163—44no—
172C23×C163C2×D412yes—
173C23×C163C2×D412yes—
174C23×C163#4112yes—
175C23×C163#4112yes—
176C23×C163#4112yes—
177C23×C163C2411yescharacteristic
178C2×C4×C163—21no—
179C2×C4×C163—21no—
180C2×C4×C163—21no—
181C2×C4×C163—21no—
182C2×C4×C163—21no—
183C2×C4×C163—21no—
184C2×C4×C163—21no—
185C2×C4×C163—21no—
186C2×C4×C163—21no—
187C2×C4×C163—21no—
188C2×C4×C163—44no—
189C2×C4×C163—44no—
190C2×C4×C163C2×D412yes—
191C2×C4×C163C2×D412yes—
192C22×C163—22no—
193C22×C163—22no—
194C22×C163—22no—
195C22×C163—22no—
196C22×C163—22no—
197C22×C163—22no—
198C22×C163—22no—
199C22×C163—22no—
200C22×C163—22no—
201C22×C163—22no—
202C22×C163—22no—
203C22×C163—22no—
204C22×C163—416no—
205C22×C163—41no—
206C22×C163—41no—
207C22×C163—41no—
208C22×C163—41no—
209C22×C163—41no—
210C22×C163—41no—
211C22×C163—41no—
212C22×C163—41no—
213C22×C163#12314yescomplemented
214C22×C163C22×D411yescharacteristic
215C22×C163C22×D411yescharacteristic
216C22×C163C2×#4111yescharacteristic
217C22×C163C2×#4111yescharacteristic
218C22×C163C2×#4111yescharacteristic
219C22×C163H(5,𝔽2)11yescharacteristic
220C22×C163H(5,𝔽2)11yescharacteristic
221C2×D163—22no—
222C2×D163—22no—
223C2×D163—22no—
224C2×D163—41no—
225C2×D163—41no—
226C2×D163—41no—
227C2×D163—41no—
228C4×C163—22no—
229C4×C163—22no—
230C4×C163—41no—
231C4×C163—41no—
232Dic163—22no—
233Dic163—41no—
234Dic163—41no—
235Dic163—41no—
236Dic163—41no—
237Dic163—41no—
238Dic163—41no—
239C2×C163—24no—
240C2×C163—42no—
241C2×C163—42no—
242C2×C163C2×#12311yescharacteristic
243C2×C163#52511yescharacteristic
244C2×C163#53811yescharacteristic
245C2×C163D4211yescharacteristic
246C2×C163#54611yescharacteristic
247C2×C163#54611yescharacteristic
248C2×C163#55111yescharacteristic
249D163—42no—
250C163#241811yescharacteristic, complemented, Sylow
251#2418—1631nomaximal, Sylow
252C2×#123—1631no—
253C2×#120—1631no—
254C2×C4×D4—1631no—
255C22×C22⋊C4D16311yescharacteristic, complemented
256#399—1631no—
257#397—1631no—
258#394—1631no—
259#394—1631no—
260#388—1631no—
261#388—1631no—
262#386—1631no—
263#386—1631no—
264#385—1631no—
265#380—1631no—
266#379—1631no—
267C25C2×D16311yescharacteristic
268C22×D4—1631no—
269C22×D4—1631no—
270C22×D4—1631no—
271C23×C4—1631no—
272C23×C4C2×D16311yescharacteristic
273C23×C4C2×D16311yescharacteristic
274#123—3264no—
275#120—3264no—
276C4×D4—3264no—
277C2×C4⋊C4—1631no—
278C2×C4⋊C4—1631no—
279C2×C4⋊C4—1631no—
280C2×C4⋊C4—1631no—
281C2×C4⋊C4—1631no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—1631no—
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⋊C4C2×D16311yescharacteristic
296C2×C22⋊C4C2×D16311yescharacteristic
297C2×C22⋊C4C2×D16311yescharacteristic
298C2×C22⋊C4C2×D16311yescharacteristic
299C2×C42—1631no—
300#95—1631no—
301#95—1631no—
302#95—1631no—
303#95—1631no—
304#95—1631no—
305#95—1631no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—41no—
312C24—41no—
313C24—1631no—
314C24C22×D16311yescharacteristic
315C24C22×D16311yescharacteristic
316C24C22×D16311yescharacteristic
317C2×D4—3262no—
318C2×D4—3262no—
319C2×D4—3262no—
320C2×D4—3262no—
321C2×D4—3262no—
322C2×D4—3262no—
323C2×D4—6521no—
324C2×D4—6521no—
325C2×D4—1634no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—3262no—
333C22×C4—3262no—
334C22×C4—6521no—
335C22×C4—1631no—
336C22×C4—1631no—
337C22×C4—1631no—
338C22×C4—1631no—
339C22×C4—1631no—
340C22×C4—1631no—
341C22×C4—1631no—
342C22×C4C22×D16311yescharacteristic
343C22×C4C22×D16311yescharacteristic
344C22×C4C22×D16311yescharacteristic
345C22×C4C22×D16311yescharacteristic
346C4⋊C4—3262no—
347C4⋊C4—3262no—
348C4⋊C4—3262no—
349C4⋊C4—6521no—
350C4⋊C4—6521no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—3262no—
355C22⋊C4—3262no—
356C22⋊C4—3262no—
357C22⋊C4—3262no—
358C22⋊C4—3262no—
359C22⋊C4—3262no—
360C22⋊C4—3262no—
361C22⋊C4—3262no—
362C22⋊C4—3262no—
363C22⋊C4—6521no—
364C22⋊C4—6521no—
365C22⋊C4—6521no—
366C22⋊C4C22⋊D16314yescomplemented
367C42—3262no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—28no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—3261no—
392C23—3261no—
393C23—3261no—
394C23—3261no—
395C23—3261no—
396C23—3261no—
397C23—1632no—
398C23D4×D16312yes—
399C23C2×C22⋊D16312yes—
400C23#1457969812yes—
401C23#1457969912yes—
402C23#1457969912yes—
403C23C23×D16311yescharacteristic
404D4—3268no—
405D4—6524no—
406D4—6524no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—3261no—
420C2×C4—3261no—
421C2×C4—3261no—
422C2×C4—3261no—
423C2×C4—3261no—
424C2×C4—3261no—
425C2×C4—3261no—
426C2×C4—3261no—
427C2×C4—3261no—
428C2×C4—3261no—
429C2×C4—3261no—
430C2×C4—3261no—
431C2×C4—3261no—
432C2×C4—3261no—
433C2×C4—3261no—
434C2×C4—3261no—
435C2×C4—3261no—
436C2×C4—3261no—
437C2×C4—3261no—
438C2×C4—3261no—
439C2×C4—6524no—
440C2×C4—1632no—
441C2×C4C2×C22⋊D16312yes—
442C2×C4C2×C22⋊D16312yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—416no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—3262no—
465C22—3262no—
466C22—3262no—
467C22—6521no—
468C22—6521no—
469C22—6521no—
470C22—6521no—
471C22#1910471714yescomplemented
472C22C2×D4×D16311yescharacteristic
473C22C22×C22⋊D16311yescharacteristic
474C22C2×#1457969811yescharacteristic
475C22C2×#1457969911yescharacteristic
476C22C2×#1457969911yescharacteristic
477C22#1910478011yescharacteristic
478C22#1910478011yescharacteristic
479C4—22no—
480C4—22no—
481C4—41no—
482C4—41no—
483C4—3262no—
484C4—6521no—
485C4—6521no—
486C4—6521no—
487C4—6521no—
488C4—6521no—
489C4—6521no—
490C2—24no—
491C2—42no—
492C2—42no—
493C2—6522no—
494C2C2×#1910471711yescharacteristic
495C2#2378217611yescharacteristic
496C2#2378225311yescharacteristic
497C2D4×C22⋊D16311yescharacteristic
498C2#2378230511yescharacteristic
499C2#2378230911yescharacteristic
500C2#2378234111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
FittingC22×C22⋊C4×C163C2
prime radicalC2×#19104704C2
hypercenterC22×C22⋊C4D163
hypocenterC163#2418
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#19104717C2semidirect
3. C2×C4×C22⋊D163C2semidirect
4. #23781601C2semidirect
5. #23781593C2semidirect
6. #23781581C2semidirect
7. #23781580C2semidirect
8. #23781559C2semidirect
9. #23781558C2semidirect
10. #23781544C2semidirect
11. #23781541C2semidirect
12. #23781538C2semidirect
13. #23781525C2semidirect
14. #23781521C2semidirect
15. C22×C22⋊C4×C163C2semidirect
19. C22×C4×D163C22semidirect
23. C2×C4⋊Dic163C22semidirect
24. C2×C4⋊Dic163C22semidirect
25. C2×Dic163⋊C4C22semidirect
26. C2×Dic163⋊C4C22semidirect
27. C2×Dic163⋊C4C22semidirect
37. C2×(C2×D163)⋊C4C22semidirect
38. C2×(C2×D163)⋊C4C22semidirect
39. C2×(C2×D163)⋊C4C22semidirect
40. C2×C4×Dic163C22semidirect
41. C163⋊#95C22semidirect
42. C163⋊#95C22semidirect
43. C163⋊#95C22semidirect
44. C163⋊#95C22semidirect
45. C163⋊#95C22semidirect
46. C163⋊#95C22semidirect
116. C22⋊C4×C163D4semidirect
213. C22×C163#123semidirect
250. C163#2418semidirect
255. C22×C22⋊C4D163semidirect
366. C22⋊C4C22⋊D163semidirect
471. C22#19104717semidirect

Maximal subgroups

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

subgroupindex
C2×#191047172
C2×#191047042
C2×C4×C22⋊D1632
#237816012
#237815932
#237815812
#237815802
#237815592
#237815582
#237815442
#237815412
#237815382
#237815252
#237815212
C22×C22⋊C4×C1632
#2418163