Finite Group Databasebeta

← #35218072

Subgroups — #35218072

Order 20864.

922 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104709C211yesmaximal, characteristic, complemented
2C2×#19104708C211yesmaximal, characteristic, complemented
3C2×#19104704C211yesmaximal, characteristic, complemented
4C2×#19104704C211yesmaximal, characteristic, complemented
5C2×C4×C22⋊D163C211yesmaximal, characteristic, complemented
6#23781588C211yesmaximal, characteristic, complemented
7#23781580C211yesmaximal, characteristic, complemented
8#23781580C211yesmaximal, characteristic, complemented
9#23781574C211yesmaximal, characteristic, complemented
10#23781565C211yesmaximal, characteristic, complemented
11#23781558C211yesmaximal, characteristic, complemented
12#23781544C211yesmaximal, characteristic, complemented
13#23781541C211yesmaximal, characteristic, complemented
14#23781538C211yesmaximal, characteristic, complemented
15C22×C22⋊C4×C163C211yesmaximal, characteristic, complemented
16C22×D652C2211yescharacteristic, complemented
17C22×C22⋊D163C2211yescharacteristic
18C22×C22⋊D163C2211yescharacteristic
19C22×C22⋊D163C2211yescharacteristic
20C22×C22⋊D163C2211yescharacteristic
21C22×C22⋊D163C2211yescharacteristic
22C22×C22⋊D163C2211yescharacteristic
23C22×C4×D163C2211yescharacteristic, complemented
24#19104709—24no—
25#19104708—24no—
26#19104704—24no—
27#19104704—24no—
28C4×C22⋊D163—24no—
29C2×C4⋊Dic163C2211yescharacteristic, complemented
30C2×Dic163⋊C4C2211yescharacteristic, complemented
31C2×Dic163⋊C4C2211yescharacteristic, complemented
32C2×Dic163⋊C4C2211yescharacteristic, complemented
33C2×C22⋊Dic163C2211yescharacteristic
34C2×C22⋊Dic163C2211yescharacteristic
35C2×C22⋊Dic163C2211yescharacteristic
36C2×C22⋊Dic163C2211yescharacteristic
37C2×C22⋊Dic163C2211yescharacteristic
38C2×C22⋊Dic163C2211yescharacteristic
39C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
40C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
41C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
42C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
44C2×C4×Dic163C2211yescharacteristic, complemented
45C163⋊#95C2211yescharacteristic, complemented
46C163⋊#95C2211yescharacteristic, complemented
47C163⋊#95C2211yescharacteristic, complemented
48C163⋊#95C2211yescharacteristic, complemented
49C25×C163C2211yescharacteristic
50C23×C4×C163C2211yescharacteristic
51C23×C4×C163C2211yescharacteristic
52C2×C22⋊C4×C163—24no—
53C2×C22⋊C4×C163C2211yescharacteristic
54C2×C22⋊C4×C163C2211yescharacteristic
55C2×C22⋊C4×C163C2211yescharacteristic
56C2×C22⋊C4×C163C2211yescharacteristic
57C23×D163C2311yescharacteristic
58C23×D163C2311yescharacteristic
59C2×D652—22no—
60C2×D652—22no—
61C2×D652—41no—
62C2×C22⋊D163—22no—
63C2×C22⋊D163—22no—
64C2×C22⋊D163—22no—
65C2×C22⋊D163—22no—
66C2×C22⋊D163—22no—
67C2×C22⋊D163—22no—
68C2×C22⋊D163—22no—
69C2×C22⋊D163—22no—
70C2×C22⋊D163—22no—
71C2×C22⋊D163—22no—
72C2×C22⋊D163—22no—
73C2×C22⋊D163—22no—
74C2×C22⋊D163—22no—
75C2×C22⋊D163—41no—
76C2×C22⋊D163—41no—
77C2×C22⋊D163—41no—
78C2×C22⋊D163—41no—
79C2×C22⋊D163D414yescomplemented
80C22×Dic163C2311yescharacteristic
81C22×Dic163C2311yescharacteristic
82C22×Dic163C2311yescharacteristic
83C22×Dic163C2311yescharacteristic
84C22×Dic163C2311yescharacteristic
85C22×Dic163C2311yescharacteristic
86C2×C4×D163—22no—
87C2×C4×D163—22no—
88C2×C4×D163—41no—
89C4⋊Dic163—22no—
90Dic163⋊C4—22no—
91Dic163⋊C4—22no—
92Dic163⋊C4—41no—
93C22⋊Dic163—22no—
94C22⋊Dic163—22no—
95C22⋊Dic163—22no—
96C22⋊Dic163—22no—
97C22⋊Dic163—22no—
98C22⋊Dic163D414yescomplemented
99(C2×D163)⋊C4—22no—
100(C2×D163)⋊C4—22no—
101(C2×D163)⋊C4—22no—
102(C2×D163)⋊C4—22no—
103(C2×D163)⋊C4—41no—
104C4×Dic163—22no—
105C24×C163—22no—
106C24×C163—22no—
107C24×C163—22no—
108C24×C163—22no—
109C24×C163—22no—
110C24×C163—41no—
111C24×C163—41no—
112C24×C163C2311yescharacteristic
113C24×C163C2311yescharacteristic
114C24×C163C2311yescharacteristic
115C22×C4×C163—22no—
116C22×C4×C163—22no—
117C22×C4×C163—22no—
118C22×C4×C163—22no—
119C22×C4×C163—41no—
120C22×C4×C163—41no—
121C22×C4×C163C2311yescharacteristic
122C22×C4×C163C2311yescharacteristic
123C22×C4×C163C2311yescharacteristic
124C22×C4×C163C2311yescharacteristic
125C22⋊C4×C163—22no—
126C22⋊C4×C163—22no—
127C22⋊C4×C163—22no—
128C22⋊C4×C163D414yescomplemented
129C22×D163—21no—
130C22×D163—21no—
131C22×D163—21no—
132C22×D163—21no—
133C22×D163—21no—
134C22×D163—21no—
135C22×D163—21no—
136C22×D163—21no—
137C22×D163—21no—
138C22×D163—21no—
139C22×D163—21no—
140C22×D163—21no—
141C22×D163—21no—
142C22×D163C2×D412yes—
143D652—44no—
144C22⋊D163—28no—
145C22⋊D163—44no—
146C22⋊D163—44no—
147C22⋊D163—44no—
148C22⋊D163—44no—
149C22⋊D163—44no—
150C2×Dic163—21no—
151C2×Dic163—21no—
152C2×Dic163—21no—
153C2×Dic163—21no—
154C2×Dic163—21no—
155C2×Dic163—21no—
156C2×Dic163—21no—
157C2×Dic163—21no—
158C2×Dic163—21no—
159C2×Dic163—21no—
160C2×Dic163—21no—
161C2×Dic163—21no—
162C2×Dic163—21no—
163C2×Dic163—21no—
164C2×Dic163—21no—
165C2×Dic163C2×D412yes—
166C2×Dic163C2×D412yes—
167C2×Dic163C2×D412yes—
168C4×D163—44no—
169C23×C163—21no—
170C23×C163—21no—
171C23×C163—21no—
172C23×C163—21no—
173C23×C163—28no—
174C23×C163—21no—
175C23×C163—21no—
176C23×C163—21no—
177C23×C163—21no—
178C23×C163—21no—
179C23×C163—21no—
180C23×C163—21no—
181C23×C163—21no—
182C23×C163—21no—
183C23×C163—21no—
184C23×C163—21no—
185C23×C163—21no—
186C23×C163—44no—
187C23×C163—44no—
188C23×C163—44no—
189C23×C163—44no—
190C23×C163—44no—
191C23×C163—44no—
192C23×C163C2×D412yes—
193C23×C163C2×D412yes—
194C23×C163C2×D412yes—
195C23×C163#4112yes—
196C23×C163#4112yes—
197C23×C163C2411yescharacteristic
198C2×C4×C163—21no—
199C2×C4×C163—21no—
200C2×C4×C163—21no—
201C2×C4×C163—21no—
202C2×C4×C163—21no—
203C2×C4×C163—21no—
204C2×C4×C163—21no—
205C2×C4×C163—21no—
206C2×C4×C163—21no—
207C2×C4×C163—21no—
208C2×C4×C163—44no—
209C2×C4×C163—44no—
210C2×C4×C163C2×D412yes—
211C2×C4×C163C2×D412yes—
212C22×C163—22no—
213C22×C163—22no—
214C22×C163—22no—
215C22×C163—22no—
216C22×C163—22no—
217C22×C163—22no—
218C22×C163—22no—
219C22×C163—22no—
220C22×C163—22no—
221C22×C163—22no—
222C22×C163—22no—
223C22×C163—22no—
224C22×C163—416no—
225C22×C163—41no—
226C22×C163—41no—
227C22×C163—41no—
228C22×C163—41no—
229C22×C163—41no—
230C22×C163—41no—
231C22×C163—41no—
232C22×C163—41no—
233C22×C163#12114yescomplemented
234C22×C163C22×D411yescharacteristic
235C22×C163C22×D411yescharacteristic
236C22×C163C22×D411yescharacteristic
237C22×C163C2×#4111yescharacteristic
238C22×C163C2×#4111yescharacteristic
239C22×C163H(5,𝔽2)11yescharacteristic
240C22×C163H(5,𝔽2)11yescharacteristic
241C2×D163—22no—
242C2×D163—22no—
243C2×D163—22no—
244C2×D163—41no—
245C2×D163—41no—
246C2×D163—41no—
247C2×D163—41no—
248C2×D163—41no—
249C2×D163—41no—
250C2×D163—41no—
251C2×D163—41no—
252C2×D163—41no—
253C2×D163—41no—
254C2×D163—41no—
255C4×C163—22no—
256C4×C163—22no—
257C4×C163—41no—
258C4×C163—41no—
259Dic163—22no—
260Dic163—22no—
261Dic163—22no—
262Dic163—41no—
263Dic163—41no—
264Dic163—41no—
265C2×C163—24no—
266C2×C163—42no—
267C2×C163—42no—
268C2×C163C2×#12111yescharacteristic
269C2×C163#52511yescharacteristic
270C2×C163#53811yescharacteristic
271C2×C163D4211yescharacteristic
272C2×C163D4211yescharacteristic
273C2×C163#54611yescharacteristic
274C2×C163#54611yescharacteristic
275D163—42no—
276D163—81no—
277C163#242411yescharacteristic, complemented, Sylow
278#2424—1631nomaximal, Sylow
279C2×#121—1631no—
280C2×#121—1631no—
281C2×#120—1631no—
282C2×#120—1631no—
283C2×C4×D4—1631no—
284C22×C22⋊C4D16311yescharacteristic, complemented
285#396—1631no—
286#394—1631no—
287#394—1631no—
288#392—1631no—
289#390—1631no—
290#388—1631no—
291#386—1631no—
292#386—1631no—
293#385—1631no—
294C25C2×D16311yescharacteristic
295C22×D4—1631no—
296C22×D4—1631no—
297C22×D4—1631no—
298C22×D4—1631no—
299C22×D4—1631no—
300C22×D4—1631no—
301C22×D4—1631no—
302C23×C4—1631no—
303C23×C4C2×D16311yescharacteristic
304C23×C4C2×D16311yescharacteristic
305#121—3264no—
306#121—3264no—
307#120—3264no—
308#120—3264no—
309C4×D4—3264no—
310C2×C4⋊C4—1631no—
311C2×C4⋊C4—1631no—
312C2×C4⋊C4—1631no—
313C2×C4⋊C4—1631no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—1631no—
316C2×C22⋊C4—1631no—
317C2×C22⋊C4—1631no—
318C2×C22⋊C4—1631no—
319C2×C22⋊C4—1631no—
320C2×C22⋊C4—1631no—
321C2×C22⋊C4—1631no—
322C2×C22⋊C4—1631no—
323C2×C22⋊C4—1631no—
324C2×C22⋊C4—1631no—
325C2×C22⋊C4—1631no—
326C2×C22⋊C4C2×D16311yescharacteristic
327C2×C22⋊C4C2×D16311yescharacteristic
328C2×C22⋊C4C2×D16311yescharacteristic
329C2×C22⋊C4C2×D16311yescharacteristic
330C2×C42—1631no—
331#95—1631no—
332#95—1631no—
333#95—1631no—
334#95—1631no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—41no—
341C24—41no—
342C24—1631no—
343C24—1631no—
344C24C22×D16311yescharacteristic
345C24C22×D16311yescharacteristic
346C24C22×D16311yescharacteristic
347C2×D4—3262no—
348C2×D4—3262no—
349C2×D4—3262no—
350C2×D4—3262no—
351C2×D4—3262no—
352C2×D4—3262no—
353C2×D4—3262no—
354C2×D4—3262no—
355C2×D4—3262no—
356C2×D4—3262no—
357C2×D4—3262no—
358C2×D4—3262no—
359C2×D4—3262no—
360C2×D4—3262no—
361C2×D4—3262no—
362C2×D4—6521no—
363C2×D4—6521no—
364C2×D4—6521no—
365C2×D4—6521no—
366C2×D4—6521no—
367C2×D4—1634no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—3262no—
375C22×C4—3262no—
376C22×C4—6521no—
377C22×C4—1631no—
378C22×C4—1631no—
379C22×C4—1631no—
380C22×C4—1631no—
381C22×C4—1631no—
382C22×C4—1631no—
383C22×C4C22×D16311yescharacteristic
384C22×C4C22×D16311yescharacteristic
385C22×C4C22×D16311yescharacteristic
386C22×C4C22×D16311yescharacteristic
387C4⋊C4—3262no—
388C4⋊C4—3262no—
389C4⋊C4—3262no—
390C4⋊C4—6521no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—3262no—
395C22⋊C4—3262no—
396C22⋊C4—3262no—
397C22⋊C4—3262no—
398C22⋊C4—3262no—
399C22⋊C4—3262no—
400C22⋊C4—3262no—
401C22⋊C4—3262no—
402C22⋊C4—3262no—
403C22⋊C4—6521no—
404C22⋊C4—1634no—
405C22⋊C4C22⋊D16314yescomplemented
406C42—3262no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—28no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—3261no—
431C23—3261no—
432C23—3261no—
433C23—3261no—
434C23—3261no—
435C23—3261no—
436C23—3261no—
437C23—3261no—
438C23—3261no—
439C23—3261no—
440C23—3261no—
441C23—3261no—
442C23—3261no—
443C23—1632no—
444C23D4×D16312yes—
445C23D4×D16312yes—
446C23C2×C22⋊D16312yes—
447C23#1457969912yes—
448C23#1457969912yes—
449C23C23×D16311yescharacteristic
450D4—3268no—
451D4—6524no—
452D4—6524no—
453D4—6524no—
454D4—6524no—
455D4—6524no—
456D4—6524no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—3261no—
470C2×C4—3261no—
471C2×C4—3261no—
472C2×C4—3261no—
473C2×C4—3261no—
474C2×C4—3261no—
475C2×C4—3261no—
476C2×C4—3261no—
477C2×C4—3261no—
478C2×C4—3261no—
479C2×C4—3261no—
480C2×C4—3261no—
481C2×C4—3261no—
482C2×C4—3261no—
483C2×C4—3261no—
484C2×C4—6524no—
485C2×C4—1632no—
486C2×C4—1632no—
487C2×C4—1632no—
488C2×C4C2×C22⋊D16312yes—
489C2×C4C2×C22⋊D16312yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—416no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—3262no—
512C22—3262no—
513C22—3262no—
514C22—6521no—
515C22—6521no—
516C22—6521no—
517C22—6521no—
518C22—6521no—
519C22—6521no—
520C22—6521no—
521C22—6521no—
522C22—6521no—
523C22—6521no—
524C22—6521no—
525C22#1910470814yescomplemented
526C22C2×D4×D16311yescharacteristic
527C22C2×D4×D16311yescharacteristic
528C22C22×C22⋊D16311yescharacteristic
529C22C2×#1457969911yescharacteristic
530C22C2×#1457969911yescharacteristic
531C22#1910478011yescharacteristic
532C22#1910478011yescharacteristic
533C4—22no—
534C4—22no—
535C4—41no—
536C4—41no—
537C4—3262no—
538C4—3262no—
539C4—3262no—
540C4—6521no—
541C4—6521no—
542C4—6521no—
543C2—24no—
544C2—42no—
545C2—42no—
546C2—13041no—
547C2—6522no—
548C2C2×#1910470811yescharacteristic
549C2#2378217611yescharacteristic
550C2#2378225311yescharacteristic
551C2D4×C22⋊D16311yescharacteristic
552C2D4×C22⋊D16311yescharacteristic
553C2#2378230511yescharacteristic
554C2#2378230511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
FittingC22×C22⋊C4×C163C2
hypercenterC22×C22⋊C4D163
hypocenterC163#2424
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×#19104709C2semidirect
2. C2×#19104708C2semidirect
3. C2×#19104704C2semidirect
4. C2×#19104704C2semidirect
5. C2×C4×C22⋊D163C2semidirect
6. #23781588C2semidirect
7. #23781580C2semidirect
8. #23781580C2semidirect
9. #23781574C2semidirect
10. #23781565C2semidirect
11. #23781558C2semidirect
12. #23781544C2semidirect
13. #23781541C2semidirect
14. #23781538C2semidirect
15. C22×C22⋊C4×C163C2semidirect
16. C22×D652C22semidirect
23. C22×C4×D163C22semidirect
29. C2×C4⋊Dic163C22semidirect
30. C2×Dic163⋊C4C22semidirect
31. C2×Dic163⋊C4C22semidirect
32. C2×Dic163⋊C4C22semidirect
39. C2×(C2×D163)⋊C4C22semidirect
40. C2×(C2×D163)⋊C4C22semidirect
41. C2×(C2×D163)⋊C4C22semidirect
42. C2×(C2×D163)⋊C4C22semidirect
43. C2×(C2×D163)⋊C4C22semidirect
44. C2×C4×Dic163C22semidirect
45. C163⋊#95C22semidirect
46. C163⋊#95C22semidirect
47. C163⋊#95C22semidirect
48. C163⋊#95C22semidirect
79. C2×C22⋊D163D4semidirect
98. C22⋊Dic163D4semidirect
128. C22⋊C4×C163D4semidirect
233. C22×C163#121semidirect
277. C163#2424semidirect
284. C22×C22⋊C4D163semidirect
405. C22⋊C4C22⋊D163semidirect
525. C22#19104708semidirect

Maximal subgroups

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

subgroupindex
C2×#191047092
C2×#191047082
C2×#191047042
C2×C4×C22⋊D1632
#237815882
#237815802
#237815742
#237815652
#237815582
#237815442
#237815412
#237815382
C22×C22⋊C4×C1632
#2424163