Finite Group Databasebeta

← #35218066

Subgroups — #35218066

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×#19104707C211yesmaximal, characteristic, complemented
3C2×#19104705C211yesmaximal, characteristic, complemented
4C2×#19104703C211yesmaximal, characteristic, complemented
5C22×(C2×D163)⋊C4C211yesmaximal, characteristic, complemented
6#23781587C211yesmaximal, characteristic, complemented
7#23781582C211yesmaximal, characteristic, complemented
8#23781581C211yesmaximal, characteristic, complemented
9#23781574C211yesmaximal, characteristic, complemented
10#23781567C211yesmaximal, characteristic, complemented
11#23781556C211yesmaximal, characteristic, complemented
12#23781544C211yesmaximal, characteristic, complemented
13#23781544C211yesmaximal, characteristic, complemented
14#23781540C211yesmaximal, characteristic, complemented
15C2×C4×D4×C163C211yesmaximal, characteristic, complemented
16C24×D163C2211yescharacteristic
17C22×D652C2211yescharacteristic
18C22×D652C2211yescharacteristic
19C22×C22⋊D163C2211yescharacteristic
20C22×C22⋊D163C2211yescharacteristic
21C22×C22⋊D163C2211yescharacteristic
22C22×C22⋊D163C2211yescharacteristic, complemented
23C23×Dic163C2211yescharacteristic
24#19104709—24no—
25#19104707—24no—
26#19104705—24no—
27#19104703—24no—
28C2×C4⋊Dic163C2211yescharacteristic, complemented
29C2×C4⋊Dic163C2211yescharacteristic, complemented
30C2×Dic163⋊C4C2211yescharacteristic, complemented
31C2×C22⋊Dic163C2211yescharacteristic
32C2×C22⋊Dic163C2211yescharacteristic, complemented
33C2×(C2×D163)⋊C4—24no—
34C2×(C2×D163)⋊C4C2211yescharacteristic
35C2×(C2×D163)⋊C4C2211yescharacteristic
36C2×(C2×D163)⋊C4C2211yescharacteristic
37C2×(C2×D163)⋊C4C2211yescharacteristic
38C2×(C2×D163)⋊C4C2211yescharacteristic
39C2×(C2×D163)⋊C4C2211yescharacteristic
40C2×(C2×D163)⋊C4C2211yescharacteristic
41C2×(C2×D163)⋊C4C2211yescharacteristic
42C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
44C2×(C2×D163)⋊C4C2211yescharacteristic, complemented
45C163⋊#95C2211yescharacteristic, complemented
46C163⋊#95C2211yescharacteristic, complemented
47C163⋊#95C2211yescharacteristic, complemented
48C163⋊#95C2211yescharacteristic, complemented
49C22×D4×C163C2211yescharacteristic
50C23×C4×C163C2211yescharacteristic
51C23×C4×C163C2211yescharacteristic, complemented
52C4×D4×C163—24no—
53C2×C4⋊C4×C163C2211yescharacteristic, complemented
54C2×C22⋊C4×C163C2211yescharacteristic
55C2×C22⋊C4×C163C2211yescharacteristic, complemented
56C2×C42×C163C2211yescharacteristic, complemented
57C23×D163—22no—
58C23×D163—22no—
59C23×D163—22no—
60C23×D163—22no—
61C23×D163—22no—
62C23×D163—41no—
63C23×D163—41no—
64C23×D163C2311yescharacteristic
65C23×D163C2311yescharacteristic
66C23×D163C2311yescharacteristic
67C2×D652—22no—
68C2×D652—22no—
69C2×D652—22no—
70C2×D652—22no—
71C2×D652—41no—
72C2×D652—41no—
73C2×C22⋊D163—22no—
74C2×C22⋊D163—22no—
75C2×C22⋊D163—22no—
76C2×C22⋊D163—22no—
77C2×C22⋊D163—22no—
78C2×C22⋊D163—22no—
79C2×C22⋊D163—22no—
80C2×C22⋊D163—22no—
81C2×C22⋊D163—22no—
82C2×C22⋊D163—41no—
83C2×C22⋊D163—41no—
84C2×C22⋊D163—41no—
85C22×Dic163—22no—
86C22×Dic163—22no—
87C22×Dic163—41no—
88C22×Dic163C2311yescharacteristic
89C22×Dic163C2311yescharacteristic
90C22×Dic163C2311yescharacteristic
91C22×Dic163C2311yescharacteristic
92C22×Dic163C2311yescharacteristic
93C4⋊Dic163—22no—
94C4⋊Dic163—41no—
95Dic163⋊C4—22no—
96C22⋊Dic163—22no—
97C22⋊Dic163—22no—
98(C2×D163)⋊C4—22no—
99(C2×D163)⋊C4—22no—
100(C2×D163)⋊C4—22no—
101(C2×D163)⋊C4—22no—
102(C2×D163)⋊C4—22no—
103(C2×D163)⋊C4—22no—
104(C2×D163)⋊C4—22no—
105(C2×D163)⋊C4—22no—
106(C2×D163)⋊C4—22no—
107(C2×D163)⋊C4—41no—
108(C2×D163)⋊C4D414yescomplemented
109C24×C163C2311yescharacteristic
110C24×C163C2311yescharacteristic
111C2×D4×C163—22no—
112C2×D4×C163—22no—
113C2×D4×C163D414yescomplemented
114C22×C4×C163—22no—
115C22×C4×C163—22no—
116C22×C4×C163—22no—
117C22×C4×C163—22no—
118C22×C4×C163—41no—
119C22×C4×C163—41no—
120C22×C4×C163C2311yescharacteristic
121C22×C4×C163C2311yescharacteristic
122C22×C4×C163C2311yescharacteristic
123C22×C4×C163C2311yescharacteristic
124C22×C4×C163C2311yescharacteristic
125C4⋊C4×C163—22no—
126C22⋊C4×C163—22no—
127C22⋊C4×C163D414yescomplemented
128C42×C163—22no—
129C22×D163—28no—
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×D163—21no—
143C22×D163—21no—
144C22×D163—21no—
145C22×D163—21no—
146C22×D163—21no—
147C22×D163—21no—
148C22×D163—21no—
149C22×D163—44no—
150C22×D163—44no—
151C22×D163—44no—
152C22×D163—44no—
153C22×D163—44no—
154C22×D163—44no—
155C22×D163C2×D412yes—
156C22×D163#4112yes—
157D652—44no—
158D652—44no—
159C22⋊D163—44no—
160C22⋊D163—44no—
161C22⋊D163—44no—
162C22⋊D163—44no—
163C2×Dic163—21no—
164C2×Dic163—21no—
165C2×Dic163—21no—
166C2×Dic163—21no—
167C2×Dic163—21no—
168C2×Dic163—21no—
169C2×Dic163—21no—
170C2×Dic163—21no—
171C2×Dic163—21no—
172C2×Dic163—21no—
173C2×Dic163—21no—
174C2×Dic163—21no—
175C2×Dic163—21no—
176C2×Dic163—21no—
177C2×Dic163—44no—
178C2×Dic163C2×D412yes—
179C23×C163—21no—
180C23×C163—21no—
181C23×C163—21no—
182C23×C163—21no—
183C23×C163—21no—
184C23×C163—21no—
185C23×C163—21no—
186C23×C163—21no—
187C23×C163—21no—
188C23×C163—21no—
189C23×C163C2×D412yes—
190C23×C163C2×D412yes—
191C23×C163C2×D412yes—
192C23×C163#4112yes—
193C23×C163C2411yescharacteristic
194D4×C163—28no—
195C2×C4×C163—21no—
196C2×C4×C163—21no—
197C2×C4×C163—21no—
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—44no—
207C2×C4×C163—44no—
208C2×C4×C163C2×D412yes—
209C2×C4×C163C2×D412yes—
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—41no—
222C22×C163—41no—
223C22×C163—41no—
224C22×C163—41no—
225C22×C163#12114yescomplemented
226C22×C163C22×D411yescharacteristic
227C22×C163C22×D411yescharacteristic
228C22×C163C22×D411yescharacteristic
229C22×C163C2×#4111yescharacteristic
230C22×C163C2×#4111yescharacteristic
231C22×C163H(5,𝔽2)11yescharacteristic
232C22×C163H(5,𝔽2)11yescharacteristic
233C2×D163—22no—
234C2×D163—22no—
235C2×D163—22no—
236C2×D163—22no—
237C2×D163—22no—
238C2×D163—22no—
239C2×D163—416no—
240C2×D163—41no—
241C2×D163—41no—
242C2×D163—41no—
243C2×D163—41no—
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—22no—
258C4×C163—22no—
259C4×C163—41no—
260Dic163—22no—
261Dic163—41no—
262Dic163—41no—
263Dic163—41no—
264Dic163—41no—
265C2×C163—24no—
266C2×C163—42no—
267C2×C163C2×#12111yescharacteristic
268C2×C163#52511yescharacteristic
269C2×C163#53811yescharacteristic
270C2×C163D4211yescharacteristic
271C2×C163D4211yescharacteristic
272C2×C163#54611yescharacteristic
273C2×C163#54611yescharacteristic
274D163—42no—
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×D4D16311yescharacteristic, complemented
284C22×C22⋊C4—1631no—
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—
294C25—1631no—
295C22×D4—1631no—
296C22×D4—1631no—
297C22×D4—1631no—
298C22×D4—1631no—
299C22×D4—1631no—
300C22×D4—1631no—
301C22×D4C2×D16311yescharacteristic
302C23×C4—1631no—
303C23×C4C2×D16311yescharacteristic
304C23×C4C2×D16311yescharacteristic, complemented
305#121—3264no—
306#121—3264no—
307#120—3264no—
308#120—3264no—
309C4×D4—24no—
310C2×C4⋊C4—1631no—
311C2×C4⋊C4—1631no—
312C2×C4⋊C4—1631no—
313C2×C4⋊C4C2×D16311yescharacteristic, complemented
314C2×C22⋊C4—3264no—
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⋊C4—1631no—
327C2×C22⋊C4—1631no—
328C2×C22⋊C4C2×D16311yescharacteristic
329C2×C22⋊C4C2×D16311yescharacteristic, complemented
330C2×C42C2×D16311yescharacteristic, complemented
331#95—1631no—
332#95—1631no—
333#95—1631no—
334#95—1631no—
335C24—3262no—
336C24—3262no—
337C24—3262no—
338C24—3262no—
339C24—3262no—
340C24—6521no—
341C24—6521no—
342C24—1631no—
343C24—1631no—
344C24—1631no—
345C24C22×D16311yescharacteristic
346C24C22×D16311yescharacteristic
347C2×D4—22no—
348C2×D4—22no—
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×D4D65214yescomplemented
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×C4C22×D16311yescharacteristic
383C22×C4C22×D16311yescharacteristic
384C22×C4C22×D16311yescharacteristic
385C22×C4C22×D16311yescharacteristic
386C22×C4C22×D16311yescharacteristic
387C4⋊C4—22no—
388C4⋊C4—3262no—
389C4⋊C4—3262no—
390C4⋊C4—6521no—
391C22⋊C4—22no—
392C22⋊C4—3262no—
393C22⋊C4—3262no—
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—22no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—3268no—
418C23—3261no—
419C23—3261no—
420C23—3261no—
421C23—3261no—
422C23—3261no—
423C23—3261no—
424C23—3261no—
425C23—3261no—
426C23—3261no—
427C23—3261no—
428C23—3261no—
429C23—3261no—
430C23—3261no—
431C23—3261no—
432C23—3261no—
433C23—3261no—
434C23—3261no—
435C23—3261no—
436C23—3261no—
437C23—6524no—
438C23—6524no—
439C23—6524no—
440C23—6524no—
441C23—6524no—
442C23—6524no—
443C23—1632no—
444C23—1632no—
445C23C2×C22⋊D16312yes—
446C23C2×D65212yes—
447C23C2×D65212yes—
448C23#1457969912yes—
449C23C23×D16311yescharacteristic
450D4—28no—
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—21no—
468C2×C4—44no—
469C2×C4—44no—
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×C4D4×D16312yes—
487C2×C4C2×C22⋊D16312yes—
488C2×C4C2×C22⋊D16312yes—
489C2×C4C2×D65212yes—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—3262no—
504C22—3262no—
505C22—3262no—
506C22—3262no—
507C22—3262no—
508C22—3262no—
509C22—65216no—
510C22—6521no—
511C22—6521no—
512C22—6521no—
513C22—6521no—
514C22—6521no—
515C22—6521no—
516C22—6521no—
517C22—6521no—
518C22—6521no—
519C22—6521no—
520C22—6521no—
521C22—6521no—
522C22—6521no—
523C22—6521no—
524C22—6521no—
525C22#1910470914yescomplemented
526C22C2×D4×D16311yescharacteristic
527C22C22×C22⋊D16311yescharacteristic
528C22C22×D65211yescharacteristic
529C22#41×D16311yescharacteristic
530C22C2×#1457969911yescharacteristic
531C22#1910478011yescharacteristic
532C22#1910478111yescharacteristic
533C4—22no—
534C4—22no—
535C4—22no—
536C4—22no—
537C4—41no—
538C4—3262no—
539C4—6521no—
540C4—6521no—
541C4—6521no—
542C4—6521no—
543C2—24no—
544C2—42no—
545C2—13041no—
546C2—6522no—
547C2—6522no—
548C2C2×#1910470911yescharacteristic
549C2#2378217411yescharacteristic
550C2#2378224811yescharacteristic
551C2D4×C22⋊D16311yescharacteristic
552C2D4×D65211yescharacteristic
553C2#2378230311yescharacteristic
554C2#2378231011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D163
commutator, socle, cosocleC23×C163C24
FittingC2×C4×D4×C163C2
hypercenterC2×C4×D4D163
hypocenterC163#2424
LCSG ▷ C23×C163 ▷ C163
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#19104709C2semidirect
2. C2×#19104707C2semidirect
3. C2×#19104705C2semidirect
4. C2×#19104703C2semidirect
5. C22×(C2×D163)⋊C4C2semidirect
6. #23781587C2semidirect
7. #23781582C2semidirect
8. #23781581C2semidirect
9. #23781574C2semidirect
10. #23781567C2semidirect
11. #23781556C2semidirect
12. #23781544C2semidirect
13. #23781544C2semidirect
14. #23781540C2semidirect
15. C2×C4×D4×C163C2semidirect
22. C22×C22⋊D163C22semidirect
28. C2×C4⋊Dic163C22semidirect
29. C2×C4⋊Dic163C22semidirect
30. C2×Dic163⋊C4C22semidirect
32. C2×C22⋊Dic163C22semidirect
42. C2×(C2×D163)⋊C4C22semidirect
43. C2×(C2×D163)⋊C4C22semidirect
44. C2×(C2×D163)⋊C4C22semidirect
45. C163⋊#95C22semidirect
46. C163⋊#95C22semidirect
47. C163⋊#95C22semidirect
48. C163⋊#95C22semidirect
51. C23×C4×C163C22semidirect
53. C2×C4⋊C4×C163C22semidirect
55. C2×C22⋊C4×C163C22semidirect
56. C2×C42×C163C22semidirect
108. (C2×D163)⋊C4D4semidirect
113. C2×D4×C163D4semidirect
127. C22⋊C4×C163D4semidirect
225. C22×C163#121semidirect
277. C163#2424semidirect
283. C2×C4×D4D163semidirect
304. C23×C4C2×D163semidirect
313. C2×C4⋊C4C2×D163semidirect
329. C2×C22⋊C4C2×D163semidirect
330. C2×C42C2×D163semidirect
367. C2×D4D652semidirect
405. C22⋊C4C22⋊D163semidirect
525. C22#19104709semidirect

Maximal subgroups

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

subgroupindex
C2×#191047092
C2×#191047072
C2×#191047052
C2×#191047032
C22×(C2×D163)⋊C42
#237815872
#237815822
#237815812
#237815742
#237815672
#237815562
#237815442
#237815402
C2×C4×D4×C1632
#2424163