Finite Group Databasebeta

← #11905983

Subgroups — #11905983

Order 1152.

3651 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64307C212yesmaximal, complemented
2#10595993C211yesmaximal, characteristic, complemented
3C3×#5525C212yesmaximal, complemented
4#10595032C211yesmaximal, characteristic, complemented
5C2×#63914C211yesmaximal, characteristic, complemented
6#10594618C214yesmaximal, complemented
7#10594527C214yesmaximal, complemented
8#10594271C218yesmaximal, complemented
9#10594010C214yesmaximal, complemented
10#10594008C212yesmaximal, complemented
11#10593971C212yesmaximal, complemented
12#86316—32nomaximal
13C2×#3828C2211yescharacteristic
14C2×C3×#235C2212yescomplemented
15C22×C3×D12C2212yescomplemented
16C2×D3×#67C2218yescomplemented
17C22×#625C2214yescomplemented
18#64307—216no—
19C2×#3802C2214yescomplemented
20C2×#3802C2218yescomplemented
21C2×D3×D12C2214yescomplemented
22C2×#3796C2214yescomplemented
23C2×#3795C2214yescomplemented
24C32×#117C2211yescharacteristic, complemented
25C2×C4×C3⋊Dic3C2211yescharacteristic, complemented
26#64102C2212yescomplemented
27#64096C2212yescomplemented
28#64090C2214yescomplemented
29#64081C2212yescomplemented
30C42×C3⋊D3C2212yescomplemented
31C3×#895C2218yescomplemented
32C3×#882C2218yescomplemented
33C3×#875C2214yescomplemented
34C3×#874C2214yescomplemented
35#63980C2214yescomplemented
36#63922C22116yescomplemented
37#63914C2214yescomplemented
38#63914C2212yescomplemented
39#63914C2212yescomplemented
40#63914C2211yescharacteristic, complemented
41#63914C2218yescomplemented
42#63912C2218yescomplemented
43#63902C2218yescomplemented
44C4×#625C2214yescomplemented
45#63885C22116yescomplemented
46#63873C2211yescharacteristic, complemented
47#63863C2212yescomplemented
48C2×#1009—32no—
49C2×#1004—32no—
50C3×#535—32no—
51#5832—32no—
52C2×#940—32no—
53#5715—34no—
54D3×#127—34no—
55#5674—34no—
56#5668—34no—
57#5609—38no—
58#5599—38no—
59#5590—316no—
60#5525D312yescomplemented
61#5522—32no—
62#5522—61no—
63C23×C3×D3C2314yes—
64C22×D32C2318yes—
65C22×C32×C4C2311yescharacteristic
66C2×#638C2312yes—
67C22×C3⋊Dic3C2312yes—
68#3828D418yescomplemented
69C2×C4⋊(C3⋊D3)C2311yescharacteristic
70C2×C4×C3⋊D3C2314yes—
71C2×C4×C3⋊D3C2312yes—
72C2×#634C2311yescharacteristic
73C2×C3×#67C2314yes—
74C3×#235—28no—
75C2×C3×D12—28no—
76C2×C3×D12C2314yes—
77C2×C3×D12C2314yes—
78C2×C3×D12C2312yes—
79C2×C3×D12C23116yes—
80C2×C3×C4×D3C2314yes—
81C2×C3×Dic6C2312yes—
82D3×#67—232no—
83C2×#626C2318yes—
84C2×#625—48no—
85C2×#625C2318yes—
86C2×#625C2314yes—
87C2×#625C2318yes—
88C2×#625C2318yes—
89C2×D3×Dic3C2318yes—
90#3802—216no—
91#3802—416no—
92D3×D12—216no—
93#3796—216no—
94#3795—216no—
95C32×C4⋊C4C2312yes—
96C32×C22⋊C4C2312yes—
97C32×C42C2312yes—
98#3752C2314yes—
99#3750C2314yes—
100C4×C3⋊Dic3C2311yescharacteristic
101C4×C3⋊Dic3C2314yes—
102C4×C3⋊Dic3C2311yescharacteristic
103C4×C3⋊Dic3C2312yes—
104C3×(C2×D3)⋊C4C23116yes—
105#3724C2314yes—
106(C2×D3)⋊Dic3C2318yes—
107#3498—91noSylow
108C2×C3×#41—32no—
109C22×C3×D4—32no—
110C22×#67—38no—
111C22×#67—38no—
112#1009—616no—
113C2×#239—34no—
114#1004—616no—
115C2×#237—34no—
116C2×#237—34no—
117C2×D3×D4—38no—
118C2×D3×D4—34no—
119C2×D3×D4—34no—
120C2×D3×D4—316no—
121C2×D3×D4—34no—
122C2×#235—32no—
123C2×#235—34no—
124C2×#235—61no—
125C2×#235C2×D312yescomplemented
126C22×D12—34no—
127C22×D12C2×D312yescomplemented
128C3×#127—34no—
129C3×#124—34no—
130C3×#121—38no—
131C3×#120—38no—
132C3×#117—21no—
133C3×#117C2×D312yescomplemented
134#947—32no—
135#944—32no—
136#940—34no—
137#940—34no—
138#940—32no—
139#940—316no—
140#940—38no—
141#939—38no—
142#939—316no—
143#937—316no—
144#936—32no—
145C4×#67—38no—
146C2×C4×Dic3—32no—
147C2×C4×Dic3—61no—
148#895C2×D318yescomplemented
149#893—34no—
150#893—62no—
151#887—34no—
152#887—62no—
153#884—316no—
154#884—316no—
155#882—316no—
156#882C2×D318yescomplemented
157#67⋊C4—38no—
158#67⋊C4—316no—
159#67⋊C4—64no—
160#875C2×D314yescomplemented
161#874C2×D314yescomplemented
162#872—34no—
163#872—62no—
164C42×D3—34no—
165C42×D3—62no—
166C23×C32C2411yescharacteristic
167C22×C3⋊D3C2412yes—
168C22×C3×D3—28no—
169C22×C3×D3—48no—
170C22×C3×D3C2414yes—
171C22×C3×D3C2418yes—
172C2×D32—216no—
173C2×D32—416no—
174C2×C32×C4C2×D414yescomplemented
175C2×C32×C4C2414yes—
176C2×C32×C4C2411yescharacteristic
177C2×C32×C4C2411yescharacteristic
178#638C2×D418yescomplemented
179C2×C3⋊Dic3—24no—
180C2×C3⋊Dic3C2412yes—
181C2×C3⋊Dic3C2412yes—
182C2×C3⋊Dic3C2412yes—
183C4⋊(C3⋊D3)C2×D414yescomplemented
184C4×C3⋊D3—28no—
185C4×C3⋊D3C2×D418yescomplemented
186#634C2×D414yescomplemented
187C3×#67—28no—
188C2×C3×Dic3C2414yes—
189C3×D12—28no—
190C3×D12—28no—
191C3×D12—24no—
192C3×D12—416no—
193C3×C4×D3—28no—
194C3×Dic6—24no—
195#626—216no—
196#625—216no—
197#625—28no—
198#625—216no—
199#625—48no—
200D3×Dic3—216no—
201C2×H(5,𝔽2)—92no—
202#550—94no—
203D42—98no—
204D42—94no—
205#535—32no—
206#535—94no—
207#532—92no—
208#532—91no—
209C2×#127—92no—
210C2×#127—91no—
211C4×#41—91no—
212C24×C3—34no—
213C23×D3—38no—
214C23×D3—38no—
215C23×D3C22×D314yes—
216C3×#41—68no—
217C2×C3×Q8—32no—
218C2×C3×D4—34no—
219C2×C3×D4—32no—
220C2×C3×D4—34no—
221C2×C3×D4—34no—
222C2×C3×D4—316no—
223C2×C3×D4—68no—
224C22×C3×C4—21no—
225C22×C3×C4—34no—
226C22×C3×C4C22×D312yes—
227C2×#67—34no—
228C2×#67—316no—
229C2×#67—38no—
230C2×#67—38no—
231C2×#67—38no—
232C2×#67—316no—
233C2×#67—38no—
234C2×#67—62no—
235C2×#67—632no—
236C2×#67—1216no—
237C2×#67C22×D314yes—
238C22×Dic3—34no—
239C22×Dic3—38no—
240C22×Dic3—62no—
241#239—616no—
242#237—616no—
243#237—616no—
244D3×D4—616no—
245D3×D4—616no—
246D3×D4—616no—
247D3×D4—632no—
248D3×D4—1232no—
249#235—28no—
250#235—316no—
251#235—68no—
252#235—616no—
253C2×D12—28no—
254C2×D12—32no—
255C2×D12—38no—
256C2×D12—61no—
257C2×D12—616no—
258C2×D12C22×D312yes—
259C2×D12C22×D314yes—
260C2×D12C22×D314yes—
261C2×D12C22×D3116yes—
262C2×C4×D3—34no—
263C2×C4×D3—38no—
264C2×C4×D3—38no—
265C2×C4×D3—62no—
266C2×C4×D3—64no—
267C2×C4×D3C22×D314yes—
268C2×Dic6—32no—
269C2×Dic6—61no—
270C2×Dic6C22×D312yes—
271C3×C4⋊C4—22no—
272C3×C4⋊C4C22×D314yes—
273C3×C22⋊C4—22no—
274C3×C22⋊C4—316no—
275C3×C22⋊C4C22×D314yes—
276C3×C42—22no—
277C3×C42C22×D314yes—
278#217—38no—
279(C2×D3)⋊C4—38no—
280(C2×D3)⋊C4—38no—
281(C2×D3)⋊C4—64no—
282(C2×D3)⋊C4C22×D3116yes—
283Dic3⋊C4—38no—
284Dic3⋊C4—38no—
285Dic3⋊C4—64no—
286C4×Dic3—32no—
287C4×Dic3—38no—
288C4×Dic3—32no—
289C4×Dic3—34no—
290C4×Dic3—61no—
291C4×Dic3—64no—
292C4×Dic3—61no—
293C4×Dic3—62no—
294C22×C32—22no—
295C22×C32C22×D412yes—
296C22×C32C2511yescharacteristic
297C2×C3⋊D3—24no—
298C2×C3⋊D3C22×D414yes—
299C2×C3×D3—28no—
300C2×C3×D3—24no—
301C2×C3×D3—216no—
302C2×C3×D3—28no—
303C2×C3×D3—88no—
304D32—816no—
305C32×C4—24no—
306C32×C4C22×D412yes—
307C32×C4C22×D412yes—
308C3⋊Dic3—22no—
309C3⋊Dic3C22×D414yes—
310C3⋊Dic3H(5,𝔽2)14yescomplemented
311C3×Dic3—24no—
312H(5,𝔽2)—1816no—
313C2×#41—32no—
314C2×#41—94no—
315C2×#41—94no—
316C2×#41—91no—
317C22×D4—32no—
318C22×D4—94no—
319C22×D4—94no—
320C22×D4—98no—
321C22×D4—98no—
322C22×D4—94no—
323C4⋊Q8—91no—
324#127—34no—
325#127—92no—
326#127—91no—
327#127—92no—
328#127—94no—
329#127—98no—
330#124—34no—
331#124—92no—
332#121—38no—
333#121—98no—
334#121—916no—
335#121—94no—
336#120—38no—
337#120—916no—
338C4×Q8—92no—
339C4×D4—98no—
340C4×D4—94no—
341C4×D4—92no—
342C4×D4—94no—
343#117—92no—
344#117D3211yescharacteristic, complemented
345C2×C42—91no—
346C2×C42—92no—
347C23×C3—21no—
348C23×C3—34no—
349C23×C3—38no—
350C23×C3—68no—
351C23×C3—128no—
352C23×C3C23×D312yes—
353C22×D3—28no—
354C22×D3—34no—
355C22×D3—48no—
356C22×D3—62no—
357C22×D3—616no—
358C22×D3—616no—
359C22×D3—1216no—
360C22×D3—1216no—
361C22×D3C23×D314yes—
362C22×D3C23×D318yes—
363C3×Q8—64no—
364C3×D4—68no—
365C3×D4—64no—
366C3×D4—68no—
367C3×D4—68no—
368C3×D4—1216no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—24no—
372C2×C3×C4—24no—
373C2×C3×C4—34no—
374C2×C3×C4—68no—
375C2×C3×C4D3×D418yescomplemented
376C2×C3×C4C23×D318yes—
377C2×C3×C4C23×D312yes—
378C2×C3×C4C23×D312yes—
379#67—28no—
380#67—316no—
381#67—68no—
382#67—616no—
383#67—616no—
384#67—616no—
385#67—616no—
386#67—632no—
387#67—1216no—
388C2×Dic3—34no—
389C2×Dic3—34no—
390C2×Dic3—34no—
391C2×Dic3—62no—
392C2×Dic3—62no—
393C2×Dic3—62no—
394C2×Dic3—616no—
395C2×Dic3—68no—
396C2×Dic3—124no—
397C2×Dic3C23×D314yes—
398D12—28no—
399D12—28no—
400D12—24no—
401D12—38no—
402D12—416no—
403D12—64no—
404D12—616no—
405C4×D3—28no—
406C4×D3—316no—
407C4×D3—68no—
408C4×D3—616no—
409C4×D3—616no—
410C4×D3—128no—
411Dic6—24no—
412Dic6—38no—
413Dic6—64no—
414C2×C32—22no—
415C2×C32C23×D411yescharacteristic
416C2×C32C2×H(5,𝔽2)12yes—
417C3⋊D3—24no—
418C3×D3—44no—
419C3×D3—48no—
420C24—34no—
421C24—98no—
422#41—68no—
423#41—98no—
424#41—1816no—
425#41—1816no—
426C2×Q8—32no—
427C2×Q8—91no—
428C2×D4—34no—
429C2×D4—34no—
430C2×D4—34no—
431C2×D4—32no—
432C2×D4—316no—
433C2×D4—68no—
434C2×D4—98no—
435C2×D4—92no—
436C2×D4—98no—
437C2×D4—94no—
438C2×D4—98no—
439C2×D4—98no—
440C2×D4—91no—
441C2×D4—1816no—
442C2×D4—1816no—
443C2×D4—1832no—
444C2×D4—3616no—
445C2×D4—368no—
446C22×C4—34no—
447C22×C4—98no—
448C22×C4—92no—
449C22×C4—92no—
450C22×C4—94no—
451C22×C4C2×D3211yescharacteristic
452C4⋊C4—94no—
453C4⋊C4—94no—
454C4⋊C4C2×D3212yes—
455C22⋊C4—316no—
456C22⋊C4—98no—
457C22⋊C4—94no—
458C22⋊C4C2×D3212yes—
459C42—91no—
460C42—91no—
461C42—94no—
462C42—92no—
463C42C2×D3212yes—
464C22×C3—24no—
465C22×C3—22no—
466C22×C3—21no—
467C22×C3—42no—
468C22×C3—68no—
469C22×C3—64no—
470C22×C3—616no—
471C22×C3—68no—
472C22×C3—248no—
473C22×C3C2×D3×D414yes—
474C22×C3C24×D312yes—
475C2×D3—24no—
476C2×D3—28no—
477C2×D3—28no—
478C2×D3—216no—
479C2×D3—38no—
480C2×D3—64no—
481C2×D3—68no—
482C2×D3—88no—
483C2×D3—124no—
484C2×D3—2416no—
485C2×D3—2416no—
486C3×C4—28no—
487C3×C4—22no—
488C3×C4—22no—
489C3×C4—44no—
490C3×C4—64no—
491C3×C4C2×D3×D414yes—
492C3×C4C2×D3×D414yes—
493Dic3—24no—
494Dic3—38no—
495Dic3—38no—
496Dic3—64no—
497Dic3—64no—
498Dic3—64no—
499Dic3—122no—
500C32#349811yescharacteristic, complemented, Sylow
501C23—38no—
502C23—34no—
503C23—68no—
504C23—92no—
505C23—128no—
506C23—1816no—
507C23—3616no—
508C23C22×D3211yescharacteristic
509Q8—64no—
510Q8—94no—
511D4—68no—
512D4—68no—
513D4—68no—
514D4—64no—
515D4—98no—
516D4—94no—
517D4—1216no—
518D4—1816no—
519D4—1816no—
520D4—188no—
521D4—1816no—
522D4—368no—
523C2×C4—34no—
524C2×C4—68no—
525C2×C4—92no—
526C2×C4—92no—
527C2×C4—98no—
528C2×C4—92no—
529C2×C4—188no—
530C2×C4—1816no—
531C2×C4—184no—
532C2×C4#380214yescomplemented
533C2×C4C22×D3214yes—
534C2×C4C22×D3211yescharacteristic
535C2×C4C22×D3211yescharacteristic
536C2×C3—24no—
537C2×C3—21no—
538C2×C3—22no—
539C2×C3—42no—
540C2×C3—124no—
541C2×C3—128no—
542C2×C3C22×D3×D412yes—
543C2×C3D3×H(5,𝔽2)14yes—
544D3—48no—
545D3—44no—
546D3—68no—
547D3—124no—
548C22—22no—
549C22—64no—
550C22—68no—
551C22—68no—
552C22—616no—
553C22—94no—
554C22—184no—
555C22—248no—
556C22—7216no—
557C22C2×#380212yes—
558C22C23×D3211yescharacteristic
559C4—24no—
560C4—64no—
561C4—94no—
562C4—94no—
563C4—182no—
564C4C2×#380212yes—
565C4C2×#380212yes—
566C3—21no—
567C3#8631612yescomplemented
568C2—22no—
569C2—128no—
570C2—124no—
571C2—184no—
572C2C22×#380211yescharacteristic
573C2#1059669212yes—
subgroupG/H
center, FrattiniC22C23×D32
commutator, socle, cosocleC22×C32C25
FittingC32×#117C22
hypercenter#117D32
hypocenterC32#3498
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C2×#64307C2semidirect
2. #10595993C2semidirect
3. C3×#5525C2semidirect
4. #10595032C2semidirect
5. C2×#63914C2semidirect
6. #10594618C2semidirect
7. #10594527C2semidirect
8. #10594271C2semidirect
9. #10594010C2semidirect
10. #10594008C2semidirect
11. #10593971C2semidirect
14. C2×C3×#235C22semidirect
15. C22×C3×D12C22semidirect
16. C2×D3×#67C22semidirect
17. C22×#625C22semidirect
19. C2×#3802C22semidirect
20. C2×#3802C22semidirect
21. C2×D3×D12C22semidirect
22. C2×#3796C22semidirect
23. C2×#3795C22semidirect
24. C32×#117C22semidirect
25. C2×C4×C3⋊Dic3C22semidirect
26. #64102C22semidirect
27. #64096C22semidirect
28. #64090C22semidirect
29. #64081C22semidirect
30. C42×C3⋊D3C22semidirect
31. C3×#895C22semidirect
32. C3×#882C22semidirect
33. C3×#875C22semidirect
34. C3×#874C22semidirect
35. #63980C22semidirect
36. #63922C22semidirect
37. #63914C22semidirect
38. #63914C22semidirect
39. #63914C22semidirect
40. #63914C22semidirect
41. #63914C22semidirect
42. #63912C22semidirect
43. #63902C22semidirect
44. C4×#625C22semidirect
45. #63885C22semidirect
46. #63873C22semidirect
47. #63863C22semidirect
60. #5525D3semidirect
68. #3828D4semidirect
125. C2×#235C2×D3semidirect
127. C22×D12C2×D3semidirect
133. C3×#117C2×D3semidirect
148. #895C2×D3semidirect
156. #882C2×D3semidirect
160. #875C2×D3semidirect
161. #874C2×D3semidirect
174. C2×C32×C4C2×D4semidirect
178. #638C2×D4semidirect
183. C4⋊(C3⋊D3)C2×D4semidirect
185. C4×C3⋊D3C2×D4semidirect
186. #634C2×D4semidirect
310. C3⋊Dic3H(5,𝔽2)semidirect
344. #117D32semidirect
375. C2×C3×C4D3×D4semidirect
500. C32#3498semidirect
532. C2×C4#3802semidirect
567. C3#86316semidirect

Maximal subgroups

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

subgroupindex
C2×#643072
#105959932
C3×#55252
#105950322
C2×#639142
#105946182
#105945272
#105942712
#105940102
#105940082
#105939712
#863163