Finite Group Databasebeta

← #11861173

Subgroups — #11861173

Order 1152.

1045 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596216C211yesmaximal, characteristic, complemented
2C3×#5745C211yesmaximal, characteristic, complemented
3#10594795C211yesmaximal, characteristic
4C3×Dic3×M16C211yesmaximal, characteristic, complemented
5#10591141C211yesmaximal, characteristic, complemented
6C2×#63570C211yesmaximal, characteristic, complemented
7C2×#63569C211yesmaximal, characteristic, complemented
8#10590924C214yesmaximal, complemented
9#10590923C214yesmaximal, complemented
10#83811—31nomaximal
11#78541—31nomaximal
12C2×#3828C2211yescharacteristic
13C2×C3×#235C2211yescharacteristic
14C2×C32×M16C2211yescharacteristic
15C2×#3748C2211yescharacteristic
16#64116—22no—
17M16×C3⋊D3—22no—
18#64110C2214yescomplemented
19C2×C3×C4×Dic3C2211yescharacteristic
20C2×C3×#208C2211yescharacteristic
21C2×C3×#208C2211yescharacteristic
22C3×#907—22no—
23C3×D3×M16—22no—
24C3×#901C2214yescomplemented
25#63833C412yescomplemented
26#63831C412yescomplemented
27#63830C412yescomplemented
28#63829C2214yes—
29#63825C412yescomplemented
30C3×#815C2212yescomplemented
31C3×C8×Dic3C2212yescomplemented
32#63577C2214yescomplemented
33#63570C2212yescomplemented
34#63570C2212yescomplemented
35#63569C2212yescomplemented
36#63569C2212yescomplemented
37C3×#568—31no—
38#5822—31no—
39#5748—31no—
40#5745—31no—
41#5745—31no—
42#5745—61no—
43#5745D311yescharacteristic, complemented
44C3×C4×M16—31no—
45C2×#837—31no—
46#5145—34no—
47#5144—34no—
48C2×#825—31no—
49#5125—31no—
50#5124—31no—
51Dic3×M16D311yescharacteristic, complemented
52C2×#805—31no—
53C2×#805—31no—
54#4717—34no—
55#4700—34no—
56C22×C32×C4C2311yescharacteristic
57C2×#638—21no—
58#3828—21no—
59#3828—21no—
60#3828C2×C412yes—
61#3828C2312yes—
62C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
63C2×C4×C3⋊D3—21no—
64C2×#634C2×C411yescharacteristic
65C2×C3×#67—21no—
66C22×C3×Dic3—21no—
67C3×#235—21no—
68C3×#235—21no—
69C3×#235C2×C412yes—
70C3×#235C2312yes—
71C2×C3×D12C2×C411yescharacteristic
72C2×C3×C4×D3—21no—
73C2×C3×Dic6C2×C411yescharacteristic
74C32×M16D412yes—
75C32×M16D412yes—
76C2×C32×C8C2312yes—
77#3748—21no—
78#3748C2312yes—
79C2×C3⋊(C3⋊C8)—21no—
80#3743—24no—
81C8×C3⋊D3—24no—
82C3×C4×Dic3—21no—
83C3×C4×Dic3C2311yescharacteristic
84C3×C4×Dic3C2311yescharacteristic
85C3×#208—21no—
86C3×#208C2×C412yes—
87C3×#208C2×C412yes—
88C3×#208C2312yes—
89C2×C3×C3⋊C8—21no—
90C2×C3×C3⋊C8C2312yes—
91C3×ΓL(1,𝔽25)—24no—
92C3×D3×C8—24no—
93#3719C2×C414yescomplemented
94#3717C2×C414yescomplemented
95#3716C2×C414yescomplemented
96#3714C2×C414yescomplemented
97#2956—91noSylow
98C2×C3×#41—31no—
99C2×#235—31no—
100C2×#235—31no—
101C2×#235—61no—
102C2×#235C2×D311yescharacteristic
103C3×#131—34no—
104C3×#131—62no—
105C2×C3×M16—21no—
106C2×C3×M16—31no—
107C2×C3×M16—31no—
108C2×C3×M16—62no—
109C2×C3×M16C2×D311yescharacteristic
110C2×C3×M16C2×D311yescharacteristic
111C2×C3×C42—31no—
112#951—32no—
113#950—34no—
114#949—32no—
115#942—32no—
116#932—32no—
117C2×C4×Dic3C2×D311yescharacteristic
118C2×#208—31no—
119C2×#208—31no—
120C2×#208—61no—
121C2×#208C2×D311yescharacteristic
122C2×#208C2×D311yescharacteristic
123#909—32no—
124#909—32no—
125#908—32no—
126#908—32no—
127#907—22no—
128#907—64no—
129#907—62no—
130#907—62no—
131D3×M16—22no—
132D3×M16—64no—
133D3×M16—62no—
134D3×M16—62no—
135#904—34no—
136#901—34no—
137#901—34no—
138#901—64no—
139#901C2×D314yescomplemented
140C3×#97—32no—
141C3×C4×C8—32no—
142#837—32no—
143#837—32no—
144#825—32no—
145#825—32no—
146#822—34no—
147#819—34no—
148#815C2×D312yescomplemented
149C8×Dic3C2×D312yescomplemented
150#805—32no—
151#805—32no—
152#805—32no—
153#805—32no—
154C23×C32#4111yescharacteristic
155C22×C3⋊D3—21no—
156C22×C3×D3—21no—
157C2×C32×C4C22×C411yescharacteristic
158C2×C32×C4C22×C411yescharacteristic
159C2×C32×C4C2×D411yescharacteristic
160C2×C32×C4C2×D411yescharacteristic
161C2×C32×C4#4111yescharacteristic
162C2×C32×C4C2411yescharacteristic
163#638—22no—
164#638—22no—
165C2×C3⋊Dic3—21no—
166C4⋊(C3⋊D3)—21no—
167C4⋊(C3⋊D3)C22×C412yes—
168C4×C3⋊D3—22no—
169C4×C3⋊D3—22no—
170#634—21no—
171#634C22×C412yes—
172C3×#67—22no—
173C3×#67—22no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—42no—
178C3×D12—21no—
179C3×D12C22×C412yes—
180C3×C4×D3—22no—
181C3×C4×D3—22no—
182C3×Dic6—21no—
183C3×Dic6C22×C412yes—
184C32×C8C2×D414yes—
185C3⋊(C3⋊C8)—22no—
186C3×C3⋊C8—22no—
187C3×C3⋊C8C22×C414yes—
188#575—91no—
189#568—31no—
190#568—91no—
191#444—94no—
192#443—94no—
193#430—91no—
194C2×#104—91no—
195C2×#104—91no—
196C4×M16—31no—
197C3×#41—32no—
198C3×#41—32no—
199C3×#41—61no—
200C3×#41—61no—
201C2×C3×Q8—31no—
202C2×C3×D4—31no—
203C2×C3×D4—61no—
204C22×C3×C4—21no—
205C22×C3×C4—61no—
206C22×C3×C4—61no—
207C22×C3×C4C22×D311yescharacteristic
208C22×C3×C4C22×D311yescharacteristic
209C2×#67—21no—
210C2×#67—62no—
211C2×#67—61no—
212C2×#67—61no—
213C22×Dic3—21no—
214#235—21no—
215#235—21no—
216#235—32no—
217#235—32no—
218#235—32no—
219#235—32no—
220#235—62no—
221#235—62no—
222#235—62no—
223#235—62no—
224#235—61no—
225#235—61no—
226#235—61no—
227#235—61no—
228#235C4×D312yes—
229#235C22×D312yes—
230C2×D12—31no—
231C2×D12—31no—
232C2×D12—61no—
233C2×D12C4×D311yescharacteristic
234C2×C4×D3—21no—
235C2×C4×D3—62no—
236C2×C4×D3—61no—
237C2×C4×D3—61no—
238C2×Dic6—31no—
239C2×Dic6—31no—
240C2×Dic6—61no—
241C2×Dic6C4×D311yescharacteristic
242C3×M16—22no—
243C3×M16—22no—
244C3×M16—32no—
245C3×M16—32no—
246C3×M16—32no—
247C3×M16—61no—
248C3×M16—64no—
249C3×M16D1212yes—
250C3×M16D1212yes—
251C3×M16#6712yes—
252C3×M16#6712yes—
253C2×C3×C8—22no—
254C2×C3×C8—32no—
255C2×C3×C8—61no—
256C2×C3×C8—64no—
257C2×C3×C8C22×D312yes—
258C2×C3×C8C22×D312yes—
259C3×C42—31no—
260C3×C42—31no—
261C3×C42—61no—
262#216—34no—
263#215—34no—
264#214—34no—
265#213—34no—
266C4×Dic3—21no—
267C4×Dic3C22×D311yescharacteristic
268C4×Dic3C22×D311yescharacteristic
269#208—21no—
270#208—32no—
271#208—32no—
272#208—62no—
273#208—62no—
274#208—61no—
275#208—61no—
276#208C4×D312yes—
277#208C4×D312yes—
278#208C22×D312yes—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—62no—
281C2×C3⋊C8—61no—
282C2×C3⋊C8—61no—
283C2×C3⋊C8C22×D312yes—
284Dic12—34no—
285D24—34no—
286#204—34no—
287#204—34no—
288ΓL(1,𝔽25)—24no—
289ΓL(1,𝔽25)—68no—
290ΓL(1,𝔽25)—64no—
291ΓL(1,𝔽25)—64no—
292D3×C8—24no—
293D3×C8—68no—
294D3×C8—64no—
295D3×C8—64no—
296C22×C32—41no—
297C22×C32C4×D411yescharacteristic
298C22×C32C4×D411yescharacteristic
299C22×C32C2×#4111yescharacteristic
300C2×C3⋊D3—22no—
301C2×C3⋊D3—81no—
302C2×C3×D3—22no—
303C2×C3×D3—81no—
304C32×C4C4×D411yescharacteristic
305C32×C4C4×D411yescharacteristic
306C32×C4C23×C411yescharacteristic
307C32×C4C22×D411yescharacteristic
308C3⋊Dic3—22no—
309C3×Dic3—22no—
310C3×Dic3—22no—
311C3×Dic3—41no—
312C2×#41—31no—
313C2×#41—91no—
314#137—92no—
315#137—92no—
316AGL(1,ℤ/8)—92no—
317AGL(1,ℤ/8)—92no—
318#135—94no—
319#131—34no—
320#131—62no—
321#131—94no—
322#131—182no—
323C2×M16—31no—
324C2×M16—31no—
325C2×M16—62no—
326C2×M16—91no—
327C2×M16—182no—
328C2×M16D3211yescharacteristic
329C2×C42—31no—
330#108—94no—
331#104—92no—
332#104—92no—
333#104—92no—
334#104—92no—
335#97—32no—
336C4×C8—32no—
337C23×C3—21no—
338C23×C3—61no—
339C23×C3#23511yescharacteristic
340C23×C3#23511yescharacteristic
341C22×D3—21no—
342C22×D3—62no—
343C22×D3—61no—
344C22×D3—61no—
345C3×Q8—32no—
346C3×Q8—61no—
347C3×D4—32no—
348C3×D4—62no—
349C3×D4—62no—
350C3×D4—61no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—61no—
358C2×C3×C4—61no—
359C2×C3×C4—62no—
360C2×C3×C4—62no—
361C2×C3×C4—61no—
362C2×C3×C4—122no—
363C2×C3×C4C2×C4×D311yescharacteristic
364C2×C3×C4C2×C4×D311yescharacteristic
365C2×C3×C4C2×C4×D311yescharacteristic
366C2×C3×C4C2×C4×D311yescharacteristic
367C2×C3×C4C2×D1211yescharacteristic
368C2×C3×C4C2×D1211yescharacteristic
369C2×C3×C4#23511yescharacteristic
370C2×C3×C4#23511yescharacteristic
371C2×C3×C4C2×#6711yescharacteristic
372C2×C3×C4C2×#6711yescharacteristic
373C2×C3×C4C23×D311yescharacteristic
374C2×C3×C4C23×D311yescharacteristic
375#67—22no—
376#67—22no—
377#67—64no—
378#67—64no—
379#67—62no—
380#67—62no—
381#67—62no—
382#67—62no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—42no—
387C2×Dic3—62no—
388C2×Dic3—61no—
389C2×Dic3—61no—
390D12—21no—
391D12—32no—
392D12—32no—
393D12—62no—
394D12—62no—
395D12—61no—
396D12—61no—
397D12C2×C4×D312yes—
398C4×D3—22no—
399C4×D3—22no—
400C4×D3—64no—
401C4×D3—64no—
402C4×D3—62no—
403C4×D3—62no—
404C4×D3—62no—
405C4×D3—62no—
406Dic6—21no—
407Dic6—32no—
408Dic6—32no—
409Dic6—62no—
410Dic6—62no—
411Dic6—61no—
412Dic6—61no—
413Dic6C2×C4×D312yes—
414C3×C8—24no—
415C3×C8—34no—
416C3×C8—62no—
417C3×C8C2×D1214yes—
418C3×C8C2×#6714yes—
419C3⋊C8—22no—
420C3⋊C8—64no—
421C3⋊C8—62no—
422C3⋊C8—62no—
423C3⋊C8C2×C4×D314yes—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32C2×C4×D411yescharacteristic
428C3⋊D3—42no—
429C3×D3—42no—
430#41—32no—
431#41—32no—
432#41—61no—
433#41—61no—
434#41—92no—
435#41—92no—
436#41—181no—
437#41—181no—
438C2×Q8—31no—
439C2×Q8—91no—
440C2×D4—31no—
441C2×D4—61no—
442C2×D4—91no—
443C2×D4—181no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—181no—
447C22×C4C2×D3211yescharacteristic
448Q16—94no—
449SD16—94no—
450SD16—94no—
451D8—94no—
452M16—32no—
453M16—32no—
454M16—32no—
455M16—61no—
456M16—64no—
457M16—92no—
458M16—181no—
459M16—184no—
460M16#62612yes—
461M16#62612yes—
462C2×C8—32no—
463C2×C8—61no—
464C2×C8—64no—
465C2×C8—181no—
466C2×C8—184no—
467C2×C8C2×D3212yes—
468C42—31no—
469C42—31no—
470C42—61no—
471C22×C3—21no—
472C22×C3—21no—
473C22×C3—21no—
474C22×C3—41no—
475C22×C3—41no—
476C22×C3—62no—
477C22×C3—81no—
478C22×C3—241no—
479C22×C3C4×D1211yescharacteristic
480C22×C3C4×D1211yescharacteristic
481C22×C3C4×#6711yescharacteristic
482C22×C3C4×#6711yescharacteristic
483C22×C3C2×#23511yescharacteristic
484C22×C3C2×#23511yescharacteristic
485C2×D3—22no—
486C2×D3—64no—
487C2×D3—62no—
488C2×D3—62no—
489C2×D3—81no—
490C2×D3—241no—
491C2×D3—241no—
492C2×D3—242no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—62no—
498C3×C4—62no—
499C3×C4—121no—
500C3×C4C4×D1211yescharacteristic
501C3×C4C4×D1211yescharacteristic
502C3×C4C4×#6711yescharacteristic
503C3×C4C4×#6711yescharacteristic
504C3×C4C22×C4×D311yescharacteristic
505C3×C4C22×C4×D311yescharacteristic
506C3×C4C22×D1211yescharacteristic
507C3×C4C22×#6711yescharacteristic
508Dic3—22no—
509Dic3—22no—
510Dic3—41no—
511Dic3—64no—
512Dic3—62no—
513Dic3—62no—
514C32#295611yescharacteristic, complemented, Sylow
515C23—61no—
516C23—181no—
517C23#379811yescharacteristic
518Q8—32no—
519Q8—61no—
520Q8—92no—
521Q8—181no—
522D4—32no—
523D4—62no—
524D4—62no—
525D4—61no—
526D4—92no—
527D4—182no—
528D4—182no—
529D4—181no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—61no—
535C2×C4—122no—
536C2×C4—181no—
537C2×C4—182no—
538C2×C4—182no—
539C2×C4#379811yescharacteristic
540C2×C4C4×D3211yescharacteristic
541C2×C4C4×D3211yescharacteristic
542C2×C4C2×#62611yescharacteristic
543C2×C4C2×#62611yescharacteristic
544C2×C4C22×D3211yescharacteristic
545C8—34no—
546C8—62no—
547C8—182no—
548C8C2×#62614yes—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—21no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—41no—
557C2×C3—41no—
558C2×C3—41no—
559C2×C3—122no—
560C2×C3C2×C4×D1211yescharacteristic
561C2×C3C2×C4×#6711yescharacteristic
562D3—42no—
563D3—124no—
564D3—122no—
565D3—122no—
566C22—41no—
567C22—62no—
568C22—182no—
569C22—241no—
570C22—721no—
571C22C4×#62611yescharacteristic
572C22C4×#62611yescharacteristic
573C22C2×#379811yescharacteristic
574C4—62no—
575C4—62no—
576C4—121no—
577C4—182no—
578C4C4×#62611yescharacteristic
579C4C4×#62611yescharacteristic
580C4C2×C4×D3211yescharacteristic
581C4C22×#62611yescharacteristic
582C3—21no—
583C3#7854111yescharacteristic, complemented
584C3#8381111yescharacteristic, complemented
585C2—21no—
586C2—21no—
587C2—21no—
588C2—122no—
589C2—362no—
590C2C2×C4×#62611yescharacteristic
subgroupG/H
centerC4C22×#626
commutatorC32×C4C23×C4
FrattiniC2×C4C22×D32
FittingC2×C32×M16C22
socleC2×C32C2×C4×D4
prime radical#10594795C2
cosocleC2×C32×C4C24
hypercenterC2×M16D32
hypocenterC32#2956
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. #10596216C2semidirect
2. C3×#5745C2semidirect
4. C3×Dic3×M16C2semidirect
5. #10591141C2semidirect
6. C2×#63570C2semidirect
7. C2×#63569C2semidirect
8. #10590924C2semidirect
9. #10590923C2semidirect
18. #64110C22semidirect
24. C3×#901C22semidirect
25. #63833C4semidirect
26. #63831C4semidirect
27. #63830C4semidirect
29. #63825C4semidirect
30. C3×#815C22semidirect
31. C3×C8×Dic3C22semidirect
32. #63577C22semidirect
33. #63570C22semidirect
34. #63570C22semidirect
35. #63569C22semidirect
36. #63569C22semidirect
43. #5745D3semidirect
51. Dic3×M16D3semidirect
93. #3719C2×C4semidirect
94. #3717C2×C4semidirect
95. #3716C2×C4semidirect
96. #3714C2×C4semidirect
139. #901C2×D3semidirect
148. #815C2×D3semidirect
149. C8×Dic3C2×D3semidirect
514. C32#2956semidirect
583. C3#78541semidirect
584. C3#83811semidirect

Maximal subgroups

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

subgroupindex
#105962162
C3×#57452
#105947952
C3×Dic3×M162
#105911412
C2×#635702
C2×#635692
#105909242
#105909232
#838113
#785413