Finite Group Databasebeta

← #13177538

Subgroups — #13177538

Order 1728.

1880 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64361C211yesmaximal, characteristic, complemented
2#11724579C211yesmaximal, characteristic, complemented
3#11723973C211yesmaximal, characteristic, complemented
4C2×C32×#213C211yesmaximal, characteristic, complemented
5#11723560C212yesmaximal, complemented
6#11723559C212yesmaximal, complemented
7#11723546C212yesmaximal, complemented
8C3⋊D3×#213C212yesmaximal, complemented
9#11723369C211yesmaximal, characteristic, complemented
10#11723364C211yesmaximal, characteristic, complemented
11#11723352C211yesmaximal, characteristic, complemented
12#10596228—31nomaximal
13#10595096—34nomaximal
14C2×C33×D4C2211yescharacteristic, complemented
15#89076C2211yescharacteristic, complemented
16C2×C3×#638—22no—
17C3×#3830—21no—
18C3×#3830C2212yescomplemented
19C3×D4×C3⋊D3—21no—
20C3×D4×C3⋊D3C2212yescomplemented
21C3×#3828C2211yescharacteristic
22C3×#3828C2211yescharacteristic, complemented
23C2×C32×D12C2211yescharacteristic, complemented
24D3×#638—22no—
25#89026—21no—
26C3⋊D3×D12C2212yescomplemented
27#89015—21no—
28#89014C2212yescomplemented
29#88861C2212yescomplemented
30#88860C2212yescomplemented
31#88855C2211yescharacteristic, complemented
32C32×#213C2212yescomplemented
33C32×#213C2212yescomplemented
34C2×C32×C3⋊C8C2211yescharacteristic, complemented
35#88800C2211yescharacteristic, complemented
36#88797C2211yescharacteristic, complemented
37#88796C2211yescharacteristic, complemented
38#88792C2212yescomplemented
39#88791C2211yescharacteristic, complemented
40#88789C2212yescomplemented
41#88787C2212yescomplemented
42C3⋊D3×C3⋊C8C2212yescomplemented
43#64361—31no—
44#64361D311yescharacteristic, complemented
45C3×#1004—34no—
46#64307—34no—
47C2×C32×D8—31no—
48#64141—31no—
49#64141—31no—
50#64141—34no—
51#64141—64no—
52#64121—32no—
53#64120—32no—
54#64120—32no—
55D8×C3⋊D3—32no—
56#64113—31no—
57#64110—31no—
58C2×C3×#213D314yescomplemented
59#63933—38no—
60#63932—38no—
61#63925—38no—
62D3×#213—38no—
63#63829—34no—
64#63823—34no—
65#63814—34no—
66C23×C33—21no—
67C22×C3×C3⋊D3—22no—
68C22×C32×D3—21no—
69C2×D3×C3⋊D3—22no—
70C33×D4—21no—
71C33×D4C2312yes—
72C2×C33×C4C2311yescharacteristic
73#6312—21no—
74C4⋊(C3⋊(C3⋊D3))C2311yescharacteristic
75C4×C3⋊(C3⋊D3)—21no—
76#6308C2311yescharacteristic
77C3×#638—21no—
78C3×#638—22no—
79C3×#638—22no—
80C3×#638D412yes—
81C2×C3×C3⋊Dic3—22no—
82C3×C4⋊(C3⋊D3)D411yescharacteristic
83C3×C4⋊(C3⋊D3)C2311yescharacteristic
84C3×C4×C3⋊D3—21no—
85C3×C4×C3⋊D3C2312yes—
86C3×#634D411yescharacteristic
87C3×#634C2311yescharacteristic
88C32×D12—21no—
89C32×D12C2312yes—
90#6291—22no—
91#6290—22no—
92D3×C3⋊Dic3—22no—
93C3⋊(C3⋊(C3⋊C8))C2312yes—
94C32×C3⋊C8C2311yescharacteristic
95C32×C3⋊C8C2311yescharacteristic
96#5835—91no—
97#5757—94no—
98C2×C32×D4—24no—
99C2×C32×D4—31no—
100C2×C32×D4C2×D314yescomplemented
101C2×C32×D4C2×D311yescharacteristic, complemented
102C2×#638—22no—
103C2×#638—62no—
104#3830—21no—
105#3830—32no—
106#3830—61no—
107#3830C2×D312yescomplemented
108D4×C3⋊D3—21no—
109D4×C3⋊D3—32no—
110D4×C3⋊D3—61no—
111D4×C3⋊D3C2×D312yescomplemented
112#3828—31no—
113#3828—34no—
114#3828—64no—
115#3828C2×D311yescharacteristic
116#3828C2×D311yescharacteristic, complemented
117C2×C3×#67—68no—
118C3×#237—38no—
119C3×#237—64no—
120C3×D3×D4—38no—
121C3×D3×D4—64no—
122C3×#235—34no—
123C3×#235—34no—
124C2×C3×D12C2×D314yescomplemented
125D3×#67—68no—
126#3802—64no—
127D3×D12—38no—
128#3796—64no—
129#3795—38no—
130C32×D8—32no—
131C32×D8—32no—
132C2×C32×C8—31no—
133#3754—32no—
134#3754—38no—
135#3754—32no—
136#3754—68no—
137#3753—32no—
138#3753—38no—
139#3753—32no—
140#3753—68no—
141#3748—31no—
142#3748—34no—
143#3748—64no—
144#3746—31no—
145#3745—31no—
146#3744—31no—
147#3744—31no—
148#3743—32no—
149C8×C3⋊D3—32no—
150C3×#213C2×D318yescomplemented
151C3×#213C2×D318yescomplemented
152C2×C3×C3⋊C8C2×D314yescomplemented
153#3719—34no—
154#3717—34no—
155#3716—34no—
156#3715—38no—
157#3714—34no—
158#3713—38no—
159#3711—38no—
160D3×C3⋊C8—38no—
161C22×C33—22no—
162C22×C33—81no—
163C22×C33C2×D411yescharacteristic
164C2×C3⋊(C3⋊D3)—21no—
165C2×C3×C3⋊D3—21no—
166C2×C3×C3⋊D3—82no—
167C2×C3×C3⋊D3C2×D412yes—
168C2×C32×D3—22no—
169C2×C32×D3—81no—
170D3×C3⋊D3—82no—
171C33×C4C2×D411yescharacteristic
172C33×C4C2411yescharacteristic
173C3⋊(C3⋊Dic3)—21no—
174C3×C3⋊Dic3—21no—
175C3×C3⋊Dic3C2×D412yes—
176C3×H(5,𝔽2)—91no—
177#1009—91no—
178#1004—34no—
179#1004—94no—
180C2×C3×D8—34no—
181#950—92no—
182#950—91no—
183#949—91no—
184#949—92no—
185#948—91no—
186#932—94no—
187#932—94no—
188#932—91no—
189#932—92no—
190#932—184no—
191C2×#213—92no—
192C2×#213C3⋊D311yescharacteristic, complemented
193#912—98no—
194#911—98no—
195#911—98no—
196D3×D8—98no—
197#904—94no—
198#901—94no—
199C23×C32—21no—
200C23×C32—24no—
201C23×C32—44no—
202C23×C32—61no—
203C22×C3⋊D3—22no—
204C22×C3⋊D3—62no—
205C22×C3×D3—24no—
206C22×C3×D3—68no—
207C2×D32—68no—
208C32×D4—21no—
209C32×D4—28no—
210C32×D4—24no—
211C32×D4—32no—
212C32×D4—44no—
213C32×D4—61no—
214C32×D4C22×D318yes—
215C32×D4C22×D312yes—
216C2×C32×C4—24no—
217C2×C32×C4C22×D314yes—
218C2×C32×C4C22×D311yescharacteristic
219#638—21no—
220#638—22no—
221#638—22no—
222#638—61no—
223#638—62no—
224#638—62no—
225#638—64no—
226#638—124no—
227#638#6712yes—
228C2×C3⋊Dic3—22no—
229C2×C3⋊Dic3—62no—
230C4⋊(C3⋊D3)—31no—
231C4⋊(C3⋊D3)—34no—
232C4⋊(C3⋊D3)—64no—
233C4⋊(C3⋊D3)#6711yescharacteristic
234C4⋊(C3⋊D3)C22×D311yescharacteristic
235C4×C3⋊D3—21no—
236C4×C3⋊D3—61no—
237C4×C3⋊D3—64no—
238C4×C3⋊D3—124no—
239C4×C3⋊D3C22×D312yes—
240#634—31no—
241#634—34no—
242#634—64no—
243#634#6711yescharacteristic
244#634C22×D311yescharacteristic
245C3×#67—38no—
246C3×#67—68no—
247C3×#67—68no—
248C3×#67—64no—
249C2×C3×Dic3—68no—
250C3×D12—24no—
251C3×D12—34no—
252C3×D12—34no—
253C3×D12C22×D318yes—
254C3×C4×D3—38no—
255C3×C4×D3—64no—
256C3×Dic6—34no—
257C3×Dic6—34no—
258#626—68no—
259#625—68no—
260D3×Dic3—68no—
261C32×C8—31no—
262C32×C8—31no—
263C3⋊(C3⋊C8)—32no—
264C3⋊(C3⋊C8)—38no—
265C3⋊(C3⋊C8)—68no—
266C3×C3⋊C8C22×D314yes—
267C3×C3⋊C8C22×D314yes—
268#576—271noSylow
269C2×C33—21no—
270C2×C33—42no—
271C2×C33C22×D411yescharacteristic
272C3⋊(C3⋊D3)—41no—
273C3×C3⋊D3—22no—
274C3×C3⋊D3—41no—
275C32×D3—42no—
276C3×#41—91no—
277C3×#41—92no—
278C3×#41—91no—
279C3×#41—181no—
280C2×C3×D4—24no—
281C2×C3×D4—34no—
282C2×C3×D4—92no—
283C2×C3×D4—181no—
284C2×C3×D4—182no—
285C2×C3×D4D3214yescomplemented
286C2×C3×D4C2×C3⋊D311yescharacteristic, complemented
287C2×#67—68no—
288C2×#67—188no—
289#239—181no—
290#237—38no—
291#237—64no—
292#237—98no—
293#237—184no—
294D3×D4—38no—
295D3×D4—64no—
296D3×D4—98no—
297D3×D4—181no—
298D3×D4—184no—
299D3×D4—182no—
300#235—34no—
301#235—34no—
302#235—94no—
303#235—91no—
304#235—92no—
305#235—184no—
306C2×D12—92no—
307C2×D12C2×C3⋊D311yescharacteristic, complemented
308C3×D8—38no—
309C3×D8—38no—
310C2×C3×C8—34no—
311#216—91no—
312#215—92no—
313#215—91no—
314#214—98no—
315#214—98no—
316#214—92no—
317#214—91no—
318#214—188no—
319#213—98no—
320#213—98no—
321#213—92no—
322#213—92no—
323#213—91no—
324#213—188no—
325#213C2×C3⋊D312yescomplemented
326#213C2×C3⋊D312yescomplemented
327#208—94no—
328#208—91no—
329#208—92no—
330#208—184no—
331C2×C3⋊C8—92no—
332C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
333Dic12—94no—
334D24—94no—
335#204—94no—
336#204—94no—
337ΓL(1,𝔽25)—98no—
338D3×C8—98no—
339C22×C32—22no—
340C22×C32—24no—
341C22×C32—28no—
342C22×C32—48no—
343C22×C32—62no—
344C22×C32—81no—
345C22×C32—84no—
346C22×C32—88no—
347C22×C32—241no—
348C22×C32D3×D414yes—
349C22×C32C2×#6711yescharacteristic
350C2×C3⋊D3—21no—
351C2×C3⋊D3—61no—
352C2×C3⋊D3—64no—
353C2×C3⋊D3—82no—
354C2×C3⋊D3—124no—
355C2×C3⋊D3—242no—
356C2×C3⋊D3C2×#6712yes—
357C2×C3×D3—28no—
358C2×C3×D3—38no—
359C2×C3×D3—64no—
360C2×C3×D3—84no—
361C2×C3×D3—248no—
362D32—248no—
363C32×C4—24no—
364C32×C4—24no—
365C32×C4D3×D414yes—
366C32×C4C2×#6711yescharacteristic
367C32×C4C23×D314yes—
368C32×C4C23×D311yescharacteristic
369C3⋊Dic3—21no—
370C3⋊Dic3—61no—
371C3⋊Dic3—64no—
372C3⋊Dic3—124no—
373C3⋊Dic3C2×#6712yes—
374C3×Dic3—38no—
375C3×Dic3—64no—
376H(5,𝔽2)—91no—
377H(5,𝔽2)—271no—
378AGL(1,ℤ/8)—272no—
379AGL(1,ℤ/8)—272no—
380AGL(1,ℤ/8)—271no—
381AGL(1,ℤ/8)—271no—
382#135—271no—
383#135—272no—
384C2×D8—31no—
385C2×D8—272no—
386#131—271no—
387C33#57611yescharacteristic, complemented, Sylow
388C23×C3—24no—
389C23×C3—21no—
390C23×C3—44no—
391C23×C3—64no—
392C23×C3—182no—
393C22×D3—21no—
394C22×D3—68no—
395C22×D3—182no—
396C22×D3—188no—
397C3×Q8—91no—
398C3×Q8—91no—
399C3×D4—24no—
400C3×D4—21no—
401C3×D4—28no—
402C3×D4—38no—
403C3×D4—44no—
404C3×D4—64no—
405C3×D4—92no—
406C3×D4—91no—
407C3×D4—91no—
408C3×D4—182no—
409C3×D4—182no—
410C3×D4—181no—
411C3×D4C2×D3218yes—
412C3×D4C22×C3⋊D312yes—
413C2×C3×C4—24no—
414C2×C3×C4—92no—
415C2×C3×C4—182no—
416C2×C3×C4—181no—
417C2×C3×C4C2×D3214yes—
418C2×C3×C4C22×C3⋊D311yescharacteristic
419#67—38no—
420#67—68no—
421#67—68no—
422#67—64no—
423#67—188no—
424#67—182no—
425#67—188no—
426#67—184no—
427#67—181no—
428#67—364no—
429C2×Dic3—68no—
430C2×Dic3—188no—
431D12—21no—
432D12—34no—
433D12—34no—
434D12—94no—
435D12—91no—
436D12—184no—
437D12—182no—
438D12C22×C3⋊D312yes—
439C4×D3—38no—
440C4×D3—64no—
441C4×D3—182no—
442C4×D3—184no—
443C4×D3—181no—
444C4×D3—364no—
445Dic6—34no—
446Dic6—34no—
447Dic6—94no—
448Dic6—91no—
449Dic6—184no—
450C3×C8—34no—
451C3×C8—34no—
452C3⋊C8—98no—
453C3⋊C8—92no—
454C3⋊C8—188no—
455C3⋊C8C22×C3⋊D311yescharacteristic
456C3⋊C8C22×C3⋊D311yescharacteristic
457C2×C32—21no—
458C2×C32—24no—
459C2×C32—24no—
460C2×C32—44no—
461C2×C32—42no—
462C2×C32—48no—
463C2×C32—88no—
464C2×C32—122no—
465C2×C32C2×D3×D414yes—
466C2×C32C22×#6711yescharacteristic
467C3⋊D3—22no—
468C3⋊D3—41no—
469C3⋊D3—121no—
470C3⋊D3—124no—
471C3⋊D3—244no—
472C3×D3—48no—
473C3×D3—68no—
474C3×D3—124no—
475#41—91no—
476#41—92no—
477#41—91no—
478#41—181no—
479#41—272no—
480#41—271no—
481#41—541no—
482C2×D4—31no—
483C2×D4—92no—
484C2×D4—181no—
485C2×D4—182no—
486C2×D4—272no—
487C2×D4—541no—
488C2×D4—542no—
489C2×D4D3×C3⋊D311yescharacteristic, complemented
490Q16—271no—
491SD16—271no—
492SD16—271no—
493SD16—272no—
494SD16—272no—
495D8—32no—
496D8—32no—
497D8—271no—
498D8—272no—
499D8—272no—
500M16—272no—
501M16—271no—
502C2×C8—31no—
503C2×C8—272no—
504C22×C3—28no—
505C22×C3—22no—
506C22×C3—24no—
507C22×C3—48no—
508C22×C3—68no—
509C22×C3—84no—
510C22×C3—81no—
511C22×C3—88no—
512C22×C3—92no—
513C22×C3—181no—
514C22×C3—244no—
515C22×C3—722no—
516C22×C3D3×#6714yes—
517C22×C3D4×C3⋊D311yescharacteristic
518C2×D3—22no—
519C2×D3—38no—
520C2×D3—64no—
521C2×D3—81no—
522C2×D3—184no—
523C2×D3—181no—
524C2×D3—248no—
525C2×D3—364no—
526C2×D3—722no—
527C2×D3—728no—
528C3×C4—24no—
529C3×C4—24no—
530C3×C4—92no—
531C3×C4—181no—
532C3×C4D3×#6714yes—
533C3×C4D4×C3⋊D311yescharacteristic
534C3×C4C22×D3214yes—
535C3×C4C23×C3⋊D311yescharacteristic
536Dic3—38no—
537Dic3—64no—
538Dic3—184no—
539Dic3—181no—
540Dic3—364no—
541C32—24no—
542C32#575714yescomplemented
543C32#583511yescharacteristic, complemented
544C23—21no—
545C23—61no—
546C23—182no—
547C23—542no—
548Q8—91no—
549Q8—91no—
550Q8—271no—
551D4—21no—
552D4—32no—
553D4—61no—
554D4—92no—
555D4—91no—
556D4—91no—
557D4—182no—
558D4—182no—
559D4—181no—
560D4—271no—
561D4—542no—
562D4—542no—
563D4—541no—
564D4C2×D3×C3⋊D312yes—
565C2×C4—92no—
566C2×C4—182no—
567C2×C4—181no—
568C2×C4—542no—
569C2×C4—541no—
570C2×C4C2×D3×C3⋊D311yescharacteristic
571C8—31no—
572C8—31no—
573C8—272no—
574C2×C3—24no—
575C2×C3—21no—
576C2×C3—24no—
577C2×C3—44no—
578C2×C3—48no—
579C2×C3—42no—
580C2×C3—88no—
581C2×C3—128no—
582C2×C3—182no—
583C2×C3—361no—
584C2×C3C2×D3×#6714yes—
585C2×C3C2×D4×C3⋊D311yescharacteristic
586D3—42no—
587D3—68no—
588D3—124no—
589D3—364no—
590D3—361no—
591D3—724no—
592C22—22no—
593C22—62no—
594C22—81no—
595C22—92no—
596C22—181no—
597C22—241no—
598C22—541no—
599C22—722no—
600C22—2162no—
601C22C3⋊D3×#6711yescharacteristic
602C4—92no—
603C4—181no—
604C4—541no—
605C4C3⋊D3×#6711yescharacteristic
606C4C22×D3×C3⋊D311yescharacteristic
607C3—24no—
608C3#1059509614yescomplemented
609C3#1059622811yescharacteristic, complemented
610C2—21no—
611C2—42no—
612C2—122no—
613C2—182no—
614C2—361no—
615C2—1081no—
616C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC2C2×C3⋊D3×#67
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC2×C33×D4C22
socleC2×C33C22×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#576
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×#64361C2semidirect
2. #11724579C2semidirect
3. #11723973C2semidirect
4. C2×C32×#213C2semidirect
5. #11723560C2semidirect
6. #11723559C2semidirect
7. #11723546C2semidirect
8. C3⋊D3×#213C2semidirect
9. #11723369C2semidirect
10. #11723364C2semidirect
11. #11723352C2semidirect
14. C2×C33×D4C22semidirect
15. #89076C22semidirect
18. C3×#3830C22semidirect
20. C3×D4×C3⋊D3C22semidirect
22. C3×#3828C22semidirect
23. C2×C32×D12C22semidirect
26. C3⋊D3×D12C22semidirect
28. #89014C22semidirect
29. #88861C22semidirect
30. #88860C22semidirect
31. #88855C22semidirect
32. C32×#213C22semidirect
33. C32×#213C22semidirect
34. C2×C32×C3⋊C8C22semidirect
35. #88800C22semidirect
36. #88797C22semidirect
37. #88796C22semidirect
38. #88792C22semidirect
39. #88791C22semidirect
40. #88789C22semidirect
41. #88787C22semidirect
42. C3⋊D3×C3⋊C8C22semidirect
44. #64361D3semidirect
58. C2×C3×#213D3semidirect
100. C2×C32×D4C2×D3semidirect
101. C2×C32×D4C2×D3semidirect
107. #3830C2×D3semidirect
111. D4×C3⋊D3C2×D3semidirect
116. #3828C2×D3semidirect
124. C2×C3×D12C2×D3semidirect
150. C3×#213C2×D3semidirect
151. C3×#213C2×D3semidirect
152. C2×C3×C3⋊C8C2×D3semidirect
192. C2×#213C3⋊D3semidirect
285. C2×C3×D4D32semidirect
286. C2×C3×D4C2×C3⋊D3semidirect
307. C2×D12C2×C3⋊D3semidirect
325. #213C2×C3⋊D3semidirect
326. #213C2×C3⋊D3semidirect
332. C2×C3⋊C8C2×C3⋊D3semidirect
387. C33#576semidirect
489. C2×D4D3×C3⋊D3semidirect
542. C32#5757semidirect
543. C32#5835semidirect
608. C3#10595096semidirect
609. C3#10596228semidirect

Maximal subgroups

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

subgroupindex
C3×#643612
#117245792
#117239732
C2×C32×#2132
#117235602
#117235592
#117235462
C3⋊D3×#2132
#117233692
#117233642
#117233522
#105962283
#105950963