Finite Group Databasebeta

← #13176705

Subgroups — #13176705

Order 1728.

1856 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11724580C211yesmaximal, characteristic, complemented
2#11724579C211yesmaximal, characteristic, complemented
3#11723948C211yesmaximal, characteristic, complemented
4C3×#64117C211yesmaximal, characteristic, complemented
5C32×#908C211yesmaximal, characteristic, complemented
6#11723363C211yesmaximal, characteristic, complemented
7C2×#88789C211yesmaximal, characteristic, complemented
8#11723342C212yesmaximal, complemented
9#11723324C212yesmaximal, complemented
10#11723323C212yesmaximal, complemented
11#11723322C212yesmaximal, complemented
12#10596245—31nomaximal
13#10594720—34nomaximal
14#89076C2211yescharacteristic
15C3×#3828C2211yescharacteristic, complemented
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
17C32×#235C2211yescharacteristic, complemented
18C2×C32×D12C2211yescharacteristic, complemented
19C3⋊D3×#67—22no—
20D3×#638—22no—
21#89026C2212yescomplemented
22#89026C2212yescomplemented
23C3⋊D3×D12—21no—
24D3×C4⋊(C3⋊D3)—21no—
25#89021—21no—
26#89017C2212yescomplemented
27#89015C2212yescomplemented
28#89014—21no—
29C33×M16C2211yescharacteristic, complemented
30C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
31#88850C2212yescomplemented
32C8×C3⋊(C3⋊D3)C2212yescomplemented
33C3×#3745C2212yescomplemented
34C3×#3744C2212yescomplemented
35C32×D24C2212yescomplemented
36C32×#204C2212yescomplemented
37#88798C2211yescharacteristic, complemented
38#88793C2211yescharacteristic, complemented
39#88792C2211yescharacteristic, complemented
40#88789C2211yescharacteristic, complemented
41#88789C2212yescomplemented
42#88789C2212yescomplemented
43#64366—31no—
44#64361—31no—
45#64307—34no—
46#64307—34no—
47C32×AGL(1,ℤ/8)—31no—
48#64159—31no—
49C2×#3753—31no—
50#64125—32no—
51#64123—32no—
52#64120—32no—
53D8×C3⋊D3—32no—
54#64117D311yescharacteristic, complemented
55#64116—31no—
56#64116—34no—
57#64116—64no—
58C3×#908—34no—
59C3×#908D314yescomplemented
60#63822—34no—
61C2×#3713—34no—
62#63809—38no—
63#63798—38no—
64#63798—38no—
65#63797—38no—
66C22×C3×C3⋊D3—21no—
67C22×C32×D3—21no—
68C2×D3×C3⋊D3—22no—
69C2×D3×C3⋊D3—22no—
70C2×C33×C4C2311yescharacteristic
71#6312D412yes—
72C4⋊(C3⋊(C3⋊D3))D411yescharacteristic
73C4×C3⋊(C3⋊D3)C2312yes—
74#6308D411yescharacteristic
75C3×#638—21no—
76C3×C4⋊(C3⋊D3)—21no—
77C3×C4⋊(C3⋊D3)C2312yes—
78C3×C4⋊(C3⋊D3)C2311yescharacteristic
79C3×C4×C3⋊D3—21no—
80C3×#634C2311yescharacteristic
81C32×#67—21no—
82C32×D12—21no—
83C32×D12C2312yes—
84C32×D12C2311yescharacteristic
85C32×C4×D3—21no—
86C32×Dic6C2311yescharacteristic
87#6292—22no—
88#6291—22no—
89#6290—22no—
90#6290—22no—
91Dic3×C3⋊D3—22no—
92D3×C3⋊Dic3—22no—
93C33×C8C2312yes—
94C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
95C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
96#5774—94no—
97#5774—91no—
98C32×#41—31no—
99C2×C32×D4—31no—
100C2×#638—62no—
101#3832—32no—
102#3830—32no—
103#3830—61no—
104D4×C3⋊D3—32no—
105D4×C3⋊D3—32no—
106D4×C3⋊D3—61no—
107D4×C3⋊D3—62no—
108#3828—31no—
109#3828—34no—
110#3828—61no—
111#3828—64no—
112#3828C2×D311yescharacteristic, complemented
113C2×C4⋊(C3⋊D3)—61no—
114C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
115C3×#235—34no—
116C3×#235C2×D314yescomplemented
117C2×C3×D12—34no—
118C2×C3×D12C2×D314yescomplemented
119D3×#67—68no—
120D3×#67—68no—
121#3802—38no—
122#3802—38no—
123D3×D12—64no—
124D3×D12—64no—
125#3796—38no—
126#3796—38no—
127#3795—64no—
128#3795—64no—
129C32×SD16—32no—
130C32×D8—32no—
131C32×M16—24no—
132C32×M16C2×D314yescomplemented
133C32×M16C2×D311yescharacteristic, complemented
134#3756—31no—
135#3755—31no—
136#3754—31no—
137#3753—31no—
138#3753—32no—
139#3753—32no—
140C2×C3⋊(C3⋊C8)—31no—
141C2×C3⋊(C3⋊C8)—34no—
142C2×C3⋊(C3⋊C8)—64no—
143#3745C2×D312yescomplemented
144#3744C2×D312yescomplemented
145#3743—32no—
146#3743—38no—
147#3743—68no—
148C8×C3⋊D3—32no—
149C8×C3⋊D3—38no—
150C8×C3⋊D3—68no—
151C3×D24—38no—
152C3×D24C2×D318yescomplemented
153C3×#204—38no—
154C3×#204C2×D318yescomplemented
155#3718—34no—
156#3715—34no—
157#3715—34no—
158#3713—34no—
159#3713—38no—
160#3713—38no—
161C22×C33C2×D411yescharacteristic
162C2×C3⋊(C3⋊D3)C2×D412yes—
163C2×C3×C3⋊D3—22no—
164C2×C3×C3⋊D3—21no—
165C2×C3×C3⋊D3—81no—
166C2×C32×D3—22no—
167C2×C32×D3—21no—
168C2×C32×D3—81no—
169D3×C3⋊D3—82no—
170D3×C3⋊D3—82no—
171C33×C4C2×D411yescharacteristic
172C33×C4C2411yescharacteristic
173C3⋊(C3⋊Dic3)C2×D412yes—
174C3×C3⋊Dic3—21no—
175C32×Dic3—21no—
176#1009—94no—
177#1009—91no—
178#1004—94no—
179#1004—91no—
180C3×AGL(1,ℤ/8)—34no—
181C3×AGL(1,ℤ/8)—91no—
182#950—94no—
183#950—91no—
184C2×#213—94no—
185C2×#213—91no—
186#916—98no—
187#916—92no—
188#914—98no—
189#914—92no—
190#911—98no—
191#911—92no—
192D3×D8—98no—
193D3×D8—92no—
194#908—34no—
195#908C3⋊D311yescharacteristic, complemented
196#907—94no—
197#907—91no—
198#907—184no—
199C23×C32—61no—
200C22×C3⋊D3—21no—
201C22×C3⋊D3—62no—
202C22×C3⋊D3—62no—
203C22×C3×D3—24no—
204C22×C3×D3—64no—
205C2×D32—68no—
206C2×D32—68no—
207C32×Q8—31no—
208C32×D4—32no—
209C32×D4—31no—
210C32×D4—61no—
211C32×D4—61no—
212C2×C32×C4—24no—
213C2×C32×C4—61no—
214C2×C32×C4C22×D314yes—
215C2×C32×C4C22×D311yescharacteristic
216#638—21no—
217#638—32no—
218#638—38no—
219#638—68no—
220#638—62no—
221#638—62no—
222#638—62no—
223C2×C3⋊Dic3—62no—
224C4⋊(C3⋊D3)—21no—
225C4⋊(C3⋊D3)—31no—
226C4⋊(C3⋊D3)—34no—
227C4⋊(C3⋊D3)—64no—
228C4⋊(C3⋊D3)—62no—
229C4⋊(C3⋊D3)C22×D312yes—
230C4⋊(C3⋊D3)C22×D311yescharacteristic
231C4×C3⋊D3—21no—
232C4×C3⋊D3—32no—
233C4×C3⋊D3—38no—
234C4×C3⋊D3—68no—
235C4×C3⋊D3—62no—
236#634—31no—
237#634—34no—
238#634—64no—
239#634C22×D311yescharacteristic
240C3×#67—24no—
241C3×#67—64no—
242C3×D12—24no—
243C3×D12—38no—
244C3×D12—34no—
245C3×D12—64no—
246C3×D12C22×D318yes—
247C3×D12C22×D314yes—
248C3×C4×D3—24no—
249C3×C4×D3—64no—
250C3×Dic6—34no—
251C3×Dic6C22×D314yes—
252#626—68no—
253#626—68no—
254#625—68no—
255#625—68no—
256D3×Dic3—68no—
257D3×Dic3—68no—
258C32×C8—28no—
259C32×C8C22×D318yes—
260C32×C8C22×D312yes—
261C3⋊(C3⋊C8)—31no—
262C3⋊(C3⋊C8)—31no—
263C3⋊(C3⋊C8)—34no—
264C3⋊(C3⋊C8)—34no—
265C3⋊(C3⋊C8)—64no—
266C3⋊(C3⋊C8)—64no—
267#576—271noSylow
268C2×C33—21no—
269C2×C33C22×D411yescharacteristic
270C3⋊(C3⋊D3)—22no—
271C3×C3⋊D3—41no—
272C3×C3⋊D3—42no—
273C32×D3—41no—
274C32×D3—42no—
275C3×#41—34no—
276C3×#41—91no—
277C2×C3×D4—34no—
278C2×C3×D4—91no—
279C2×#67—182no—
280C2×#67—188no—
281#239—98no—
282#239—92no—
283#237—98no—
284#237—92no—
285#237—181no—
286#237—184no—
287D3×D4—98no—
288D3×D4—98no—
289D3×D4—92no—
290D3×D4—92no—
291D3×D4—181no—
292D3×D4—184no—
293D3×D4—188no—
294D3×D4—182no—
295#235—34no—
296#235—94no—
297#235—91no—
298#235—184no—
299#235—181no—
300#235—184no—
301#235C2×C3⋊D311yescharacteristic, complemented
302C2×D12—34no—
303C2×D12—181no—
304C2×D12—184no—
305C2×D12C2×C3⋊D311yescharacteristic, complemented
306C3×SD16—38no—
307C3×SD16—92no—
308C3×D8—38no—
309C3×D8—92no—
310C3×M16—24no—
311C3×M16D3214yescomplemented
312C3×M16C2×C3⋊D311yescharacteristic, complemented
313#216—94no—
314#216—91no—
315#215—94no—
316#215—91no—
317#214—94no—
318#214—91no—
319#213—94no—
320#213—98no—
321#213—98no—
322#213—91no—
323#213—92no—
324#213—92no—
325C2×C3⋊C8—94no—
326C2×C3⋊C8—91no—
327C2×C3⋊C8—184no—
328D24—38no—
329D24C2×C3⋊D312yescomplemented
330#204—38no—
331#204C2×C3⋊D312yescomplemented
332ΓL(1,𝔽25)—98no—
333ΓL(1,𝔽25)—92no—
334ΓL(1,𝔽25)—188no—
335D3×C8—98no—
336D3×C8—92no—
337D3×C8—188no—
338C22×C32—24no—
339C22×C32—62no—
340C22×C32—61no—
341C22×C32—241no—
342C22×C32D3×D414yes—
343C22×C32D3×D411yescharacteristic
344C2×C3⋊D3—21no—
345C2×C3⋊D3—22no—
346C2×C3⋊D3—32no—
347C2×C3⋊D3—38no—
348C2×C3⋊D3—68no—
349C2×C3⋊D3—81no—
350C2×C3⋊D3—242no—
351C2×C3⋊D3—242no—
352C2×C3×D3—28no—
353C2×C3×D3—24no—
354C2×C3×D3—68no—
355C2×C3×D3—64no—
356C2×C3×D3—84no—
357C2×C3×D3—244no—
358D32—248no—
359D32—248no—
360C32×C4—24no—
361C32×C4—24no—
362C32×C4—61no—
363C32×C4D3×D414yes—
364C32×C4D3×D411yescharacteristic
365C32×C4C23×D314yes—
366C32×C4C23×D311yescharacteristic
367C3⋊Dic3—21no—
368C3⋊Dic3—32no—
369C3⋊Dic3—38no—
370C3⋊Dic3—68no—
371C3×Dic3—24no—
372C3×Dic3—64no—
373H(5,𝔽2)—271no—
374H(5,𝔽2)—271no—
375AGL(1,ℤ/8)—31no—
376AGL(1,ℤ/8)—91no—
377AGL(1,ℤ/8)—272no—
378AGL(1,ℤ/8)—272no—
379#135—272no—
380#135—271no—
381C2×D8—272no—
382C2×D8—271no—
383#131—271no—
384C33#57611yescharacteristic, complemented, Sylow
385C23×C3—64no—
386C23×C3—181no—
387C22×D3—21no—
388C22×D3—64no—
389C22×D3—182no—
390C22×D3—182no—
391C22×D3—188no—
392C22×D3—188no—
393C3×Q8—34no—
394C3×Q8—91no—
395C3×D4—38no—
396C3×D4—34no—
397C3×D4—64no—
398C3×D4—64no—
399C3×D4—92no—
400C3×D4—91no—
401C3×D4—181no—
402C3×D4—181no—
403C2×C3×C4—24no—
404C2×C3×C4—64no—
405C2×C3×C4—181no—
406C2×C3×C4C2×D3214yes—
407C2×C3×C4C22×C3⋊D311yescharacteristic
408#67—21no—
409#67—64no—
410#67—98no—
411#67—92no—
412#67—188no—
413#67—182no—
414#67—188no—
415#67—182no—
416#67—182no—
417#67—188no—
418#67—188no—
419C2×Dic3—182no—
420C2×Dic3—188no—
421D12—21no—
422D12—38no—
423D12—34no—
424D12—64no—
425D12—94no—
426D12—91no—
427D12—184no—
428D12—188no—
429D12—182no—
430D12C22×C3⋊D312yes—
431D12C22×C3⋊D311yescharacteristic
432C4×D3—21no—
433C4×D3—64no—
434C4×D3—98no—
435C4×D3—92no—
436C4×D3—188no—
437C4×D3—188no—
438C4×D3—182no—
439Dic6—34no—
440Dic6—94no—
441Dic6—91no—
442Dic6—184no—
443Dic6C22×C3⋊D311yescharacteristic
444C3×C8—28no—
445C3×C8C2×D3218yes—
446C3×C8C22×C3⋊D312yes—
447C3⋊C8—94no—
448C3⋊C8—94no—
449C3⋊C8—91no—
450C3⋊C8—91no—
451C3⋊C8—184no—
452C3⋊C8—184no—
453C2×C32—21no—
454C2×C32—24no—
455C2×C32—24no—
456C2×C32—44no—
457C2×C32—121no—
458C2×C32—122no—
459C2×C32C2×D3×D414yes—
460C2×C32C2×D3×D411yescharacteristic
461C3⋊D3—42no—
462C3⋊D3—41no—
463C3⋊D3—62no—
464C3⋊D3—68no—
465C3⋊D3—128no—
466C3×D3—44no—
467C3×D3—48no—
468C3×D3—124no—
469C3×D3—128no—
470#41—31no—
471#41—91no—
472#41—271no—
473#41—272no—
474#41—272no—
475#41—541no—
476#41—541no—
477C2×D4—31no—
478C2×D4—91no—
479C2×D4—272no—
480C2×D4—272no—
481C2×D4—541no—
482C2×D4—541no—
483C2×D4—542no—
484C2×D4—542no—
485Q16—271no—
486SD16—32no—
487SD16—92no—
488SD16—271no—
489SD16—271no—
490D8—32no—
491D8—92no—
492D8—271no—
493D8—272no—
494D8—272no—
495M16—272no—
496M16D3×C3⋊D311yescharacteristic, complemented
497C2×C8—271no—
498C2×C8—272no—
499C22×C3—24no—
500C22×C3—68no—
501C22×C3—64no—
502C22×C3—182no—
503C22×C3—181no—
504C22×C3—244no—
505C22×C3—721no—
506C22×C3#380214yes—
507C22×C3D4×C3⋊D311yescharacteristic
508C2×D3—21no—
509C2×D3—22no—
510C2×D3—64no—
511C2×D3—68no—
512C2×D3—81no—
513C2×D3—98no—
514C2×D3—92no—
515C2×D3—188no—
516C2×D3—244no—
517C2×D3—722no—
518C2×D3—722no—
519C2×D3—728no—
520C2×D3—728no—
521C3×C4—24no—
522C3×C4—24no—
523C3×C4—64no—
524C3×C4—181no—
525C3×C4#380214yes—
526C3×C4D4×C3⋊D311yescharacteristic
527C3×C4C22×D3214yes—
528C3×C4C23×C3⋊D311yescharacteristic
529Dic3—21no—
530Dic3—64no—
531Dic3—98no—
532Dic3—92no—
533Dic3—188no—
534C32—24no—
535C32#577414yescomplemented
536C32#577411yescharacteristic, complemented
537C23—61no—
538C23—181no—
539C23—542no—
540C23—542no—
541Q8—31no—
542Q8—91no—
543Q8—271no—
544D4—31no—
545D4—32no—
546D4—61no—
547D4—61no—
548D4—91no—
549D4—92no—
550D4—181no—
551D4—181no—
552D4—272no—
553D4—271no—
554D4—542no—
555D4—542no—
556D4—542no—
557D4—542no—
558C2×C4—61no—
559C2×C4—181no—
560C2×C4—272no—
561C2×C4—542no—
562C2×C4—542no—
563C2×C4C2×D3×C3⋊D311yescharacteristic
564C8—271no—
565C8—271no—
566C8C2×D3×C3⋊D312yes—
567C2×C3—21no—
568C2×C3—24no—
569C2×C3—24no—
570C2×C3—44no—
571C2×C3—124no—
572C2×C3—128no—
573C2×C3—361no—
574C2×C3—362no—
575C2×C3C2×#380214yes—
576C2×C3C2×D4×C3⋊D311yescharacteristic
577D3—42no—
578D3—41no—
579D3—128no—
580D3—124no—
581D3—182no—
582D3—188no—
583D3—368no—
584C22—61no—
585C22—62no—
586C22—181no—
587C22—182no—
588C22—241no—
589C22—272no—
590C22—721no—
591C22—2162no—
592C22—2162no—
593C22#8902611yescharacteristic
594C4—61no—
595C4—181no—
596C4—272no—
597C4#8902611yescharacteristic
598C4C22×D3×C3⋊D311yescharacteristic
599C3—24no—
600C3#1059472014yescomplemented
601C3#1059624511yescharacteristic, complemented
602C2—21no—
603C2—122no—
604C2—121no—
605C2—362no—
606C2—361no—
607C2—542no—
608C2C2×#8902611yescharacteristic
subgroupG/H
centerC2C2×#89026
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×M16C22
socleC2×C33C22×D4
hypercenterM16D3×C3⋊D3
hypocenterC33#576
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. #11724580C2semidirect
2. #11724579C2semidirect
3. #11723948C2semidirect
4. C3×#64117C2semidirect
5. C32×#908C2semidirect
6. #11723363C2semidirect
7. C2×#88789C2semidirect
8. #11723342C2semidirect
9. #11723324C2semidirect
10. #11723323C2semidirect
11. #11723322C2semidirect
15. C3×#3828C22semidirect
16. C2×C3×C4⋊(C3⋊D3)C22semidirect
17. C32×#235C22semidirect
18. C2×C32×D12C22semidirect
21. #89026C22semidirect
22. #89026C22semidirect
26. #89017C22semidirect
27. #89015C22semidirect
29. C33×M16C22semidirect
30. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
31. #88850C22semidirect
32. C8×C3⋊(C3⋊D3)C22semidirect
33. C3×#3745C22semidirect
34. C3×#3744C22semidirect
35. C32×D24C22semidirect
36. C32×#204C22semidirect
37. #88798C22semidirect
38. #88793C22semidirect
39. #88792C22semidirect
40. #88789C22semidirect
41. #88789C22semidirect
42. #88789C22semidirect
54. #64117D3semidirect
59. C3×#908D3semidirect
112. #3828C2×D3semidirect
114. C2×C4⋊(C3⋊D3)C2×D3semidirect
116. C3×#235C2×D3semidirect
118. C2×C3×D12C2×D3semidirect
132. C32×M16C2×D3semidirect
133. C32×M16C2×D3semidirect
143. #3745C2×D3semidirect
144. #3744C2×D3semidirect
152. C3×D24C2×D3semidirect
154. C3×#204C2×D3semidirect
195. #908C3⋊D3semidirect
301. #235C2×C3⋊D3semidirect
305. C2×D12C2×C3⋊D3semidirect
311. C3×M16D32semidirect
312. C3×M16C2×C3⋊D3semidirect
329. D24C2×C3⋊D3semidirect
331. #204C2×C3⋊D3semidirect
384. C33#576semidirect
496. M16D3×C3⋊D3semidirect
535. C32#5774semidirect
536. C32#5774semidirect
600. C3#10594720semidirect
601. C3#10596245semidirect

Maximal subgroups

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

subgroupindex
#117245802
#117245792
#117239482
C3×#641172
C32×#9082
#117233632
C2×#887892
#117233422
#117233242
#117233232
#117233222
#105962453
#105947203