Finite Group Databasebeta

← #13177984

Subgroups — #13177984

Order 1728.

1796 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64367C211yesmaximal, characteristic, complemented
2#11724579C211yesmaximal, characteristic, complemented
3#11723991C211yesmaximal, characteristic, complemented
4C32×#949C211yesmaximal, characteristic, complemented
5#11723584C212yesmaximal, complemented
6C3⋊D3×#215C212yesmaximal, complemented
7#11723559C212yesmaximal, complemented
8#11723546C212yesmaximal, complemented
9#11723387C211yesmaximal, characteristic, complemented
10C2×#88792C211yesmaximal, characteristic, complemented
11#11723354C211yesmaximal, characteristic, complemented
12#10596246—31nomaximal
13#10595455—34nomaximal
14C33×#41C2211yescharacteristic, complemented
15#89076C2211yescharacteristic
16C3×Q8×C3⋊D3C2212yescomplemented
17C3×#3830—22no—
18C3×#3830C2212yescomplemented
19C3×#3828—21no—
20C3×#3828C2211yescharacteristic
21C2×C3×#634—21no—
22C2×C3×#634C2211yescharacteristic, complemented
23C2×C32×D12C2211yescharacteristic
24D3×#638—22no—
25#89026—21no—
26C3⋊D3×D12C2212yes—
27#89015—21no—
28#89014C2212yes—
29#88863C2211yescharacteristic, complemented
30#88862C2211yescharacteristic, complemented
31#88861C2211yescharacteristic, complemented
32#88860C2211yescharacteristic, complemented
33C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
34C32×#215C2212yescomplemented
35C32×#213C2212yescomplemented
36C32×#208C2211yescharacteristic, complemented
37#88800C2212yescomplemented
38#88797C2212yescomplemented
39#88792C2212yescomplemented
40#88792C2212yescomplemented
41#88787C2212yescomplemented
42C3⋊D3×C3⋊C8C2212yescomplemented
43#64367D311yescharacteristic, complemented
44#64361—31no—
45C3×#1010—34no—
46#64307—34no—
47C32×AGL(1,ℤ/8)—31no—
48#64159—31no—
49#64159—34no—
50#64159—64no—
51C2×#3754—31no—
52#64124—32no—
53SD16×C3⋊D3—32no—
54#64121—32no—
55#64120—32no—
56#64118—31no—
57#64116—31no—
58C3×#949D314yescomplemented
59#63946—38no—
60D3×#215—38no—
61#63933—38no—
62#63925—38no—
63#63833—34no—
64C2×#3715—34no—
65#63815—34no—
66C22×C32×D3—21no—
67C2×D3×C3⋊D3—22no—
68C33×Q8C2311yescharacteristic
69C33×D4—21no—
70C33×D4C2311yescharacteristic
71C2×C33×C4—21no—
72C2×C33×C4C2311yescharacteristic
73#6312—21no—
74C4⋊(C3⋊(C3⋊D3))C2311yescharacteristic
75C4×C3⋊(C3⋊D3)—21no—
76#6308C2311yescharacteristic
77C3×#638—22no—
78C3×#638D412yes—
79C2×C3×C3⋊Dic3—21no—
80C2×C3×C3⋊Dic3—22no—
81C3×C4⋊(C3⋊D3)D411yescharacteristic
82C3×C4×C3⋊D3—22no—
83C3×C4×C3⋊D3C2312yes—
84C3×#634—21no—
85C3×#634—22no—
86C3×#634D411yescharacteristic
87C3×#634C2312yes—
88C32×D12—21no—
89C32×D12C2312yes—
90#6291—22no—
91#6290—22no—
92D3×C3⋊Dic3—22no—
93C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
94C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
95C32×C3⋊C8C2312yes—
96#5837—91no—
97#5775—94no—
98C32×#41—24no—
99C32×#41C2×D314yescomplemented
100C32×#41C2×D311yescharacteristic, complemented
101C2×C32×D4—31no—
102C2×#638—62no—
103Q8×C3⋊D3C2×D312yescomplemented
104#3830—22no—
105#3830—32no—
106#3830—61no—
107#3830C2×D312yescomplemented
108D4×C3⋊D3—32no—
109D4×C3⋊D3—61no—
110#3828—21no—
111#3828—31no—
112#3828—34no—
113#3828—64no—
114#3828C2×D311yescharacteristic
115C2×#634—21no—
116C2×#634C2×D311yescharacteristic, complemented
117C3×D3×Q8—38no—
118C3×#237—38no—
119C3×#237—68no—
120C3×#235—34no—
121C3×#235—64no—
122C2×C3×D12C2×D314yes—
123C2×C3×Dic6—34no—
124C2×C3×Dic6—64no—
125D3×#67—68no—
126#3802—64no—
127D3×D12—38no—
128#3796—64no—
129#3795—38no—
130C32×SD16—32no—
131C32×D8—32no—
132C32×M16—31no—
133#3756—31no—
134#3756—34no—
135#3756—64no—
136#3755—31no—
137#3755—34no—
138#3755—64no—
139#3754—31no—
140#3754—34no—
141#3754—32no—
142#3754—32no—
143#3754—64no—
144#3753—31no—
145#3753—34no—
146#3753—64no—
147C2×C3⋊(C3⋊C8)—31no—
148C2×C3⋊(C3⋊C8)—34no—
149C2×C3⋊(C3⋊C8)—64no—
150#3746—32no—
151#3744—32no—
152#3743—32no—
153C8×C3⋊D3—32no—
154C3×#215C2×D318yescomplemented
155C3×#213C2×D318yescomplemented
156C3×#208C2×D314yescomplemented
157#3719—38no—
158#3717—38no—
159#3715—38no—
160#3715—38no—
161#3711—38no—
162D3×C3⋊C8—38no—
163C22×C33—21no—
164C22×C33C2×D411yescharacteristic
165C2×C3⋊(C3⋊D3)—21no—
166C2×C3×C3⋊D3C2×D412yes—
167C2×C32×D3—22no—
168C2×C32×D3—81no—
169D3×C3⋊D3—82no—
170C33×C4—21no—
171C33×C4C2×D411yescharacteristic
172C33×C4C2411yescharacteristic
173C3⋊(C3⋊Dic3)—21no—
174C3×C3⋊Dic3—22no—
175C3×C3⋊Dic3C2×D412yes—
176C3×2−1+4—91no—
177#1010—34no—
178#1009—91no—
179#1004—94no—
180C3×AGL(1,ℤ/8)—34no—
181#950—94no—
182#950—91no—
183#950—92no—
184#950—184no—
185#949—92no—
186#949C3⋊D311yescharacteristic, complemented
187#948—91no—
188#942—92no—
189#942—91no—
190C2×#215—91no—
191C2×#215—92no—
192C2×#214—94no—
193#915—98no—
194D3×SD16—98no—
195#912—98no—
196#911—98no—
197#909—94no—
198#907—94no—
199C23×C32—61no—
200C22×C3⋊D3—62no—
201C22×C3×D3—24no—
202C2×D32—68no—
203C32×Q8—24no—
204C32×Q8C22×D314yes—
205C32×Q8C22×D311yescharacteristic
206C32×D4—21no—
207C32×D4—24no—
208C32×D4—24no—
209C32×D4—32no—
210C32×D4—44no—
211C32×D4—61no—
212C32×D4C22×D314yes—
213C32×D4C22×D311yescharacteristic
214C2×C32×C4—21no—
215C2×C32×C4—24no—
216C2×C32×C4—24no—
217C2×C32×C4—44no—
218C2×C32×C4C22×D314yes—
219C2×C32×C4C22×D311yescharacteristic
220#638—22no—
221#638—61no—
222#638—62no—
223#638—62no—
224#638—64no—
225#638—124no—
226#638#6712yes—
227C2×C3⋊Dic3—21no—
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—22no—
235C4×C3⋊D3—61no—
236C4×C3⋊D3—64no—
237C4×C3⋊D3—124no—
238C4×C3⋊D3C22×D312yes—
239#634—21no—
240#634—22no—
241#634—31no—
242#634—34no—
243#634—64no—
244#634#6711yescharacteristic
245#634C22×D312yes—
246C3×#67—38no—
247C3×#67—68no—
248C2×C3×Dic3—68no—
249C2×C3×Dic3—64no—
250C3×D12—24no—
251C3×D12—34no—
252C3×D12C22×D318yes—
253C3×C4×D3—38no—
254C3×C4×D3—68no—
255C3×Dic6—34no—
256C3×Dic6—38no—
257C3×Dic6—68no—
258C3×Dic6—64no—
259#626—68no—
260#625—68no—
261D3×Dic3—68no—
262C32×C8—32no—
263C3⋊(C3⋊C8)—31no—
264C3⋊(C3⋊C8)—31no—
265C3⋊(C3⋊C8)—34no—
266C3⋊(C3⋊C8)—34no—
267C3⋊(C3⋊C8)—64no—
268C3⋊(C3⋊C8)—64no—
269C3×C3⋊C8C22×D318yes—
270#577—271noSylow
271C2×C33—21no—
272C2×C33—41no—
273C2×C33C22×D411yescharacteristic
274C3⋊(C3⋊D3)—41no—
275C3×C3⋊D3—22no—
276C32×D3—42no—
277C3×#41—24no—
278C3×#41—91no—
279C3×#41—92no—
280C3×#41—181no—
281C3×#41—182no—
282C3×#41D3214yescomplemented
283C3×#41C2×C3⋊D311yescharacteristic, complemented
284C2×C3×Q8—92no—
285C2×C3×Q8—91no—
286C2×C3×Q8—181no—
287C2×C3×D4—34no—
288C2×#67—188no—
289#239—181no—
290D3×Q8—38no—
291#237—38no—
292#237—68no—
293#237—98no—
294#237—184no—
295D3×D4—98no—
296D3×D4—181no—
297D3×D4—184no—
298D3×D4—182no—
299#235—34no—
300#235—64no—
301#235—94no—
302#235—91no—
303#235—92no—
304#235—184no—
305C2×D12—92no—
306C2×D12C2×C3⋊D311yescharacteristic
307C2×Dic6—34no—
308C2×Dic6—64no—
309C3×SD16—38no—
310C3×D8—38no—
311C3×M16—34no—
312#216—94no—
313#216—91no—
314#216—92no—
315#216—184no—
316#215—94no—
317#215—91no—
318#215—92no—
319#215—92no—
320#215—92no—
321#215—184no—
322#215C2×C3⋊D312yescomplemented
323#214—98no—
324#214—98no—
325#214—94no—
326#214—91no—
327#214—184no—
328#213—94no—
329#213—91no—
330#213—184no—
331#213C2×C3⋊D312yescomplemented
332#208—92no—
333#208C2×C3⋊D311yescharacteristic, complemented
334C2×C3⋊C8—94no—
335C2×C3⋊C8—91no—
336C2×C3⋊C8—92no—
337C2×C3⋊C8—184no—
338Dic12—98no—
339#204—98no—
340ΓL(1,𝔽25)—98no—
341D3×C8—98no—
342C22×C32—21no—
343C22×C32—24no—
344C22×C32—24no—
345C22×C32—44no—
346C22×C32—62no—
347C22×C32—241no—
348C22×C32D3×D414yes—
349C22×C32C2×#6711yescharacteristic
350C2×C3⋊D3—61no—
351C2×C3⋊D3—64no—
352C2×C3⋊D3—124no—
353C2×C3⋊D3—242no—
354C2×C3⋊D3C2×#6712yes—
355C2×C3×D3—28no—
356C2×C3×D3—38no—
357C2×C3×D3—84no—
358D32—248no—
359C32×C4—21no—
360C32×C4—24no—
361C32×C4—24no—
362C32×C4—24no—
363C32×C4—44no—
364C32×C4D3×D414yes—
365C32×C4C2×#6711yescharacteristic
366C32×C4C23×D314yes—
367C32×C4C23×D311yescharacteristic
368C3⋊Dic3—22no—
369C3⋊Dic3—61no—
370C3⋊Dic3—64no—
371C3⋊Dic3—124no—
372C3⋊Dic3C2×#6712yes—
373C3×Dic3—38no—
374C3×Dic3—68no—
3752−1+4—91no—
376H(5,𝔽2)—271no—
377#137—271no—
378#137—272no—
379AGL(1,ℤ/8)—31no—
380AGL(1,ℤ/8)—272no—
381#135—272no—
382#135—271no—
383C2×SD16—272no—
384C2×SD16—271no—
385#131—271no—
386C33#57711yescharacteristic, complemented, Sylow
387C23×C3—64no—
388C22×D3—21no—
389C22×D3—182no—
390C22×D3—188no—
391C3×Q8—24no—
392C3×Q8—91no—
393C3×Q8—92no—
394C3×Q8—182no—
395C3×Q8—181no—
396C3×Q8C2×D3214yes—
397C3×Q8C22×C3⋊D311yescharacteristic
398C3×D4—24no—
399C3×D4—21no—
400C3×D4—24no—
401C3×D4—38no—
402C3×D4—44no—
403C3×D4—64no—
404C3×D4—92no—
405C3×D4—91no—
406C3×D4—182no—
407C3×D4C2×D3214yes—
408C3×D4C22×C3⋊D311yescharacteristic
409C2×C3×C4—24no—
410C2×C3×C4—21no—
411C2×C3×C4—24no—
412C2×C3×C4—44no—
413C2×C3×C4—92no—
414C2×C3×C4—182no—
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—188no—
422#67—182no—
423#67—188no—
424#67—184no—
425#67—181no—
426#67—364no—
427C2×Dic3—68no—
428C2×Dic3—64no—
429C2×Dic3—188no—
430D12—21no—
431D12—34no—
432D12—94no—
433D12—91no—
434D12—184no—
435D12—182no—
436D12C22×C3⋊D312yes—
437C4×D3—38no—
438C4×D3—68no—
439C4×D3—182no—
440C4×D3—184no—
441C4×D3—181no—
442C4×D3—364no—
443Dic6—34no—
444Dic6—38no—
445Dic6—68no—
446Dic6—64no—
447Dic6—94no—
448Dic6—91no—
449Dic6—184no—
450C3×C8—38no—
451C3⋊C8—94no—
452C3⋊C8—94no—
453C3⋊C8—91no—
454C3⋊C8—91no—
455C3⋊C8—184no—
456C3⋊C8—184no—
457C3⋊C8C22×C3⋊D312yes—
458C2×C32—21no—
459C2×C32—24no—
460C2×C32—24no—
461C2×C32—44no—
462C2×C32—41no—
463C2×C32—44no—
464C2×C32—84no—
465C2×C32—122no—
466C2×C32C2×D3×D414yes—
467C2×C32C22×#6711yescharacteristic
468C3⋊D3—22no—
469C3⋊D3—121no—
470C3⋊D3—124no—
471C3⋊D3—244no—
472C3×D3—48no—
473C3×D3—68no—
474#41—91no—
475#41—92no—
476#41—181no—
477#41—182no—
478#41—272no—
479#41—271no—
480#41—541no—
481#41D3×C3⋊D311yescharacteristic, complemented
482C2×Q8—92no—
483C2×Q8—91no—
484C2×Q8—181no—
485C2×D4—31no—
486C2×D4—272no—
487C2×D4—541no—
488C2×D4—542no—
489Q16—272no—
490Q16—271no—
491SD16—32no—
492SD16—272no—
493SD16—272no—
494SD16—272no—
495SD16—271no—
496SD16—271no—
497D8—32no—
498D8—271no—
499M16—31no—
500M16—272no—
501C2×C8—272no—
502C2×C8—271no—
503C22×C3—24no—
504C22×C3—21no—
505C22×C3—24no—
506C22×C3—44no—
507C22×C3—68no—
508C22×C3—92no—
509C22×C3—244no—
510C22×C3D3×#6714yes—
511C22×C3D4×C3⋊D311yescharacteristic
512C2×D3—22no—
513C2×D3—38no—
514C2×D3—81no—
515C2×D3—184no—
516C2×D3—181no—
517C2×D3—364no—
518C2×D3—722no—
519C2×D3—728no—
520C3×C4—24no—
521C3×C4—21no—
522C3×C4—24no—
523C3×C4—24no—
524C3×C4—44no—
525C3×C4—92no—
526C3×C4—182no—
527C3×C4D3×#6714yes—
528C3×C4D4×C3⋊D311yescharacteristic
529C3×C4C22×D3214yes—
530C3×C4C23×C3⋊D311yescharacteristic
531Dic3—38no—
532Dic3—68no—
533Dic3—184no—
534Dic3—181no—
535Dic3—364no—
536C32—24no—
537C32#577514yescomplemented
538C32#583711yescharacteristic, complemented
539C23—61no—
540C23—542no—
541Q8—91no—
542Q8—92no—
543Q8—182no—
544Q8—181no—
545Q8—271no—
546Q8C2×D3×C3⋊D311yescharacteristic
547D4—21no—
548D4—32no—
549D4—61no—
550D4—92no—
551D4—91no—
552D4—182no—
553D4—271no—
554D4—542no—
555D4—542no—
556D4—541no—
557D4C2×D3×C3⋊D311yescharacteristic
558C2×C4—21no—
559C2×C4—92no—
560C2×C4—182no—
561C2×C4—182no—
562C2×C4—181no—
563C2×C4—542no—
564C2×C4—541no—
565C2×C4C2×D3×C3⋊D311yescharacteristic
566C8—32no—
567C8—271no—
568C8—271no—
569C2×C3—24no—
570C2×C3—21no—
571C2×C3—24no—
572C2×C3—44no—
573C2×C3—44no—
574C2×C3—41no—
575C2×C3—84no—
576C2×C3—128no—
577C2×C3—182no—
578C2×C3C2×D3×#6714yes—
579C2×C3C2×D4×C3⋊D311yescharacteristic
580D3—42no—
581D3—68no—
582D3—364no—
583D3—361no—
584D3—724no—
585C22—21no—
586C22—62no—
587C22—92no—
588C22—241no—
589C22—541no—
590C22—2162no—
591C22C3⋊D3×#6711yescharacteristic
592C4—21no—
593C4—92no—
594C4—182no—
595C4—541no—
596C4C3⋊D3×#6711yescharacteristic
597C4C22×D3×C3⋊D311yescharacteristic
598C3—24no—
599C3#1059545514yescomplemented
600C3#1059624611yescharacteristic, complemented
601C2—21no—
602C2—41no—
603C2—122no—
604C2—182no—
605C2—1081no—
606C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC2C2×C3⋊D3×#67
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33C22×D4
hypercenter#41D3×C3⋊D3
hypocenterC33#577
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. C3×#64367C2semidirect
2. #11724579C2semidirect
3. #11723991C2semidirect
4. C32×#949C2semidirect
5. #11723584C2semidirect
6. C3⋊D3×#215C2semidirect
7. #11723559C2semidirect
8. #11723546C2semidirect
9. #11723387C2semidirect
10. C2×#88792C2semidirect
11. #11723354C2semidirect
14. C33×#41C22semidirect
16. C3×Q8×C3⋊D3C22semidirect
18. C3×#3830C22semidirect
22. C2×C3×#634C22semidirect
29. #88863C22semidirect
30. #88862C22semidirect
31. #88861C22semidirect
32. #88860C22semidirect
33. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
34. C32×#215C22semidirect
35. C32×#213C22semidirect
36. C32×#208C22semidirect
37. #88800C22semidirect
38. #88797C22semidirect
39. #88792C22semidirect
40. #88792C22semidirect
41. #88787C22semidirect
42. C3⋊D3×C3⋊C8C22semidirect
43. #64367D3semidirect
58. C3×#949D3semidirect
99. C32×#41C2×D3semidirect
100. C32×#41C2×D3semidirect
103. Q8×C3⋊D3C2×D3semidirect
107. #3830C2×D3semidirect
116. C2×#634C2×D3semidirect
154. C3×#215C2×D3semidirect
155. C3×#213C2×D3semidirect
156. C3×#208C2×D3semidirect
186. #949C3⋊D3semidirect
282. C3×#41D32semidirect
283. C3×#41C2×C3⋊D3semidirect
322. #215C2×C3⋊D3semidirect
331. #213C2×C3⋊D3semidirect
333. #208C2×C3⋊D3semidirect
386. C33#577semidirect
481. #41D3×C3⋊D3semidirect
537. C32#5775semidirect
538. C32#5837semidirect
599. C3#10595455semidirect
600. C3#10596246semidirect

Maximal subgroups

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

subgroupindex
C3×#643672
#117245792
#117239912
C32×#9492
#117235842
C3⋊D3×#2152
#117235592
#117235462
#117233872
C2×#887922
#117233542
#105962463
#105954553