Finite Group Databasebeta

← #13177991

Subgroups — #13177991

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
1C32×#1010C211yesmaximal, characteristic, complemented
2#11724580C211yesmaximal, characteristic, complemented
3#11723991C211yesmaximal, characteristic, complemented
4C3×#64158C211yesmaximal, characteristic, complemented
5#11723579C212yesmaximal, complemented
6D3×#3755C212yesmaximal, complemented
7#11723556C212yesmaximal, complemented
8#11723544C212yesmaximal, complemented
9#11723381C211yesmaximal, characteristic, complemented
10C2×#88793C211yesmaximal, characteristic, complemented
11#11723350C211yesmaximal, characteristic, complemented
12#10596308—31nomaximal
13#10595455—34nomaximal
14C33×#41C2211yescharacteristic, complemented
15#89076C2211yescharacteristic
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C32×D3×Q8C2212yescomplemented
18C32×#237—22no—
19C32×#237C2212yescomplemented
20C32×#235—21no—
21C32×#235C2211yescharacteristic
22C2×C32×Dic6—21no—
23C2×C32×Dic6C2211yescharacteristic, complemented
24C3⋊D3×#67—22no—
25#89026—21no—
26D3×C4⋊(C3⋊D3)C2212yes—
27#89021C2212yes—
28#89017—21no—
29#88863C2211yescharacteristic, complemented
30#88862C2211yescharacteristic, complemented
31#88861C2211yescharacteristic, complemented
32#88860C2211yescharacteristic, complemented
33C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
34C3×#3755C2212yescomplemented
35C3×#3753C2212yescomplemented
36C3×#3748C2211yescharacteristic, complemented
37#88799C2212yescomplemented
38#88795C2212yescomplemented
39#88793C2212yescomplemented
40#88793C2212yescomplemented
41#88786C2212yescomplemented
42D3×C3⋊(C3⋊C8)C2212yescomplemented
43C32×2−1+4—31no—
44#64366—31no—
45C3×#1010D314yescomplemented
46#64307—34no—
47#64159—31no—
48#64159—34no—
49#64159—32no—
50#64159—64no—
51#64158—32no—
52#64158D311yescharacteristic, complemented
53#64157—31no—
54#64151—32no—
55#64151—31no—
56C2×#3755—31no—
57C2×#3755—32no—
58C3×#949—34no—
59#63946—38no—
60D3×#215—38no—
61#63933—38no—
62#63925—38no—
63#63833—34no—
64C2×#3715—34no—
65#63815—34no—
66C22×C3×C3⋊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×C4⋊(C3⋊D3)—21no—
78C3×C4⋊(C3⋊D3)C2312yes—
79C32×#67—22no—
80C32×#67D412yes—
81C2×C32×Dic3—21no—
82C2×C32×Dic3—22no—
83C32×D12D411yescharacteristic
84C32×C4×D3—22no—
85C32×C4×D3C2312yes—
86C32×Dic6—21no—
87C32×Dic6—22no—
88C32×Dic6D411yescharacteristic
89C32×Dic6C2312yes—
90#6292—22no—
91#6290—22no—
92Dic3×C3⋊D3—22no—
93C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
94C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
95C3×C3⋊(C3⋊C8)C2312yes—
96#5837—94no—
97#5775—91no—
98C32×#41—24no—
99C32×#41—31no—
100C32×#41—32no—
101C32×#41—61no—
102C32×#41—62no—
103C32×#41C2×D314yescomplemented
104C32×#41C2×D311yescharacteristic, complemented
105C2×C32×Q8—32no—
106C2×C32×Q8—31no—
107C2×C32×Q8—61no—
108#3832—61no—
109D4×C3⋊D3—61no—
110D4×C3⋊D3—62no—
111#3828—31no—
112#3828—34no—
113#3828—32no—
114#3828—64no—
115C2×C4⋊(C3⋊D3)—32no—
116C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
117C3×D3×Q8C2×D318yescomplemented
118C3×#237—28no—
119C3×#237C2×D318yescomplemented
120C3×#235—24no—
121C3×#235C2×D314yes—
122C2×C3×D12—34no—
123C2×C3×Dic6—24no—
124C2×C3×Dic6C2×D314yescomplemented
125D3×#67—68no—
126#3802—64no—
127D3×D12—38no—
128#3796—64no—
129#3795—38no—
130#3756—31no—
131#3756—34no—
132#3756—32no—
133#3756—64no—
134#3755—31no—
135#3755—34no—
136#3755—32no—
137#3755—32no—
138#3755—32no—
139#3755—64no—
140#3755C2×D312yescomplemented
141#3754—31no—
142#3754—34no—
143#3754—64no—
144#3753—31no—
145#3753—34no—
146#3753—64no—
147#3753C2×D312yescomplemented
148#3748—32no—
149#3748C2×D311yescharacteristic, complemented
150C2×C3⋊(C3⋊C8)—31no—
151C2×C3⋊(C3⋊C8)—34no—
152C2×C3⋊(C3⋊C8)—32no—
153C2×C3⋊(C3⋊C8)—64no—
154C3×#215—38no—
155C3×#213—38no—
156C3×#208—34no—
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⋊D3—22no—
167C2×C3×C3⋊D3—81no—
168C2×C32×D3C2×D412yes—
169D3×C3⋊D3—82no—
170C33×C4—21no—
171C33×C4C2×D411yescharacteristic
172C33×C4C2411yescharacteristic
173C3⋊(C3⋊Dic3)—21no—
174C32×Dic3—22no—
175C32×Dic3C2×D412yes—
176C3×2−1+4—34no—
177#1010C3⋊D311yescharacteristic, complemented
178#1009—94no—
179#1004—91no—
180C3×AGL(1,ℤ/8)—91no—
181#950—98no—
182#950—94no—
183#950—91no—
184#950—184no—
185#949—34no—
186#949—98no—
187#948—94no—
188#942—94no—
189#942—98no—
190C2×#215—98no—
191C2×#215—94no—
192C2×#214—91no—
193#915—92no—
194D3×SD16—92no—
195#912—92no—
196#911—92no—
197#909—91no—
198#907—91no—
199C22×C3⋊D3—21no—
200C22×C3⋊D3—62no—
201C22×C3×D3—64no—
202C2×D32—68no—
203C32×Q8—24no—
204C32×Q8—31no—
205C32×Q8—32no—
206C32×Q8—62no—
207C32×Q8—61no—
208C32×Q8C22×D314yes—
209C32×Q8C22×D311yescharacteristic
210C32×D4—21no—
211C32×D4—24no—
212C32×D4—24no—
213C32×D4—32no—
214C32×D4—31no—
215C32×D4—44no—
216C32×D4—62no—
217C32×D4C22×D314yes—
218C32×D4C22×D311yescharacteristic
219C2×C32×C4—21no—
220C2×C32×C4—24no—
221C2×C32×C4—24no—
222C2×C32×C4—32no—
223C2×C32×C4—44no—
224C2×C32×C4—62no—
225C2×C32×C4—62no—
226C2×C32×C4—61no—
227C2×C32×C4C22×D314yes—
228C2×C32×C4C22×D311yescharacteristic
229#638—62no—
230#638—61no—
231#638—64no—
232#638—124no—
233C4⋊(C3⋊D3)—21no—
234C4⋊(C3⋊D3)—31no—
235C4⋊(C3⋊D3)—34no—
236C4⋊(C3⋊D3)—64no—
237C4⋊(C3⋊D3)—62no—
238C4⋊(C3⋊D3)C22×D312yes—
239C4×C3⋊D3—62no—
240C4×C3⋊D3—61no—
241C4×C3⋊D3—64no—
242C4×C3⋊D3—124no—
243#634—31no—
244#634—34no—
245#634—64no—
246C3×#67—28no—
247C3×#67#6718yes—
248C2×C3×Dic3—24no—
249C2×C3×Dic3—28no—
250C3×D12—38no—
251C3×D12—64no—
252C3×D12#6714yes—
253C3×C4×D3—28no—
254C3×C4×D3C22×D318yes—
255C3×Dic6—24no—
256C3×Dic6—28no—
257C3×Dic6#6714yes—
258C3×Dic6C22×D318yes—
259#626—68no—
260#625—68no—
261D3×Dic3—68no—
262C3⋊(C3⋊C8)—31no—
263C3⋊(C3⋊C8)—31no—
264C3⋊(C3⋊C8)—34no—
265C3⋊(C3⋊C8)—34no—
266C3⋊(C3⋊C8)—64no—
267C3⋊(C3⋊C8)—64no—
268C3⋊(C3⋊C8)C22×D312yes—
269C3×C3⋊C8—38no—
270#577—271noSylow
271C2×C33—21no—
272C2×C33—41no—
273C2×C33C22×D411yescharacteristic
274C3⋊(C3⋊D3)—41no—
275C3×C3⋊D3—42no—
276C32×D3—22no—
277C3×#41—24no—
278C3×#41—34no—
279C3×#41—38no—
280C3×#41—64no—
281C3×#41—68no—
282C3×#41D3214yescomplemented
283C3×#41C2×C3⋊D311yescharacteristic, complemented
284C2×C3×Q8—38no—
285C2×C3×Q8—34no—
286C2×C3×Q8—64no—
287C2×C3×D4—91no—
288C2×#67—182no—
289#239—184no—
290D3×Q8C2×C3⋊D312yescomplemented
291#237—22no—
292#237—92no—
293#237—181no—
294#237C2×C3⋊D312yescomplemented
295D3×D4—92no—
296D3×D4—184no—
297D3×D4—188no—
298D3×D4—181no—
299#235—21no—
300#235—98no—
301#235—94no—
302#235—91no—
303#235—184no—
304#235C2×C3⋊D311yescharacteristic
305C2×D12—34no—
306C2×D12—98no—
307C2×Dic6—21no—
308C2×Dic6C2×C3⋊D311yescharacteristic, complemented
309C3×SD16—92no—
310C3×D8—92no—
311C3×M16—91no—
312#216—98no—
313#216—94no—
314#216—91no—
315#216—184no—
316#215—38no—
317#215—98no—
318#215—98no—
319#215—98no—
320#215—94no—
321#215—91no—
322#215—184no—
323#214—94no—
324#214—91no—
325#214—92no—
326#214—92no—
327#214—184no—
328#213—38no—
329#213—94no—
330#213—91no—
331#213—184no—
332#208—34no—
333#208—98no—
334C2×C3⋊C8—98no—
335C2×C3⋊C8—94no—
336C2×C3⋊C8—91no—
337C2×C3⋊C8—184no—
338Dic12—92no—
339#204—92no—
340ΓL(1,𝔽25)—92no—
341D3×C8—92no—
342C22×C32—21no—
343C22×C32—24no—
344C22×C32—24no—
345C22×C32—32no—
346C22×C32—44no—
347C22×C32D3×D411yescharacteristic
348C22×C32C2×#6714yes—
349C2×C3⋊D3—22no—
350C2×C3⋊D3—61no—
351C2×C3⋊D3—64no—
352C2×C3⋊D3—81no—
353C2×C3⋊D3—124no—
354C2×C3⋊D3—242no—
355C2×C3×D3—68no—
356C2×C3×D3—244no—
357C2×C3×D3C2×#6718yes—
358D32—248no—
359C32×C4—21no—
360C32×C4—24no—
361C32×C4—24no—
362C32×C4—24no—
363C32×C4—32no—
364C32×C4—44no—
365C32×C4—62no—
366C32×C4D3×D411yescharacteristic
367C32×C4C2×#6714yes—
368C32×C4C23×D314yes—
369C32×C4C23×D311yescharacteristic
370C3⋊Dic3—61no—
371C3⋊Dic3—64no—
372C3⋊Dic3—124no—
373C3×Dic3—28no—
374C3×Dic3C2×#6718yes—
3752−1+4—31no—
376H(5,𝔽2)—271no—
377#137—271no—
378#137—272no—
379AGL(1,ℤ/8)—91no—
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—181no—
388C22×D3—64no—
389C22×D3—188no—
390C22×D3—182no—
391C3×Q8—24no—
392C3×Q8—34no—
393C3×Q8—38no—
394C3×Q8—68no—
395C3×Q8—64no—
396C3×Q8C2×D3214yes—
397C3×Q8C22×C3⋊D311yescharacteristic
398C3×D4—24no—
399C3×D4—21no—
400C3×D4—24no—
401C3×D4—38no—
402C3×D4—34no—
403C3×D4—44no—
404C3×D4—68no—
405C3×D4—92no—
406C3×D4—181no—
407C3×D4C2×D3214yes—
408C3×D4C22×C3⋊D311yescharacteristic
409C2×C3×C4—24no—
410C2×C3×C4—21no—
411C2×C3×C4—24no—
412C2×C3×C4—38no—
413C2×C3×C4—44no—
414C2×C3×C4—68no—
415C2×C3×C4—68no—
416C2×C3×C4—64no—
417C2×C3×C4C2×D3214yes—
418C2×C3×C4C22×C3⋊D311yescharacteristic
419#67—22no—
420#67—188no—
421#67—184no—
422#67—181no—
423#67—182no—
424#67—182no—
425#67—364no—
426#67#63812yes—
427C2×Dic3—21no—
428C2×Dic3—22no—
429C2×Dic3—182no—
430D12—38no—
431D12—64no—
432D12—94no—
433D12—91no—
434D12—184no—
435D12—188no—
436D12#63811yescharacteristic
437C4×D3—22no—
438C4×D3—188no—
439C4×D3—184no—
440C4×D3—181no—
441C4×D3—364no—
442C4×D3C22×C3⋊D312yes—
443Dic6—21no—
444Dic6—22no—
445Dic6—94no—
446Dic6—91no—
447Dic6—184no—
448Dic6#63811yescharacteristic
449Dic6C22×C3⋊D312yes—
450C3×C8—92no—
451C3⋊C8—38no—
452C3⋊C8—94no—
453C3⋊C8—94no—
454C3⋊C8—91no—
455C3⋊C8—91no—
456C3⋊C8—184no—
457C3⋊C8—184no—
458C2×C32—21no—
459C2×C32—24no—
460C2×C32—24no—
461C2×C32—44no—
462C2×C32—41no—
463C2×C32—44no—
464C2×C32—62no—
465C2×C32—84no—
466C2×C32C2×D3×D411yescharacteristic
467C2×C32C22×#6714yes—
468C3⋊D3—42no—
469C3⋊D3—121no—
470C3⋊D3—124no—
471C3⋊D3—244no—
472C3×D3—28no—
473C3×D3—128no—
474#41—31no—
475#41—32no—
476#41—61no—
477#41—62no—
478#41—272no—
479#41—271no—
480#41—541no—
481#41D3×C3⋊D311yescharacteristic, complemented
482C2×Q8—32no—
483C2×Q8—31no—
484C2×Q8—61no—
485C2×D4—91no—
486C2×D4—272no—
487C2×D4—541no—
488C2×D4—542no—
489Q16—272no—
490Q16—271no—
491SD16—92no—
492SD16—272no—
493SD16—272no—
494SD16—272no—
495SD16—271no—
496SD16—271no—
497D8—92no—
498D8—271no—
499M16—91no—
500M16—272no—
501C2×C8—272no—
502C2×C8—271no—
503C22×C3—24no—
504C22×C3—21no—
505C22×C3—24no—
506C22×C3—38no—
507C22×C3—44no—
508C22×C3—182no—
509C22×C3—721no—
510C22×C3D3×#6714yes—
511C22×C3C2×#63811yescharacteristic
512C2×D3—68no—
513C2×D3—184no—
514C2×D3—181no—
515C2×D3—244no—
516C2×D3—364no—
517C2×D3—722no—
518C2×D3—728no—
519C2×D3C2×#63812yes—
520C3×C4—24no—
521C3×C4—21no—
522C3×C4—24no—
523C3×C4—24no—
524C3×C4—38no—
525C3×C4—44no—
526C3×C4—68no—
527C3×C4D3×#6714yes—
528C3×C4C2×#63811yescharacteristic
529C3×C4C22×D3214yes—
530C3×C4C23×C3⋊D311yescharacteristic
531Dic3—22no—
532Dic3—184no—
533Dic3—181no—
534Dic3—364no—
535Dic3C2×#63812yes—
536C32—24no—
537C32#577511yescharacteristic, complemented
538C32#583714yescomplemented
539C23—181no—
540C23—542no—
541Q8—31no—
542Q8—32no—
543Q8—62no—
544Q8—61no—
545Q8—271no—
546Q8C2×D3×C3⋊D311yescharacteristic
547D4—21no—
548D4—32no—
549D4—31no—
550D4—62no—
551D4—92no—
552D4—181no—
553D4—271no—
554D4—542no—
555D4—542no—
556D4—541no—
557D4C2×D3×C3⋊D311yescharacteristic
558C2×C4—21no—
559C2×C4—32no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—61no—
563C2×C4—542no—
564C2×C4—541no—
565C2×C4C2×D3×C3⋊D311yescharacteristic
566C8—92no—
567C8—271no—
568C8—271no—
569C2×C3—24no—
570C2×C3—21no—
571C2×C3—24no—
572C2×C3—44no—
573C2×C3—44no—
574C2×C3—41no—
575C2×C3—68no—
576C2×C3—84no—
577C2×C3—362no—
578C2×C3C2×D3×#6714yes—
579C2×C3C22×#63811yescharacteristic
580D3—22no—
581D3—128no—
582D3—364no—
583D3—361no—
584D3—724no—
585C22—21no—
586C22—32no—
587C22—182no—
588C22—541no—
589C22—721no—
590C22—2162no—
591C22D3×#63811yescharacteristic
592C4—21no—
593C4—32no—
594C4—62no—
595C4—541no—
596C4D3×#63811yescharacteristic
597C4C22×D3×C3⋊D311yescharacteristic
598C3—24no—
599C3#1059545514yescomplemented
600C3#1059630811yescharacteristic, complemented
601C2—21no—
602C2—41no—
603C2—62no—
604C2—362no—
605C2—1081no—
606C2C2×D3×#63811yescharacteristic
subgroupG/H
centerC2C2×D3×#638
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. C32×#1010C2semidirect
2. #11724580C2semidirect
3. #11723991C2semidirect
4. C3×#64158C2semidirect
5. #11723579C2semidirect
6. D3×#3755C2semidirect
7. #11723556C2semidirect
8. #11723544C2semidirect
9. #11723381C2semidirect
10. C2×#88793C2semidirect
11. #11723350C2semidirect
14. C33×#41C22semidirect
17. C32×D3×Q8C22semidirect
19. C32×#237C22semidirect
23. C2×C32×Dic6C22semidirect
29. #88863C22semidirect
30. #88862C22semidirect
31. #88861C22semidirect
32. #88860C22semidirect
33. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
34. C3×#3755C22semidirect
35. C3×#3753C22semidirect
36. C3×#3748C22semidirect
37. #88799C22semidirect
38. #88795C22semidirect
39. #88793C22semidirect
40. #88793C22semidirect
41. #88786C22semidirect
42. D3×C3⋊(C3⋊C8)C22semidirect
45. C3×#1010D3semidirect
52. #64158D3semidirect
103. C32×#41C2×D3semidirect
104. C32×#41C2×D3semidirect
117. C3×D3×Q8C2×D3semidirect
119. C3×#237C2×D3semidirect
124. C2×C3×Dic6C2×D3semidirect
140. #3755C2×D3semidirect
147. #3753C2×D3semidirect
149. #3748C2×D3semidirect
177. #1010C3⋊D3semidirect
282. C3×#41D32semidirect
283. C3×#41C2×C3⋊D3semidirect
290. D3×Q8C2×C3⋊D3semidirect
294. #237C2×C3⋊D3semidirect
308. C2×Dic6C2×C3⋊D3semidirect
386. C33#577semidirect
481. #41D3×C3⋊D3semidirect
537. C32#5775semidirect
538. C32#5837semidirect
599. C3#10595455semidirect
600. C3#10596308semidirect

Maximal subgroups

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

subgroupindex
C32×#10102
#117245802
#117239912
C3×#641582
#117235792
D3×#37552
#117235562
#117235442
#117233812
C2×#887932
#117233502
#105963083
#105954553