Finite Group Databasebeta

← #13177539

Subgroups — #13177539

Order 1728.

1912 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#11724581C211yesmaximal, characteristic, complemented
3C2×#88860C211yesmaximal, characteristic, complemented
4C32×#932C211yesmaximal, characteristic, complemented
5#11723562C212yesmaximal, complemented
6#11723561C212yesmaximal, complemented
7#11723546C212yesmaximal, complemented
8C3⋊D3×#213C212yesmaximal, complemented
9#11723371C211yesmaximal, characteristic, complemented
10#11723363C211yesmaximal, characteristic, complemented
11#11723354C211yesmaximal, characteristic, complemented
12#10596245—31nomaximal
13#10595097—34nomaximal
14C2×C33×D4C2211yescharacteristic, complemented
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
16C2×C3×#638—22no—
17C3×#3830—21no—
18C3×#3830C2212yescomplemented
19C3×D4×C3⋊D3—21no—
20C3×D4×C3⋊D3C2212yescomplemented
21C3×#3828C2211yescharacteristic
22C3×#3828C2211yescharacteristic, complemented
23C32×#235C2211yescharacteristic, complemented
24#89039—22no—
25C3⋊D3×D12C2212yescomplemented
26D3×C4⋊(C3⋊D3)—21no—
27#89022—21no—
28#89018C2212yescomplemented
29#88860C2212yescomplemented
30#88860C2212yescomplemented
31C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
32C32×#214C2212yescomplemented
33C32×#213C2212yescomplemented
34C32×#208C2211yescharacteristic, complemented
35#88798C2211yescharacteristic, complemented
36#88797C2212yescomplemented
37#88793C2211yescharacteristic, complemented
38#88792C2211yescharacteristic, complemented
39#88791C2212yescomplemented
40#88789C2211yescharacteristic, complemented
41#88787C2212yescomplemented
42C3⋊D3×C3⋊C8C2212yescomplemented
43#64366—31no—
44#64361D311yescharacteristic, complemented
45C3×#1004—34no—
46#64308—34no—
47C32×AGL(1,ℤ/8)—31no—
48#64159—31no—
49C2×#3753—31no—
50C2×#3753—34no—
51C2×#3753—64no—
52#64125—32no—
53#64123—32no—
54#64120—32no—
55D8×C3⋊D3—32no—
56#64117—31no—
57#64116—31no—
58C3×#932D314yescomplemented
59#63935—38no—
60#63934—38no—
61#63925—38no—
62D3×#213—38no—
63#63825—34no—
64#63822—34no—
65#63815—34no—
66C23×C33—21no—
67C22×C3⋊(C3⋊D3)—21no—
68C22×C3×C3⋊D3—22no—
69C2×D3×C3⋊D3—22no—
70C33×D4—21no—
71C33×D4C2312yes—
72C2×C33×C4C2311yescharacteristic
73C4⋊(C3⋊(C3⋊D3))—21no—
74C4⋊(C3⋊(C3⋊D3))C2312yes—
75C3×#638—21no—
76C3×#638—22no—
77C3×#638—22no—
78C3×#638D412yes—
79C2×C3×C3⋊Dic3—22no—
80C3×C4⋊(C3⋊D3)D411yescharacteristic
81C3×C4⋊(C3⋊D3)C2311yescharacteristic
82C3×C4×C3⋊D3—21no—
83C3×C4×C3⋊D3C2312yes—
84C3×#634D411yescharacteristic
85C3×#634C2311yescharacteristic
86C32×#67—21no—
87C32×D12C2311yescharacteristic
88C32×C4×D3—21no—
89C32×Dic6C2311yescharacteristic
90#6292—22no—
91#6291—22no—
92#6289—22no—
93C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
94C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
95C32×C3⋊C8C2312yes—
96#5835—91no—
97#5774—94no—
98C32×#41—31no—
99C2×C32×D4—24no—
100C2×C32×D4C2×D314yescomplemented
101C2×C32×D4C2×D311yescharacteristic, complemented
102C2×#638—22no—
103#3832—32no—
104#3830—21no—
105#3830C2×D312yescomplemented
106D4×C3⋊D3—21no—
107D4×C3⋊D3—32no—
108D4×C3⋊D3—62no—
109D4×C3⋊D3C2×D312yescomplemented
110#3828—61no—
111#3828C2×D311yescharacteristic
112#3828C2×D311yescharacteristic, complemented
113C2×C4⋊(C3⋊D3)—31no—
114C2×C4⋊(C3⋊D3)—34no—
115C2×C4⋊(C3⋊D3)—61no—
116C2×C4⋊(C3⋊D3)—64no—
117C2×C3×#67—68no—
118C3×#237—38no—
119C3×#237—64no—
120C3×D3×D4—38no—
121C3×D3×D4—64no—
122C3×#235—34no—
123C3×#235—34no—
124C3×#235C2×D314yescomplemented
125#3811—68no—
126D3×D12—38no—
127D3×D12—64no—
128#3799—38no—
129#3799—64no—
130C32×SD16—32no—
131C32×D8—32no—
132C32×M16—31no—
133#3756—31no—
134#3755—31no—
135#3754—31no—
136#3753—32no—
137#3753—32no—
138#3753—38no—
139#3753—38no—
140#3753—31no—
141#3753—68no—
142#3753—68no—
143C2×C3⋊(C3⋊C8)—31no—
144C2×C3⋊(C3⋊C8)—34no—
145C2×C3⋊(C3⋊C8)—64no—
146#3745—32no—
147#3744—32no—
148#3743—32no—
149C8×C3⋊D3—32no—
150C3×#214C2×D318yescomplemented
151C3×#213C2×D318yescomplemented
152C3×#208C2×D314yescomplemented
153#3718—34no—
154#3717—38no—
155#3715—34no—
156#3715—34no—
157#3714—38no—
158#3713—34no—
159#3711—38no—
160D3×C3⋊C8—38no—
161C22×C33—22no—
162C22×C33—81no—
163C22×C33C2×D411yescharacteristic
164C2×C3⋊(C3⋊D3)—22no—
165C2×C3⋊(C3⋊D3)—81no—
166C2×C3×C3⋊D3—21no—
167C2×C3×C3⋊D3—82no—
168C2×C3×C3⋊D3C2×D412yes—
169C2×C32×D3—21no—
170D3×C3⋊D3—82no—
171C33×C4C2×D411yescharacteristic
172C33×C4C2411yescharacteristic
173C3×C3⋊Dic3—21no—
174C3×C3⋊Dic3C2×D412yes—
175C32×Dic3—21no—
176C3×H(5,𝔽2)—91no—
177#1009—91no—
178#1009—94no—
179#1004—34no—
180C3×AGL(1,ℤ/8)—34no—
181#950—94no—
182#950—91no—
183#950—92no—
184#949—92no—
185#949—91no—
186#948—91no—
187#932—92no—
188#932C3⋊D311yescharacteristic, complemented
189C2×#213—94no—
190C2×#213—91no—
191C2×#213—92no—
192C2×#213—184no—
193#916—98no—
194#914—98no—
195#911—98no—
196D3×D8—98no—
197#908—94no—
198#907—94no—
199C23×C32—21no—
200C23×C32—24no—
201C23×C32—44no—
202C22×C3⋊D3—22no—
203C22×C3⋊D3—61no—
204C22×C3⋊D3—62no—
205C22×C3⋊D3—64no—
206C22×C3⋊D3—124no—
207C22×C3×D3—68no—
208C2×D32—68no—
209C32×Q8—31no—
210C32×D4—21no—
211C32×D4—28no—
212C32×D4—24no—
213C32×D4—31no—
214C32×D4—44no—
215C32×D4—61no—
216C32×D4C22×D318yes—
217C32×D4C22×D312yes—
218C2×C32×C4—24no—
219C2×C32×C4—61no—
220C2×C32×C4C22×D314yes—
221C2×C32×C4C22×D311yescharacteristic
222#638—21no—
223#638—22no—
224#638—22no—
225#638—62no—
226#638#6712yes—
227C2×C3⋊Dic3—22no—
228C4⋊(C3⋊D3)—32no—
229C4⋊(C3⋊D3)—38no—
230C4⋊(C3⋊D3)—68no—
231C4⋊(C3⋊D3)—62no—
232C4⋊(C3⋊D3)—61no—
233C4⋊(C3⋊D3)—64no—
234C4⋊(C3⋊D3)—124no—
235C4⋊(C3⋊D3)#6711yescharacteristic
236C4⋊(C3⋊D3)C22×D311yescharacteristic
237C4×C3⋊D3—21no—
238C4×C3⋊D3—62no—
239C4×C3⋊D3C22×D312yes—
240#634#6711yescharacteristic
241#634C22×D311yescharacteristic
242C3×#67—24no—
243C3×#67—38no—
244C3×#67—68no—
245C3×#67—68no—
246C3×#67—64no—
247C2×C3×Dic3—68no—
248C3×D12—34no—
249C3×D12—34no—
250C3×D12C22×D314yes—
251C3×C4×D3—24no—
252C3×C4×D3—38no—
253C3×C4×D3—64no—
254C3×Dic6—34no—
255C3×Dic6—34no—
256C3×Dic6C22×D314yes—
257#626—68no—
258#626—68no—
259#624—68no—
260C32×C8—32no—
261C3⋊(C3⋊C8)—31no—
262C3⋊(C3⋊C8)—31no—
263C3⋊(C3⋊C8)—34no—
264C3⋊(C3⋊C8)—34no—
265C3⋊(C3⋊C8)—64no—
266C3⋊(C3⋊C8)—64no—
267C3×C3⋊C8C22×D318yes—
268#576—271noSylow
269C2×C33—21no—
270C2×C33—42no—
271C2×C33C22×D411yescharacteristic
272C3⋊(C3⋊D3)—42no—
273C3×C3⋊D3—22no—
274C3×C3⋊D3—41no—
275C32×D3—41no—
276C3×#41—34no—
277C3×#41—91no—
278C3×#41—92no—
279C3×#41—91no—
280C3×#41—181no—
281C2×C3×D4—24no—
282C2×C3×D4—92no—
283C2×C3×D4—181no—
284C2×C3×D4—182no—
285C2×C3×D4D3214yescomplemented
286C2×C3×D4C2×C3⋊D311yescharacteristic, complemented
287C2×#67—68no—
288#239—98no—
289#239—181no—
290#237—38no—
291#237—64no—
292D3×D4—38no—
293D3×D4—64no—
294D3×D4—98no—
295D3×D4—181no—
296D3×D4—182no—
297D3×D4—188no—
298#235—34no—
299#235—34no—
300#235—92no—
301#235—184no—
302#235C2×C3⋊D311yescharacteristic, complemented
303C2×D12—94no—
304C2×D12—91no—
305C2×D12—92no—
306C2×D12—184no—
307C2×D12—184no—
308C3×SD16—38no—
309C3×D8—38no—
310C3×M16—34no—
311#216—94no—
312#216—91no—
313#215—94no—
314#215—91no—
315#215—92no—
316#214—94no—
317#214—91no—
318#214C2×C3⋊D312yescomplemented
319#213—94no—
320#213—98no—
321#213—98no—
322#213—92no—
323#213—92no—
324#213—91no—
325#213—92no—
326#213—188no—
327#213—188no—
328#213C2×C3⋊D312yescomplemented
329#208—92no—
330#208C2×C3⋊D311yescharacteristic, complemented
331C2×C3⋊C8—94no—
332C2×C3⋊C8—91no—
333C2×C3⋊C8—92no—
334C2×C3⋊C8—184no—
335D24—98no—
336#204—98no—
337ΓL(1,𝔽25)—98no—
338D3×C8—98no—
339C22×C32—22no—
340C22×C32—24no—
341C22×C32—28no—
342C22×C32—48no—
343C22×C32—61no—
344C22×C32—81no—
345C22×C32—84no—
346C22×C32—88no—
347C22×C32D3×D414yes—
348C22×C32C2×#6711yescharacteristic
349C2×C3⋊D3—21no—
350C2×C3⋊D3—62no—
351C2×C3⋊D3—68no—
352C2×C3⋊D3—82no—
353C2×C3⋊D3—128no—
354C2×C3⋊D3—242no—
355C2×C3⋊D3—241no—
356C2×C3⋊D3—244no—
357C2×C3⋊D3—248no—
358C2×C3⋊D3C2×#6712yes—
359C2×C3×D3—24no—
360C2×C3×D3—38no—
361C2×C3×D3—64no—
362C2×C3×D3—248no—
363D32—248no—
364C32×C4—24no—
365C32×C4—24no—
366C32×C4—61no—
367C32×C4D3×D414yes—
368C32×C4C2×#6711yescharacteristic
369C32×C4C23×D314yes—
370C32×C4C23×D311yescharacteristic
371C3⋊Dic3—21no—
372C3⋊Dic3C2×#6712yes—
373C3×Dic3—24no—
374C3×Dic3—38no—
375C3×Dic3—64no—
376H(5,𝔽2)—91no—
377H(5,𝔽2)—271no—
378AGL(1,ℤ/8)—31no—
379AGL(1,ℤ/8)—271no—
380AGL(1,ℤ/8)—272no—
381AGL(1,ℤ/8)—272no—
382#135—272no—
383#135—271no—
384C2×D8—272no—
385C2×D8—271no—
386#131—271no—
387C33#57611yescharacteristic, complemented, Sylow
388C23×C3—24no—
389C23×C3—21no—
390C23×C3—44no—
391C23×C3—182no—
392C22×D3—68no—
393C22×D3—182no—
394C22×D3—188no—
395C22×D3—184no—
396C22×D3—181no—
397C22×D3—364no—
398C3×Q8—34no—
399C3×Q8—91no—
400C3×Q8—91no—
401C3×D4—24no—
402C3×D4—21no—
403C3×D4—28no—
404C3×D4—34no—
405C3×D4—44no—
406C3×D4—64no—
407C3×D4—92no—
408C3×D4—91no—
409C3×D4—91no—
410C3×D4—182no—
411C3×D4—182no—
412C3×D4—181no—
413C3×D4C2×D3218yes—
414C3×D4C22×C3⋊D312yes—
415C2×C3×C4—24no—
416C2×C3×C4—64no—
417C2×C3×C4—92no—
418C2×C3×C4—182no—
419C2×C3×C4—181no—
420C2×C3×C4C2×D3214yes—
421C2×C3×C4C22×C3⋊D311yescharacteristic
422#67—21no—
423#67—38no—
424#67—68no—
425#67—68no—
426#67—64no—
427#67—182no—
428#67—188no—
429C2×Dic3—68no—
430D12—34no—
431D12—34no—
432D12—98no—
433D12—92no—
434D12—188no—
435D12—188no—
436D12—182no—
437D12—184no—
438D12—181no—
439D12—364no—
440D12C22×C3⋊D311yescharacteristic
441C4×D3—21no—
442C4×D3—38no—
443C4×D3—64no—
444C4×D3—182no—
445C4×D3—188no—
446Dic6—34no—
447Dic6—34no—
448Dic6C22×C3⋊D311yescharacteristic
449C3×C8—38no—
450C3⋊C8—94no—
451C3⋊C8—94no—
452C3⋊C8—91no—
453C3⋊C8—91no—
454C3⋊C8—184no—
455C3⋊C8—184no—
456C3⋊C8C22×C3⋊D312yes—
457C2×C32—28no—
458C2×C32—24no—
459C2×C32—21no—
460C2×C32—24no—
461C2×C32—42no—
462C2×C32—48no—
463C2×C32—88no—
464C2×C32—121no—
465C2×C32C2×D3×D414yes—
466C2×C32C22×#6711yescharacteristic
467C3⋊D3—22no—
468C3⋊D3—41no—
469C3⋊D3—122no—
470C3⋊D3—128no—
471C3⋊D3—248no—
472C3×D3—44no—
473C3×D3—68no—
474C3×D3—124no—
475#41—31no—
476#41—91no—
477#41—92no—
478#41—91no—
479#41—181no—
480#41—272no—
481#41—541no—
482C2×D4—92no—
483C2×D4—181no—
484C2×D4—182no—
485C2×D4—272no—
486C2×D4—271no—
487C2×D4—541no—
488C2×D4—542no—
489C2×D4D3×C3⋊D311yescharacteristic, complemented
490Q16—271no—
491SD16—32no—
492SD16—272no—
493SD16—271no—
494SD16—271no—
495D8—32no—
496D8—272no—
497D8—271no—
498D8—272no—
499D8—272no—
500M16—31no—
501M16—272no—
502C2×C8—272no—
503C2×C8—271no—
504C22×C3—28no—
505C22×C3—22no—
506C22×C3—24no—
507C22×C3—48no—
508C22×C3—64no—
509C22×C3—84no—
510C22×C3—81no—
511C22×C3—88no—
512C22×C3—92no—
513C22×C3—181no—
514C22×C3—722no—
515C22×C3D3×#6714yes—
516C22×C3D4×C3⋊D311yescharacteristic
517C2×D3—21no—
518C2×D3—38no—
519C2×D3—64no—
520C2×D3—188no—
521C2×D3—182no—
522C2×D3—248no—
523C2×D3—368no—
524C2×D3—722no—
525C2×D3—728no—
526C2×D3—724no—
527C2×D3—721no—
528C2×D3—728no—
529C3×C4—24no—
530C3×C4—24no—
531C3×C4—64no—
532C3×C4—92no—
533C3×C4—181no—
534C3×C4D3×#6714yes—
535C3×C4D4×C3⋊D311yescharacteristic
536C3×C4C22×D3214yes—
537C3×C4C23×C3⋊D311yescharacteristic
538Dic3—21no—
539Dic3—38no—
540Dic3—64no—
541C32—24no—
542C32#577414yescomplemented
543C32#583511yescharacteristic, complemented
544C23—21no—
545C23—182no—
546C23—542no—
547C23—541no—
548Q8—31no—
549Q8—91no—
550Q8—91no—
551D4—21no—
552D4—31no—
553D4—61no—
554D4—92no—
555D4—91no—
556D4—91no—
557D4—182no—
558D4—182no—
559D4—181no—
560D4—272no—
561D4—542no—
562D4—542no—
563D4—541no—
564D4C2×D3×C3⋊D312yes—
565C2×C4—61no—
566C2×C4—92no—
567C2×C4—182no—
568C2×C4—181no—
569C2×C4—542no—
570C2×C4C2×D3×C3⋊D311yescharacteristic
571C8—32no—
572C8—271no—
573C8—271no—
574C2×C3—28no—
575C2×C3—24no—
576C2×C3—21no—
577C2×C3—24no—
578C2×C3—48no—
579C2×C3—42no—
580C2×C3—88no—
581C2×C3—124no—
582C2×C3—182no—
583C2×C3—361no—
584C2×C3C2×D3×#6714yes—
585C2×C3C2×D4×C3⋊D311yescharacteristic
586D3—41no—
587D3—68no—
588D3—124no—
589D3—368no—
590D3—362no—
591D3—728no—
592C22—22no—
593C22—61no—
594C22—81no—
595C22—92no—
596C22—181no—
597C22—542no—
598C22—722no—
599C22—2161no—
600C22—2162no—
601C22C3⋊D3×#6711yescharacteristic
602C4—61no—
603C4—92no—
604C4—181no—
605C4C3⋊D3×#6711yescharacteristic
606C4C22×D3×C3⋊D311yescharacteristic
607C3—24no—
608C3#1059509714yescomplemented
609C3#1059624511yescharacteristic, complemented
610C2—21no—
611C2—42no—
612C2—121no—
613C2—182no—
614C2—361no—
615C2—1082no—
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. #11724581C2semidirect
3. C2×#88860C2semidirect
4. C32×#932C2semidirect
5. #11723562C2semidirect
6. #11723561C2semidirect
7. #11723546C2semidirect
8. C3⋊D3×#213C2semidirect
9. #11723371C2semidirect
10. #11723363C2semidirect
11. #11723354C2semidirect
14. C2×C33×D4C22semidirect
15. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
18. C3×#3830C22semidirect
20. C3×D4×C3⋊D3C22semidirect
22. C3×#3828C22semidirect
23. C32×#235C22semidirect
25. C3⋊D3×D12C22semidirect
28. #89018C22semidirect
29. #88860C22semidirect
30. #88860C22semidirect
31. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
32. C32×#214C22semidirect
33. C32×#213C22semidirect
34. C32×#208C22semidirect
35. #88798C22semidirect
36. #88797C22semidirect
37. #88793C22semidirect
38. #88792C22semidirect
39. #88791C22semidirect
40. #88789C22semidirect
41. #88787C22semidirect
42. C3⋊D3×C3⋊C8C22semidirect
44. #64361D3semidirect
58. C3×#932D3semidirect
100. C2×C32×D4C2×D3semidirect
101. C2×C32×D4C2×D3semidirect
105. #3830C2×D3semidirect
109. D4×C3⋊D3C2×D3semidirect
112. #3828C2×D3semidirect
124. C3×#235C2×D3semidirect
150. C3×#214C2×D3semidirect
151. C3×#213C2×D3semidirect
152. C3×#208C2×D3semidirect
188. #932C3⋊D3semidirect
285. C2×C3×D4D32semidirect
286. C2×C3×D4C2×C3⋊D3semidirect
302. #235C2×C3⋊D3semidirect
318. #214C2×C3⋊D3semidirect
328. #213C2×C3⋊D3semidirect
330. #208C2×C3⋊D3semidirect
387. C33#576semidirect
489. C2×D4D3×C3⋊D3semidirect
542. C32#5774semidirect
543. C32#5835semidirect
608. C3#10595097semidirect
609. C3#10596245semidirect

Maximal subgroups

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

subgroupindex
C3×#643612
#117245812
C2×#888602
C32×#9322
#117235622
#117235612
#117235462
C3⋊D3×#2132
#117233712
#117233632
#117233542
#105962453
#105950973