Finite Group Databasebeta

← #13177990

Subgroups — #13177990

Order 1728.

1856 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#1009C211yesmaximal, characteristic, complemented
2#11724580C211yesmaximal, characteristic, complemented
3#11723991C211yesmaximal, characteristic, complemented
4C3×#64158C211yesmaximal, characteristic, complemented
5#11723580C212yesmaximal, complemented
6#11723568C212yesmaximal, complemented
7#11723555C212yesmaximal, complemented
8D3×#3753C212yesmaximal, complemented
9#11723378C211yesmaximal, characteristic, complemented
10C2×#88789C211yesmaximal, characteristic, complemented
11#11723350C211yesmaximal, characteristic, complemented
12#10596306—31nomaximal
13#10595454—34nomaximal
14C33×#41C2211yescharacteristic, complemented
15#89076C2211yescharacteristic, complemented
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
17C32×#239C2212yescomplemented
18C32×D3×D4—22no—
19C32×D3×D4C2212yescomplemented
20C32×#235—21no—
21C32×#235C2211yescharacteristic
22C2×C32×D12—21no—
23C2×C32×D12C2211yescharacteristic, complemented
24C3⋊D3×#67—22no—
25#89026—21no—
26D3×C4⋊(C3⋊D3)C2212yescomplemented
27#89021C2212yescomplemented
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#88794C2212yescomplemented
38#88790C2212yescomplemented
39#88789C2212yescomplemented
40#88789C2212yescomplemented
41#88786C2212yescomplemented
42D3×C3⋊(C3⋊C8)C2212yescomplemented
43C32×H(5,𝔽2)—31no—
44#64366—31no—
45C3×#1009D314yescomplemented
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#64141—32no—
55#64141—31no—
56C2×#3753—31no—
57C2×#3753—32no—
58C3×#949—34no—
59#63947—38no—
60#63939—38no—
61#63932—38no—
62D3×#213—38no—
63#63830—34no—
64C2×#3713—34no—
65#63815—34no—
66C22×C3×C3⋊D3—21no—
67C22×C32×D3—21no—
68C22×C32×D3—22no—
69C2×D3×C3⋊D3—22no—
70C33×Q8C2311yescharacteristic
71C33×D4—21no—
72C33×D4C2311yescharacteristic
73C2×C33×C4—21no—
74C2×C33×C4C2311yescharacteristic
75#6312—21no—
76C4⋊(C3⋊(C3⋊D3))C2311yescharacteristic
77C4×C3⋊(C3⋊D3)—21no—
78#6308C2311yescharacteristic
79C3×C4⋊(C3⋊D3)—21no—
80C3×C4⋊(C3⋊D3)C2312yes—
81C32×#67—22no—
82C32×#67D412yes—
83C32×D12—22no—
84C32×D12—21no—
85C32×D12D411yescharacteristic
86C32×D12C2312yes—
87C32×C4×D3—22no—
88C32×C4×D3C2312yes—
89C32×Dic6D411yescharacteristic
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#5835—94no—
97#5774—91no—
98C32×#41—24no—
99C32×#41—31no—
100C32×#41—32no—
101C32×#41—61no—
102C32×#41C2×D314yescomplemented
103C32×#41C2×D311yescharacteristic, complemented
104C2×C32×D4—32no—
105C2×C32×D4—31no—
106C2×C32×D4—61no—
107C2×C32×D4—62no—
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, complemented
117C3×#239C2×D318yescomplemented
118C3×D3×D4—28no—
119C3×D3×D4C2×D318yescomplemented
120C3×#235—24no—
121C3×#235C2×D314yes—
122C2×C3×D12—24no—
123C2×C3×D12—34no—
124C2×C3×D12C2×D314yescomplemented
125D3×#67—68no—
126#3802—64no—
127D3×D12—38no—
128#3796—64no—
129#3795—38no—
130#3756—31no—
131#3756—34no—
132#3756—64no—
133#3755—31no—
134#3755—34no—
135#3755—64no—
136#3755C2×D312yescomplemented
137#3754—31no—
138#3754—34no—
139#3754—32no—
140#3754—64no—
141#3753—31no—
142#3753—34no—
143#3753—32no—
144#3753—32no—
145#3753—32no—
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#3716—38no—
158#3714—38no—
159#3713—38no—
160#3713—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×D3—22no—
169C2×C32×D3—82no—
170C2×C32×D3—81no—
171C2×C32×D3C2×D412yes—
172D3×C3⋊D3—82no—
173C33×C4—21no—
174C33×C4C2×D411yescharacteristic
175C33×C4C2411yescharacteristic
176C3⋊(C3⋊Dic3)—21no—
177C32×Dic3C2×D412yes—
178C3×H(5,𝔽2)—34no—
179#1009—94no—
180#1009C3⋊D311yescharacteristic, complemented
181#1004—91no—
182C3×AGL(1,ℤ/8)—91no—
183#950—98no—
184#950—94no—
185#950—91no—
186#950—184no—
187#949—34no—
188#949—98no—
189#948—94no—
190#932—94no—
191#932—98no—
192C2×#213—98no—
193C2×#213—94no—
194C2×#213—91no—
195#916—92no—
196#914—92no—
197#911—92no—
198D3×D8—92no—
199#908—91no—
200#907—91no—
201C23×C32—62no—
202C23×C32—61no—
203C22×C3⋊D3—21no—
204C22×C3⋊D3—62no—
205C22×C3×D3—24no—
206C22×C3×D3—28no—
207C22×C3×D3—64no—
208C2×D32—68no—
209C32×Q8—24no—
210C32×Q8—31no—
211C32×Q8C22×D314yes—
212C32×Q8C22×D311yescharacteristic
213C32×D4—21no—
214C32×D4—24no—
215C32×D4—24no—
216C32×D4—32no—
217C32×D4—31no—
218C32×D4—32no—
219C32×D4—44no—
220C32×D4—62no—
221C32×D4—62no—
222C32×D4—61no—
223C32×D4C22×D314yes—
224C32×D4C22×D311yescharacteristic
225C2×C32×C4—21no—
226C2×C32×C4—24no—
227C2×C32×C4—24no—
228C2×C32×C4—32no—
229C2×C32×C4—44no—
230C2×C32×C4—62no—
231C2×C32×C4C22×D314yes—
232C2×C32×C4C22×D311yescharacteristic
233#638—62no—
234#638—61no—
235#638—64no—
236#638—124no—
237C4⋊(C3⋊D3)—21no—
238C4⋊(C3⋊D3)—31no—
239C4⋊(C3⋊D3)—34no—
240C4⋊(C3⋊D3)—64no—
241C4⋊(C3⋊D3)—62no—
242C4⋊(C3⋊D3)C22×D312yes—
243C4×C3⋊D3—62no—
244C4×C3⋊D3—61no—
245C4×C3⋊D3—64no—
246C4×C3⋊D3—124no—
247#634—31no—
248#634—34no—
249#634—64no—
250C3×#67—28no—
251C3×#67#6718yes—
252C3×D12—28no—
253C3×D12—24no—
254C3×D12—38no—
255C3×D12—64no—
256C3×D12#6714yes—
257C3×D12C22×D318yes—
258C3×C4×D3—28no—
259C3×C4×D3C22×D318yes—
260C3×Dic6#6714yes—
261#626—68no—
262#625—68no—
263D3×Dic3—68no—
264C3⋊(C3⋊C8)—31no—
265C3⋊(C3⋊C8)—31no—
266C3⋊(C3⋊C8)—34no—
267C3⋊(C3⋊C8)—34no—
268C3⋊(C3⋊C8)—64no—
269C3⋊(C3⋊C8)—64no—
270C3⋊(C3⋊C8)C22×D312yes—
271C3×C3⋊C8—38no—
272#576—271noSylow
273C2×C33—21no—
274C2×C33—41no—
275C2×C33C22×D411yescharacteristic
276C3⋊(C3⋊D3)—41no—
277C3×C3⋊D3—42no—
278C32×D3—22no—
279C32×D3—42no—
280C3×#41—24no—
281C3×#41—34no—
282C3×#41—38no—
283C3×#41—64no—
284C3×#41D3214yescomplemented
285C3×#41C2×C3⋊D311yescharacteristic, complemented
286C2×C3×D4—38no—
287C2×C3×D4—34no—
288C2×C3×D4—64no—
289C2×C3×D4—68no—
290C2×C3×D4—91no—
291C2×#67—182no—
292#239—184no—
293#239C2×C3⋊D312yescomplemented
294#237—92no—
295#237—181no—
296D3×D4—22no—
297D3×D4—92no—
298D3×D4—184no—
299D3×D4—188no—
300D3×D4—181no—
301D3×D4C2×C3⋊D312yescomplemented
302#235—21no—
303#235—98no—
304#235—94no—
305#235—91no—
306#235—184no—
307#235C2×C3⋊D311yescharacteristic
308C2×D12—21no—
309C2×D12—34no—
310C2×D12—98no—
311C2×D12C2×C3⋊D311yescharacteristic, complemented
312C3×SD16—92no—
313C3×D8—92no—
314C3×M16—91no—
315#216—94no—
316#216—91no—
317#216—184no—
318#215—38no—
319#215—94no—
320#215—91no—
321#215—184no—
322#214—98no—
323#214—94no—
324#214—91no—
325#214—184no—
326#213—38no—
327#213—98no—
328#213—98no—
329#213—98no—
330#213—94no—
331#213—91no—
332#213—92no—
333#213—92no—
334#213—184no—
335#208—34no—
336#208—98no—
337C2×C3⋊C8—98no—
338C2×C3⋊C8—94no—
339C2×C3⋊C8—91no—
340C2×C3⋊C8—184no—
341D24—92no—
342#204—92no—
343ΓL(1,𝔽25)—92no—
344D3×C8—92no—
345C22×C32—21no—
346C22×C32—24no—
347C22×C32—24no—
348C22×C32—32no—
349C22×C32—44no—
350C22×C32—62no—
351C22×C32—241no—
352C22×C32—242no—
353C22×C32D3×D411yescharacteristic
354C22×C32C2×#6714yes—
355C2×C3⋊D3—22no—
356C2×C3⋊D3—61no—
357C2×C3⋊D3—64no—
358C2×C3⋊D3—81no—
359C2×C3⋊D3—124no—
360C2×C3⋊D3—242no—
361C2×C3×D3—28no—
362C2×C3×D3—68no—
363C2×C3×D3—88no—
364C2×C3×D3—84no—
365C2×C3×D3—244no—
366C2×C3×D3C2×#6718yes—
367D32—248no—
368C32×C4—21no—
369C32×C4—24no—
370C32×C4—24no—
371C32×C4—24no—
372C32×C4—32no—
373C32×C4—44no—
374C32×C4D3×D411yescharacteristic
375C32×C4C2×#6714yes—
376C32×C4C23×D314yes—
377C32×C4C23×D311yescharacteristic
378C3⋊Dic3—61no—
379C3⋊Dic3—64no—
380C3⋊Dic3—124no—
381C3×Dic3C2×#6718yes—
382H(5,𝔽2)—31no—
383H(5,𝔽2)—271no—
384AGL(1,ℤ/8)—91no—
385AGL(1,ℤ/8)—271no—
386AGL(1,ℤ/8)—272no—
387AGL(1,ℤ/8)—272no—
388#135—272no—
389#135—271no—
390C2×D8—272no—
391C2×D8—271no—
392#131—271no—
393C33#57611yescharacteristic, complemented, Sylow
394C23×C3—68no—
395C23×C3—64no—
396C23×C3—181no—
397C22×D3—21no—
398C22×D3—22no—
399C22×D3—64no—
400C22×D3—188no—
401C22×D3—182no—
402C3×Q8—24no—
403C3×Q8—34no—
404C3×Q8C2×D3214yes—
405C3×Q8C22×C3⋊D311yescharacteristic
406C3×D4—24no—
407C3×D4—21no—
408C3×D4—24no—
409C3×D4—38no—
410C3×D4—38no—
411C3×D4—34no—
412C3×D4—44no—
413C3×D4—68no—
414C3×D4—68no—
415C3×D4—64no—
416C3×D4—92no—
417C3×D4—181no—
418C3×D4C2×D3214yes—
419C3×D4C22×C3⋊D311yescharacteristic
420C2×C3×C4—24no—
421C2×C3×C4—21no—
422C2×C3×C4—24no—
423C2×C3×C4—38no—
424C2×C3×C4—44no—
425C2×C3×C4—68no—
426C2×C3×C4C2×D3214yes—
427C2×C3×C4C22×C3⋊D311yescharacteristic
428#67—22no—
429#67—188no—
430#67—184no—
431#67—181no—
432#67—182no—
433#67—182no—
434#67—364no—
435#67#63812yes—
436C2×Dic3—182no—
437D12—22no—
438D12—21no—
439D12—38no—
440D12—64no—
441D12—94no—
442D12—91no—
443D12—184no—
444D12—188no—
445D12#63811yescharacteristic
446D12C22×C3⋊D312yes—
447C4×D3—22no—
448C4×D3—188no—
449C4×D3—184no—
450C4×D3—181no—
451C4×D3—364no—
452C4×D3C22×C3⋊D312yes—
453Dic6—94no—
454Dic6—91no—
455Dic6—184no—
456Dic6#63811yescharacteristic
457C3×C8—92no—
458C3⋊C8—38no—
459C3⋊C8—94no—
460C3⋊C8—94no—
461C3⋊C8—91no—
462C3⋊C8—91no—
463C3⋊C8—184no—
464C3⋊C8—184no—
465C2×C32—21no—
466C2×C32—24no—
467C2×C32—24no—
468C2×C32—44no—
469C2×C32—41no—
470C2×C32—44no—
471C2×C32—62no—
472C2×C32—84no—
473C2×C32—122no—
474C2×C32C2×D3×D411yescharacteristic
475C2×C32C22×#6714yes—
476C3⋊D3—42no—
477C3⋊D3—121no—
478C3⋊D3—124no—
479C3⋊D3—244no—
480C3×D3—28no—
481C3×D3—48no—
482C3×D3—128no—
483#41—31no—
484#41—32no—
485#41—61no—
486#41—272no—
487#41—271no—
488#41—541no—
489#41D3×C3⋊D311yescharacteristic, complemented
490C2×D4—32no—
491C2×D4—31no—
492C2×D4—61no—
493C2×D4—62no—
494C2×D4—91no—
495C2×D4—272no—
496C2×D4—541no—
497C2×D4—542no—
498Q16—271no—
499SD16—92no—
500SD16—272no—
501SD16—271no—
502SD16—271no—
503D8—92no—
504D8—272no—
505D8—272no—
506D8—272no—
507D8—271no—
508M16—91no—
509M16—272no—
510C2×C8—272no—
511C2×C8—271no—
512C22×C3—24no—
513C22×C3—21no—
514C22×C3—24no—
515C22×C3—38no—
516C22×C3—44no—
517C22×C3—68no—
518C22×C3—182no—
519C22×C3—244no—
520C22×C3—248no—
521C22×C3—721no—
522C22×C3D3×#6714yes—
523C22×C3C2×#63811yescharacteristic
524C2×D3—22no—
525C2×D3—68no—
526C2×D3—82no—
527C2×D3—81no—
528C2×D3—184no—
529C2×D3—181no—
530C2×D3—244no—
531C2×D3—364no—
532C2×D3—722no—
533C2×D3—728no—
534C2×D3C2×#63812yes—
535C3×C4—24no—
536C3×C4—21no—
537C3×C4—24no—
538C3×C4—24no—
539C3×C4—38no—
540C3×C4—44no—
541C3×C4D3×#6714yes—
542C3×C4C2×#63811yescharacteristic
543C3×C4C22×D3214yes—
544C3×C4C23×C3⋊D311yescharacteristic
545Dic3—184no—
546Dic3—181no—
547Dic3—364no—
548Dic3C2×#63812yes—
549C32—24no—
550C32#577411yescharacteristic, complemented
551C32#583514yescomplemented
552C23—62no—
553C23—61no—
554C23—181no—
555C23—542no—
556Q8—31no—
557Q8—271no—
558Q8C2×D3×C3⋊D311yescharacteristic
559D4—21no—
560D4—32no—
561D4—32no—
562D4—31no—
563D4—62no—
564D4—62no—
565D4—61no—
566D4—92no—
567D4—181no—
568D4—271no—
569D4—542no—
570D4—542no—
571D4—541no—
572D4C2×D3×C3⋊D311yescharacteristic
573C2×C4—21no—
574C2×C4—32no—
575C2×C4—62no—
576C2×C4—542no—
577C2×C4—541no—
578C2×C4C2×D3×C3⋊D311yescharacteristic
579C8—92no—
580C8—271no—
581C8—271no—
582C2×C3—24no—
583C2×C3—21no—
584C2×C3—24no—
585C2×C3—44no—
586C2×C3—44no—
587C2×C3—41no—
588C2×C3—68no—
589C2×C3—84no—
590C2×C3—128no—
591C2×C3—362no—
592C2×C3C2×D3×#6714yes—
593C2×C3C22×#63811yescharacteristic
594D3—22no—
595D3—42no—
596D3—128no—
597D3—364no—
598D3—361no—
599D3—724no—
600C22—21no—
601C22—32no—
602C22—62no—
603C22—182no—
604C22—241no—
605C22—242no—
606C22—541no—
607C22—721no—
608C22—2162no—
609C22D3×#63811yescharacteristic
610C4—21no—
611C4—32no—
612C4—541no—
613C4D3×#63811yescharacteristic
614C4C22×D3×C3⋊D311yescharacteristic
615C3—24no—
616C3#1059545414yescomplemented
617C3#1059630611yescharacteristic, complemented
618C2—21no—
619C2—41no—
620C2—62no—
621C2—122no—
622C2—362no—
623C2—1081no—
624C2C2×D3×#63811yescharacteristic
subgroupG/H
centerC2C2×D3×#638
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33C22×D4
hypercenter#41D3×C3⋊D3
hypocenterC33#576
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×#1009C2semidirect
2. #11724580C2semidirect
3. #11723991C2semidirect
4. C3×#64158C2semidirect
5. #11723580C2semidirect
6. #11723568C2semidirect
7. #11723555C2semidirect
8. D3×#3753C2semidirect
9. #11723378C2semidirect
10. C2×#88789C2semidirect
11. #11723350C2semidirect
14. C33×#41C22semidirect
15. #89076C22semidirect
16. C2×C3×C4⋊(C3⋊D3)C22semidirect
17. C32×#239C22semidirect
19. C32×D3×D4C22semidirect
23. C2×C32×D12C22semidirect
26. D3×C4⋊(C3⋊D3)C22semidirect
27. #89021C22semidirect
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. #88794C22semidirect
38. #88790C22semidirect
39. #88789C22semidirect
40. #88789C22semidirect
41. #88786C22semidirect
42. D3×C3⋊(C3⋊C8)C22semidirect
45. C3×#1009D3semidirect
52. #64158D3semidirect
102. C32×#41C2×D3semidirect
103. C32×#41C2×D3semidirect
116. C2×C4⋊(C3⋊D3)C2×D3semidirect
117. C3×#239C2×D3semidirect
119. C3×D3×D4C2×D3semidirect
124. C2×C3×D12C2×D3semidirect
136. #3755C2×D3semidirect
147. #3753C2×D3semidirect
149. #3748C2×D3semidirect
180. #1009C3⋊D3semidirect
284. C3×#41D32semidirect
285. C3×#41C2×C3⋊D3semidirect
293. #239C2×C3⋊D3semidirect
301. D3×D4C2×C3⋊D3semidirect
311. C2×D12C2×C3⋊D3semidirect
393. C33#576semidirect
489. #41D3×C3⋊D3semidirect
550. C32#5774semidirect
551. C32#5835semidirect
616. C3#10595454semidirect
617. C3#10596306semidirect

Maximal subgroups

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

subgroupindex
C32×#10092
#117245802
#117239912
C3×#641582
#117235802
#117235682
#117235552
D3×#37532
#117233782
C2×#887892
#117233502
#105963063
#105954543