Finite Group Databasebeta

← #13176698

Subgroups — #13176698

Order 1728.

1880 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
1#11724581C211yesmaximal, characteristic, complemented
2#11724580C211yesmaximal, characteristic, complemented
3#11723949C211yesmaximal, characteristic, complemented
4C3×#64117C211yesmaximal, characteristic, complemented
5C32×#907C211yesmaximal, characteristic, complemented
6#11723369C211yesmaximal, characteristic, complemented
7C2×#88791C211yesmaximal, characteristic, complemented
8#11723335C212yesmaximal, complemented
9#11723316C212yesmaximal, complemented
10#11723315C212yesmaximal, complemented
11D3×#3745C212yesmaximal, complemented
12#10596223—31nomaximal
13#10594714—34nomaximal
14#89076C2211yescharacteristic, complemented
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
16C3×#3828C2211yescharacteristic, complemented
17C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
18C32×#235C2211yescharacteristic
19#89039—22no—
20C3⋊D3×#67—22no—
21#89026—21no—
22C3⋊D3×D12—21no—
23D3×C4⋊(C3⋊D3)C2212yescomplemented
24D3×C4⋊(C3⋊D3)C2212yescomplemented
25#89022C2212yescomplemented
26#89021C2212yescomplemented
27#89018—21no—
28#89017—21no—
29C33×M16C2211yescharacteristic, complemented
30#88852C2212yescomplemented
31#88851C2212yescomplemented
32C3×#3745C2212yescomplemented
33C3×#3744C2212yescomplemented
34C2×C32×C3⋊C8C2211yescharacteristic, complemented
35C32×ΓL(1,𝔽25)C2212yescomplemented
36C32×D3×C8C2212yescomplemented
37#88800C2211yescharacteristic, complemented
38#88797C2211yescharacteristic, complemented
39#88796C2211yescharacteristic, complemented
40#88791C2211yescharacteristic, complemented
41#88791C2212yescomplemented
42#88791C2212yescomplemented
43#64366—31no—
44#64366—31no—
45#64308—34no—
46#64307—34no—
47C32×#131—31no—
48#64117—31no—
49#64117—32no—
50#64117—32no—
51#64117—34no—
52#64117—64no—
53#64117D311yescharacteristic, complemented
54#64113—31no—
55#64113—32no—
56C2×#3745—31no—
57C2×#3745—32no—
58C3×#908—34no—
59C3×#907D314yescomplemented
60#63829—34no—
61C2×#3714—34no—
62#63806—38no—
63#63795—38no—
64#63794—38no—
65D3×D24—38no—
66C22×C3⋊(C3⋊D3)—21no—
67C22×C3×C3⋊D3—21no—
68C2×D3×C3⋊D3—22no—
69C2×D3×C3⋊D3—22no—
70C2×C33×C4C2311yescharacteristic
71#6312—21no—
72C4⋊(C3⋊(C3⋊D3))—21no—
73C4⋊(C3⋊(C3⋊D3))C2312yes—
74C4⋊(C3⋊(C3⋊D3))C2311yescharacteristic
75C4×C3⋊(C3⋊D3)—21no—
76#6308C2311yescharacteristic
77C3×#638—21no—
78C3×C4⋊(C3⋊D3)—21no—
79C3×C4⋊(C3⋊D3)C2312yes—
80C3×C4⋊(C3⋊D3)C2311yescharacteristic
81C3×C4×C3⋊D3—21no—
82C3×#634C2311yescharacteristic
83C32×#67D412yes—
84C32×D12D411yescharacteristic
85C32×C4×D3C2312yes—
86C32×Dic6D411yescharacteristic
87#6292—22no—
88#6292—22no—
89#6291—22no—
90#6290—22no—
91#6289—22no—
92Dic3×C3⋊D3—22no—
93C33×C8C2312yes—
94C32×C3⋊C8C2311yescharacteristic
95C32×C3⋊C8C2311yescharacteristic
96#5774—91no—
97#5752—94no—
98C32×#41—31no—
99#3832—61no—
100#3832—61no—
101D4×C3⋊D3—61no—
102D4×C3⋊D3—61no—
103D4×C3⋊D3—62no—
104D4×C3⋊D3—62no—
105#3828—31no—
106#3828—34no—
107#3828—32no—
108#3828—32no—
109#3828—64no—
110#3828C2×D311yescharacteristic, complemented
111C2×C4⋊(C3⋊D3)—31no—
112C2×C4⋊(C3⋊D3)—34no—
113C2×C4⋊(C3⋊D3)—32no—
114C2×C4⋊(C3⋊D3)—32no—
115C2×C4⋊(C3⋊D3)—64no—
116C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
117C3×#235—34no—
118C3×#235C2×D314yes—
119C2×C3×D12—34no—
120#3811—68no—
121D3×#67—68no—
122#3802—64no—
123D3×D12—38no—
124D3×D12—38no—
125D3×D12—64no—
126#3799—38no—
127#3799—64no—
128#3796—64no—
129#3795—38no—
130C32×M16—24no—
131C32×M16—32no—
132C32×M16C2×D314yescomplemented
133C32×M16C2×D311yescharacteristic, complemented
134C2×C32×C8—31no—
135C2×C32×C8—32no—
136#3746—31no—
137#3745—31no—
138#3745—32no—
139#3745—32no—
140#3745—32no—
141#3745—38no—
142#3745—68no—
143#3745C2×D312yescomplemented
144#3744—31no—
145#3744—31no—
146#3744—32no—
147#3744—38no—
148#3744—68no—
149#3744C2×D312yescomplemented
150C2×C3×C3⋊C8C2×D314yescomplemented
151C3×D24—38no—
152C3×#204—38no—
153C3×ΓL(1,𝔽25)C2×D318yescomplemented
154C3×D3×C8C2×D318yescomplemented
155#3719—34no—
156#3717—34no—
157#3716—34no—
158#3714—34no—
159#3714—38no—
160#3714—38no—
161C22×C33C2×D411yescharacteristic
162C2×C3⋊(C3⋊D3)—22no—
163C2×C3⋊(C3⋊D3)—21no—
164C2×C3⋊(C3⋊D3)—81no—
165C2×C3×C3⋊D3—22no—
166C2×C3×C3⋊D3—21no—
167C2×C3×C3⋊D3—81no—
168C2×C32×D3C2×D412yes—
169D3×C3⋊D3—82no—
170D3×C3⋊D3—82no—
171C33×C4C2×D411yescharacteristic
172C33×C4C2411yescharacteristic
173C3⋊(C3⋊Dic3)—21no—
174C3×C3⋊Dic3—21no—
175C32×Dic3C2×D412yes—
176#1009—94no—
177#1009—94no—
178#1009—91no—
179#1004—91no—
180C3×AGL(1,ℤ/8)—91no—
181C3×#131—34no—
182#950—91no—
183C2×#213—91no—
184#916—92no—
185#914—92no—
186#911—92no—
187D3×D8—92no—
188#908—34no—
189#908—98no—
190#908—98no—
191#908—94no—
192#908—91no—
193#908—184no—
194#907C3⋊D311yescharacteristic, complemented
195#904—98no—
196#904—94no—
197C2×D24—98no—
198C2×D24—94no—
199C22×C3⋊D3—21no—
200C22×C3⋊D3—62no—
201C22×C3⋊D3—62no—
202C22×C3⋊D3—61no—
203C22×C3⋊D3—64no—
204C22×C3⋊D3—124no—
205C22×C3×D3—64no—
206C2×D32—68no—
207C2×D32—68no—
208C32×Q8—31no—
209C32×D4—32no—
210C32×D4—31no—
211C2×C32×C4—24no—
212C2×C32×C4—32no—
213C2×C32×C4C22×D314yes—
214C2×C32×C4C22×D311yescharacteristic
215#638—21no—
216#638—62no—
217#638—62no—
218#638—61no—
219#638—64no—
220#638—124no—
221C4⋊(C3⋊D3)—21no—
222C4⋊(C3⋊D3)—31no—
223C4⋊(C3⋊D3)—32no—
224C4⋊(C3⋊D3)—38no—
225C4⋊(C3⋊D3)—34no—
226C4⋊(C3⋊D3)—68no—
227C4⋊(C3⋊D3)—64no—
228C4⋊(C3⋊D3)—62no—
229C4⋊(C3⋊D3)—62no—
230C4⋊(C3⋊D3)—61no—
231C4⋊(C3⋊D3)—64no—
232C4⋊(C3⋊D3)—124no—
233C4⋊(C3⋊D3)C22×D311yescharacteristic
234C4⋊(C3⋊D3)C22×D312yes—
235C4×C3⋊D3—21no—
236C4×C3⋊D3—62no—
237C4×C3⋊D3—62no—
238C4×C3⋊D3—61no—
239C4×C3⋊D3—64no—
240C4×C3⋊D3—124no—
241#634—31no—
242#634—34no—
243#634—64no—
244#634C22×D311yescharacteristic
245C3×#67—64no—
246C3×#67D1218yes—
247C3×D12—38no—
248C3×D12—34no—
249C3×D12—64no—
250C3×D12D1214yes—
251C3×C4×D3—64no—
252C3×C4×D3C22×D318yes—
253C3×Dic6—34no—
254C3×Dic6D1214yes—
255#626—68no—
256#626—68no—
257#626—68no—
258#625—68no—
259#624—68no—
260D3×Dic3—68no—
261C32×C8—28no—
262C32×C8—31no—
263C32×C8—31no—
264C32×C8C22×D318yes—
265C32×C8C22×D312yes—
266C3×C3⋊C8C22×D314yes—
267C3×C3⋊C8C22×D314yes—
268#576—271noSylow
269C2×C33—21no—
270C2×C33C22×D411yescharacteristic
271C3⋊(C3⋊D3)—41no—
272C3⋊(C3⋊D3)—42no—
273C3×C3⋊D3—41no—
274C3×C3⋊D3—42no—
275C32×D3—22no—
276C3×#41—34no—
277C3×#41—91no—
278C2×C3×D4—91no—
279C2×#67—182no—
280#239—92no—
281#239—184no—
282#239—184no—
283#237—92no—
284#237—181no—
285D3×D4—92no—
286D3×D4—92no—
287D3×D4—184no—
288D3×D4—184no—
289D3×D4—188no—
290D3×D4—188no—
291D3×D4—182no—
292D3×D4—181no—
293#235—34no—
294#235—98no—
295#235—98no—
296#235—94no—
297#235—91no—
298#235—184no—
299#235—181no—
300#235C2×C3⋊D311yescharacteristic
301C2×D12—34no—
302C2×D12—98no—
303C2×D12—98no—
304C2×D12—94no—
305C2×D12—91no—
306C2×D12—184no—
307C2×D12—181no—
308C3×SD16—92no—
309C3×D8—92no—
310C3×M16—24no—
311C3×M16—38no—
312C3×M16D3214yescomplemented
313C3×M16C2×C3⋊D311yescharacteristic, complemented
314C2×C3×C8—34no—
315C2×C3×C8—38no—
316#216—91no—
317#215—91no—
318#214—91no—
319#213—91no—
320#213—92no—
321#213—92no—
322C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
323Dic12—94no—
324D24—38no—
325D24—98no—
326D24—94no—
327D24—98no—
328D24—98no—
329D24—92no—
330D24—188no—
331#204—38no—
332#204—94no—
333#204—94no—
334#204—98no—
335#204—92no—
336#204—188no—
337ΓL(1,𝔽25)C2×C3⋊D312yescomplemented
338D3×C8C2×C3⋊D312yescomplemented
339C22×C32—24no—
340C22×C32—32no—
341C22×C32C2×D1214yes—
342C22×C32D3×D411yescharacteristic
343C2×C3⋊D3—21no—
344C2×C3⋊D3—22no—
345C2×C3⋊D3—62no—
346C2×C3⋊D3—61no—
347C2×C3⋊D3—68no—
348C2×C3⋊D3—64no—
349C2×C3⋊D3—81no—
350C2×C3⋊D3—124no—
351C2×C3⋊D3—128no—
352C2×C3⋊D3—242no—
353C2×C3⋊D3—242no—
354C2×C3⋊D3—241no—
355C2×C3⋊D3—244no—
356C2×C3⋊D3—248no—
357C2×C3×D3—68no—
358C2×C3×D3—64no—
359C2×C3×D3—244no—
360C2×C3×D3C2×D1218yes—
361D32—248no—
362D32—248no—
363C32×C4—24no—
364C32×C4—24no—
365C32×C4—32no—
366C32×C4C2×D1214yes—
367C32×C4D3×D411yescharacteristic
368C32×C4C23×D314yes—
369C32×C4C23×D311yescharacteristic
370C3⋊Dic3—21no—
371C3⋊Dic3—61no—
372C3⋊Dic3—64no—
373C3⋊Dic3—124no—
374C3×Dic3—64no—
375C3×Dic3C2×D1218yes—
376H(5,𝔽2)—271no—
377H(5,𝔽2)—271no—
378AGL(1,ℤ/8)—91no—
379AGL(1,ℤ/8)—272no—
380AGL(1,ℤ/8)—272no—
381AGL(1,ℤ/8)—271no—
382#135—272no—
383#135—271no—
384C2×D8—272no—
385C2×D8—271no—
386#131—31no—
387C33#57611yescharacteristic, complemented, Sylow
388C23×C3—181no—
389C22×D3—64no—
390C22×D3—188no—
391C22×D3—188no—
392C22×D3—184no—
393C22×D3—181no—
394C22×D3—182no—
395C22×D3—182no—
396C22×D3—364no—
397C3×Q8—34no—
398C3×Q8—91no—
399C3×D4—38no—
400C3×D4—34no—
401C3×D4—92no—
402C3×D4—91no—
403C3×D4—181no—
404C3×D4—181no—
405C2×C3×C4—24no—
406C2×C3×C4—38no—
407C2×C3×C4—181no—
408C2×C3×C4C2×D3214yes—
409C2×C3×C4C22×C3⋊D311yescharacteristic
410#67—64no—
411#67—188no—
412#67—188no—
413#67—184no—
414#67—181no—
415#67—182no—
416#67—182no—
417#67—182no—
418#67—364no—
419#67C4⋊(C3⋊D3)12yes—
420C2×Dic3—182no—
421D12—38no—
422D12—34no—
423D12—64no—
424D12—98no—
425D12—94no—
426D12—92no—
427D12—91no—
428D12—188no—
429D12—184no—
430D12—188no—
431D12—188no—
432D12—182no—
433D12—184no—
434D12—181no—
435D12—364no—
436D12C4⋊(C3⋊D3)11yescharacteristic
437C4×D3—64no—
438C4×D3—188no—
439C4×D3—188no—
440C4×D3—184no—
441C4×D3—181no—
442C4×D3—182no—
443C4×D3—364no—
444C4×D3C22×C3⋊D312yes—
445Dic6—34no—
446Dic6—94no—
447Dic6—91no—
448Dic6—184no—
449Dic6C4⋊(C3⋊D3)11yescharacteristic
450C3×C8—28no—
451C3×C8—34no—
452C3×C8—34no—
453C3×C8C2×D3218yes—
454C3×C8C22×C3⋊D312yes—
455C3⋊C8C22×C3⋊D311yescharacteristic
456C3⋊C8C22×C3⋊D311yescharacteristic
457C2×C32—21no—
458C2×C32—24no—
459C2×C32—24no—
460C2×C32—44no—
461C2×C32—62no—
462C2×C32C22×D1214yes—
463C2×C32C2×D3×D411yescharacteristic
464C3⋊D3—42no—
465C3⋊D3—41no—
466C3⋊D3—121no—
467C3⋊D3—122no—
468C3⋊D3—124no—
469C3⋊D3—128no—
470C3⋊D3—244no—
471C3⋊D3—248no—
472C3×D3—28no—
473C3×D3—124no—
474C3×D3—128no—
475#41—31no—
476#41—91no—
477#41—272no—
478#41—272no—
479#41—271no—
480#41—541no—
481#41—541no—
482C2×D4—91no—
483C2×D4—272no—
484C2×D4—272no—
485C2×D4—271no—
486C2×D4—541no—
487C2×D4—542no—
488C2×D4—541no—
489C2×D4—542no—
490Q16—271no—
491SD16—92no—
492SD16—271no—
493SD16—271no—
494SD16—272no—
495D8—92no—
496D8—271no—
497D8—272no—
498D8—272no—
499D8—272no—
500M16—32no—
501M16D3×C3⋊D311yescharacteristic, complemented
502C2×C8—31no—
503C2×C8—32no—
504C22×C3—24no—
505C22×C3—38no—
506C22×C3—182no—
507C22×C3—181no—
508C22×C3—721no—
509C22×C3D3×D1214yes—
510C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
511C2×D3—64no—
512C2×D3—68no—
513C2×D3—188no—
514C2×D3—184no—
515C2×D3—182no—
516C2×D3—181no—
517C2×D3—244no—
518C2×D3—364no—
519C2×D3—368no—
520C2×D3—722no—
521C2×D3—722no—
522C2×D3—728no—
523C2×D3—728no—
524C2×D3—721no—
525C2×D3—724no—
526C2×D3—728no—
527C2×D3C2×C4⋊(C3⋊D3)12yes—
528C3×C4—24no—
529C3×C4—24no—
530C3×C4—38no—
531C3×C4—181no—
532C3×C4D3×D1214yes—
533C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
534C3×C4C22×D3214yes—
535C3×C4C23×C3⋊D311yescharacteristic
536Dic3—64no—
537Dic3—184no—
538Dic3—181no—
539Dic3—364no—
540Dic3C2×C4⋊(C3⋊D3)12yes—
541C32—24no—
542C32#575214yescomplemented
543C32#577411yescharacteristic, complemented
544C23—181no—
545C23—542no—
546C23—542no—
547C23—541no—
548Q8—31no—
549Q8—91no—
550Q8—271no—
551D4—32no—
552D4—31no—
553D4—91no—
554D4—92no—
555D4—181no—
556D4—181no—
557D4—272no—
558D4—271no—
559D4—542no—
560D4—542no—
561D4—542no—
562D4—542no—
563D4—541no—
564D4—541no—
565C2×C4—32no—
566C2×C4—181no—
567C2×C4—542no—
568C2×C4—542no—
569C2×C4—541no—
570C2×C4C2×D3×C3⋊D311yescharacteristic
571C8—31no—
572C8—31no—
573C8C2×D3×C3⋊D312yes—
574C2×C3—21no—
575C2×C3—24no—
576C2×C3—24no—
577C2×C3—44no—
578C2×C3—68no—
579C2×C3—361no—
580C2×C3—362no—
581C2×C3C2×D3×D1214yes—
582C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
583D3—22no—
584D3—128no—
585D3—124no—
586D3—362no—
587D3—361no—
588D3—364no—
589D3—368no—
590D3—728no—
591D3—724no—
592C22—32no—
593C22—181no—
594C22—182no—
595C22—542no—
596C22—541no—
597C22—721no—
598C22—2161no—
599C22—2162no—
600C22—2162no—
601C22D3×C4⋊(C3⋊D3)11yescharacteristic
602C4—32no—
603C4—181no—
604C4—541no—
605C4D3×C4⋊(C3⋊D3)11yescharacteristic
606C4C22×D3×C3⋊D311yescharacteristic
607C3—24no—
608C3#1059471414yescomplemented
609C3#1059622311yescharacteristic, complemented
610C2—21no—
611C2—62no—
612C2—362no—
613C2—361no—
614C2—1082no—
615C2—1081no—
616C2C2×D3×C4⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC2C2×D3×C4⋊(C3⋊D3)
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. #11724581C2semidirect
2. #11724580C2semidirect
3. #11723949C2semidirect
4. C3×#64117C2semidirect
5. C32×#907C2semidirect
6. #11723369C2semidirect
7. C2×#88791C2semidirect
8. #11723335C2semidirect
9. #11723316C2semidirect
10. #11723315C2semidirect
11. D3×#3745C2semidirect
14. #89076C22semidirect
15. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
16. C3×#3828C22semidirect
17. C2×C3×C4⋊(C3⋊D3)C22semidirect
23. D3×C4⋊(C3⋊D3)C22semidirect
24. D3×C4⋊(C3⋊D3)C22semidirect
25. #89022C22semidirect
26. #89021C22semidirect
29. C33×M16C22semidirect
30. #88852C22semidirect
31. #88851C22semidirect
32. C3×#3745C22semidirect
33. C3×#3744C22semidirect
34. C2×C32×C3⋊C8C22semidirect
35. C32×ΓL(1,𝔽25)C22semidirect
36. C32×D3×C8C22semidirect
37. #88800C22semidirect
38. #88797C22semidirect
39. #88796C22semidirect
40. #88791C22semidirect
41. #88791C22semidirect
42. #88791C22semidirect
53. #64117D3semidirect
59. C3×#907D3semidirect
110. #3828C2×D3semidirect
116. C2×C4⋊(C3⋊D3)C2×D3semidirect
132. C32×M16C2×D3semidirect
133. C32×M16C2×D3semidirect
143. #3745C2×D3semidirect
149. #3744C2×D3semidirect
150. C2×C3×C3⋊C8C2×D3semidirect
153. C3×ΓL(1,𝔽25)C2×D3semidirect
154. C3×D3×C8C2×D3semidirect
194. #907C3⋊D3semidirect
312. C3×M16D32semidirect
313. C3×M16C2×C3⋊D3semidirect
322. C2×C3⋊C8C2×C3⋊D3semidirect
337. ΓL(1,𝔽25)C2×C3⋊D3semidirect
338. D3×C8C2×C3⋊D3semidirect
387. C33#576semidirect
501. M16D3×C3⋊D3semidirect
542. C32#5752semidirect
543. C32#5774semidirect
608. C3#10594714semidirect
609. C3#10596223semidirect

Maximal subgroups

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

subgroupindex
#117245812
#117245802
#117239492
C3×#641172
C32×#9072
#117233692
C2×#887912
#117233352
#117233162
#117233152
D3×#37452
#105962233
#105947143