Finite Group Databasebeta

← #13177998

Subgroups — #13177998

Order 1728.

2132 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11724685C211yesmaximal, characteristic, complemented
2#11724580C211yesmaximal, characteristic, complemented
3C3×#64158C211yesmaximal, characteristic, complemented
4C32×#950C211yesmaximal, characteristic, complemented
5#11723590C212yesmaximal, complemented
6#11723571C212yesmaximal, complemented
7#11723565C212yesmaximal, complemented
8#11723548C212yesmaximal, complemented
9C2×#88796C211yesmaximal, characteristic, complemented
10#11723379C211yesmaximal, characteristic, complemented
11#11723356C211yesmaximal, characteristic, complemented
12#10596241—31nomaximal
13#10595462—34nomaximal
14C33×#41C2211yescharacteristic, complemented
15Q8×C3⋊(C3⋊D3)C2212yescomplemented
16#89078—22no—
17#89078C2212yescomplemented
18#89076—21no—
19#89076C2211yescharacteristic
20C2×#6308—21no—
21C2×#6308C2211yescharacteristic, complemented
22C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
23C32×#235C2211yescharacteristic
24C3⋊D3×#67—22no—
25#89026C2212yes—
26D3×C4⋊(C3⋊D3)—21no—
27#89021—21no—
28#89017C2212yes—
29C3×#3755C2212yescomplemented
30C3×#3753C2212yescomplemented
31C3×#3748C2211yescharacteristic, complemented
32C32×#216C2211yescharacteristic, complemented
33C32×#215C2211yescharacteristic, complemented
34C32×#214C2211yescharacteristic, complemented
35C32×#213C2211yescharacteristic, complemented
36C2×C32×C3⋊C8C2211yescharacteristic, complemented
37#88799C2212yescomplemented
38#88796C2212yescomplemented
39#88796C2212yescomplemented
40#88794C2212yescomplemented
41#88788C2212yescomplemented
42#88785C2212yescomplemented
43#64367—31no—
44#64367—34no—
45#64367—64no—
46#64366—31no—
47#64307—34no—
48C32×#135—31no—
49#64160—31no—
50#64158D311yescharacteristic, complemented
51#64127—32no—
52#64125—32no—
53SD16×C3⋊D3—32no—
54#64120—32no—
55C2×#3744—31no—
56#64110—31no—
57C3×#950D314yescomplemented
58C3×#949—34no—
59#63951—38no—
60#63940—38no—
61#63936—38no—
62#63927—38no—
63#63831—34no—
64C2×#3716—34no—
65#63817—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—22no—
74#6312D412yes—
75C2×C3⋊(C3⋊Dic3)—21no—
76C2×C3⋊(C3⋊Dic3)—22no—
77C4⋊(C3⋊(C3⋊D3))D411yescharacteristic
78C4×C3⋊(C3⋊D3)—22no—
79C4×C3⋊(C3⋊D3)C2312yes—
80#6308—21no—
81#6308—22no—
82#6308D411yescharacteristic
83#6308C2312yes—
84C3×C4⋊(C3⋊D3)—21no—
85C3×C4⋊(C3⋊D3)C2312yes—
86C32×#67—21no—
87C32×D12C2311yescharacteristic
88C32×C4×D3—21no—
89C32×Dic6C2311yescharacteristic
90#6292—22no—
91#6290—22no—
92Dic3×C3⋊D3—22no—
93C3×C3⋊(C3⋊C8)C2312yes—
94C32×C3⋊C8C2311yescharacteristic
95C32×C3⋊C8C2311yescharacteristic
96#5775—91no—
97#5770—94no—
98C32×#41—24no—
99C32×#41—31no—
100C32×#41C2×D314yescomplemented
101C32×#41C2×D311yescharacteristic, complemented
102#3832—32no—
103Q8×C3⋊D3—32no—
104Q8×C3⋊D3—38no—
105Q8×C3⋊D3—68no—
106#3830—32no—
107#3830—38no—
108#3830—62no—
109#3830—68no—
110#3830—68no—
111#3830—616no—
112D4×C3⋊D3—32no—
113D4×C3⋊D3—62no—
114#3828—31no—
115#3828—34no—
116#3828—61no—
117#3828—61no—
118#3828—64no—
119#3828—64no—
120#3828—68no—
121C2×C4⋊(C3⋊D3)—61no—
122C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
123C2×#634—31no—
124C2×#634—34no—
125C2×#634—61no—
126C2×#634—64no—
127C2×#634—64no—
128C2×#634—68no—
129C3×#235C2×D314yes—
130C2×C3×D12—34no—
131D3×#67—68no—
132#3802—38no—
133D3×D12—64no—
134#3796—38no—
135#3795—64no—
136C32×Q16—31no—
137C32×SD16—31no—
138C32×SD16—31no—
139C32×D8—31no—
140C2×C32×C8—31no—
141#3756—32no—
142#3755C2×D312yescomplemented
143#3754—32no—
144#3753C2×D312yescomplemented
145#3748C2×D311yescharacteristic, complemented
146#3744—32no—
147#3744—32no—
148#3743—32no—
149C8×C3⋊D3—32no—
150C3×#216C2×D314yescomplemented
151C3×#215—38no—
152C3×#215C2×D314yescomplemented
153C3×#214C2×D314yescomplemented
154C3×#213—38no—
155C3×#213C2×D314yescomplemented
156C3×#208—34no—
157C2×C3×C3⋊C8C2×D314yescomplemented
158#3719—38no—
159#3716—38no—
160#3716—38no—
161#3716—38no—
162#3712—38no—
163#3710—38no—
164C22×C33—21no—
165C22×C33C2×D411yescharacteristic
166C2×C3⋊(C3⋊D3)C2×D412yes—
167C2×C3×C3⋊D3—22no—
168C2×C3×C3⋊D3—81no—
169C2×C32×D3—21no—
170D3×C3⋊D3—82no—
171C33×C4—21no—
172C33×C4C2×D411yescharacteristic
173C33×C4C2411yescharacteristic
174C3⋊(C3⋊Dic3)—22no—
175C3⋊(C3⋊Dic3)C2×D412yes—
176C32×Dic3—21no—
177#1010—94no—
178#1010—91no—
179#1010—184no—
180#1009—94no—
181#1004—91no—
182C3×AGL(1,ℤ/8)—91no—
183C3×#135—34no—
184#951—94no—
185#950C3⋊D311yescharacteristic, complemented
186#949—34no—
187C2×#214—91no—
188#918—98no—
189#916—98no—
190#915—92no—
191D3×SD16—98no—
192D3×SD16—92no—
193#912—92no—
194#911—98no—
195#911—92no—
196#909—91no—
197#907—91no—
198C2×#204—94no—
199#901—94no—
200C22×C3⋊D3—21no—
201C22×C3⋊D3—62no—
202C22×C3×D3—64no—
203C2×D32—68no—
204C32×Q8—24no—
205C32×Q8—31no—
206C32×Q8C22×D314yes—
207C32×Q8C22×D311yescharacteristic
208C32×D4—28no—
209C32×D4—24no—
210C32×D4—21no—
211C32×D4—24no—
212C32×D4—31no—
213C32×D4—61no—
214C32×D4C22×D314yes—
215C32×D4C22×D311yescharacteristic
216C2×C32×C4—28no—
217C2×C32×C4—24no—
218C2×C32×C4—21no—
219C2×C32×C4—24no—
220C2×C32×C4—61no—
221C2×C32×C4C22×D314yes—
222C2×C32×C4C22×D311yescharacteristic
223#638—32no—
224#638—38no—
225#638—68no—
226#638—62no—
227#638—62no—
228#638—68no—
229#638—616no—
230C2×C3⋊Dic3—62no—
231C2×C3⋊Dic3—68no—
232C2×C3⋊Dic3—616no—
233C2×C3⋊Dic3—68no—
234C2×C3⋊Dic3—64no—
235C2×C3⋊Dic3—61no—
236C4⋊(C3⋊D3)—21no—
237C4⋊(C3⋊D3)—31no—
238C4⋊(C3⋊D3)—34no—
239C4⋊(C3⋊D3)—64no—
240C4⋊(C3⋊D3)—62no—
241C4⋊(C3⋊D3)C22×D312yes—
242C4×C3⋊D3—32no—
243C4×C3⋊D3—38no—
244C4×C3⋊D3—68no—
245C4×C3⋊D3—62no—
246C4×C3⋊D3—62no—
247C4×C3⋊D3—68no—
248C4×C3⋊D3—616no—
249#634—31no—
250#634—32no—
251#634—34no—
252#634—38no—
253#634—64no—
254#634—68no—
255#634—62no—
256#634—68no—
257#634—68no—
258#634—616no—
259#634—64no—
260#634—61no—
261C3×#67—24no—
262C3×D12—38no—
263C3×D12—64no—
264C3×D12C22×D314yes—
265C3×C4×D3—24no—
266C3×Dic6C22×D314yes—
267#626—68no—
268#625—68no—
269D3×Dic3—68no—
270C32×C8—31no—
271C32×C8—31no—
272C3⋊(C3⋊C8)C22×D312yes—
273C3×C3⋊C8—38no—
274C3×C3⋊C8C22×D314yes—
275C3×C3⋊C8C22×D314yes—
276#577—271noSylow
277C2×C33—21no—
278C2×C33—41no—
279C2×C33C22×D411yescharacteristic
280C3⋊(C3⋊D3)—22no—
281C3×C3⋊D3—42no—
282C32×D3—41no—
283C3×#41—24no—
284C3×#41—34no—
285C3×#41D3214yescomplemented
286C3×#41C2×C3⋊D311yescharacteristic, complemented
287C2×C3×D4—91no—
288C2×#67—182no—
289#239—98no—
290D3×Q8—98no—
291D3×Q8—92no—
292D3×Q8—188no—
293#237—98no—
294#237—92no—
295#237—92no—
296#237—188no—
297#237—181no—
298#237—188no—
299#237—182no—
300#237—1816no—
301D3×D4—98no—
302D3×D4—92no—
303D3×D4—181no—
304D3×D4—188no—
305#235—94no—
306#235—91no—
307#235—184no—
308#235—184no—
309#235—184no—
310#235—181no—
311#235—188no—
312#235C2×C3⋊D311yescharacteristic
313C2×D12—34no—
314C2×D12—184no—
315C2×Dic6—94no—
316C2×Dic6—91no—
317C2×Dic6—184no—
318C2×Dic6—184no—
319C2×Dic6—181no—
320C2×Dic6—188no—
321C3×Q16—34no—
322C3×SD16—34no—
323C3×SD16—34no—
324C3×SD16—92no—
325C3×D8—34no—
326C3×D8—92no—
327C3×M16—91no—
328C2×C3×C8—34no—
329#216—98no—
330#216C2×C3⋊D311yescharacteristic, complemented
331#215—38no—
332#215C2×C3⋊D311yescharacteristic, complemented
333#214—98no—
334#214—92no—
335#214—92no—
336#214C2×C3⋊D311yescharacteristic, complemented
337#213—38no—
338#213C2×C3⋊D311yescharacteristic, complemented
339#208—34no—
340C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
341Dic12—92no—
342#204—98no—
343#204—98no—
344#204—92no—
345ΓL(1,𝔽25)—98no—
346ΓL(1,𝔽25)—92no—
347D3×C8—98no—
348D3×C8—92no—
349C22×C32—28no—
350C22×C32—24no—
351C22×C32—21no—
352C22×C32—24no—
353C22×C32—61no—
354C22×C32D3×D414yes—
355C22×C32D3×D411yescharacteristic
356C2×C3⋊D3—22no—
357C2×C3⋊D3—32no—
358C2×C3⋊D3—38no—
359C2×C3⋊D3—68no—
360C2×C3⋊D3—81no—
361C2×C3⋊D3—242no—
362C2×C3×D3—24no—
363C2×C3×D3—68no—
364C2×C3×D3—244no—
365D32—248no—
366C32×C4—28no—
367C32×C4—24no—
368C32×C4—21no—
369C32×C4—24no—
370C32×C4—24no—
371C32×C4—61no—
372C32×C4D3×D414yes—
373C32×C4D3×D411yescharacteristic
374C32×C4C23×D314yes—
375C32×C4C23×D311yescharacteristic
376C3⋊Dic3—32no—
377C3⋊Dic3—38no—
378C3⋊Dic3—68no—
379C3⋊Dic3—616no—
380C3⋊Dic3—68no—
381C3⋊Dic3—62no—
382C3×Dic3—24no—
3832−1+4—271no—
384H(5,𝔽2)—271no—
385#137—271no—
386#137—272no—
387AGL(1,ℤ/8)—91no—
388AGL(1,ℤ/8)—272no—
389#135—31no—
390#135—272no—
391C2×SD16—271no—
392C2×SD16—272no—
393#131—271no—
394C33#57711yescharacteristic, complemented, Sylow
395C23×C3—181no—
396C22×D3—64no—
397C22×D3—182no—
398C22×D3—188no—
399C3×Q8—24no—
400C3×Q8—34no—
401C3×Q8C2×D3214yes—
402C3×Q8C22×C3⋊D311yescharacteristic
403C3×D4—28no—
404C3×D4—24no—
405C3×D4—21no—
406C3×D4—24no—
407C3×D4—34no—
408C3×D4—64no—
409C3×D4—92no—
410C3×D4—181no—
411C3×D4C2×D3214yes—
412C3×D4C22×C3⋊D311yescharacteristic
413C2×C3×C4—28no—
414C2×C3×C4—24no—
415C2×C3×C4—21no—
416C2×C3×C4—24no—
417C2×C3×C4—64no—
418C2×C3×C4C2×D3214yes—
419C2×C3×C4C22×C3⋊D311yescharacteristic
420#67—21no—
421#67—98no—
422#67—92no—
423#67—188no—
424#67—182no—
425#67—182no—
426#67—188no—
427#67—188no—
428#67—182no—
429#67—1816no—
430C2×Dic3—182no—
431C2×Dic3—188no—
432C2×Dic3—182no—
433C2×Dic3—1816no—
434C2×Dic3—188no—
435C2×Dic3—184no—
436C2×Dic3—181no—
437D12—38no—
438D12—64no—
439D12—94no—
440D12—91no—
441D12—184no—
442D12—188no—
443D12C22×C3⋊D311yescharacteristic
444C4×D3—21no—
445C4×D3—98no—
446C4×D3—92no—
447C4×D3—188no—
448C4×D3—188no—
449C4×D3—188no—
450C4×D3—182no—
451C4×D3—1816no—
452Dic6—94no—
453Dic6—91no—
454Dic6—98no—
455Dic6—92no—
456Dic6—184no—
457Dic6—188no—
458Dic6—188no—
459Dic6—182no—
460Dic6—188no—
461Dic6—1816no—
462Dic6—184no—
463Dic6—181no—
464Dic6C22×C3⋊D311yescharacteristic
465C3×C8—34no—
466C3×C8—34no—
467C3×C8—92no—
468C3⋊C8—38no—
469C3⋊C8C22×C3⋊D311yescharacteristic
470C3⋊C8C22×C3⋊D311yescharacteristic
471C2×C32—21no—
472C2×C32—24no—
473C2×C32—24no—
474C2×C32—44no—
475C2×C32—48no—
476C2×C32—44no—
477C2×C32—41no—
478C2×C32—121no—
479C2×C32C2×D3×D414yes—
480C2×C32C2×D3×D411yescharacteristic
481C3⋊D3—42no—
482C3⋊D3—62no—
483C3⋊D3—68no—
484C3⋊D3—128no—
485C3×D3—44no—
486C3×D3—128no—
487#41—31no—
488#41—271no—
489#41—272no—
490#41—272no—
491#41—541no—
492#41—541no—
493#41—542no—
494#41D3×C3⋊D311yescharacteristic, complemented
495C2×Q8—272no—
496C2×Q8—271no—
497C2×Q8—541no—
498C2×D4—91no—
499C2×D4—272no—
500C2×D4—541no—
501C2×D4—542no—
502Q16—31no—
503Q16—272no—
504SD16—31no—
505SD16—31no—
506SD16—92no—
507SD16—272no—
508SD16—272no—
509SD16—272no—
510D8—31no—
511D8—92no—
512M16—91no—
513M16—272no—
514C2×C8—31no—
515C2×C8—272no—
516C22×C3—28no—
517C22×C3—24no—
518C22×C3—21no—
519C22×C3—24no—
520C22×C3—64no—
521C22×C3—182no—
522C22×C3—721no—
523C22×C3#381114yes—
524C22×C3D4×C3⋊D311yescharacteristic
525C2×D3—21no—
526C2×D3—68no—
527C2×D3—98no—
528C2×D3—92no—
529C2×D3—188no—
530C2×D3—244no—
531C2×D3—722no—
532C2×D3—728no—
533C3×C4—28no—
534C3×C4—24no—
535C3×C4—21no—
536C3×C4—24no—
537C3×C4—24no—
538C3×C4—64no—
539C3×C4#381114yes—
540C3×C4D4×C3⋊D311yescharacteristic
541C3×C4C22×D3214yes—
542C3×C4C23×C3⋊D311yescharacteristic
543Dic3—21no—
544Dic3—98no—
545Dic3—92no—
546Dic3—188no—
547Dic3—1816no—
548Dic3—188no—
549Dic3—182no—
550C32—24no—
551C32#577014yescomplemented
552C32#577511yescharacteristic, complemented
553C23—181no—
554C23—542no—
555Q8—31no—
556Q8—271no—
557Q8—272no—
558Q8—542no—
559Q8—541no—
560Q8C2×D3×C3⋊D311yescharacteristic
561D4—21no—
562D4—31no—
563D4—61no—
564D4—92no—
565D4—181no—
566D4—272no—
567D4—271no—
568D4—542no—
569D4—542no—
570D4—542no—
571D4C2×D3×C3⋊D311yescharacteristic
572C2×C4—21no—
573C2×C4—61no—
574C2×C4—272no—
575C2×C4—542no—
576C2×C4—542no—
577C2×C4—542no—
578C2×C4—541no—
579C2×C4C2×D3×C3⋊D311yescharacteristic
580C8—31no—
581C8—31no—
582C8—92no—
583C2×C3—24no—
584C2×C3—21no—
585C2×C3—24no—
586C2×C3—44no—
587C2×C3—48no—
588C2×C3—44no—
589C2×C3—41no—
590C2×C3—124no—
591C2×C3—362no—
592C2×C3C2×#381114yes—
593C2×C3C2×D4×C3⋊D311yescharacteristic
594D3—41no—
595D3—128no—
596D3—188no—
597D3—182no—
598D3—368no—
599C22—21no—
600C22—61no—
601C22—182no—
602C22—272no—
603C22—721no—
604C22—2162no—
605C22#8903911yescharacteristic
606C4—21no—
607C4—61no—
608C4—272no—
609C4—542no—
610C4#8903911yescharacteristic
611C4C22×D3×C3⋊D311yescharacteristic
612C3—24no—
613C3#1059546214yescomplemented
614C3#1059624111yescharacteristic, complemented
615C2—21no—
616C2—41no—
617C2—121no—
618C2—362no—
619C2—542no—
620C2C2×#8903911yescharacteristic
subgroupG/H
centerC2C2×#89039
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. #11724685C2semidirect
2. #11724580C2semidirect
3. C3×#64158C2semidirect
4. C32×#950C2semidirect
5. #11723590C2semidirect
6. #11723571C2semidirect
7. #11723565C2semidirect
8. #11723548C2semidirect
9. C2×#88796C2semidirect
10. #11723379C2semidirect
11. #11723356C2semidirect
14. C33×#41C22semidirect
15. Q8×C3⋊(C3⋊D3)C22semidirect
17. #89078C22semidirect
21. C2×#6308C22semidirect
29. C3×#3755C22semidirect
30. C3×#3753C22semidirect
31. C3×#3748C22semidirect
32. C32×#216C22semidirect
33. C32×#215C22semidirect
34. C32×#214C22semidirect
35. C32×#213C22semidirect
36. C2×C32×C3⋊C8C22semidirect
37. #88799C22semidirect
38. #88796C22semidirect
39. #88796C22semidirect
40. #88794C22semidirect
41. #88788C22semidirect
42. #88785C22semidirect
50. #64158D3semidirect
57. C3×#950D3semidirect
100. C32×#41C2×D3semidirect
101. C32×#41C2×D3semidirect
142. #3755C2×D3semidirect
144. #3753C2×D3semidirect
145. #3748C2×D3semidirect
150. C3×#216C2×D3semidirect
152. C3×#215C2×D3semidirect
153. C3×#214C2×D3semidirect
155. C3×#213C2×D3semidirect
157. C2×C3×C3⋊C8C2×D3semidirect
185. #950C3⋊D3semidirect
285. C3×#41D32semidirect
286. C3×#41C2×C3⋊D3semidirect
330. #216C2×C3⋊D3semidirect
332. #215C2×C3⋊D3semidirect
336. #214C2×C3⋊D3semidirect
338. #213C2×C3⋊D3semidirect
340. C2×C3⋊C8C2×C3⋊D3semidirect
394. C33#577semidirect
494. #41D3×C3⋊D3semidirect
551. C32#5770semidirect
552. C32#5775semidirect
613. C3#10595462semidirect
614. C3#10596241semidirect

Maximal subgroups

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

subgroupindex
#117246852
#117245802
C3×#641582
C32×#9502
#117235902
#117235712
#117235652
#117235482
C2×#887962
#117233792
#117233562
#105962413
#105954623