Finite Group Databasebeta

← #13177986

Subgroups — #13177986

Order 1728.

1820 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64367C211yesmaximal, characteristic, complemented
2#11724581C211yesmaximal, characteristic, complemented
3#11723990C211yesmaximal, characteristic, complemented
4C32×#950C211yesmaximal, characteristic, complemented
5#11723586C212yesmaximal, complemented
6C3⋊D3×#215C212yesmaximal, complemented
7#11723561C212yesmaximal, complemented
8#11723546C212yesmaximal, complemented
9C2×#88797C211yesmaximal, characteristic, complemented
10#11723373C211yesmaximal, characteristic, complemented
11#11723352C211yesmaximal, characteristic, complemented
12#10596241—31nomaximal
13#10595457—34nomaximal
14C33×#41C2211yescharacteristic, complemented
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
16C3×Q8×C3⋊D3C2212yescomplemented
17C3×#3830—22no—
18C3×#3830C2212yescomplemented
19C3×#3828—21no—
20C3×#3828C2211yescharacteristic
21C2×C3×#634—21no—
22C2×C3×#634C2211yescharacteristic, complemented
23C32×#235C2211yescharacteristic
24#89039—22no—
25C3⋊D3×D12C2212yes—
26D3×C4⋊(C3⋊D3)—21no—
27#89022—21no—
28#89018C2212yes—
29#88862C2212yescomplemented
30#88860C2212yescomplemented
31#88855C2211yescharacteristic, complemented
32C32×#216C2211yescharacteristic, complemented
33C32×#215C2211yescharacteristic, complemented
34C32×#214C2211yescharacteristic, complemented
35C32×#213C2211yescharacteristic, complemented
36C2×C32×C3⋊C8C2211yescharacteristic, complemented
37#88798C2212yescomplemented
38#88797C2212yescomplemented
39#88797C2212yescomplemented
40#88792C2212yescomplemented
41#88787C2212yescomplemented
42C3⋊D3×C3⋊C8C2212yescomplemented
43#64367D311yescharacteristic, complemented
44#64366—31no—
45C3×#1010—34no—
46#64308—34no—
47C32×#135—31no—
48#64160—31no—
49#64158—31no—
50#64158—34no—
51#64158—64no—
52#64127—32no—
53#64125—32no—
54SD16×C3⋊D3—32no—
55#64120—32no—
56C2×#3744—31no—
57#64110—31no—
58C3×#950D314yescomplemented
59#63948—38no—
60D3×#215—38no—
61#63935—38no—
62#63925—38no—
63C2×#3717—34no—
64#63827—34no—
65#63814—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
73C4⋊(C3⋊(C3⋊D3))—21no—
74C4⋊(C3⋊(C3⋊D3))C2312yes—
75C3×#638—22no—
76C3×#638D412yes—
77C2×C3×C3⋊Dic3—21no—
78C2×C3×C3⋊Dic3—22no—
79C3×C4⋊(C3⋊D3)D411yescharacteristic
80C3×C4×C3⋊D3—22no—
81C3×C4×C3⋊D3C2312yes—
82C3×#634—21no—
83C3×#634—22no—
84C3×#634D411yescharacteristic
85C3×#634C2312yes—
86C32×#67—21no—
87C32×D12C2311yescharacteristic
88C32×C4×D3—21no—
89C32×Dic6C2311yescharacteristic
90#6292—22no—
91#6291—22no—
92#6289—22no—
93C3⋊(C3⋊(C3⋊C8))C2312yes—
94C32×C3⋊C8C2311yescharacteristic
95C32×C3⋊C8C2311yescharacteristic
96#5837—91no—
97#5770—94no—
98C32×#41—24no—
99C32×#41—31no—
100C32×#41C2×D314yescomplemented
101C32×#41C2×D311yescharacteristic, complemented
102#3832—32no—
103Q8×C3⋊D3C2×D312yescomplemented
104#3830—22no—
105#3830C2×D312yescomplemented
106D4×C3⋊D3—32no—
107D4×C3⋊D3—62no—
108#3828—21no—
109#3828—61no—
110#3828C2×D311yescharacteristic
111C2×C4⋊(C3⋊D3)—31no—
112C2×C4⋊(C3⋊D3)—34no—
113C2×C4⋊(C3⋊D3)—61no—
114C2×C4⋊(C3⋊D3)—64no—
115C2×#634—21no—
116C2×#634C2×D311yescharacteristic, complemented
117C3×D3×Q8—38no—
118C3×#237—38no—
119C3×#237—68no—
120C3×#235—34no—
121C3×#235—64no—
122C3×#235C2×D314yes—
123C2×C3×Dic6—34no—
124C2×C3×Dic6—64no—
125#3811—68no—
126D3×D12—38no—
127D3×D12—64no—
128#3799—38no—
129#3799—64no—
130C32×Q16—31no—
131C32×SD16—31no—
132C32×SD16—31no—
133C32×D8—31no—
134C2×C32×C8—31no—
135#3756—32no—
136#3755—32no—
137#3755—38no—
138#3755—68no—
139#3754—32no—
140#3753—32no—
141#3753—38no—
142#3753—68no—
143#3748—31no—
144#3748—34no—
145#3748—64no—
146#3744—32no—
147#3744—32no—
148#3743—32no—
149C8×C3⋊D3—32no—
150C3×#216C2×D314yescomplemented
151C3×#215C2×D314yescomplemented
152C3×#214C2×D314yescomplemented
153C3×#213C2×D314yescomplemented
154C2×C3×C3⋊C8C2×D314yescomplemented
155#3718—38no—
156#3717—38no—
157#3717—38no—
158#3715—38no—
159#3711—38no—
160D3×C3⋊C8—38no—
161C22×C33—21no—
162C22×C33C2×D411yescharacteristic
163C2×C3⋊(C3⋊D3)—22no—
164C2×C3⋊(C3⋊D3)—81no—
165C2×C3×C3⋊D3C2×D412yes—
166C2×C32×D3—21no—
167D3×C3⋊D3—82no—
168C33×C4—21no—
169C33×C4C2×D411yescharacteristic
170C33×C4C2411yescharacteristic
171C3×C3⋊Dic3—22no—
172C3×C3⋊Dic3C2×D412yes—
173C32×Dic3—21no—
174C3×2−1+4—91no—
175#1010—34no—
176#1009—91no—
177#1009—94no—
178C3×#135—34no—
179#951—94no—
180#950—92no—
181#950C3⋊D311yescharacteristic, complemented
182#949—94no—
183#949—91no—
184#949—92no—
185#949—184no—
186#948—91no—
187#942—91no—
188#942—92no—
189C2×#215—92no—
190C2×#215—91no—
191#918—98no—
192#916—98no—
193D3×SD16—98no—
194#911—98no—
195C2×#204—94no—
196#901—94no—
197C22×C3⋊D3—61no—
198C22×C3⋊D3—62no—
199C22×C3⋊D3—64no—
200C22×C3⋊D3—124no—
201C2×D32—68no—
202C32×Q8—24no—
203C32×Q8—31no—
204C32×Q8C22×D314yes—
205C32×Q8C22×D311yescharacteristic
206C32×D4—21no—
207C32×D4—24no—
208C32×D4—24no—
209C32×D4—31no—
210C32×D4—44no—
211C32×D4—61no—
212C32×D4C22×D314yes—
213C32×D4C22×D311yescharacteristic
214C2×C32×C4—21no—
215C2×C32×C4—24no—
216C2×C32×C4—24no—
217C2×C32×C4—44no—
218C2×C32×C4—61no—
219C2×C32×C4C22×D314yes—
220C2×C32×C4C22×D311yescharacteristic
221#638—22no—
222#638—62no—
223#638#6712yes—
224C2×C3⋊Dic3—21no—
225C2×C3⋊Dic3—22no—
226C4⋊(C3⋊D3)—32no—
227C4⋊(C3⋊D3)—38no—
228C4⋊(C3⋊D3)—68no—
229C4⋊(C3⋊D3)—62no—
230C4⋊(C3⋊D3)—61no—
231C4⋊(C3⋊D3)—64no—
232C4⋊(C3⋊D3)—124no—
233C4⋊(C3⋊D3)#6711yescharacteristic
234C4×C3⋊D3—22no—
235C4×C3⋊D3—62no—
236C4×C3⋊D3C22×D312yes—
237#634—21no—
238#634—22no—
239#634#6711yescharacteristic
240#634C22×D312yes—
241C3×#67—24no—
242C3×#67—38no—
243C3×#67—68no—
244C2×C3×Dic3—68no—
245C2×C3×Dic3—64no—
246C3×D12—34no—
247C3×D12C22×D314yes—
248C3×C4×D3—24no—
249C3×C4×D3—38no—
250C3×C4×D3—68no—
251C3×Dic6—34no—
252C3×Dic6—38no—
253C3×Dic6—68no—
254C3×Dic6—64no—
255C3×Dic6C22×D314yes—
256#626—68no—
257#626—68no—
258#624—68no—
259C32×C8—31no—
260C32×C8—31no—
261C3⋊(C3⋊C8)—32no—
262C3⋊(C3⋊C8)—38no—
263C3⋊(C3⋊C8)—68no—
264C3×C3⋊C8C22×D314yes—
265C3×C3⋊C8C22×D314yes—
266#577—271noSylow
267C2×C33—21no—
268C2×C33—41no—
269C2×C33C22×D411yescharacteristic
270C3⋊(C3⋊D3)—42no—
271C3×C3⋊D3—22no—
272C32×D3—41no—
273C3×#41—24no—
274C3×#41—34no—
275C3×#41—91no—
276C3×#41—92no—
277C3×#41—181no—
278C3×#41—182no—
279C3×#41D3214yescomplemented
280C3×#41C2×C3⋊D311yescharacteristic, complemented
281C2×C3×Q8—92no—
282C2×C3×Q8—91no—
283C2×C3×Q8—181no—
284#239—98no—
285#239—181no—
286D3×Q8—38no—
287#237—38no—
288#237—68no—
289D3×D4—98no—
290D3×D4—181no—
291D3×D4—182no—
292D3×D4—188no—
293#235—34no—
294#235—64no—
295#235—92no—
296#235—184no—
297#235C2×C3⋊D311yescharacteristic
298C2×D12—94no—
299C2×D12—91no—
300C2×D12—92no—
301C2×D12—184no—
302C2×D12—184no—
303C2×Dic6—34no—
304C2×Dic6—64no—
305C3×Q16—34no—
306C3×SD16—34no—
307C3×SD16—34no—
308C3×D8—34no—
309C2×C3×C8—34no—
310#216—98no—
311#216—92no—
312#216C2×C3⋊D311yescharacteristic, complemented
313#215—98no—
314#215—92no—
315#215—92no—
316#215—92no—
317#215—92no—
318#215—188no—
319#215C2×C3⋊D311yescharacteristic, complemented
320#214—98no—
321#214C2×C3⋊D311yescharacteristic, complemented
322#213—98no—
323#213—92no—
324#213—188no—
325#213C2×C3⋊D311yescharacteristic, complemented
326#208—94no—
327#208—91no—
328#208—92no—
329#208—184no—
330C2×C3⋊C8—92no—
331C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
332#204—98no—
333#204—98no—
334ΓL(1,𝔽25)—98no—
335D3×C8—98no—
336C22×C32—21no—
337C22×C32—24no—
338C22×C32—24no—
339C22×C32—44no—
340C22×C32—61no—
341C22×C32D3×D414yes—
342C22×C32C2×#6711yescharacteristic
343C2×C3⋊D3—62no—
344C2×C3⋊D3—68no—
345C2×C3⋊D3—128no—
346C2×C3⋊D3—242no—
347C2×C3⋊D3—241no—
348C2×C3⋊D3—244no—
349C2×C3⋊D3—248no—
350C2×C3⋊D3C2×#6712yes—
351C2×C3×D3—24no—
352C2×C3×D3—38no—
353D32—248no—
354C32×C4—21no—
355C32×C4—24no—
356C32×C4—24no—
357C32×C4—24no—
358C32×C4—44no—
359C32×C4—61no—
360C32×C4D3×D414yes—
361C32×C4C2×#6711yescharacteristic
362C32×C4C23×D314yes—
363C32×C4C23×D311yescharacteristic
364C3⋊Dic3—22no—
365C3⋊Dic3C2×#6712yes—
366C3×Dic3—24no—
367C3×Dic3—38no—
368C3×Dic3—68no—
3692−1+4—91no—
370H(5,𝔽2)—271no—
371#137—272no—
372#137—271no—
373AGL(1,ℤ/8)—272no—
374AGL(1,ℤ/8)—271no—
375#135—31no—
376#135—272no—
377C2×SD16—271no—
378C2×SD16—272no—
379#131—271no—
380C33#57711yescharacteristic, complemented, Sylow
381C22×D3—182no—
382C22×D3—188no—
383C22×D3—184no—
384C22×D3—181no—
385C22×D3—364no—
386C3×Q8—24no—
387C3×Q8—34no—
388C3×Q8—91no—
389C3×Q8—92no—
390C3×Q8—182no—
391C3×Q8—181no—
392C3×Q8C2×D3214yes—
393C3×Q8C22×C3⋊D311yescharacteristic
394C3×D4—24no—
395C3×D4—21no—
396C3×D4—24no—
397C3×D4—34no—
398C3×D4—44no—
399C3×D4—64no—
400C3×D4—92no—
401C3×D4—91no—
402C3×D4—182no—
403C3×D4C2×D3214yes—
404C3×D4C22×C3⋊D311yescharacteristic
405C2×C3×C4—24no—
406C2×C3×C4—21no—
407C2×C3×C4—24no—
408C2×C3×C4—44no—
409C2×C3×C4—64no—
410C2×C3×C4—92no—
411C2×C3×C4—182no—
412C2×C3×C4—182no—
413C2×C3×C4—181no—
414C2×C3×C4C2×D3214yes—
415C2×C3×C4C22×C3⋊D311yescharacteristic
416#67—21no—
417#67—38no—
418#67—68no—
419#67—182no—
420#67—188no—
421C2×Dic3—68no—
422C2×Dic3—64no—
423D12—34no—
424D12—98no—
425D12—92no—
426D12—188no—
427D12—188no—
428D12—182no—
429D12—184no—
430D12—181no—
431D12—364no—
432D12C22×C3⋊D311yescharacteristic
433C4×D3—21no—
434C4×D3—38no—
435C4×D3—68no—
436C4×D3—182no—
437C4×D3—188no—
438Dic6—34no—
439Dic6—38no—
440Dic6—68no—
441Dic6—64no—
442Dic6C22×C3⋊D311yescharacteristic
443C3×C8—34no—
444C3×C8—34no—
445C3⋊C8—98no—
446C3⋊C8—92no—
447C3⋊C8—188no—
448C3⋊C8C22×C3⋊D311yescharacteristic
449C3⋊C8C22×C3⋊D311yescharacteristic
450C2×C32—21no—
451C2×C32—24no—
452C2×C32—24no—
453C2×C32—44no—
454C2×C32—41no—
455C2×C32—44no—
456C2×C32—84no—
457C2×C32—121no—
458C2×C32C2×D3×D414yes—
459C2×C32C22×#6711yescharacteristic
460C3⋊D3—22no—
461C3⋊D3—122no—
462C3⋊D3—128no—
463C3⋊D3—248no—
464C3×D3—44no—
465C3×D3—68no—
466#41—31no—
467#41—91no—
468#41—92no—
469#41—181no—
470#41—182no—
471#41—272no—
472#41—541no—
473#41D3×C3⋊D311yescharacteristic, complemented
474C2×Q8—92no—
475C2×Q8—91no—
476C2×Q8—181no—
477C2×D4—272no—
478C2×D4—271no—
479C2×D4—541no—
480C2×D4—542no—
481Q16—31no—
482Q16—272no—
483SD16—31no—
484SD16—31no—
485SD16—272no—
486SD16—272no—
487SD16—272no—
488SD16—272no—
489D8—31no—
490D8—272no—
491M16—272no—
492M16—271no—
493C2×C8—31no—
494C2×C8—272no—
495C22×C3—24no—
496C22×C3—21no—
497C22×C3—24no—
498C22×C3—44no—
499C22×C3—64no—
500C22×C3—92no—
501C22×C3D3×#6714yes—
502C22×C3D4×C3⋊D311yescharacteristic
503C2×D3—21no—
504C2×D3—38no—
505C2×D3—188no—
506C2×D3—182no—
507C2×D3—368no—
508C2×D3—722no—
509C2×D3—728no—
510C2×D3—724no—
511C2×D3—721no—
512C2×D3—728no—
513C3×C4—24no—
514C3×C4—21no—
515C3×C4—24no—
516C3×C4—24no—
517C3×C4—44no—
518C3×C4—64no—
519C3×C4—92no—
520C3×C4—182no—
521C3×C4D3×#6714yes—
522C3×C4D4×C3⋊D311yescharacteristic
523C3×C4C22×D3214yes—
524C3×C4C23×C3⋊D311yescharacteristic
525Dic3—21no—
526Dic3—38no—
527Dic3—68no—
528C32—24no—
529C32#577014yescomplemented
530C32#583711yescharacteristic, complemented
531C23—542no—
532C23—541no—
533Q8—31no—
534Q8—91no—
535Q8—92no—
536Q8—182no—
537Q8—181no—
538Q8C2×D3×C3⋊D311yescharacteristic
539D4—21no—
540D4—31no—
541D4—61no—
542D4—92no—
543D4—91no—
544D4—182no—
545D4—272no—
546D4—542no—
547D4—542no—
548D4—541no—
549D4C2×D3×C3⋊D311yescharacteristic
550C2×C4—21no—
551C2×C4—61no—
552C2×C4—92no—
553C2×C4—182no—
554C2×C4—182no—
555C2×C4—181no—
556C2×C4—542no—
557C2×C4C2×D3×C3⋊D311yescharacteristic
558C8—31no—
559C8—31no—
560C8—272no—
561C2×C3—24no—
562C2×C3—21no—
563C2×C3—24no—
564C2×C3—44no—
565C2×C3—44no—
566C2×C3—41no—
567C2×C3—84no—
568C2×C3—124no—
569C2×C3—182no—
570C2×C3C2×D3×#6714yes—
571C2×C3C2×D4×C3⋊D311yescharacteristic
572D3—41no—
573D3—68no—
574D3—368no—
575D3—362no—
576D3—728no—
577C22—21no—
578C22—61no—
579C22—92no—
580C22—542no—
581C22—2161no—
582C22—2162no—
583C22C3⋊D3×#6711yescharacteristic
584C4—21no—
585C4—61no—
586C4—92no—
587C4—182no—
588C4C3⋊D3×#6711yescharacteristic
589C4C22×D3×C3⋊D311yescharacteristic
590C3—24no—
591C3#1059545714yescomplemented
592C3#1059624111yescharacteristic, complemented
593C2—21no—
594C2—41no—
595C2—121no—
596C2—182no—
597C2—1082no—
598C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC2C2×C3⋊D3×#67
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. C3×#64367C2semidirect
2. #11724581C2semidirect
3. #11723990C2semidirect
4. C32×#950C2semidirect
5. #11723586C2semidirect
6. C3⋊D3×#215C2semidirect
7. #11723561C2semidirect
8. #11723546C2semidirect
9. C2×#88797C2semidirect
10. #11723373C2semidirect
11. #11723352C2semidirect
14. C33×#41C22semidirect
16. C3×Q8×C3⋊D3C22semidirect
18. C3×#3830C22semidirect
22. C2×C3×#634C22semidirect
29. #88862C22semidirect
30. #88860C22semidirect
31. #88855C22semidirect
32. C32×#216C22semidirect
33. C32×#215C22semidirect
34. C32×#214C22semidirect
35. C32×#213C22semidirect
36. C2×C32×C3⋊C8C22semidirect
37. #88798C22semidirect
38. #88797C22semidirect
39. #88797C22semidirect
40. #88792C22semidirect
41. #88787C22semidirect
42. C3⋊D3×C3⋊C8C22semidirect
43. #64367D3semidirect
58. C3×#950D3semidirect
100. C32×#41C2×D3semidirect
101. C32×#41C2×D3semidirect
103. Q8×C3⋊D3C2×D3semidirect
105. #3830C2×D3semidirect
116. C2×#634C2×D3semidirect
150. C3×#216C2×D3semidirect
151. C3×#215C2×D3semidirect
152. C3×#214C2×D3semidirect
153. C3×#213C2×D3semidirect
154. C2×C3×C3⋊C8C2×D3semidirect
181. #950C3⋊D3semidirect
279. C3×#41D32semidirect
280. C3×#41C2×C3⋊D3semidirect
312. #216C2×C3⋊D3semidirect
319. #215C2×C3⋊D3semidirect
321. #214C2×C3⋊D3semidirect
325. #213C2×C3⋊D3semidirect
331. C2×C3⋊C8C2×C3⋊D3semidirect
380. C33#577semidirect
473. #41D3×C3⋊D3semidirect
529. C32#5770semidirect
530. C32#5837semidirect
591. C3#10595457semidirect
592. C3#10596241semidirect

Maximal subgroups

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

subgroupindex
C3×#643672
#117245812
#117239902
C32×#9502
#117235862
C3⋊D3×#2152
#117235612
#117235462
C2×#887972
#117233732
#117233522
#105962413
#105954573