Finite Group Databasebeta

← #13177993

Subgroups — #13177993

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
1C32×#1010C211yesmaximal, characteristic, complemented
2#11724581C211yesmaximal, characteristic, complemented
3#11723990C211yesmaximal, characteristic, complemented
4C3×#64159C211yesmaximal, characteristic, complemented
5#11723581C212yesmaximal, complemented
6D3×#3755C212yesmaximal, complemented
7#11723558C212yesmaximal, complemented
8#11723544C212yesmaximal, complemented
9C2×#88795C211yesmaximal, characteristic, complemented
10#11723375C211yesmaximal, characteristic, complemented
11#11723349C211yesmaximal, characteristic, complemented
12#10596308—31nomaximal
13#10595457—34nomaximal
14C33×#41C2211yescharacteristic, complemented
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
16C3×#3828C2211yescharacteristic
17C32×D3×Q8C2212yescomplemented
18C32×#237—22no—
19C32×#237C2212yescomplemented
20C32×#235—21no—
21C32×#235C2211yescharacteristic
22C2×C32×Dic6—21no—
23C2×C32×Dic6C2211yescharacteristic, complemented
24#89039—22no—
25C3⋊D3×D12—21no—
26D3×C4⋊(C3⋊D3)C2212yes—
27#89022C2212yes—
28#89018—21no—
29#88862C2212yescomplemented
30#88860C2212yescomplemented
31#88855C2211yescharacteristic, complemented
32C3×#3756C2211yescharacteristic, complemented
33C3×#3755C2211yescharacteristic, complemented
34C3×#3754C2211yescharacteristic, complemented
35C3×#3753C2211yescharacteristic, complemented
36C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
37#88798C2212yescomplemented
38#88795C2212yescomplemented
39#88795C2212yescomplemented
40#88793C2212yescomplemented
41#88786C2212yescomplemented
42D3×C3⋊(C3⋊C8)C2212yescomplemented
43C32×2−1+4—31no—
44#64366—31no—
45C3×#1010D314yescomplemented
46#64308—34no—
47#64159—32no—
48#64159D311yescharacteristic, complemented
49#64158—31no—
50#64158—34no—
51#64158—32no—
52#64158—64no—
53#64157—31no—
54#64151—31no—
55#64151—32no—
56C2×#3755—32no—
57C2×#3755—31no—
58C3×#950—34no—
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—21no—
76C3×C4⋊(C3⋊D3)C2311yescharacteristic
77C3×C4×C3⋊D3—21no—
78C3×#634C2311yescharacteristic
79C32×#67—22no—
80C32×#67D412yes—
81C2×C32×Dic3—21no—
82C2×C32×Dic3—22no—
83C32×D12D411yescharacteristic
84C32×C4×D3—22no—
85C32×C4×D3C2312yes—
86C32×Dic6—21no—
87C32×Dic6—22no—
88C32×Dic6D411yescharacteristic
89C32×Dic6C2312yes—
90#6292—22no—
91#6291—22no—
92#6289—22no—
93C3⋊(C3⋊(C3⋊C8))C2312yes—
94C3×C3⋊(C3⋊C8)C2311yescharacteristic
95C3×C3⋊(C3⋊C8)C2311yescharacteristic
96#5837—94no—
97#5770—91no—
98C32×#41—24no—
99C32×#41—31no—
100C32×#41—32no—
101C32×#41—61no—
102C32×#41—62no—
103C32×#41C2×D314yescomplemented
104C32×#41C2×D311yescharacteristic, complemented
105C2×C32×Q8—32no—
106C2×C32×Q8—31no—
107C2×C32×Q8—61no—
108#3832—61no—
109D4×C3⋊D3—61no—
110D4×C3⋊D3—62no—
111#3828—32no—
112#3828C2×D311yescharacteristic
113C2×C4⋊(C3⋊D3)—31no—
114C2×C4⋊(C3⋊D3)—34no—
115C2×C4⋊(C3⋊D3)—32no—
116C2×C4⋊(C3⋊D3)—64no—
117C3×D3×Q8C2×D318yescomplemented
118C3×#237—28no—
119C3×#237C2×D318yescomplemented
120C3×#235—24no—
121C3×#235—34no—
122C3×#235C2×D314yes—
123C2×C3×Dic6—24no—
124C2×C3×Dic6C2×D314yescomplemented
125#3811—68no—
126D3×D12—38no—
127D3×D12—64no—
128#3799—38no—
129#3799—64no—
130#3756—32no—
131#3756C2×D311yescharacteristic, complemented
132#3755—32no—
133#3755—38no—
134#3755—32no—
135#3755—32no—
136#3755—32no—
137#3755—68no—
138#3755C2×D311yescharacteristic, complemented
139#3754C2×D311yescharacteristic, complemented
140#3753—32no—
141#3753—38no—
142#3753—68no—
143#3753C2×D311yescharacteristic, complemented
144#3748—31no—
145#3748—34no—
146#3748—32no—
147#3748—64no—
148C2×C3⋊(C3⋊C8)—32no—
149C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
150C3×#216—34no—
151C3×#215—34no—
152C3×#214—34no—
153C3×#213—34no—
154C2×C3×C3⋊C8—34no—
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⋊D3—21no—
166C2×C32×D3C2×D412yes—
167D3×C3⋊D3—82no—
168C33×C4—21no—
169C33×C4C2×D411yescharacteristic
170C33×C4C2411yescharacteristic
171C3×C3⋊Dic3—21no—
172C32×Dic3—22no—
173C32×Dic3C2×D412yes—
174C3×2−1+4—34no—
175#1010C3⋊D311yescharacteristic, complemented
176#1009—94no—
177#1009—91no—
178C3×#135—91no—
179#951—91no—
180#950—34no—
181#950—98no—
182#949—98no—
183#949—94no—
184#949—91no—
185#949—184no—
186#948—94no—
187#942—98no—
188#942—94no—
189C2×#215—94no—
190C2×#215—98no—
191#918—92no—
192#916—92no—
193D3×SD16—92no—
194#911—92no—
195C2×#204—91no—
196#901—91no—
197C22×C3⋊D3—62no—
198C22×C3⋊D3—61no—
199C22×C3⋊D3—64no—
200C22×C3⋊D3—124no—
201C2×D32—68no—
202C32×Q8—24no—
203C32×Q8—31no—
204C32×Q8—32no—
205C32×Q8—62no—
206C32×Q8—61no—
207C32×Q8C22×D314yes—
208C32×Q8C22×D311yescharacteristic
209C32×D4—21no—
210C32×D4—24no—
211C32×D4—24no—
212C32×D4—32no—
213C32×D4—31no—
214C32×D4—44no—
215C32×D4—62no—
216C32×D4C22×D314yes—
217C32×D4C22×D311yescharacteristic
218C2×C32×C4—21no—
219C2×C32×C4—24no—
220C2×C32×C4—24no—
221C2×C32×C4—32no—
222C2×C32×C4—44no—
223C2×C32×C4—62no—
224C2×C32×C4—62no—
225C2×C32×C4—61no—
226C2×C32×C4C22×D314yes—
227C2×C32×C4C22×D311yescharacteristic
228#638—21no—
229#638—62no—
230C4⋊(C3⋊D3)—32no—
231C4⋊(C3⋊D3)—38no—
232C4⋊(C3⋊D3)—68no—
233C4⋊(C3⋊D3)—62no—
234C4⋊(C3⋊D3)—61no—
235C4⋊(C3⋊D3)—64no—
236C4⋊(C3⋊D3)—124no—
237C4⋊(C3⋊D3)C22×D311yescharacteristic
238C4×C3⋊D3—21no—
239C4×C3⋊D3—62no—
240#634C22×D311yescharacteristic
241C3×#67—28no—
242C3×#67—64no—
243C3×#67#6718yes—
244C2×C3×Dic3—24no—
245C2×C3×Dic3—28no—
246C3×D12—34no—
247C3×D12#6714yes—
248C3×C4×D3—28no—
249C3×C4×D3—64no—
250C3×C4×D3C22×D318yes—
251C3×Dic6—24no—
252C3×Dic6—28no—
253C3×Dic6—34no—
254C3×Dic6#6714yes—
255C3×Dic6C22×D318yes—
256#626—68no—
257#626—68no—
258#624—68no—
259C3⋊(C3⋊C8)—32no—
260C3⋊(C3⋊C8)—38no—
261C3⋊(C3⋊C8)—68no—
262C3⋊(C3⋊C8)C22×D311yescharacteristic
263C3⋊(C3⋊C8)C22×D311yescharacteristic
264C3×C3⋊C8—34no—
265C3×C3⋊C8—34no—
266#577—271noSylow
267C2×C33—21no—
268C2×C33—41no—
269C2×C33C22×D411yescharacteristic
270C3⋊(C3⋊D3)—42no—
271C3×C3⋊D3—41no—
272C32×D3—22no—
273C3×#41—24no—
274C3×#41—34no—
275C3×#41—38no—
276C3×#41—64no—
277C3×#41—68no—
278C3×#41—91no—
279C3×#41D3214yescomplemented
280C3×#41C2×C3⋊D311yescharacteristic, complemented
281C2×C3×Q8—38no—
282C2×C3×Q8—34no—
283C2×C3×Q8—64no—
284#239—92no—
285#239—184no—
286D3×Q8C2×C3⋊D312yescomplemented
287#237—22no—
288#237C2×C3⋊D312yescomplemented
289D3×D4—92no—
290D3×D4—184no—
291D3×D4—188no—
292D3×D4—182no—
293#235—21no—
294#235—34no—
295#235—98no—
296#235—181no—
297#235C2×C3⋊D311yescharacteristic
298C2×D12—98no—
299C2×D12—94no—
300C2×D12—91no—
301C2×D12—184no—
302C2×D12—181no—
303C2×Dic6—21no—
304C2×Dic6C2×C3⋊D311yescharacteristic, complemented
305C3×Q16—91no—
306C3×SD16—91no—
307C3×SD16—91no—
308C3×D8—91no—
309C2×C3×C8—91no—
310#216—34no—
311#216—98no—
312#216—92no—
313#215—34no—
314#215—98no—
315#215—98no—
316#215—98no—
317#215—98no—
318#215—92no—
319#215—188no—
320#214—34no—
321#214—92no—
322#213—34no—
323#213—98no—
324#213—92no—
325#213—188no—
326#208—98no—
327#208—94no—
328#208—91no—
329#208—184no—
330C2×C3⋊C8—34no—
331C2×C3⋊C8—98no—
332#204—92no—
333#204—92no—
334ΓL(1,𝔽25)—92no—
335D3×C8—92no—
336C22×C32—21no—
337C22×C32—24no—
338C22×C32—24no—
339C22×C32—32no—
340C22×C32—44no—
341C22×C32D3×D411yescharacteristic
342C22×C32C2×#6714yes—
343C2×C3⋊D3—21no—
344C2×C3⋊D3—62no—
345C2×C3⋊D3—68no—
346C2×C3⋊D3—128no—
347C2×C3⋊D3—242no—
348C2×C3⋊D3—241no—
349C2×C3⋊D3—244no—
350C2×C3⋊D3—248no—
351C2×C3×D3—64no—
352C2×C3×D3C2×#6718yes—
353D32—248no—
354C32×C4—21no—
355C32×C4—24no—
356C32×C4—24no—
357C32×C4—24no—
358C32×C4—32no—
359C32×C4—44no—
360C32×C4—62no—
361C32×C4D3×D411yescharacteristic
362C32×C4C2×#6714yes—
363C32×C4C23×D314yes—
364C32×C4C23×D311yescharacteristic
365C3⋊Dic3—21no—
366C3×Dic3—28no—
367C3×Dic3—64no—
368C3×Dic3C2×#6718yes—
3692−1+4—31no—
370H(5,𝔽2)—271no—
371#137—272no—
372#137—271no—
373AGL(1,ℤ/8)—272no—
374AGL(1,ℤ/8)—271no—
375#135—91no—
376#135—272no—
377C2×SD16—271no—
378C2×SD16—272no—
379#131—271no—
380C33#57711yescharacteristic, complemented, Sylow
381C22×D3—188no—
382C22×D3—184no—
383C22×D3—181no—
384C22×D3—182no—
385C22×D3—364no—
386C3×Q8—24no—
387C3×Q8—34no—
388C3×Q8—38no—
389C3×Q8—68no—
390C3×Q8—64no—
391C3×Q8—91no—
392C3×Q8C2×D3214yes—
393C3×Q8C22×C3⋊D311yescharacteristic
394C3×D4—24no—
395C3×D4—21no—
396C3×D4—24no—
397C3×D4—38no—
398C3×D4—34no—
399C3×D4—44no—
400C3×D4—68no—
401C3×D4—91no—
402C3×D4—181no—
403C3×D4C2×D3214yes—
404C3×D4C22×C3⋊D311yescharacteristic
405C2×C3×C4—24no—
406C2×C3×C4—21no—
407C2×C3×C4—24no—
408C2×C3×C4—38no—
409C2×C3×C4—44no—
410C2×C3×C4—68no—
411C2×C3×C4—68no—
412C2×C3×C4—64no—
413C2×C3×C4—181no—
414C2×C3×C4C2×D3214yes—
415C2×C3×C4C22×C3⋊D311yescharacteristic
416#67—22no—
417#67—64no—
418#67—188no—
419#67—182no—
420#67#63812yes—
421C2×Dic3—21no—
422C2×Dic3—22no—
423D12—34no—
424D12—98no—
425D12—92no—
426D12—188no—
427D12—188no—
428D12—182no—
429D12—184no—
430D12—181no—
431D12—364no—
432D12#63811yescharacteristic
433C4×D3—22no—
434C4×D3—64no—
435C4×D3—188no—
436C4×D3—182no—
437C4×D3C22×C3⋊D312yes—
438Dic6—21no—
439Dic6—22no—
440Dic6—34no—
441Dic6#63811yescharacteristic
442Dic6C22×C3⋊D312yes—
443C3×C8—91no—
444C3×C8—91no—
445C3⋊C8—34no—
446C3⋊C8—34no—
447C3⋊C8—98no—
448C3⋊C8—92no—
449C3⋊C8—188no—
450C2×C32—21no—
451C2×C32—24no—
452C2×C32—24no—
453C2×C32—44no—
454C2×C32—41no—
455C2×C32—44no—
456C2×C32—62no—
457C2×C32—84no—
458C2×C32C2×D3×D411yescharacteristic
459C2×C32C22×#6714yes—
460C3⋊D3—41no—
461C3⋊D3—122no—
462C3⋊D3—128no—
463C3⋊D3—248no—
464C3×D3—28no—
465C3×D3—124no—
466#41—31no—
467#41—32no—
468#41—61no—
469#41—62no—
470#41—91no—
471#41—272no—
472#41—541no—
473#41D3×C3⋊D311yescharacteristic, complemented
474C2×Q8—32no—
475C2×Q8—31no—
476C2×Q8—61no—
477C2×D4—272no—
478C2×D4—271no—
479C2×D4—541no—
480C2×D4—542no—
481Q16—91no—
482Q16—272no—
483SD16—91no—
484SD16—91no—
485SD16—272no—
486SD16—272no—
487SD16—272no—
488SD16—272no—
489D8—91no—
490D8—272no—
491M16—272no—
492M16—271no—
493C2×C8—91no—
494C2×C8—272no—
495C22×C3—24no—
496C22×C3—21no—
497C22×C3—24no—
498C22×C3—38no—
499C22×C3—44no—
500C22×C3—181no—
501C22×C3D3×#6714yes—
502C22×C3C2×#63811yescharacteristic
503C2×D3—64no—
504C2×D3—188no—
505C2×D3—182no—
506C2×D3—368no—
507C2×D3—722no—
508C2×D3—728no—
509C2×D3—724no—
510C2×D3—721no—
511C2×D3—728no—
512C2×D3C2×#63812yes—
513C3×C4—24no—
514C3×C4—21no—
515C3×C4—24no—
516C3×C4—24no—
517C3×C4—38no—
518C3×C4—44no—
519C3×C4—68no—
520C3×C4—181no—
521C3×C4D3×#6714yes—
522C3×C4C2×#63811yescharacteristic
523C3×C4C22×D3214yes—
524C3×C4C23×C3⋊D311yescharacteristic
525Dic3—22no—
526Dic3—64no—
527Dic3C2×#63812yes—
528C32—24no—
529C32#577011yescharacteristic, complemented
530C32#583714yescomplemented
531C23—542no—
532C23—541no—
533Q8—31no—
534Q8—32no—
535Q8—62no—
536Q8—61no—
537Q8—91no—
538Q8C2×D3×C3⋊D311yescharacteristic
539D4—21no—
540D4—32no—
541D4—31no—
542D4—62no—
543D4—91no—
544D4—181no—
545D4—272no—
546D4—542no—
547D4—542no—
548D4—541no—
549D4C2×D3×C3⋊D311yescharacteristic
550C2×C4—21no—
551C2×C4—32no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—61no—
555C2×C4—181no—
556C2×C4—542no—
557C2×C4C2×D3×C3⋊D311yescharacteristic
558C8—91no—
559C8—91no—
560C8—272no—
561C2×C3—24no—
562C2×C3—21no—
563C2×C3—24no—
564C2×C3—44no—
565C2×C3—44no—
566C2×C3—41no—
567C2×C3—68no—
568C2×C3—84no—
569C2×C3—361no—
570C2×C3C2×D3×#6714yes—
571C2×C3C22×#63811yescharacteristic
572D3—22no—
573D3—124no—
574D3—368no—
575D3—362no—
576D3—728no—
577C22—21no—
578C22—32no—
579C22—181no—
580C22—542no—
581C22—2161no—
582C22—2162no—
583C22D3×#63811yescharacteristic
584C4—21no—
585C4—32no—
586C4—62no—
587C4—181no—
588C4D3×#63811yescharacteristic
589C4C22×D3×C3⋊D311yescharacteristic
590C3—24no—
591C3#1059545714yescomplemented
592C3#1059630811yescharacteristic, complemented
593C2—21no—
594C2—41no—
595C2—62no—
596C2—361no—
597C2—1082no—
598C2C2×D3×#63811yescharacteristic
subgroupG/H
centerC2C2×D3×#638
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. C32×#1010C2semidirect
2. #11724581C2semidirect
3. #11723990C2semidirect
4. C3×#64159C2semidirect
5. #11723581C2semidirect
6. D3×#3755C2semidirect
7. #11723558C2semidirect
8. #11723544C2semidirect
9. C2×#88795C2semidirect
10. #11723375C2semidirect
11. #11723349C2semidirect
14. C33×#41C22semidirect
17. C32×D3×Q8C22semidirect
19. C32×#237C22semidirect
23. C2×C32×Dic6C22semidirect
29. #88862C22semidirect
30. #88860C22semidirect
31. #88855C22semidirect
32. C3×#3756C22semidirect
33. C3×#3755C22semidirect
34. C3×#3754C22semidirect
35. C3×#3753C22semidirect
36. C2×C3×C3⋊(C3⋊C8)C22semidirect
37. #88798C22semidirect
38. #88795C22semidirect
39. #88795C22semidirect
40. #88793C22semidirect
41. #88786C22semidirect
42. D3×C3⋊(C3⋊C8)C22semidirect
45. C3×#1010D3semidirect
48. #64159D3semidirect
103. C32×#41C2×D3semidirect
104. C32×#41C2×D3semidirect
117. C3×D3×Q8C2×D3semidirect
119. C3×#237C2×D3semidirect
124. C2×C3×Dic6C2×D3semidirect
131. #3756C2×D3semidirect
138. #3755C2×D3semidirect
139. #3754C2×D3semidirect
143. #3753C2×D3semidirect
149. C2×C3⋊(C3⋊C8)C2×D3semidirect
175. #1010C3⋊D3semidirect
279. C3×#41D32semidirect
280. C3×#41C2×C3⋊D3semidirect
286. D3×Q8C2×C3⋊D3semidirect
288. #237C2×C3⋊D3semidirect
304. C2×Dic6C2×C3⋊D3semidirect
380. C33#577semidirect
473. #41D3×C3⋊D3semidirect
529. C32#5770semidirect
530. C32#5837semidirect
591. C3#10595457semidirect
592. C3#10596308semidirect

Maximal subgroups

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

subgroupindex
C32×#10102
#117245812
#117239902
C3×#641592
#117235812
D3×#37552
#117235582
#117235442
C2×#887952
#117233752
#117233492
#105963083
#105954573