Finite Group Databasebeta

← #13178004

Subgroups — #13178004

Order 1728.

1788 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64366C211yesmaximal, characteristic, complemented
2#11724563C211yesmaximal, characteristic, complemented
3#11723992C211yesmaximal, characteristic, complemented
4C32×#950C211yesmaximal, characteristic, complemented
5#11723583C212yesmaximal, complemented
6#11723576C212yesmaximal, complemented
7#11723560C212yesmaximal, complemented
8C3⋊D3×#214C212yesmaximal, complemented
9C2×#88796C211yesmaximal, characteristic, complemented
10#11723365C211yesmaximal, characteristic, complemented
11#11723352C211yesmaximal, characteristic, complemented
12#10596241—31nomaximal
13#10595466—34nomaximal
14C33×#41C2211yescharacteristic
15C2×#6308C2211yescharacteristic, complemented
16C3×#3832C2212yes—
17C3×D4×C3⋊D3—22no—
18C3×D4×C3⋊D3C2212yes—
19C3×#3828—21no—
20C3×#3828C2211yescharacteristic
21C2×C3×C4⋊(C3⋊D3)—21no—
22C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
23C32×#235C2211yescharacteristic, complemented
24#89029—22no—
25#89021—21no—
26C3⋊D3×Dic6C2212yescomplemented
27#89014C2212yescomplemented
28D3×#634—21no—
29#88863C2212yescomplemented
30#88861C2212yescomplemented
31#88855C2211yescharacteristic, complemented
32C32×#216C2211yescharacteristic, complemented
33C32×#215C2211yescharacteristic, complemented
34C32×#214C2211yescharacteristic, complemented
35C32×#213C2211yescharacteristic, complemented
36C2×C32×C3⋊C8C2211yescharacteristic, complemented
37#88796C2212yescomplemented
38#88796C2212yescomplemented
39#88793C2212yescomplemented
40#88789C2212yescomplemented
41#88787C2212yescomplemented
42C3⋊D3×C3⋊C8C2212yescomplemented
43#64367—31no—
44#64366D311yescharacteristic, complemented
45C3×#1009—34no—
46#64298—34no—
47C32×#135—31no—
48#64160—31no—
49#64160—34no—
50#64160—64no—
51#64158—31no—
52#64127—32no—
53#64125—32no—
54SD16×C3⋊D3—32no—
55#64120—32no—
56C2×#3744—31no—
57#64110—31no—
58C3×#950D314yescomplemented
59#63947—38no—
60#63943—38no—
61#63932—38no—
62D3×#214—38no—
63C2×#3716—34no—
64#63823—34no—
65#63814—34no—
66C22×C3×C3⋊D3—21no—
67C22×C3×C3⋊D3—22no—
68C33×Q8C2311yescharacteristic
69C33×D4—21no—
70C33×D4C2311yescharacteristic
71C2×C33×C4—21no—
72C2×C33×C4C2311yescharacteristic
73C2×C3⋊(C3⋊Dic3)—21no—
74#6308—21no—
75#6308C2312yes—
76C3×#638—22no—
77C3×#638D412yes—
78C3×C4⋊(C3⋊D3)—22no—
79C3×C4⋊(C3⋊D3)—21no—
80C3×C4⋊(C3⋊D3)D411yescharacteristic
81C3×C4⋊(C3⋊D3)C2312yes—
82C3×C4×C3⋊D3—22no—
83C3×C4×C3⋊D3C2312yes—
84C3×#634D411yescharacteristic
85C32×#67—21no—
86C32×D12C2311yescharacteristic
87C32×C4×D3—21no—
88C32×Dic6C2311yescharacteristic
89#6293—22no—
90#6290—22no—
91Dic3×C3⋊D3—22no—
92D3×C3⋊Dic3—22no—
93C3⋊(C3⋊(C3⋊C8))C2312yes—
94C32×C3⋊C8C2311yescharacteristic
95C32×C3⋊C8C2311yescharacteristic
96#5836—91no—
97#5770—94no—
98C32×#41—24no—
99C32×#41—31no—
100C32×#41C2×D314yes—
101C32×#41C2×D311yescharacteristic
102#3832C2×D312yes—
103Q8×C3⋊D3—32no—
104#3830—32no—
105#3830—62no—
106D4×C3⋊D3—22no—
107D4×C3⋊D3C2×D312yes—
108#3828—21no—
109#3828—61no—
110#3828C2×D311yescharacteristic
111C2×C4⋊(C3⋊D3)—21no—
112C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
113C2×#634—31no—
114C2×#634—34no—
115C2×#634—61no—
116C2×#634—64no—
117C3×#239—38no—
118C3×D3×D4—38no—
119C3×D3×D4—68no—
120C3×#235—34no—
121C3×#235—64no—
122C3×#235C2×D314yescomplemented
123C2×C3×D12—34no—
124C2×C3×D12—64no—
125#3805—68no—
126#3795—38no—
127#3795—64no—
128D3×Dic6—38no—
129D3×Dic6—64no—
130C32×Q16—31no—
131C32×SD16—31no—
132C32×SD16—31no—
133C32×D8—31no—
134C2×C32×C8—31no—
135#3756—32no—
136#3756—38no—
137#3756—68no—
138#3755—32no—
139#3754—32no—
140#3754—38no—
141#3754—68no—
142#3753—32no—
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#3716—38no—
156#3716—38no—
157#3715—38no—
158#3713—38no—
159#3711—38no—
160D3×C3⋊C8—38no—
161C22×C33—21no—
162C22×C33C2×D411yescharacteristic
163C2×C3×C3⋊D3—22no—
164C2×C3×C3⋊D3—82no—
165C2×C3×C3⋊D3—81no—
166C2×C3×C3⋊D3C2×D412yes—
167C2×C32×D3—21no—
168C33×C4—21no—
169C33×C4C2×D411yescharacteristic
170C33×C4C2411yescharacteristic
171C3⋊(C3⋊Dic3)—22no—
172C3×C3⋊Dic3C2×D412yes—
173C32×Dic3—21no—
174C3×H(5,𝔽2)—91no—
175#1010—91no—
176#1010—94no—
177#1009—34no—
178C3×#135—34no—
179#951—94no—
180#951—91no—
181#951—92no—
182#951—184no—
183#950—92no—
184#950C3⋊D311yescharacteristic, complemented
185#949—94no—
186#948—91no—
187C2×#214—92no—
188C2×#214—91no—
189#932—91no—
190#932—92no—
191#918—98no—
192#916—98no—
193D3×SD16—98no—
194#911—98no—
195C2×#204—94no—
196#901—94no—
197C22×C3⋊D3—21no—
198C22×C3⋊D3—22no—
199C22×C3×D3—68no—
200C22×C3×D3—64no—
201C32×Q8—24no—
202C32×Q8—31no—
203C32×Q8C22×D314yes—
204C32×Q8C22×D311yescharacteristic
205C32×D4—21no—
206C32×D4—24no—
207C32×D4—24no—
208C32×D4—31no—
209C32×D4—44no—
210C32×D4—61no—
211C32×D4C22×D314yes—
212C32×D4C22×D311yescharacteristic
213C2×C32×C4—21no—
214C2×C32×C4—24no—
215C2×C32×C4—24no—
216C2×C32×C4—44no—
217C2×C32×C4—61no—
218C2×C32×C4C22×D314yes—
219C2×C32×C4C22×D311yescharacteristic
220#638—22no—
221#638—62no—
222#638#6712yes—
223C2×C3⋊Dic3—61no—
224C2×C3⋊Dic3—62no—
225C2×C3⋊Dic3—64no—
226C2×C3⋊Dic3—124no—
227C4⋊(C3⋊D3)—22no—
228C4⋊(C3⋊D3)—21no—
229C4⋊(C3⋊D3)#6711yescharacteristic
230C4⋊(C3⋊D3)C22×D312yes—
231C4×C3⋊D3—22no—
232C4×C3⋊D3—62no—
233C4×C3⋊D3C22×D312yes—
234#634—32no—
235#634—38no—
236#634—68no—
237#634—61no—
238#634—62no—
239#634—64no—
240#634—124no—
241#634#6711yescharacteristic
242C3×#67—24no—
243C3×#67—38no—
244C3×#67—68no—
245C3×D12—34no—
246C3×D12—38no—
247C3×D12—68no—
248C3×D12—64no—
249C3×D12C22×D314yes—
250C3×C4×D3—24no—
251C3×C4×D3—38no—
252C3×C4×D3—68no—
253C3×Dic6—34no—
254C3×Dic6C22×D314yes—
255#627—68no—
256#625—68no—
257D3×Dic3—68no—
258D3×Dic3—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—22no—
271C3×C3⋊D3—42no—
272C32×D3—41no—
273C3×#41—24no—
274C3×#41—34no—
275C3×#41—91no—
276C3×#41—92no—
277C3×#41—181no—
278C3×#41D3214yes—
279C3×#41C2×C3⋊D311yescharacteristic
280C2×C3×D4—92no—
281C2×C3×D4—91no—
282C2×C3×D4—181no—
283C2×C3×D4—182no—
284#239—38no—
285D3×Q8—98no—
286D3×Q8—181no—
287#237—98no—
288#237—181no—
289#237—182no—
290#237—188no—
291D3×D4—38no—
292D3×D4—68no—
293#235—34no—
294#235—64no—
295#235—92no—
296#235—184no—
297#235C2×C3⋊D311yescharacteristic, complemented
298C2×D12—34no—
299C2×D12—64no—
300C2×Dic6—94no—
301C2×Dic6—91no—
302C2×Dic6—92no—
303C2×Dic6—184no—
304C2×Dic6—184no—
305C3×Q16—34no—
306C3×SD16—34no—
307C3×SD16—34no—
308C3×D8—34no—
309C2×C3×C8—34no—
310#216—98no—
311#216—92no—
312#216—188no—
313#216C2×C3⋊D311yescharacteristic, complemented
314#215—98no—
315#215C2×C3⋊D311yescharacteristic, complemented
316#214—98no—
317#214—92no—
318#214—92no—
319#214—92no—
320#214—92no—
321#214—188no—
322#214C2×C3⋊D311yescharacteristic, complemented
323#213—98no—
324#213—92no—
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—22no—
344C2×C3⋊D3—82no—
345C2×C3⋊D3—81no—
346C2×C3⋊D3C2×#6712yes—
347C2×C3×D3—24no—
348C2×C3×D3—38no—
349C2×C3×D3—68no—
350C2×C3×D3—248no—
351C2×C3×D3—244no—
352C32×C4—21no—
353C32×C4—24no—
354C32×C4—24no—
355C32×C4—24no—
356C32×C4—44no—
357C32×C4—61no—
358C32×C4D3×D414yes—
359C32×C4C2×#6711yescharacteristic
360C32×C4C23×D314yes—
361C32×C4C23×D311yescharacteristic
362C3⋊Dic3—62no—
363C3⋊Dic3—68no—
364C3⋊Dic3—128no—
365C3⋊Dic3C2×#6712yes—
366C3×Dic3—24no—
367C3×Dic3—38no—
3682−1+4—271no—
369H(5,𝔽2)—91no—
370#137—272no—
371#137—271no—
372AGL(1,ℤ/8)—272no—
373AGL(1,ℤ/8)—271no—
374#135—31no—
375#135—272no—
376C2×SD16—271no—
377C2×SD16—272no—
378#131—271no—
379C33#57711yescharacteristic, complemented, Sylow
380C23×C3—182no—
381C23×C3—181no—
382C22×D3—68no—
383C22×D3—64no—
384C3×Q8—24no—
385C3×Q8—34no—
386C3×Q8—91no—
387C3×Q8C2×D3214yes—
388C3×Q8C22×C3⋊D311yescharacteristic
389C3×D4—24no—
390C3×D4—21no—
391C3×D4—24no—
392C3×D4—34no—
393C3×D4—44no—
394C3×D4—64no—
395C3×D4—92no—
396C3×D4—92no—
397C3×D4—91no—
398C3×D4—182no—
399C3×D4—182no—
400C3×D4—181no—
401C3×D4C2×D3214yes—
402C3×D4C22×C3⋊D311yescharacteristic
403C2×C3×C4—24no—
404C2×C3×C4—21no—
405C2×C3×C4—24no—
406C2×C3×C4—44no—
407C2×C3×C4—64no—
408C2×C3×C4—92no—
409C2×C3×C4—182no—
410C2×C3×C4C2×D3214yes—
411C2×C3×C4C22×C3⋊D311yescharacteristic
412#67—21no—
413#67—38no—
414#67—68no—
415#67—182no—
416#67—188no—
417C2×Dic3—182no—
418C2×Dic3—188no—
419C2×Dic3—184no—
420C2×Dic3—181no—
421C2×Dic3—364no—
422D12—38no—
423D12—34no—
424D12—68no—
425D12—64no—
426D12C22×C3⋊D311yescharacteristic
427C4×D3—21no—
428C4×D3—38no—
429C4×D3—68no—
430C4×D3—188no—
431C4×D3—182no—
432Dic6—34no—
433Dic6—98no—
434Dic6—92no—
435Dic6—188no—
436Dic6—182no—
437Dic6—188no—
438Dic6—184no—
439Dic6—181no—
440Dic6—364no—
441Dic6C22×C3⋊D311yescharacteristic
442C3×C8—34no—
443C3×C8—34no—
444C3⋊C8—98no—
445C3⋊C8—92no—
446C3⋊C8—188no—
447C3⋊C8C22×C3⋊D311yescharacteristic
448C3⋊C8C22×C3⋊D311yescharacteristic
449C2×C32—21no—
450C2×C32—24no—
451C2×C32—24no—
452C2×C32—44no—
453C2×C32—41no—
454C2×C32—44no—
455C2×C32—84no—
456C2×C32—121no—
457C2×C32C2×D3×D414yes—
458C2×C32C22×#6711yescharacteristic
459C3⋊D3—22no—
460C3⋊D3—42no—
461C3×D3—44no—
462C3×D3—68no—
463C3×D3—128no—
464#41—31no—
465#41—91no—
466#41—92no—
467#41—181no—
468#41—272no—
469#41—541no—
470#41—542no—
471#41D3×C3⋊D311yescharacteristic
472C2×Q8—272no—
473C2×Q8—271no—
474C2×Q8—541no—
475C2×D4—92no—
476C2×D4—91no—
477C2×D4—181no—
478C2×D4—182no—
479Q16—31no—
480Q16—272no—
481SD16—31no—
482SD16—31no—
483SD16—272no—
484SD16—272no—
485SD16—272no—
486SD16—272no—
487D8—31no—
488D8—272no—
489M16—272no—
490M16—271no—
491C2×C8—31no—
492C2×C8—272no—
493C22×C3—24no—
494C22×C3—21no—
495C22×C3—24no—
496C22×C3—44no—
497C22×C3—64no—
498C22×C3—92no—
499C22×C3—182no—
500C22×C3—721no—
501C22×C3—722no—
502C22×C3D3×#6714yes—
503C22×C3D4×C3⋊D311yescharacteristic
504C2×D3—21no—
505C2×D3—38no—
506C2×D3—68no—
507C2×D3—248no—
508C2×D3—244no—
509C3×C4—24no—
510C3×C4—21no—
511C3×C4—24no—
512C3×C4—24no—
513C3×C4—44no—
514C3×C4—64no—
515C3×C4—92no—
516C3×C4D3×#6714yes—
517C3×C4D4×C3⋊D311yescharacteristic
518C3×C4C22×D3214yes—
519C3×C4C23×C3⋊D311yescharacteristic
520Dic3—21no—
521Dic3—38no—
522Dic3—188no—
523Dic3—182no—
524Dic3—368no—
525C32—24no—
526C32#577014yescomplemented
527C32#583611yescharacteristic, complemented
528C23—182no—
529C23—181no—
530Q8—31no—
531Q8—91no—
532Q8—272no—
533Q8—542no—
534Q8—541no—
535Q8C2×D3×C3⋊D311yescharacteristic
536D4—21no—
537D4—31no—
538D4—61no—
539D4—92no—
540D4—92no—
541D4—91no—
542D4—182no—
543D4—182no—
544D4—181no—
545D4—542no—
546D4C2×D3×C3⋊D311yescharacteristic
547C2×C4—21no—
548C2×C4—61no—
549C2×C4—92no—
550C2×C4—182no—
551C2×C4—542no—
552C2×C4—542no—
553C2×C4—541no—
554C2×C4C2×D3×C3⋊D311yescharacteristic
555C8—31no—
556C8—31no—
557C8—272no—
558C2×C3—24no—
559C2×C3—21no—
560C2×C3—24no—
561C2×C3—44no—
562C2×C3—44no—
563C2×C3—41no—
564C2×C3—84no—
565C2×C3—124no—
566C2×C3—182no—
567C2×C3—362no—
568C2×C3C2×D3×#6714yes—
569C2×C3C2×D4×C3⋊D311yescharacteristic
570D3—41no—
571D3—68no—
572D3—128no—
573C22—21no—
574C22—61no—
575C22—92no—
576C22—182no—
577C22—721no—
578C22—722no—
579C22C3⋊D3×#6711yescharacteristic
580C4—21no—
581C4—61no—
582C4—92no—
583C4—542no—
584C4C3⋊D3×#6711yescharacteristic
585C4C22×D3×C3⋊D311yescharacteristic
586C3—24no—
587C3#1059546614yescomplemented
588C3#1059624111yescharacteristic, complemented
589C2—21no—
590C2—41no—
591C2—121no—
592C2—182no—
593C2—362no—
594C2C2×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×#64366C2semidirect
2. #11724563C2semidirect
3. #11723992C2semidirect
4. C32×#950C2semidirect
5. #11723583C2semidirect
6. #11723576C2semidirect
7. #11723560C2semidirect
8. C3⋊D3×#214C2semidirect
9. C2×#88796C2semidirect
10. #11723365C2semidirect
11. #11723352C2semidirect
15. C2×#6308C22semidirect
23. C32×#235C22semidirect
26. C3⋊D3×Dic6C22semidirect
27. #89014C22semidirect
29. #88863C22semidirect
30. #88861C22semidirect
31. #88855C22semidirect
32. C32×#216C22semidirect
33. C32×#215C22semidirect
34. C32×#214C22semidirect
35. C32×#213C22semidirect
36. C2×C32×C3⋊C8C22semidirect
37. #88796C22semidirect
38. #88796C22semidirect
39. #88793C22semidirect
40. #88789C22semidirect
41. #88787C22semidirect
42. C3⋊D3×C3⋊C8C22semidirect
44. #64366D3semidirect
58. C3×#950D3semidirect
122. C3×#235C2×D3semidirect
150. C3×#216C2×D3semidirect
151. C3×#215C2×D3semidirect
152. C3×#214C2×D3semidirect
153. C3×#213C2×D3semidirect
154. C2×C3×C3⋊C8C2×D3semidirect
184. #950C3⋊D3semidirect
297. #235C2×C3⋊D3semidirect
313. #216C2×C3⋊D3semidirect
315. #215C2×C3⋊D3semidirect
322. #214C2×C3⋊D3semidirect
325. #213C2×C3⋊D3semidirect
331. C2×C3⋊C8C2×C3⋊D3semidirect
379. C33#577semidirect
526. C32#5770semidirect
527. C32#5836semidirect
587. C3#10595466semidirect
588. C3#10596241semidirect

Maximal subgroups

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

subgroupindex
C3×#643662
#117245632
#117239922
C32×#9502
#117235832
#117235762
#117235602
C3⋊D3×#2142
C2×#887962
#117233652
#117233522
#105962413
#105954663