Finite Group Databasebeta

← #13176649

Subgroups — #13176649

Order 1728.

1796 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11724579C211yesmaximal, characteristic, complemented
2#11724566C211yesmaximal, characteristic, complemented
3#11723942C211yesmaximal, characteristic, complemented
4C3×#64113C211yesmaximal, characteristic, complemented
5C2×C32×#204C211yesmaximal, characteristic, complemented
6#11723375C211yesmaximal, characteristic, complemented
7#11723364C211yesmaximal, characteristic, complemented
8#11723344C212yesmaximal, complemented
9#11723342C212yesmaximal, complemented
10#11723323C212yesmaximal, complemented
11#11723321C212yesmaximal, complemented
12#10596233—31nomaximal
13#10594686—34nomaximal
14#89076C2211yescharacteristic
15C3×#3828C2211yescharacteristic
16C3×#3828C2211yescharacteristic, complemented
17C2×C32×D12C2211yescharacteristic
18C2×C32×Dic6C2211yescharacteristic, complemented
19D3×#638—22no—
20#89031—22no—
21#89026C2212yes—
22C3⋊D3×D12—21no—
23#89019C2212yescomplemented
24#89018—21no—
25#89017C2212yescomplemented
26C3⋊D3×Dic6—21no—
27#89015C2212yes—
28#89014—21no—
29C2×C33×C8C2211yescharacteristic, complemented
30#88855C2211yescharacteristic, complemented
31#88850C2212yescomplemented
32C8×C3⋊(C3⋊D3)C2212yescomplemented
33C3×#3746C2211yescharacteristic, complemented
34C3×#3745C2211yescharacteristic, complemented
35C3×#3744C2211yescharacteristic, complemented
36C3×#3744C2211yescharacteristic, complemented
37C32×#204C2212yescomplemented
38C32×#204C2212yescomplemented
39#88798C2212yescomplemented
40#88793C2212yescomplemented
41#88792C2212yescomplemented
42#88789C2212yescomplemented
43#64364—31no—
44#64361—31no—
45#64307—34no—
46#64297—34no—
47C2×C32×SD16—31no—
48#64151—31no—
49#64141—31no—
50#64125—32no—
51#64124—32no—
52#64123—32no—
53SD16×C3⋊D3—32no—
54#64113D311yescharacteristic, complemented
55#64110—31no—
56#64110—34no—
57#64110—64no—
58C3×#904—34no—
59C2×C3×#204D314yescomplemented
60#63827—34no—
61#63823—34no—
62#63811—38no—
63#63809—38no—
64#63798—38no—
65#63796—38no—
66C22×C32×D3—21no—
67C2×D3×C3⋊D3—22no—
68C2×C33×C4C2311yescharacteristic
69#6312D412yes—
70C4⋊(C3⋊(C3⋊D3))D411yescharacteristic
71C4×C3⋊(C3⋊D3)C2312yes—
72#6308D411yescharacteristic
73C3×#638—21no—
74C3×#638—21no—
75C3×C4⋊(C3⋊D3)C2311yescharacteristic
76C3×C4⋊(C3⋊D3)C2311yescharacteristic
77C3×C4×C3⋊D3—21no—
78C3×C4×C3⋊D3—21no—
79C3×#634C2311yescharacteristic
80C3×#634C2311yescharacteristic
81C2×C32×Dic3—21no—
82C32×D12—21no—
83C32×D12C2312yes—
84C32×Dic6—21no—
85C32×Dic6C2312yes—
86#6293—22no—
87#6292—22no—
88#6291—22no—
89#6290—22no—
90#6289—22no—
91Dic3×C3⋊D3—22no—
92D3×C3⋊Dic3—22no—
93C33×C8C2311yescharacteristic
94C33×C8C2311yescharacteristic
95C3⋊(C3⋊(C3⋊C8))C2312yes—
96#5770—91no—
97#5762—94no—
98C2×C32×Q8—31no—
99C2×C32×D4—31no—
100C2×#638—62no—
101#3832—32no—
102#3832—61no—
103Q8×C3⋊D3—32no—
104Q8×C3⋊D3—61no—
105#3830—32no—
106#3830—61no—
107D4×C3⋊D3—32no—
108D4×C3⋊D3—61no—
109#3828—31no—
110#3828—34no—
111#3828—62no—
112#3828—64no—
113#3828C2×D311yescharacteristic
114#3828C2×D311yescharacteristic, complemented
115C3×#235—34no—
116C3×#235—34no—
117C2×C3×D12C2×D314yes—
118C2×C3×Dic6C2×D314yescomplemented
119D3×#67—68no—
120#3804—68no—
121#3802—38no—
122D3×D12—64no—
123#3799—64no—
124#3797—38no—
125#3796—38no—
126#3796—38no—
127#3795—64no—
128D3×Dic6—64no—
129C32×SD16—32no—
130C32×SD16—32no—
131C2×C32×C8—24no—
132C2×C32×C8C2×D314yescomplemented
133C2×C32×C8C2×D311yescharacteristic, complemented
134#3756—32no—
135#3755—32no—
136#3754—32no—
137#3753—32no—
138#3748—31no—
139#3748—34no—
140#3748—64no—
141#3746C2×D311yescharacteristic, complemented
142#3745C2×D311yescharacteristic, complemented
143#3744C2×D311yescharacteristic, complemented
144#3744C2×D311yescharacteristic, complemented
145#3743—32no—
146#3743—38no—
147#3743—68no—
148C8×C3⋊D3—32no—
149C8×C3⋊D3—38no—
150C8×C3⋊D3—68no—
151C3×Dic12—34no—
152C3×D24—34no—
153C3×#204—34no—
154C3×#204—34no—
155C3×#204C2×D318yescomplemented
156C3×#204C2×D318yescomplemented
157#3718—38no—
158#3715—38no—
159#3715—38no—
160#3713—38no—
161C22×C33C2×D411yescharacteristic
162C2×C3⋊(C3⋊D3)C2×D412yes—
163C2×C3×C3⋊D3—21no—
164C2×C3×C3⋊D3—21no—
165C2×C32×D3—22no—
166C2×C32×D3—81no—
167D3×C3⋊D3—82no—
168C33×C4C2×D411yescharacteristic
169C33×C4C2411yescharacteristic
170C3⋊(C3⋊Dic3)C2×D412yes—
171C3×C3⋊Dic3—21no—
172C3×C3⋊Dic3—21no—
173C32×Dic3—22no—
174#1010—91no—
175#1009—91no—
176#1007—94no—
177#1004—94no—
178C3×#135—91no—
179C2×C3×SD16—34no—
180#951—91no—
181#949—91no—
182#942—94no—
183#932—94no—
184#918—92no—
185#916—98no—
186#916—92no—
187#915—98no—
188#914—98no—
189D3×SD16—98no—
190D3×SD16—92no—
191#911—92no—
192#904—34no—
193C2×#204C3⋊D311yescharacteristic, complemented
194#901—94no—
195#901—91no—
196#901—184no—
197C23×C32—61no—
198C22×C3⋊D3—62no—
199C22×C3×D3—24no—
200C2×D32—68no—
201C32×Q8—32no—
202C32×Q8—61no—
203C32×D4—32no—
204C32×D4—61no—
205C2×C32×C4—24no—
206C2×C32×C4—61no—
207C2×C32×C4C22×D314yes—
208C2×C32×C4C22×D311yescharacteristic
209#638—21no—
210#638—21no—
211#638—32no—
212#638—38no—
213#638—68no—
214#638—62no—
215#638—62no—
216C2×C3⋊Dic3—62no—
217C4⋊(C3⋊D3)—31no—
218C4⋊(C3⋊D3)—34no—
219C4⋊(C3⋊D3)—64no—
220C4⋊(C3⋊D3)—62no—
221C4⋊(C3⋊D3)C22×D311yescharacteristic
222C4⋊(C3⋊D3)C22×D311yescharacteristic
223C4×C3⋊D3—21no—
224C4×C3⋊D3—21no—
225C4×C3⋊D3—32no—
226C4×C3⋊D3—38no—
227C4×C3⋊D3—68no—
228C4×C3⋊D3—62no—
229C4×C3⋊D3—62no—
230#634—31no—
231#634—34no—
232#634—64no—
233#634—62no—
234#634C22×D311yescharacteristic
235#634C22×D311yescharacteristic
236C3×#67—64no—
237C3×#67—64no—
238C2×C3×Dic3—24no—
239C3×D12—24no—
240C3×D12—34no—
241C3×D12—34no—
242C3×D12C22×D318yes—
243C3×C4×D3—64no—
244C3×C4×D3—64no—
245C3×Dic6—24no—
246C3×Dic6—34no—
247C3×Dic6—34no—
248C3×Dic6C22×D318yes—
249#627—68no—
250#626—68no—
251#626—68no—
252#625—68no—
253#624—68no—
254D3×Dic3—68no—
255D3×Dic3—68no—
256C32×C8—24no—
257C32×C8—24no—
258C32×C8C22×D314yes—
259C32×C8C22×D314yes—
260C32×C8C22×D311yescharacteristic
261C32×C8C22×D311yescharacteristic
262C3⋊(C3⋊C8)—32no—
263C3⋊(C3⋊C8)—38no—
264C3⋊(C3⋊C8)—68no—
265#577—271noSylow
266C2×C33—21no—
267C2×C33C22×D411yescharacteristic
268C3⋊(C3⋊D3)—22no—
269C3×C3⋊D3—41no—
270C3×C3⋊D3—41no—
271C32×D3—42no—
272C3×#41—91no—
273C3×#41—91no—
274C2×C3×Q8—34no—
275C2×C3×D4—34no—
276C2×#67—188no—
277#239—98no—
278#239—92no—
279#239—184no—
280D3×Q8—98no—
281D3×Q8—92no—
282D3×Q8—184no—
283#237—98no—
284#237—92no—
285#237—184no—
286#237—182no—
287D3×D4—98no—
288D3×D4—92no—
289D3×D4—184no—
290D3×D4—182no—
291#235—34no—
292#235—34no—
293#235—94no—
294#235—91no—
295#235—184no—
296#235—181no—
297#235—181no—
298#235—188no—
299C2×D12—181no—
300C2×D12C2×C3⋊D311yescharacteristic
301C2×Dic6—181no—
302C2×Dic6C2×C3⋊D311yescharacteristic, complemented
303C3×Q16—91no—
304C3×SD16—38no—
305C3×SD16—38no—
306C3×SD16—91no—
307C3×SD16—91no—
308C3×D8—91no—
309C2×C3×C8—24no—
310C2×C3×C8D3214yescomplemented
311C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
312#216—98no—
313#216—92no—
314#215—98no—
315#215—92no—
316#214—98no—
317#214—92no—
318#213—98no—
319#213—92no—
320#208—94no—
321#208—91no—
322#208—184no—
323Dic12—34no—
324D24—34no—
325#204—34no—
326#204—34no—
327#204C2×C3⋊D312yescomplemented
328#204C2×C3⋊D312yescomplemented
329ΓL(1,𝔽25)—98no—
330ΓL(1,𝔽25)—92no—
331ΓL(1,𝔽25)—188no—
332D3×C8—98no—
333D3×C8—92no—
334D3×C8—188no—
335C22×C32—24no—
336C22×C32—62no—
337C22×C32—241no—
338C22×C32D3×D414yes—
339C22×C32D3×D411yescharacteristic
340C2×C3⋊D3—21no—
341C2×C3⋊D3—21no—
342C2×C3⋊D3—32no—
343C2×C3⋊D3—38no—
344C2×C3⋊D3—68no—
345C2×C3⋊D3—242no—
346C2×C3×D3—28no—
347C2×C3×D3—64no—
348C2×C3×D3—64no—
349C2×C3×D3—84no—
350D32—248no—
351C32×C4—24no—
352C32×C4—24no—
353C32×C4—62no—
354C32×C4D3×D414yes—
355C32×C4D3×D411yescharacteristic
356C32×C4C23×D314yes—
357C32×C4C23×D311yescharacteristic
358C3⋊Dic3—21no—
359C3⋊Dic3—21no—
360C3⋊Dic3—32no—
361C3⋊Dic3—38no—
362C3⋊Dic3—68no—
363C3×Dic3—28no—
364C3×Dic3—64no—
365C3×Dic3—64no—
3662−1+4—271no—
367H(5,𝔽2)—271no—
368#137—272no—
369#137—271no—
370AGL(1,ℤ/8)—272no—
371AGL(1,ℤ/8)—271no—
372#135—91no—
373#135—272no—
374C2×SD16—31no—
375C2×SD16—272no—
376#131—271no—
377C33#57711yescharacteristic, complemented, Sylow
378C23×C3—64no—
379C22×D3—21no—
380C22×D3—182no—
381C22×D3—188no—
382C3×Q8—38no—
383C3×Q8—64no—
384C3×Q8—91no—
385C3×Q8—91no—
386C3×D4—38no—
387C3×D4—64no—
388C3×D4—91no—
389C3×D4—91no—
390C3×D4—181no—
391C3×D4—181no—
392C2×C3×C4—24no—
393C2×C3×C4—64no—
394C2×C3×C4—181no—
395C2×C3×C4—181no—
396C2×C3×C4C2×D3214yes—
397C2×C3×C4C22×C3⋊D311yescharacteristic
398#67—64no—
399#67—64no—
400#67—98no—
401#67—92no—
402#67—188no—
403#67—182no—
404#67—188no—
405#67—182no—
406#67—188no—
407C2×Dic3—21no—
408C2×Dic3—182no—
409C2×Dic3—188no—
410D12—21no—
411D12—34no—
412D12—34no—
413D12—94no—
414D12—91no—
415D12—184no—
416D12—188no—
417D12—182no—
418D12C22×C3⋊D312yes—
419C4×D3—64no—
420C4×D3—64no—
421C4×D3—98no—
422C4×D3—92no—
423C4×D3—188no—
424C4×D3—182no—
425C4×D3—188no—
426C4×D3—188no—
427C4×D3—182no—
428Dic6—21no—
429Dic6—34no—
430Dic6—34no—
431Dic6—94no—
432Dic6—91no—
433Dic6—184no—
434Dic6—182no—
435Dic6—188no—
436Dic6C22×C3⋊D312yes—
437C3×C8—24no—
438C3×C8—24no—
439C3×C8C2×D3214yes—
440C3×C8C2×D3214yes—
441C3×C8C22×C3⋊D311yescharacteristic
442C3×C8C22×C3⋊D311yescharacteristic
443C3⋊C8—98no—
444C3⋊C8—92no—
445C3⋊C8—188no—
446C2×C32—21no—
447C2×C32—24no—
448C2×C32—24no—
449C2×C32—44no—
450C2×C32—122no—
451C2×C32C2×D3×D414yes—
452C2×C32C2×D3×D411yescharacteristic
453C3⋊D3—41no—
454C3⋊D3—41no—
455C3⋊D3—62no—
456C3⋊D3—68no—
457C3⋊D3—128no—
458C3×D3—48no—
459C3×D3—124no—
460C3×D3—124no—
461#41—91no—
462#41—91no—
463#41—271no—
464#41—272no—
465#41—272no—
466#41—542no—
467#41—541no—
468#41—541no—
469C2×Q8—31no—
470C2×Q8—272no—
471C2×Q8—541no—
472C2×D4—31no—
473C2×D4—272no—
474C2×D4—541no—
475C2×D4—542no—
476Q16—91no—
477Q16—272no—
478SD16—32no—
479SD16—32no—
480SD16—91no—
481SD16—91no—
482SD16—272no—
483SD16—272no—
484D8—91no—
485D8—272no—
486M16—271no—
487M16—272no—
488C2×C8—272no—
489C2×C8D3×C3⋊D311yescharacteristic, complemented
490C22×C3—24no—
491C22×C3—68no—
492C22×C3—181no—
493C22×C3—181no—
494C22×C3—244no—
495C22×C3#380214yes—
496C22×C3D4×C3⋊D311yescharacteristic
497C2×D3—22no—
498C2×D3—64no—
499C2×D3—64no—
500C2×D3—81no—
501C2×D3—98no—
502C2×D3—92no—
503C2×D3—188no—
504C2×D3—722no—
505C2×D3—728no—
506C3×C4—24no—
507C3×C4—24no—
508C3×C4—68no—
509C3×C4—181no—
510C3×C4—181no—
511C3×C4#380214yes—
512C3×C4D4×C3⋊D311yescharacteristic
513C3×C4C22×D3214yes—
514C3×C4C23×C3⋊D311yescharacteristic
515Dic3—22no—
516Dic3—64no—
517Dic3—64no—
518Dic3—98no—
519Dic3—92no—
520Dic3—188no—
521C32—24no—
522C32#576214yescomplemented
523C32#577011yescharacteristic, complemented
524C23—61no—
525C23—542no—
526Q8—32no—
527Q8—61no—
528Q8—91no—
529Q8—91no—
530Q8—271no—
531Q8—542no—
532D4—32no—
533D4—61no—
534D4—91no—
535D4—91no—
536D4—181no—
537D4—181no—
538D4—272no—
539D4—271no—
540D4—542no—
541D4—542no—
542D4—542no—
543C2×C4—61no—
544C2×C4—181no—
545C2×C4—181no—
546C2×C4—272no—
547C2×C4—542no—
548C2×C4—542no—
549C2×C4—542no—
550C2×C4C2×D3×C3⋊D311yescharacteristic
551C8—272no—
552C8C2×D3×C3⋊D311yescharacteristic
553C8C2×D3×C3⋊D311yescharacteristic
554C2×C3—24no—
555C2×C3—21no—
556C2×C3—24no—
557C2×C3—44no—
558C2×C3—128no—
559C2×C3—361no—
560C2×C3—361no—
561C2×C3C2×#380214yes—
562C2×C3C2×D4×C3⋊D311yescharacteristic
563D3—42no—
564D3—124no—
565D3—124no—
566D3—188no—
567D3—182no—
568D3—368no—
569C22—62no—
570C22—181no—
571C22—181no—
572C22—241no—
573C22—272no—
574C22—2162no—
575C22#8902611yescharacteristic
576C4—62no—
577C4—181no—
578C4—181no—
579C4—272no—
580C4#8902611yescharacteristic
581C4C22×D3×C3⋊D311yescharacteristic
582C3—24no—
583C3#1059468614yescomplemented
584C3#1059623311yescharacteristic, complemented
585C2—21no—
586C2—122no—
587C2—361no—
588C2—361no—
589C2—542no—
590C2C2×#8902611yescharacteristic
subgroupG/H
centerC2C2×#89026
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC2×C33×C8C22
socleC2×C33C22×D4
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#577
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #11724579C2semidirect
2. #11724566C2semidirect
3. #11723942C2semidirect
4. C3×#64113C2semidirect
5. C2×C32×#204C2semidirect
6. #11723375C2semidirect
7. #11723364C2semidirect
8. #11723344C2semidirect
9. #11723342C2semidirect
10. #11723323C2semidirect
11. #11723321C2semidirect
16. C3×#3828C22semidirect
18. C2×C32×Dic6C22semidirect
23. #89019C22semidirect
25. #89017C22semidirect
29. C2×C33×C8C22semidirect
30. #88855C22semidirect
31. #88850C22semidirect
32. C8×C3⋊(C3⋊D3)C22semidirect
33. C3×#3746C22semidirect
34. C3×#3745C22semidirect
35. C3×#3744C22semidirect
36. C3×#3744C22semidirect
37. C32×#204C22semidirect
38. C32×#204C22semidirect
39. #88798C22semidirect
40. #88793C22semidirect
41. #88792C22semidirect
42. #88789C22semidirect
54. #64113D3semidirect
59. C2×C3×#204D3semidirect
114. #3828C2×D3semidirect
118. C2×C3×Dic6C2×D3semidirect
132. C2×C32×C8C2×D3semidirect
133. C2×C32×C8C2×D3semidirect
141. #3746C2×D3semidirect
142. #3745C2×D3semidirect
143. #3744C2×D3semidirect
144. #3744C2×D3semidirect
155. C3×#204C2×D3semidirect
156. C3×#204C2×D3semidirect
193. C2×#204C3⋊D3semidirect
302. C2×Dic6C2×C3⋊D3semidirect
310. C2×C3×C8D32semidirect
311. C2×C3×C8C2×C3⋊D3semidirect
327. #204C2×C3⋊D3semidirect
328. #204C2×C3⋊D3semidirect
377. C33#577semidirect
489. C2×C8D3×C3⋊D3semidirect
522. C32#5762semidirect
523. C32#5770semidirect
583. C3#10594686semidirect
584. C3#10596233semidirect

Maximal subgroups

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

subgroupindex
#117245792
#117245662
#117239422
C3×#641132
C2×C32×#2042
#117233752
#117233642
#117233442
#117233422
#117233232
#117233212
#105962333
#105946863