Finite Group Databasebeta

← #13176639

Subgroups — #13176639

Order 1728.

1820 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
1#11724581C211yesmaximal, characteristic, complemented
2#11724563C211yesmaximal, characteristic, complemented
3C2×#88851C211yesmaximal, characteristic, complemented
4C3×#64113C211yesmaximal, characteristic, complemented
5C32×#901C211yesmaximal, characteristic, complemented
6#11723388C211yesmaximal, characteristic, complemented
7#11723371C211yesmaximal, characteristic, complemented
8#11723335C212yesmaximal, complemented
9#11723327C212yesmaximal, complemented
10#11723316C212yesmaximal, complemented
11D3×#3744C212yesmaximal, complemented
12#10596222—31nomaximal
13#10594679—34nomaximal
14C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
15C2×#6308C2211yescharacteristic, complemented
16C3×#3828C2211yescharacteristic
17C3×#3828C2211yescharacteristic, complemented
18C32×#235C2211yescharacteristic
19#89039—22no—
20#89029—22no—
21C3⋊D3×D12—21no—
22D3×C4⋊(C3⋊D3)C2212yes—
23#89022C2212yes—
24#89021C2212yescomplemented
25#89018—21no—
26C3⋊D3×Dic6—21no—
27#89014—21no—
28D3×#634C2212yescomplemented
29C2×C33×C8C2211yescharacteristic, complemented
30#88851C2212yescomplemented
31#88851C2212yescomplemented
32C3×#3746C2211yescharacteristic, complemented
33C3×#3745C2211yescharacteristic, complemented
34C3×#3744C2211yescharacteristic, complemented
35C3×#3744C2211yescharacteristic, complemented
36C32×#208C2211yescharacteristic, complemented
37C32×ΓL(1,𝔽25)C2212yescomplemented
38C32×D3×C8C2212yescomplemented
39#88800C2212yescomplemented
40#88797C2212yescomplemented
41#88796C2212yescomplemented
42#88791C2212yescomplemented
43#64367—31no—
44#64366—31no—
45#64308—34no—
46#64298—34no—
47C32×#131—31no—
48#64118—31no—
49#64118—32no—
50#64117—31no—
51#64117—32no—
52#64113—32no—
53#64113D311yescharacteristic, complemented
54C2×#3744—31no—
55C2×#3744—32no—
56C2×#3744—34no—
57C2×#3744—64no—
58C3×#904—34no—
59C3×#901D314yescomplemented
60#63831—34no—
61#63825—34no—
62#63806—38no—
63#63801—38no—
64#63795—38no—
65D3×#204—38no—
66C22×C3⋊(C3⋊D3)—21no—
67C2×D3×C3⋊D3—22no—
68C2×C33×C4C2311yescharacteristic
69C2×C3⋊(C3⋊Dic3)—21no—
70C4⋊(C3⋊(C3⋊D3))—21no—
71C4⋊(C3⋊(C3⋊D3))C2312yes—
72#6308—21no—
73#6308C2312yes—
74C3×#638—21no—
75C3×#638—21no—
76C3×C4⋊(C3⋊D3)C2311yescharacteristic
77C3×C4⋊(C3⋊D3)C2311yescharacteristic
78C3×C4×C3⋊D3—21no—
79C3×C4×C3⋊D3—21no—
80C3×#634C2311yescharacteristic
81C3×#634C2311yescharacteristic
82C32×#67D412yes—
83C32×D12D411yescharacteristic
84C32×C4×D3C2312yes—
85C32×Dic6D411yescharacteristic
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
95C32×C3⋊C8C2312yes—
96#5770—91no—
97#5751—94no—
98C32×#41—31no—
99#3832—61no—
100Q8×C3⋊D3—61no—
101#3830—61no—
102#3830—62no—
103D4×C3⋊D3—61no—
104D4×C3⋊D3—62no—
105#3828—32no—
106#3828—32no—
107#3828C2×D311yescharacteristic
108#3828C2×D311yescharacteristic, complemented
109C2×C4⋊(C3⋊D3)—31no—
110C2×C4⋊(C3⋊D3)—34no—
111C2×C4⋊(C3⋊D3)—32no—
112C2×C4⋊(C3⋊D3)—64no—
113C2×#634—31no—
114C2×#634—34no—
115C2×#634—32no—
116C2×#634—64no—
117C3×#235—34no—
118C3×#235—34no—
119C3×#235C2×D314yes—
120#3811—68no—
121#3805—68no—
122D3×D12—38no—
123D3×D12—64no—
124#3799—38no—
125#3799—64no—
126#3795—38no—
127#3795—64no—
128D3×Dic6—38no—
129D3×Dic6—64no—
130C32×M16—31no—
131C32×M16—32no—
132C2×C32×C8—24no—
133C2×C32×C8—32no—
134C2×C32×C8C2×D314yescomplemented
135C2×C32×C8C2×D311yescharacteristic, complemented
136#3746—32no—
137#3746C2×D311yescharacteristic, complemented
138#3745—32no—
139#3745C2×D311yescharacteristic, complemented
140#3744—32no—
141#3744—32no—
142#3744—32no—
143#3744—32no—
144#3744—38no—
145#3744—38no—
146#3744—68no—
147#3744—68no—
148#3744C2×D311yescharacteristic, complemented
149#3744C2×D311yescharacteristic, complemented
150C3×#208C2×D314yescomplemented
151C3×Dic12—34no—
152C3×D24—34no—
153C3×#204—34no—
154C3×#204—34no—
155C3×ΓL(1,𝔽25)C2×D318yescomplemented
156C3×D3×C8C2×D318yescomplemented
157#3719—38no—
158#3717—38no—
159#3716—38no—
160#3714—38no—
161C22×C33C2×D411yescharacteristic
162C2×C3⋊(C3⋊D3)—22no—
163C2×C3⋊(C3⋊D3)—81no—
164C2×C3×C3⋊D3—21no—
165C2×C3×C3⋊D3—21no—
166C2×C32×D3C2×D412yes—
167D3×C3⋊D3—82no—
168C33×C4C2×D411yescharacteristic
169C33×C4C2411yescharacteristic
170C3⋊(C3⋊Dic3)—22no—
171C3×C3⋊Dic3—21no—
172C3×C3⋊Dic3—21no—
173C32×Dic3C2×D412yes—
174#1010—94no—
175#1010—91no—
176#1009—94no—
177#1009—91no—
178C3×#135—91no—
179C3×#131—34no—
180#951—91no—
181#949—91no—
182#918—92no—
183#916—92no—
184D3×SD16—92no—
185#911—92no—
186#909—98no—
187#909—94no—
188#908—98no—
189#908—94no—
190#904—34no—
191#904—98no—
192C2×#204—98no—
193C2×#204—94no—
194C2×#204—91no—
195C2×#204—184no—
196#901C3⋊D311yescharacteristic, complemented
197C22×C3⋊D3—62no—
198C22×C3⋊D3—61no—
199C22×C3⋊D3—64no—
200C22×C3⋊D3—124no—
201C2×D32—68no—
202C32×Q8—31no—
203C32×D4—32no—
204C32×D4—31no—
205C2×C32×C4—24no—
206C2×C32×C4—32no—
207C2×C32×C4C22×D314yes—
208C2×C32×C4C22×D311yescharacteristic
209#638—21no—
210#638—21no—
211#638—62no—
212#638—62no—
213C2×C3⋊Dic3—62no—
214C2×C3⋊Dic3—61no—
215C2×C3⋊Dic3—64no—
216C2×C3⋊Dic3—124no—
217C4⋊(C3⋊D3)—32no—
218C4⋊(C3⋊D3)—38no—
219C4⋊(C3⋊D3)—68no—
220C4⋊(C3⋊D3)—62no—
221C4⋊(C3⋊D3)—61no—
222C4⋊(C3⋊D3)—64no—
223C4⋊(C3⋊D3)—124no—
224C4⋊(C3⋊D3)C22×D311yescharacteristic
225C4⋊(C3⋊D3)C22×D311yescharacteristic
226C4×C3⋊D3—21no—
227C4×C3⋊D3—21no—
228C4×C3⋊D3—62no—
229C4×C3⋊D3—62no—
230#634—32no—
231#634—38no—
232#634—68no—
233#634—62no—
234#634—61no—
235#634—64no—
236#634—124no—
237#634C22×D311yescharacteristic
238#634C22×D311yescharacteristic
239C3×#67—64no—
240C3×#67—64no—
241C3×#67D1218yes—
242C3×D12—34no—
243C3×D12—34no—
244C3×D12D1214yes—
245C3×C4×D3—64no—
246C3×C4×D3—64no—
247C3×C4×D3C22×D318yes—
248C3×Dic6—34no—
249C3×Dic6—34no—
250C3×Dic6D1214yes—
251#627—68no—
252#626—68no—
253#626—68no—
254#625—68no—
255#624—68no—
256D3×Dic3—68no—
257D3×Dic3—68no—
258C32×C8—24no—
259C32×C8—24no—
260C32×C8—32no—
261C32×C8C22×D314yes—
262C32×C8C22×D314yes—
263C32×C8C22×D311yescharacteristic
264C32×C8C22×D311yescharacteristic
265C3×C3⋊C8C22×D318yes—
266#577—271noSylow
267C2×C33—21no—
268C2×C33C22×D411yescharacteristic
269C3⋊(C3⋊D3)—42no—
270C3×C3⋊D3—41no—
271C3×C3⋊D3—41no—
272C32×D3—22no—
273C3×#41—34no—
274C3×#41—91no—
275C3×#41—91no—
276#239—92no—
277#239—184no—
278D3×Q8—92no—
279D3×Q8—184no—
280#237—92no—
281#237—184no—
282#237—188no—
283#237—182no—
284D3×D4—92no—
285D3×D4—184no—
286D3×D4—188no—
287D3×D4—182no—
288#235—34no—
289#235—34no—
290#235—98no—
291#235—98no—
292#235—181no—
293#235—181no—
294#235C2×C3⋊D311yescharacteristic
295C2×D12—98no—
296C2×D12—94no—
297C2×D12—91no—
298C2×D12—184no—
299C2×D12—181no—
300C2×Dic6—98no—
301C2×Dic6—94no—
302C2×Dic6—91no—
303C2×Dic6—184no—
304C2×Dic6—181no—
305C3×Q16—91no—
306C3×SD16—91no—
307C3×SD16—91no—
308C3×D8—91no—
309C3×M16—34no—
310C3×M16—38no—
311C2×C3×C8—24no—
312C2×C3×C8—38no—
313C2×C3×C8D3214yescomplemented
314C2×C3×C8C2×C3⋊D311yescharacteristic, complemented
315#216—92no—
316#215—92no—
317#214—92no—
318#213—92no—
319#208C2×C3⋊D311yescharacteristic, complemented
320Dic12—34no—
321Dic12—98no—
322D24—34no—
323D24—98no—
324#204—34no—
325#204—34no—
326#204—98no—
327#204—98no—
328#204—98no—
329#204—98no—
330#204—92no—
331#204—92no—
332#204—188no—
333#204—188no—
334ΓL(1,𝔽25)C2×C3⋊D312yescomplemented
335D3×C8C2×C3⋊D312yescomplemented
336C22×C32—24no—
337C22×C32—32no—
338C22×C32C2×D1214yes—
339C22×C32D3×D411yescharacteristic
340C2×C3⋊D3—21no—
341C2×C3⋊D3—21no—
342C2×C3⋊D3—62no—
343C2×C3⋊D3—68no—
344C2×C3⋊D3—128no—
345C2×C3⋊D3—242no—
346C2×C3⋊D3—241no—
347C2×C3⋊D3—244no—
348C2×C3⋊D3—248no—
349C2×C3×D3—64no—
350C2×C3×D3—64no—
351C2×C3×D3C2×D1218yes—
352D32—248no—
353C32×C4—24no—
354C32×C4—24no—
355C32×C4—32no—
356C32×C4C2×D1214yes—
357C32×C4D3×D411yescharacteristic
358C32×C4C23×D314yes—
359C32×C4C23×D311yescharacteristic
360C3⋊Dic3—21no—
361C3⋊Dic3—21no—
362C3⋊Dic3—62no—
363C3⋊Dic3—68no—
364C3⋊Dic3—128no—
365C3×Dic3—64no—
366C3×Dic3—64no—
367C3×Dic3C2×D1218yes—
3682−1+4—271no—
369H(5,𝔽2)—271no—
370#137—272no—
371#137—271no—
372AGL(1,ℤ/8)—272no—
373AGL(1,ℤ/8)—271no—
374#135—91no—
375#135—272no—
376C2×SD16—272no—
377C2×SD16—271no—
378#131—31no—
379C33#57711yescharacteristic, complemented, Sylow
380C22×D3—188no—
381C22×D3—184no—
382C22×D3—181no—
383C22×D3—182no—
384C22×D3—364no—
385C3×Q8—34no—
386C3×Q8—91no—
387C3×Q8—91no—
388C3×D4—38no—
389C3×D4—34no—
390C3×D4—91no—
391C3×D4—91no—
392C3×D4—181no—
393C3×D4—181no—
394C2×C3×C4—24no—
395C2×C3×C4—38no—
396C2×C3×C4—181no—
397C2×C3×C4—181no—
398C2×C3×C4C2×D3214yes—
399C2×C3×C4C22×C3⋊D311yescharacteristic
400#67—64no—
401#67—64no—
402#67—188no—
403#67—188no—
404#67—182no—
405#67—182no—
406#67C4⋊(C3⋊D3)12yes—
407C2×Dic3—188no—
408C2×Dic3—184no—
409C2×Dic3—181no—
410C2×Dic3—182no—
411C2×Dic3—364no—
412D12—34no—
413D12—34no—
414D12—98no—
415D12—92no—
416D12—188no—
417D12—188no—
418D12—182no—
419D12—184no—
420D12—181no—
421D12—364no—
422D12C4⋊(C3⋊D3)11yescharacteristic
423C4×D3—64no—
424C4×D3—64no—
425C4×D3—188no—
426C4×D3—188no—
427C4×D3—182no—
428C4×D3—182no—
429C4×D3C22×C3⋊D312yes—
430Dic6—34no—
431Dic6—34no—
432Dic6—98no—
433Dic6—92no—
434Dic6—188no—
435Dic6—188no—
436Dic6—184no—
437Dic6—181no—
438Dic6—182no—
439Dic6—364no—
440Dic6C4⋊(C3⋊D3)11yescharacteristic
441C3×C8—24no—
442C3×C8—24no—
443C3×C8—38no—
444C3×C8C2×D3214yes—
445C3×C8C2×D3214yes—
446C3×C8C22×C3⋊D311yescharacteristic
447C3×C8C22×C3⋊D311yescharacteristic
448C3⋊C8C22×C3⋊D312yes—
449C2×C32—28no—
450C2×C32—24no—
451C2×C32—21no—
452C2×C32—24no—
453C2×C32—62no—
454C2×C32C22×D1214yes—
455C2×C32C2×D3×D411yescharacteristic
456C3⋊D3—41no—
457C3⋊D3—41no—
458C3⋊D3—122no—
459C3⋊D3—128no—
460C3⋊D3—248no—
461C3×D3—28no—
462C3×D3—124no—
463C3×D3—124no—
464#41—31no—
465#41—91no—
466#41—91no—
467#41—272no—
468#41—272no—
469#41—541no—
470#41—541no—
471#41—542no—
472C2×Q8—272no—
473C2×Q8—271no—
474C2×Q8—541no—
475C2×D4—272no—
476C2×D4—271no—
477C2×D4—541no—
478C2×D4—542no—
479Q16—91no—
480Q16—272no—
481SD16—91no—
482SD16—91no—
483SD16—272no—
484SD16—272no—
485SD16—272no—
486SD16—272no—
487D8—91no—
488D8—272no—
489M16—31no—
490M16—32no—
491C2×C8—32no—
492C2×C8D3×C3⋊D311yescharacteristic, complemented
493C22×C3—24no—
494C22×C3—38no—
495C22×C3—181no—
496C22×C3—181no—
497C22×C3D3×D1214yes—
498C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
499C2×D3—64no—
500C2×D3—64no—
501C2×D3—188no—
502C2×D3—182no—
503C2×D3—368no—
504C2×D3—722no—
505C2×D3—728no—
506C2×D3—724no—
507C2×D3—721no—
508C2×D3—728no—
509C2×D3C2×C4⋊(C3⋊D3)12yes—
510C3×C4—24no—
511C3×C4—24no—
512C3×C4—38no—
513C3×C4—181no—
514C3×C4—181no—
515C3×C4D3×D1214yes—
516C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
517C3×C4C22×D3214yes—
518C3×C4C23×C3⋊D311yescharacteristic
519Dic3—64no—
520Dic3—64no—
521Dic3—188no—
522Dic3—182no—
523Dic3—368no—
524Dic3C2×C4⋊(C3⋊D3)12yes—
525C32—24no—
526C32#575114yescomplemented
527C32#577011yescharacteristic, complemented
528C23—542no—
529C23—541no—
530Q8—31no—
531Q8—91no—
532Q8—91no—
533Q8—272no—
534Q8—542no—
535Q8—541no—
536D4—32no—
537D4—31no—
538D4—91no—
539D4—91no—
540D4—181no—
541D4—181no—
542D4—272no—
543D4—542no—
544D4—542no—
545D4—542no—
546D4—541no—
547C2×C4—32no—
548C2×C4—181no—
549C2×C4—181no—
550C2×C4—542no—
551C2×C4—542no—
552C2×C4—542no—
553C2×C4—541no—
554C2×C4C2×D3×C3⋊D311yescharacteristic
555C8—32no—
556C8C2×D3×C3⋊D311yescharacteristic
557C8C2×D3×C3⋊D311yescharacteristic
558C2×C3—28no—
559C2×C3—24no—
560C2×C3—21no—
561C2×C3—24no—
562C2×C3—68no—
563C2×C3—361no—
564C2×C3—361no—
565C2×C3C2×D3×D1214yes—
566C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
567D3—22no—
568D3—124no—
569D3—124no—
570D3—368no—
571D3—362no—
572D3—728no—
573C22—32no—
574C22—181no—
575C22—181no—
576C22—542no—
577C22—2161no—
578C22—2162no—
579C22D3×C4⋊(C3⋊D3)11yescharacteristic
580C4—32no—
581C4—181no—
582C4—181no—
583C4—542no—
584C4D3×C4⋊(C3⋊D3)11yescharacteristic
585C4C22×D3×C3⋊D311yescharacteristic
586C3—24no—
587C3#1059467914yescomplemented
588C3#1059622211yescharacteristic, complemented
589C2—21no—
590C2—62no—
591C2—361no—
592C2—361no—
593C2—1082no—
594C2C2×D3×C4⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC2C2×D3×C4⋊(C3⋊D3)
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. #11724581C2semidirect
2. #11724563C2semidirect
3. C2×#88851C2semidirect
4. C3×#64113C2semidirect
5. C32×#901C2semidirect
6. #11723388C2semidirect
7. #11723371C2semidirect
8. #11723335C2semidirect
9. #11723327C2semidirect
10. #11723316C2semidirect
11. D3×#3744C2semidirect
15. C2×#6308C22semidirect
17. C3×#3828C22semidirect
24. #89021C22semidirect
28. D3×#634C22semidirect
29. C2×C33×C8C22semidirect
30. #88851C22semidirect
31. #88851C22semidirect
32. C3×#3746C22semidirect
33. C3×#3745C22semidirect
34. C3×#3744C22semidirect
35. C3×#3744C22semidirect
36. C32×#208C22semidirect
37. C32×ΓL(1,𝔽25)C22semidirect
38. C32×D3×C8C22semidirect
39. #88800C22semidirect
40. #88797C22semidirect
41. #88796C22semidirect
42. #88791C22semidirect
53. #64113D3semidirect
59. C3×#901D3semidirect
108. #3828C2×D3semidirect
134. C2×C32×C8C2×D3semidirect
135. C2×C32×C8C2×D3semidirect
137. #3746C2×D3semidirect
139. #3745C2×D3semidirect
148. #3744C2×D3semidirect
149. #3744C2×D3semidirect
150. C3×#208C2×D3semidirect
155. C3×ΓL(1,𝔽25)C2×D3semidirect
156. C3×D3×C8C2×D3semidirect
196. #901C3⋊D3semidirect
313. C2×C3×C8D32semidirect
314. C2×C3×C8C2×C3⋊D3semidirect
319. #208C2×C3⋊D3semidirect
334. ΓL(1,𝔽25)C2×C3⋊D3semidirect
335. D3×C8C2×C3⋊D3semidirect
379. C33#577semidirect
492. C2×C8D3×C3⋊D3semidirect
526. C32#5751semidirect
527. C32#5770semidirect
587. C3#10594679semidirect
588. C3#10596222semidirect

Maximal subgroups

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

subgroupindex
#117245812
#117245632
C2×#888512
C3×#641132
C32×#9012
#117233882
#117233712
#117233352
#117233272
#117233162
D3×#37442
#105962223
#105946793