Finite Group Databasebeta

← #13176725

Subgroups — #13176725

Order 1728.

1912 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11724589C212yesmaximal, complemented
2D8×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C3×D8×C3⋊D3C211yesmaximal, characteristic, complemented
4C32×#912C211yesmaximal, characteristic, complemented
5#11723566C212yesmaximal, complemented
6#11723562C212yesmaximal, complemented
7#11723544C212yesmaximal, complemented
8#11723345C211yesmaximal, characteristic, complemented
9#11723341C211yesmaximal, characteristic, complemented
10#11723331C211yesmaximal, characteristic, complemented
11D3×#3745C211yesmaximal, characteristic, complemented
12#10596240—31nomaximal
13#10594734—34nomaximal
14D4×C3⋊(C3⋊D3)C2212yescomplemented
15C3×D4×C3⋊D3C2212yescomplemented
16C32×#237C2212yescomplemented
17#89039—22no—
18C3⋊D3×#67—22no—
19C2×#6292—22no—
20#89031—22no—
21D3×C4⋊(C3⋊D3)C2212yescomplemented
22#89020C2211yescharacteristic
23#89018C2212yescomplemented
24#89017C2212yescomplemented
25C33×D8C2211yescharacteristic, complemented
26#88860C2212yescomplemented
27#88852C2211yescharacteristic, complemented
28C8×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
29C3×#3753C2212yescomplemented
30C3×#3745C2211yescharacteristic, complemented
31C3×C8×C3⋊D3C2211yescharacteristic, complemented
32C32×#214C2212yescomplemented
33C32×Dic12C2211yescharacteristic, complemented
34C32×D3×C8C2211yescharacteristic, complemented
35#88795C2212yescomplemented
36#88793C2212yescomplemented
37#88791C2212yescomplemented
38#88788C2211yescharacteristic, complemented
39#88787C2211yescharacteristic, complemented
40#88786C2211yescharacteristic, complemented
41#64366—32no—
42#64314—38no—
43C32×#135—31no—
44#64158—32no—
45#64128—31no—
46#64128—31no—
47#64123—32no—
48#64123—32no—
49D8×C3⋊D3—31no—
50D8×C3⋊D3—34no—
51D8×C3⋊D3—64no—
52D8×C3⋊D3D311yescharacteristic, complemented
53C2×#3745—31no—
54#64110—31no—
55C3×#912D314yescomplemented
56C3×D3×D8—34no—
57#63937—38no—
58#63934—38no—
59#63925—38no—
60#63810—34no—
61#63808—34no—
62#63803—34no—
63D3×D24—34no—
64C22×C3⋊(C3⋊D3)—22no—
65C22×C3×C3⋊D3—22no—
66C2×D3×C3⋊D3—22no—
67C33×D4C2312yes—
68#6312—22no—
69C4⋊(C3⋊(C3⋊D3))C2312yes—
70C4×C3⋊(C3⋊D3)C2311yescharacteristic
71C3×#638—22no—
72C3×C4⋊(C3⋊D3)C2312yes—
73C3×C4×C3⋊D3C2311yescharacteristic
74C32×#67—22no—
75C2×C32×Dic3—22no—
76C32×C4×D3C2311yescharacteristic
77C32×Dic6C2312yes—
78#6293D411yescharacteristic
79#6292—22no—
80#6292—22no—
81#6292—22no—
82#6292D411yescharacteristic
83#6291D411yescharacteristic
84#6290D411yescharacteristic
85#6289—22no—
86Dic3×C3⋊D3—22no—
87C33×C8C2311yescharacteristic
88C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
89C3×C3⋊(C3⋊C8)C2311yescharacteristic
90C32×C3⋊C8C2311yescharacteristic
91#5769—94no—
92#5757—91no—
93C32×#41—32no—
94#3832—32no—
95#3832—32no—
96D4×C3⋊D3—32no—
97D4×C3⋊D3—38no—
98D4×C3⋊D3—62no—
99D4×C3⋊D3—62no—
100D4×C3⋊D3—68no—
101D4×C3⋊D3C2×D312yescomplemented
102#3828—31no—
103#3828—62no—
104C2×C4⋊(C3⋊D3)—32no—
105C2×C4⋊(C3⋊D3)—62no—
106C3×#237C2×D318yescomplemented
107C3×D3×D4—38no—
108#3811—68no—
109D3×#67—68no—
110C2×#626—68no—
111#3804—68no—
112D3×D12—38no—
113#3799—38no—
114#3798—34no—
115#3796—38no—
116C32×Q16—31no—
117C32×SD16—32no—
118C32×D8—24no—
119C32×D8C2×D314yescomplemented
120C32×D8C2×D311yescharacteristic, complemented
121C2×C32×C8—31no—
122#3755—32no—
123#3755—32no—
124#3753—32no—
125#3753—38no—
126#3753—68no—
127#3753C2×D312yescomplemented
128#3748—31no—
129#3745—32no—
130#3745—31no—
131#3745—34no—
132#3745—64no—
133#3745C2×D311yescharacteristic, complemented
134#3743—31no—
135#3743—31no—
136C8×C3⋊D3—31no—
137C8×C3⋊D3—34no—
138C8×C3⋊D3—64no—
139C8×C3⋊D3C2×D311yescharacteristic, complemented
140C3×#214C2×D318yescomplemented
141C3×#213—38no—
142C3×Dic12C2×D314yescomplemented
143C3×D24—34no—
144C3×D3×C8—34no—
145C3×D3×C8C2×D314yescomplemented
146#3717—38no—
147#3715—38no—
148#3714—38no—
149#3712—34no—
150#3711—34no—
151#3711—34no—
152C22×C33—22no—
153C2×C3⋊(C3⋊D3)—22no—
154C2×C3⋊(C3⋊D3)—82no—
155C2×C3⋊(C3⋊D3)C2×D411yescharacteristic
156C2×C3×C3⋊D3—22no—
157C2×C3×C3⋊D3—82no—
158C2×C3×C3⋊D3C2×D411yescharacteristic
159C2×C32×D3C2×D411yescharacteristic
160D3×C3⋊D3—82no—
161C33×C4C2411yescharacteristic
162C3⋊(C3⋊Dic3)C2×D411yescharacteristic
163C3×C3⋊Dic3C2×D411yescharacteristic
164C32×Dic3—22no—
165C32×Dic3C2×D411yescharacteristic
166#1009—98no—
167#1004—92no—
168C3×#135—34no—
169C2×C3×D8—91no—
170#949—98no—
171#932—92no—
172#919—94no—
173#919—94no—
174#914—98no—
175#914—98no—
176#912—91no—
177#912C3⋊D311yescharacteristic, complemented
178#911—92no—
179#911—92no—
180D3×D8—34no—
181D3×D8—94no—
182D3×D8—91no—
183D3×D8—91no—
184D3×D8—184no—
185#904—91no—
186C2×D24—94no—
187#901—94no—
188#901—91no—
189C22×C3⋊D3—22no—
190C22×C3⋊D3—62no—
191C22×C3⋊D3—62no—
192C22×C3⋊D3—68no—
193C22×C3⋊D3—128no—
194C22×C3×D3—68no—
195C2×D32—68no—
196C32×Q8—32no—
197C32×D4—28no—
198C32×D4—62no—
199C32×D4C22×D318yes—
200C32×D4C22×D312yes—
201C2×C32×C4—31no—
202C2×C32×C4—62no—
203#638—22no—
204#638—31no—
205#638—31no—
206#638—62no—
207#638—68no—
208#638—128no—
209C4⋊(C3⋊D3)—31no—
210C4⋊(C3⋊D3)—32no—
211C4⋊(C3⋊D3)—38no—
212C4⋊(C3⋊D3)—68no—
213C4⋊(C3⋊D3)—62no—
214C4⋊(C3⋊D3)—62no—
215C4⋊(C3⋊D3)—62no—
216C4⋊(C3⋊D3)C22×D312yes—
217C4×C3⋊D3—31no—
218C4×C3⋊D3—34no—
219C4×C3⋊D3—64no—
220C4×C3⋊D3—62no—
221C4×C3⋊D3—62no—
222C4×C3⋊D3C22×D311yescharacteristic
223#634—31no—
224C3×#67—28no—
225C3×#67—68no—
226C2×C3×Dic3—28no—
227C3×D12—38no—
228C3×C4×D3—34no—
229C3×C4×D3C22×D314yes—
230C3×Dic6C22×D318yes—
231#627—34no—
232#626—34no—
233#626—34no—
234#626—68no—
235#626—68no—
236#626—68no—
237#625—34no—
238#624—68no—
239D3×Dic3—68no—
240C32×C8—24no—
241C32×C8—31no—
242C32×C8C22×D314yes—
243C32×C8C22×D311yescharacteristic
244C3⋊(C3⋊C8)—31no—
245C3⋊(C3⋊C8)—34no—
246C3⋊(C3⋊C8)—64no—
247C3⋊(C3⋊C8)C22×D311yescharacteristic
248C3×C3⋊C8—34no—
249C3×C3⋊C8C22×D314yes—
250#576—271noSylow
251C2×C33—42no—
252C2×C33C22×D411yescharacteristic
253C3⋊(C3⋊D3)—21no—
254C3⋊(C3⋊D3)—42no—
255C3×C3⋊D3—21no—
256C3×C3⋊D3—42no—
257C32×D3—21no—
258C3×#41—38no—
259C2×C3×D4—92no—
260C2×#67—182no—
261C2×#67—182no—
262#239—98no—
263#239—98no—
264#237—92no—
265#237—182no—
266#237C2×C3⋊D312yescomplemented
267D3×D4—38no—
268D3×D4—98no—
269D3×D4—92no—
270D3×D4—92no—
271D3×D4—188no—
272D3×D4—182no—
273D3×D4—188no—
274D3×D4—188no—
275#235—94no—
276#235—91no—
277#235—92no—
278#235—188no—
279C2×D12—98no—
280C2×D12—188no—
281C3×Q16—34no—
282C3×SD16—38no—
283C3×D8—24no—
284C3×D8—92no—
285C3×D8—91no—
286C3×D8D3214yescomplemented
287C3×D8C2×C3⋊D311yescharacteristic, complemented
288C2×C3×C8—34no—
289C2×C3×C8—91no—
290#215—98no—
291#215—98no—
292#214—92no—
293#214C2×C3⋊D312yescomplemented
294#213—38no—
295#213—98no—
296#213—92no—
297#213—92no—
298#213—188no—
299#208—94no—
300#208—91no—
301Dic12C2×C3⋊D311yescharacteristic, complemented
302D24—34no—
303D24—98no—
304D24—94no—
305D24—91no—
306D24—184no—
307#204—92no—
308ΓL(1,𝔽25)—94no—
309ΓL(1,𝔽25)—94no—
310ΓL(1,𝔽25)—91no—
311ΓL(1,𝔽25)—91no—
312D3×C8—34no—
313D3×C8—94no—
314D3×C8—91no—
315D3×C8—184no—
316D3×C8C2×C3⋊D311yescharacteristic, complemented
317C22×C32—22no—
318C22×C32—28no—
319C22×C32—31no—
320C22×C32—48no—
321C2×C3⋊D3—22no—
322C2×C3⋊D3—31no—
323C2×C3⋊D3—34no—
324C2×C3⋊D3—64no—
325C2×C3⋊D3—62no—
326C2×C3⋊D3—68no—
327C2×C3⋊D3—82no—
328C2×C3⋊D3—128no—
329C2×C3⋊D3—242no—
330C2×C3⋊D3—242no—
331C2×C3⋊D3—248no—
332C2×C3⋊D3—2416no—
333C2×C3⋊D3D3×D411yescharacteristic
334C2×C3×D3—34no—
335C2×C3×D3—68no—
336C2×C3×D3—248no—
337C2×C3×D3D3×D414yes—
338D32—248no—
339C32×C4—24no—
340C32×C4—31no—
341C32×C4—62no—
342C32×C4C23×D314yes—
343C32×C4C23×D311yescharacteristic
344C3⋊Dic3—31no—
345C3⋊Dic3—34no—
346C3⋊Dic3—64no—
347C3⋊Dic3D3×D411yescharacteristic
348C3×Dic3—28no—
349C3×Dic3—34no—
350C3×Dic3D3×D414yes—
351H(5,𝔽2)—272no—
352AGL(1,ℤ/8)—272no—
353AGL(1,ℤ/8)—272no—
354AGL(1,ℤ/8)—272no—
355#135—31no—
356#135—271no—
357#135—271no—
358C2×D8—91no—
359C2×D8—271no—
360C2×D8—271no—
361#131—271no—
362C33#57611yescharacteristic, complemented, Sylow
363C23×C3—182no—
364C22×D3—68no—
365C22×D3—188no—
366C22×D3—182no—
367C22×D3—188no—
368C22×D3—182no—
369C22×D3—368no—
370C3×Q8—38no—
371C3×D4—28no—
372C3×D4—68no—
373C3×D4—92no—
374C3×D4—182no—
375C3×D4C2×D3218yes—
376C3×D4C22×C3⋊D312yes—
377C2×C3×C4—34no—
378C2×C3×C4—68no—
379C2×C3×C4—91no—
380#67—22no—
381#67—68no—
382#67—94no—
383#67—94no—
384#67—91no—
385#67—91no—
386#67—182no—
387#67—182no—
388#67—182no—
389#67—188no—
390#67—182no—
391#67—368no—
392C2×Dic3—22no—
393C2×Dic3—182no—
394D12—38no—
395D12—94no—
396D12—91no—
397D12—98no—
398D12—92no—
399D12—188no—
400D12—188no—
401D12—188no—
402D12—188no—
403C4×D3—34no—
404C4×D3—94no—
405C4×D3—91no—
406C4×D3—184no—
407C4×D3—182no—
408C4×D3—188no—
409C4×D3—188no—
410C4×D3C22×C3⋊D311yescharacteristic
411Dic6—94no—
412Dic6—91no—
413Dic6C22×C3⋊D312yes—
414C3×C8—24no—
415C3×C8—34no—
416C3×C8—91no—
417C3×C8C2×D3214yes—
418C3×C8C22×C3⋊D311yescharacteristic
419C3⋊C8—34no—
420C3⋊C8—94no—
421C3⋊C8—91no—
422C3⋊C8—184no—
423C3⋊C8C22×C3⋊D311yescharacteristic
424C2×C32—24no—
425C2×C32—42no—
426C2×C32—48no—
427C2×C32—61no—
428C2×C32—88no—
429C2×C32C2×D3×D414yes—
430C2×C32C2×D3×D411yescharacteristic
431C3⋊D3—21no—
432C3⋊D3—42no—
433C3⋊D3—68no—
434C3⋊D3—61no—
435C3⋊D3—64no—
436C3⋊D3—122no—
437C3⋊D3—128no—
438C3⋊D3—248no—
439C3×D3—24no—
440C3×D3—64no—
441C3×D3—128no—
442#41—32no—
443#41—271no—
444#41—272no—
445#41—272no—
446#41—542no—
447C2×D4—92no—
448C2×D4—272no—
449C2×D4—272no—
450C2×D4—542no—
451C2×D4—542no—
452C2×D4—542no—
453Q16—31no—
454SD16—32no—
455SD16—272no—
456SD16—272no—
457D8—92no—
458D8—91no—
459D8—272no—
460D8—272no—
461D8—271no—
462D8D3×C3⋊D311yescharacteristic, complemented
463M16—271no—
464M16—271no—
465M16—271no—
466C2×C8—31no—
467C2×C8—91no—
468C2×C8—271no—
469C22×C3—28no—
470C22×C3—22no—
471C22×C3—34no—
472C22×C3—48no—
473C22×C3—91no—
474C22×C3—182no—
475C22×C3—722no—
476C2×D3—34no—
477C2×D3—68no—
478C2×D3—94no—
479C2×D3—91no—
480C2×D3—184no—
481C2×D3—188no—
482C2×D3—182no—
483C2×D3—248no—
484C2×D3—368no—
485C2×D3—722no—
486C2×D3—728no—
487C2×D3—722no—
488C2×D3—728no—
489C2×D3—7216no—
490C2×D3D4×C3⋊D311yescharacteristic
491C3×C4—24no—
492C3×C4—34no—
493C3×C4—68no—
494C3×C4—91no—
495C3×C4C22×D3214yes—
496C3×C4C23×C3⋊D311yescharacteristic
497Dic3—22no—
498Dic3—34no—
499Dic3—94no—
500Dic3—91no—
501Dic3—184no—
502Dic3D4×C3⋊D311yescharacteristic
503C32—24no—
504C32#575711yescharacteristic, complemented
505C32#576914yescomplemented
506C23—182no—
507C23—542no—
508C23—542no—
509Q8—32no—
510Q8—271no—
511D4—62no—
512D4—92no—
513D4—182no—
514D4—271no—
515D4—271no—
516D4—271no—
517D4—272no—
518D4—542no—
519D4—542no—
520D4—542no—
521D4—542no—
522D4C2×D3×C3⋊D312yes—
523C2×C4—31no—
524C2×C4—62no—
525C2×C4—91no—
526C2×C4—271no—
527C2×C4—542no—
528C2×C4—542no—
529C8—31no—
530C8—91no—
531C8—271no—
532C8C2×D3×C3⋊D311yescharacteristic
533C2×C3—24no—
534C2×C3—42no—
535C2×C3—48no—
536C2×C3—64no—
537C2×C3—88no—
538C2×C3—181no—
539C2×C3—362no—
540C2×C3D32×D414yes—
541C2×C3C2×D4×C3⋊D311yescharacteristic
542D3—21no—
543D3—64no—
544D3—128no—
545D3—188no—
546D3—181no—
547D3—184no—
548D3—362no—
549D3—368no—
550D3—728no—
551C22—22no—
552C22—31no—
553C22—91no—
554C22—182no—
555C22—271no—
556C22—542no—
557C22—722no—
558C22—2162no—
559C22—2162no—
560C4—31no—
561C4—62no—
562C4—91no—
563C4—271no—
564C4C22×D3×C3⋊D311yescharacteristic
565C3—24no—
566C3#1059473414yescomplemented
567C3#1059624011yescharacteristic, complemented
568C2—42no—
569C2—61no—
570C2—181no—
571C2—362no—
572C2—541no—
573C2—1082no—
574C2D3×D4×C3⋊D311yescharacteristic
subgroupG/H
centerC2D3×D4×C3⋊D3
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×D8C22
socleC2×C33C22×D4
hypercenterD8D3×C3⋊D3
hypocenterC33#576
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ D8

Complements

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

NH ≅ G/Ntype
1. #11724589C2semidirect
2. D8×C3⋊(C3⋊D3)C2semidirect
3. C3×D8×C3⋊D3C2semidirect
4. C32×#912C2semidirect
5. #11723566C2semidirect
6. #11723562C2semidirect
7. #11723544C2semidirect
8. #11723345C2semidirect
9. #11723341C2semidirect
10. #11723331C2semidirect
11. D3×#3745C2semidirect
14. D4×C3⋊(C3⋊D3)C22semidirect
15. C3×D4×C3⋊D3C22semidirect
16. C32×#237C22semidirect
21. D3×C4⋊(C3⋊D3)C22semidirect
23. #89018C22semidirect
24. #89017C22semidirect
25. C33×D8C22semidirect
26. #88860C22semidirect
27. #88852C22semidirect
28. C8×C3⋊(C3⋊D3)C22semidirect
29. C3×#3753C22semidirect
30. C3×#3745C22semidirect
31. C3×C8×C3⋊D3C22semidirect
32. C32×#214C22semidirect
33. C32×Dic12C22semidirect
34. C32×D3×C8C22semidirect
35. #88795C22semidirect
36. #88793C22semidirect
37. #88791C22semidirect
38. #88788C22semidirect
39. #88787C22semidirect
40. #88786C22semidirect
52. D8×C3⋊D3D3semidirect
55. C3×#912D3semidirect
101. D4×C3⋊D3C2×D3semidirect
106. C3×#237C2×D3semidirect
119. C32×D8C2×D3semidirect
120. C32×D8C2×D3semidirect
127. #3753C2×D3semidirect
133. #3745C2×D3semidirect
139. C8×C3⋊D3C2×D3semidirect
140. C3×#214C2×D3semidirect
142. C3×Dic12C2×D3semidirect
145. C3×D3×C8C2×D3semidirect
177. #912C3⋊D3semidirect
266. #237C2×C3⋊D3semidirect
286. C3×D8D32semidirect
287. C3×D8C2×C3⋊D3semidirect
293. #214C2×C3⋊D3semidirect
301. Dic12C2×C3⋊D3semidirect
316. D3×C8C2×C3⋊D3semidirect
362. C33#576semidirect
462. D8D3×C3⋊D3semidirect
504. C32#5757semidirect
505. C32#5769semidirect
566. C3#10594734semidirect
567. C3#10596240semidirect

Maximal subgroups

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

subgroupindex
#117245892
D8×C3⋊(C3⋊D3)2
C3×D8×C3⋊D32
C32×#9122
#117235662
#117235622
#117235442
#117233452
#117233412
#117233312
D3×#37452
#105962403
#105947343