Finite Group Databasebeta

← #13176721

Subgroups — #13176721

Order 1728.

1856 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11724587C212yesmaximal, complemented
2#11723953C211yesmaximal, characteristic, complemented
3C3×D8×C3⋊D3C211yesmaximal, characteristic, complemented
4C32×D3×D8C211yesmaximal, characteristic, complemented
5#11723560C212yesmaximal, complemented
6#11723555C212yesmaximal, complemented
7#11723548C212yesmaximal, complemented
8#11723339C211yesmaximal, characteristic, complemented
9#11723336C211yesmaximal, characteristic, complemented
10#11723331C211yesmaximal, characteristic, complemented
11#11723322C211yesmaximal, characteristic, complemented
12#10596228—31nomaximal
13#10594731—34nomaximal
14#89078C2212yescomplemented
15C3×D4×C3⋊D3C2212yescomplemented
16C32×D3×D4C2212yescomplemented
17C3⋊D3×#67—22no—
18D3×#638—22no—
19C2×#6290—22no—
20#89029—22no—
21#89026C2212yescomplemented
22#89021C2212yescomplemented
23#89020C2211yescharacteristic
24#89014C2212yescomplemented
25C33×D8C2211yescharacteristic, complemented
26#88861C2212yescomplemented
27#88853C2211yescharacteristic, complemented
28C8×C3⋊(C3⋊D3)C2211yescharacteristic, complemented
29C3×#3753C2212yescomplemented
30C3×#3745C2211yescharacteristic, complemented
31C3×C8×C3⋊D3C2211yescharacteristic, complemented
32C32×#213C2212yescomplemented
33C32×D24C2211yescharacteristic, complemented
34C32×D3×C8C2211yescharacteristic, complemented
35#88796C2212yescomplemented
36#88794C2212yescomplemented
37#88789C2212yescomplemented
38#88788C2211yescharacteristic, complemented
39#88787C2211yescharacteristic, complemented
40#88786C2211yescharacteristic, complemented
41#64361—32no—
42#64313—38no—
43C2×C32×D8—31no—
44#64141—32no—
45#64121—31no—
46#64121—31no—
47#64121—34no—
48#64121—64no—
49#64120—32no—
50#64120—32no—
51D8×C3⋊D3—31no—
52D8×C3⋊D3D311yescharacteristic, complemented
53#64113—31no—
54#64110—31no—
55C3×D3×D8—34no—
56C3×D3×D8D314yescomplemented
57#63932—38no—
58#63932—38no—
59#63927—38no—
60#63807—34no—
61#63807—34no—
62#63803—34no—
63#63797—34no—
64C22×C3×C3⋊D3—22no—
65C22×C32×D3—22no—
66C2×D3×C3⋊D3—22no—
67C33×D4C2312yes—
68#6312—22no—
69C2×C3⋊(C3⋊Dic3)—22no—
70C4×C3⋊(C3⋊D3)C2311yescharacteristic
71#6308C2312yes—
72C3×#638—22no—
73C3×C4⋊(C3⋊D3)C2312yes—
74C3×C4×C3⋊D3C2311yescharacteristic
75C32×#67—22no—
76C32×D12C2312yes—
77C32×C4×D3C2311yescharacteristic
78#6293D411yescharacteristic
79#6292D411yescharacteristic
80#6291D411yescharacteristic
81#6290—22no—
82#6290—22no—
83#6290—22no—
84#6290D411yescharacteristic
85Dic3×C3⋊D3—22no—
86D3×C3⋊Dic3—22no—
87C33×C8C2311yescharacteristic
88C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
89C3×C3⋊(C3⋊C8)C2311yescharacteristic
90C32×C3⋊C8C2311yescharacteristic
91#5757—94no—
92#5757—91no—
93C2×C32×D4—32no—
94C2×#638—62no—
95C2×#638—62no—
96#3830—32no—
97#3830—38no—
98#3830—32no—
99#3830—62no—
100#3830—68no—
101D4×C3⋊D3—32no—
102D4×C3⋊D3—62no—
103D4×C3⋊D3C2×D312yescomplemented
104#3828—31no—
105#3828—32no—
106C3×D3×D4—38no—
107C3×D3×D4C2×D318yescomplemented
108D3×#67—68no—
109D3×#67—68no—
110C2×#625—68no—
111#3805—68no—
112#3802—38no—
113#3798—34no—
114#3795—38no—
115#3795—38no—
116C32×D8—24no—
117C32×D8—32no—
118C32×D8—31no—
119C32×D8C2×D314yescomplemented
120C32×D8C2×D311yescharacteristic, complemented
121C2×C32×C8—31no—
122#3754—32no—
123#3754—38no—
124#3754—32no—
125#3754—68no—
126#3753—32no—
127#3753C2×D312yescomplemented
128#3748—31no—
129#3746—31no—
130#3746—34no—
131#3746—64no—
132#3745C2×D311yescharacteristic, complemented
133#3744—32no—
134#3743—31no—
135#3743—31no—
136C8×C3⋊D3—31no—
137C8×C3⋊D3—34no—
138C8×C3⋊D3—64no—
139C8×C3⋊D3C2×D311yescharacteristic, complemented
140C3×#213—38no—
141C3×#213C2×D318yescomplemented
142C3×D24—34no—
143C3×D24C2×D314yescomplemented
144C3×D3×C8—34no—
145C3×D3×C8C2×D314yescomplemented
146#3716—38no—
147#3716—38no—
148#3713—38no—
149#3712—34no—
150#3711—34no—
151#3711—34no—
152C22×C33—22no—
153C2×C3⋊(C3⋊D3)C2×D411yescharacteristic
154C2×C3×C3⋊D3—22no—
155C2×C3×C3⋊D3—82no—
156C2×C3×C3⋊D3C2×D411yescharacteristic
157C2×C32×D3—22no—
158C2×C32×D3—82no—
159C2×C32×D3C2×D411yescharacteristic
160D3×C3⋊D3—82no—
161C33×C4C2411yescharacteristic
162C3⋊(C3⋊Dic3)—22no—
163C3⋊(C3⋊Dic3)C2×D411yescharacteristic
164C3×C3⋊Dic3C2×D411yescharacteristic
165C32×Dic3C2×D411yescharacteristic
166#1004—98no—
167#1004—92no—
168C2×C3×D8—34no—
169C2×C3×D8—91no—
170#932—98no—
171#932—92no—
172#912—94no—
173#912—94no—
174#912—91no—
175#912—91no—
176#912—184no—
177#911—98no—
178#911—98no—
179#911—92no—
180#911—92no—
181D3×D8—34no—
182D3×D8—94no—
183D3×D8—91no—
184D3×D8C3⋊D311yescharacteristic, complemented
185#904—94no—
186#904—91no—
187#901—94no—
188#901—91no—
189C23×C32—62no—
190C22×C3⋊D3—22no—
191C22×C3⋊D3—62no—
192C22×C3×D3—28no—
193C22×C3×D3—68no—
194C2×D32—68no—
195C32×D4—28no—
196C32×D4—32no—
197C32×D4—62no—
198C32×D4C22×D318yes—
199C32×D4C22×D312yes—
200C2×C32×C4—31no—
201#638—22no—
202#638—31no—
203#638—31no—
204#638—62no—
205#638—62no—
206#638—62no—
207#638—62no—
208#638—68no—
209#638—128no—
210C2×C3⋊Dic3—62no—
211C2×C3⋊Dic3—62no—
212C2×C3⋊Dic3—68no—
213C2×C3⋊Dic3—128no—
214C4⋊(C3⋊D3)—31no—
215C4⋊(C3⋊D3)C22×D312yes—
216C4×C3⋊D3—31no—
217C4×C3⋊D3—34no—
218C4×C3⋊D3—64no—
219C4×C3⋊D3—62no—
220C4×C3⋊D3C22×D311yescharacteristic
221#634—31no—
222#634—32no—
223#634—38no—
224#634—68no—
225C3×#67—28no—
226C3×#67—68no—
227C3×D12—38no—
228C3×D12C22×D318yes—
229C3×C4×D3—34no—
230C3×C4×D3C22×D314yes—
231#627—34no—
232#626—34no—
233#626—34no—
234#625—34no—
235#625—68no—
236#625—68no—
237#625—68no—
238D3×Dic3—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—21no—
255C3×C3⋊D3—42no—
256C32×D3—21no—
257C32×D3—42no—
258C2×C3×D4—38no—
259C2×C3×D4—92no—
260C2×#67—182no—
261C2×#67—188no—
262C2×#67—182no—
263C2×#67—188no—
264#237—98no—
265#237—98no—
266#237—92no—
267#237—92no—
268#237—188no—
269#237—182no—
270#237—188no—
271D3×D4—38no—
272D3×D4—98no—
273D3×D4—92no—
274D3×D4—188no—
275D3×D4—182no—
276D3×D4C2×C3⋊D312yescomplemented
277#235—94no—
278#235—91no—
279#235—98no—
280#235—92no—
281C3×D8—24no—
282C3×D8—38no—
283C3×D8—34no—
284C3×D8—92no—
285C3×D8—91no—
286C3×D8D3214yescomplemented
287C3×D8C2×C3⋊D311yescharacteristic, complemented
288C2×C3×C8—34no—
289C2×C3×C8—91no—
290#214—98no—
291#214—98no—
292#214—92no—
293#214—92no—
294#214—188no—
295#213—38no—
296#213—98no—
297#213—92no—
298#213C2×C3⋊D312yescomplemented
299#208—94no—
300#208—91no—
301Dic12—94no—
302Dic12—91no—
303Dic12—184no—
304D24—34no—
305D24C2×C3⋊D311yescharacteristic, complemented
306#204—98no—
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—
321C22×C32—62no—
322C22×C32—242no—
323C2×C3⋊D3—22no—
324C2×C3⋊D3—31no—
325C2×C3⋊D3—34no—
326C2×C3⋊D3—64no—
327C2×C3⋊D3—82no—
328C2×C3⋊D3—242no—
329C2×C3⋊D3D3×D411yescharacteristic
330C2×C3×D3—28no—
331C2×C3×D3—34no—
332C2×C3×D3—68no—
333C2×C3×D3—88no—
334C2×C3×D3—248no—
335C2×C3×D3D3×D414yes—
336D32—248no—
337C32×C4—24no—
338C32×C4—31no—
339C32×C4C23×D314yes—
340C32×C4C23×D311yescharacteristic
341C3⋊Dic3—31no—
342C3⋊Dic3—34no—
343C3⋊Dic3—64no—
344C3⋊Dic3—62no—
345C3⋊Dic3—68no—
346C3⋊Dic3—128no—
347C3⋊Dic3D3×D411yescharacteristic
348C3×Dic3—34no—
349C3×Dic3D3×D414yes—
350H(5,𝔽2)—272no—
351AGL(1,ℤ/8)—272no—
352AGL(1,ℤ/8)—272no—
353AGL(1,ℤ/8)—272no—
354#135—271no—
355#135—271no—
356#135—271no—
357C2×D8—31no—
358C2×D8—91no—
359C2×D8—271no—
360#131—271no—
361C33#57611yescharacteristic, complemented, Sylow
362C23×C3—68no—
363C23×C3—182no—
364C22×D3—22no—
365C22×D3—68no—
366C22×D3—182no—
367C22×D3—188no—
368C3×D4—28no—
369C3×D4—38no—
370C3×D4—68no—
371C3×D4—92no—
372C3×D4—182no—
373C3×D4C2×D3218yes—
374C3×D4C22×C3⋊D312yes—
375C2×C3×C4—34no—
376C2×C3×C4—91no—
377#67—22no—
378#67—68no—
379#67—94no—
380#67—94no—
381#67—91no—
382#67—91no—
383#67—182no—
384#67—188no—
385#67—182no—
386#67—188no—
387#67—188no—
388#67—182no—
389#67—188no—
390#67—182no—
391#67—368no—
392C2×Dic3—188no—
393C2×Dic3—182no—
394C2×Dic3—188no—
395C2×Dic3—182no—
396C2×Dic3—368no—
397D12—38no—
398D12—94no—
399D12—91no—
400D12C22×C3⋊D312yes—
401C4×D3—34no—
402C4×D3—94no—
403C4×D3—91no—
404C4×D3—184no—
405C4×D3—188no—
406C4×D3—182no—
407C4×D3C22×C3⋊D311yescharacteristic
408Dic6—94no—
409Dic6—91no—
410Dic6—98no—
411Dic6—92no—
412Dic6—188no—
413C3×C8—24no—
414C3×C8—34no—
415C3×C8—91no—
416C3×C8C2×D3214yes—
417C3×C8C22×C3⋊D311yescharacteristic
418C3⋊C8—34no—
419C3⋊C8—94no—
420C3⋊C8—91no—
421C3⋊C8—184no—
422C3⋊C8C22×C3⋊D311yescharacteristic
423C2×C32—24no—
424C2×C32—42no—
425C2×C32—48no—
426C2×C32—61no—
427C2×C32—88no—
428C2×C32—122no—
429C2×C32C2×D3×D414yes—
430C2×C32C2×D3×D411yescharacteristic
431C3⋊D3—21no—
432C3⋊D3—42no—
433C3⋊D3—61no—
434C3⋊D3—64no—
435C3⋊D3—124no—
436C3×D3—24no—
437C3×D3—48no—
438C3×D3—64no—
439C3×D3—128no—
440#41—271no—
441#41—272no—
442#41—272no—
443#41—272no—
444#41—542no—
445C2×D4—32no—
446C2×D4—92no—
447C2×D4—272no—
448C2×D4—542no—
449C2×D4—542no—
450C2×D4—542no—
451Q16—271no—
452SD16—272no—
453SD16—272no—
454SD16—272no—
455D8—32no—
456D8—31no—
457D8—92no—
458D8—91no—
459D8—272no—
460D8D3×C3⋊D311yescharacteristic, complemented
461M16—271no—
462M16—271no—
463M16—271no—
464C2×C8—31no—
465C2×C8—91no—
466C2×C8—271no—
467C22×C3—28no—
468C22×C3—22no—
469C22×C3—34no—
470C22×C3—48no—
471C22×C3—68no—
472C22×C3—91no—
473C22×C3—182no—
474C22×C3—248no—
475C22×C3—722no—
476C2×D3—22no—
477C2×D3—34no—
478C2×D3—68no—
479C2×D3—82no—
480C2×D3—94no—
481C2×D3—91no—
482C2×D3—184no—
483C2×D3—248no—
484C2×D3—722no—
485C2×D3—728no—
486C2×D3D4×C3⋊D311yescharacteristic
487C3×C4—24no—
488C3×C4—34no—
489C3×C4—91no—
490C3×C4C22×D3214yes—
491C3×C4C23×C3⋊D311yescharacteristic
492Dic3—34no—
493Dic3—94no—
494Dic3—91no—
495Dic3—184no—
496Dic3—188no—
497Dic3—182no—
498Dic3—368no—
499Dic3D4×C3⋊D311yescharacteristic
500C32—24no—
501C32#575714yescomplemented
502C32#575711yescharacteristic, complemented
503C23—62no—
504C23—182no—
505C23—542no—
506Q8—271no—
507Q8—272no—
508D4—32no—
509D4—62no—
510D4—92no—
511D4—182no—
512D4—271no—
513D4—271no—
514D4—271no—
515D4—542no—
516D4—542no—
517D4—542no—
518D4—542no—
519D4C2×D3×C3⋊D312yes—
520C2×C4—31no—
521C2×C4—91no—
522C2×C4—271no—
523C2×C4—542no—
524C2×C4—542no—
525C2×C4—542no—
526C8—31no—
527C8—91no—
528C8—271no—
529C8C2×D3×C3⋊D311yescharacteristic
530C2×C3—24no—
531C2×C3—42no—
532C2×C3—48no—
533C2×C3—64no—
534C2×C3—88no—
535C2×C3—128no—
536C2×C3—181no—
537C2×C3—362no—
538C2×C3D32×D414yes—
539C2×C3C2×D4×C3⋊D311yescharacteristic
540D3—21no—
541D3—42no—
542D3—64no—
543D3—128no—
544D3—181no—
545D3—184no—
546D3—364no—
547C22—22no—
548C22—31no—
549C22—62no—
550C22—91no—
551C22—182no—
552C22—242no—
553C22—271no—
554C22—722no—
555C22—2162no—
556C4—31no—
557C4—91no—
558C4—271no—
559C4—542no—
560C4C22×D3×C3⋊D311yescharacteristic
561C3—24no—
562C3#1059473114yescomplemented
563C3#1059622811yescharacteristic, complemented
564C2—42no—
565C2—61no—
566C2—122no—
567C2—181no—
568C2—362no—
569C2—541no—
570C2D3×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. #11724587C2semidirect
2. #11723953C2semidirect
3. C3×D8×C3⋊D3C2semidirect
4. C32×D3×D8C2semidirect
5. #11723560C2semidirect
6. #11723555C2semidirect
7. #11723548C2semidirect
8. #11723339C2semidirect
9. #11723336C2semidirect
10. #11723331C2semidirect
11. #11723322C2semidirect
14. #89078C22semidirect
15. C3×D4×C3⋊D3C22semidirect
16. C32×D3×D4C22semidirect
21. #89026C22semidirect
22. #89021C22semidirect
24. #89014C22semidirect
25. C33×D8C22semidirect
26. #88861C22semidirect
27. #88853C22semidirect
28. C8×C3⋊(C3⋊D3)C22semidirect
29. C3×#3753C22semidirect
30. C3×#3745C22semidirect
31. C3×C8×C3⋊D3C22semidirect
32. C32×#213C22semidirect
33. C32×D24C22semidirect
34. C32×D3×C8C22semidirect
35. #88796C22semidirect
36. #88794C22semidirect
37. #88789C22semidirect
38. #88788C22semidirect
39. #88787C22semidirect
40. #88786C22semidirect
52. D8×C3⋊D3D3semidirect
56. C3×D3×D8D3semidirect
103. D4×C3⋊D3C2×D3semidirect
107. C3×D3×D4C2×D3semidirect
119. C32×D8C2×D3semidirect
120. C32×D8C2×D3semidirect
127. #3753C2×D3semidirect
132. #3745C2×D3semidirect
139. C8×C3⋊D3C2×D3semidirect
141. C3×#213C2×D3semidirect
143. C3×D24C2×D3semidirect
145. C3×D3×C8C2×D3semidirect
184. D3×D8C3⋊D3semidirect
276. D3×D4C2×C3⋊D3semidirect
286. C3×D8D32semidirect
287. C3×D8C2×C3⋊D3semidirect
298. #213C2×C3⋊D3semidirect
305. D24C2×C3⋊D3semidirect
316. D3×C8C2×C3⋊D3semidirect
361. C33#576semidirect
460. D8D3×C3⋊D3semidirect
501. C32#5757semidirect
502. C32#5757semidirect
562. C3#10594731semidirect
563. C3#10596228semidirect

Maximal subgroups

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

subgroupindex
#117245872
#117239532
C3×D8×C3⋊D32
C32×D3×D82
#117235602
#117235552
#117235482
#117233392
#117233362
#117233312
#117233222
#105962283
#105947313