Finite Group Databasebeta

← #13149999

Subgroups — #13149999

Order 1728.

866 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#11722515C212yesmaximal, complemented
2#11721805C211yesmaximal, characteristic, complemented
3C9×#919C211yesmaximal, characteristic, complemented
4C3×#63481C211yesmaximal, characteristic, complemented
5#11721213C212yesmaximal, complemented
6#11721211C212yesmaximal, complemented
7#11721205C212yesmaximal, complemented
8#11720818C211yesmaximal, characteristic, complemented
9D9×D24C211yesmaximal, characteristic, complemented
10#11720806C211yesmaximal, characteristic, complemented
11D3×D72C211yesmaximal, characteristic, complemented
12#10594771—31nomaximal
13#10589871—31nomaximal
14#88746C2212yescomplemented
15C9×#239C2212yescomplemented
16C3×#3701C2212yescomplemented
17#88646—22no—
18#88646C2212yescomplemented
19D9×D12—22no—
20D9×D12C2212yescomplemented
21D3×D36—22no—
22D3×D36C2212yescomplemented
23#88640—22no—
24#88640C2211yescharacteristic
25C3×C9×Q16C2211yescharacteristic, complemented
26#88548C2212yescomplemented
27#88525C2211yescharacteristic, complemented
28C8×C3⋊D9C2211yescharacteristic, complemented
29C9×#215C2212yescomplemented
30C3×#3675C2212yescomplemented
31C9×D24C2211yescharacteristic, complemented
32D3×C8×C9C2211yescharacteristic, complemented
33C3×D72C2211yescharacteristic, complemented
34C3×C8×D9C2211yescharacteristic, complemented
35#88422C2212yescomplemented
36#88421C2212yescomplemented
37#88420C2211yescharacteristic, complemented
38#88417C2212yescomplemented
39#88414C2211yescharacteristic, complemented
40#88413C2211yescharacteristic, complemented
41#64320—32no—
42#64128—31no—
43C3×#919—31no—
44C3×#919D311yescharacteristic
45#63941—32no—
46#63939—32no—
47#63939—32no—
48#63803—31no—
49#63797—31no—
50D3×D24—31no—
51D3×D24—31no—
52#63715—32no—
53C9×#135—31no—
54#63512—32no—
55#63481—31no—
56#63481—61no—
57#63481D311yescharacteristic, complemented
58#63476—32no—
59#63476—32no—
60D8×D9—31no—
61D8×D9—31no—
62C2×D72—31no—
63#63464—31no—
64C2×D3×D9—22no—
65C2×D3×D9—22no—
66C2×D3×D9—22no—
67C3×Q8×C9C2312yes—
68C4⋊(C3⋊D9)—22no—
69C4⋊(C3⋊D9)C2312yes—
70C4×C3⋊D9—22no—
71C4×C3⋊D9C2311yescharacteristic
72C9×D12—22no—
73C9×D12C2312yes—
74C4×D3×C9—22no—
75C4×D3×C9C2311yescharacteristic
76C3×D36—22no—
77C3×D36C2312yes—
78C3×C4×D9—22no—
79C3×C4×D9C2311yescharacteristic
80#6195—22no—
81#6195D411yescharacteristic
82#6194—22no—
83#6194D411yescharacteristic
84#6192—22no—
85#6192D411yescharacteristic
86#6189D411yescharacteristic
87C3×C8×C9C2311yescharacteristic
88C3⋊(C9⋊C8)C2311yescharacteristic
89C9×C3⋊C8C2311yescharacteristic
90C3×C9⋊C8C2311yescharacteristic
91#5769—91no—
92#5769—91no—
93#3832—32no—
94C3×#239—32no—
95C3×#239C2×D312yes—
96#3802—32no—
97#3802—62no—
98D3×D12—32no—
99D3×D12—32no—
100D3×D12—62no—
101D3×D12—62no—
102#3798—31no—
103#3798—62no—
104C32×Q16C2×D311yescharacteristic
105#3755—32no—
106#3745—31no—
107C8×C3⋊D3—31no—
108C3×#215—32no—
109C3×#215C2×D312yes—
110C3×D24—31no—
111C3×D24C2×D311yescharacteristic
112C3×D3×C8—31no—
113C3×D3×C8C2×D311yescharacteristic
114#3714—32no—
115#3714—32no—
116#3713—32no—
117#3712—31no—
118#3711—31no—
119#3711—31no—
120C9×#41—32no—
121#3701—32no—
122#3701—62no—
123#3701C2×D312yescomplemented
124D4×D9—32no—
125D4×D9—32no—
126D4×D9—62no—
127D4×D9—62no—
128#3697—31no—
129#3697—62no—
130C2×D36—32no—
131C2×D36—62no—
132C9×Q16—21no—
133C9×Q16C2×D311yescharacteristic, complemented
134C9×SD16—32no—
135C9×D8—31no—
136C2×C8×C9—31no—
137#3675—32no—
138#3675—62no—
139#3675C2×D312yescomplemented
140#3673—32no—
141#3673—32no—
142#3667—31no—
143D72—32no—
144D72—31no—
145D72—61no—
146D72C2×D311yescharacteristic, complemented
147#3663—31no—
148#3663—31no—
149C8×D9—31no—
150C8×D9—61no—
151C8×D9C2×D311yescharacteristic, complemented
152C2×C3⋊D9—22no—
153C2×C3⋊D9C2×D411yescharacteristic
154C2×D3×C9—22no—
155C2×D3×C9C2×D411yescharacteristic
156C2×C3×D9—22no—
157C2×C3×D9C2×D411yescharacteristic
158D3×D9—82no—
159D3×D9—82no—
160D3×D9—82no—
161C3×C4×C9—22no—
162C3×C4×C9C2411yescharacteristic
163C3⋊Dic9C2×D411yescharacteristic
164C9×Dic3C2×D411yescharacteristic
165C3×Dic9C2×D411yescharacteristic
166#1009—92no—
167#1009—92no—
168C3×#135—31no—
169C3×#135—91no—
170#949—92no—
171#949—92no—
172#919—31no—
173#919—91no—
174#919—91no—
175#919—181no—
176#919D911yescharacteristic, complemented
177#914—92no—
178#914—92no—
179#914—92no—
180#914—92no—
181D3×D8—91no—
182D3×D8—91no—
183D3×D8—91no—
184D3×D8—91no—
185C2×D24—91no—
186C2×D24—91no—
187#901—91no—
188#901—91no—
189C2×D32—62no—
190C2×D32—62no—
191C2×D32—62no—
192C32×Q8C22×D312yes—
193C4⋊(C3⋊D3)—32no—
194C4⋊(C3⋊D3)—62no—
195C4×C3⋊D3—31no—
196C4×C3⋊D3—62no—
197C3×D12—22no—
198C3×D12—32no—
199C3×D12—62no—
200C3×D12C22×D312yes—
201C3×C4×D3—22no—
202C3×C4×D3—31no—
203C3×C4×D3—62no—
204C3×C4×D3C22×D311yescharacteristic
205#627—31no—
206#626—31no—
207#626—31no—
208#626—62no—
209#626—62no—
210#625—31no—
211#625—62no—
212C22×D9—62no—
213C22×D9—62no—
214C22×D9—62no—
215C32×C8C22×D311yescharacteristic
216C3⋊(C3⋊C8)—31no—
217C3×C3⋊C8—31no—
218C3×C3⋊C8C22×D311yescharacteristic
219Q8×C9—22no—
220Q8×C9C22×D312yes—
221D4×C9—32no—
222D4×C9—62no—
223C2×C4×C9—31no—
224C2×C4×C9—62no—
225#611—31no—
226#611—31no—
227#611—62no—
228#611—62no—
229D36—22no—
230D36—31no—
231D36—32no—
232D36—62no—
233D36—62no—
234D36—62no—
235D36—122no—
236D36C22×D312yes—
237C4×D9—22no—
238C4×D9—31no—
239C4×D9—61no—
240C4×D9—62no—
241C4×D9—122no—
242C4×D9C22×D311yescharacteristic
243Dic18—31no—
244C8×C9—21no—
245C8×C9—31no—
246C8×C9C22×D311yescharacteristic
247C9⋊C8—31no—
248C9⋊C8—61no—
249C9⋊C8C22×D311yescharacteristic
250#576—271noSylow
251C2×C3×C9C22×D411yescharacteristic
252C3⋊D9—21no—
253C3⋊D9—42no—
254D3×C9—21no—
255D3×C9—42no—
256C3×D9—21no—
257C3×D9—42no—
258C3×#41—32no—
259C3×#41—92no—
260#239—32no—
261#239—92no—
262#239—92no—
263#239—182no—
264#239C2×D912yescomplemented
265D3×D4—92no—
266D3×D4—92no—
267D3×D4—92no—
268D3×D4—92no—
269D3×D4—182no—
270D3×D4—182no—
271D3×D4—182no—
272D3×D4—182no—
273#235—91no—
274#235—91no—
275#235—182no—
276#235—182no—
277C2×D12—92no—
278C2×D12—92no—
279C2×D12—182no—
280C2×D12—182no—
281C3×Q16—21no—
282C3×Q16C2×D911yescharacteristic, complemented
283C3×Q16D3211yescharacteristic
284C3×SD16—32no—
285C3×SD16—92no—
286C3×D8—31no—
287C3×D8—91no—
288C2×C3×C8—31no—
289C2×C3×C8—91no—
290#215—32no—
291#215—92no—
292#215—92no—
293#215—182no—
294#215C2×D912yescomplemented
295#213—92no—
296#213—92no—
297#213—92no—
298#213—92no—
299#208—91no—
300#208—91no—
301D24—31no—
302D24—92no—
303D24—92no—
304D24—91no—
305D24—91no—
306D24—181no—
307D24C2×D911yescharacteristic, complemented
308ΓL(1,𝔽25)—91no—
309ΓL(1,𝔽25)—91no—
310ΓL(1,𝔽25)—91no—
311ΓL(1,𝔽25)—91no—
312D3×C8—31no—
313D3×C8—91no—
314D3×C8—91no—
315D3×C8—181no—
316D3×C8C2×D911yescharacteristic, complemented
317C2×C3⋊D3—31no—
318C2×C3⋊D3—62no—
319C2×C3×D3—22no—
320C2×C3×D3—31no—
321C2×C3×D3—62no—
322C2×C3×D3D3×D411yescharacteristic
323D32—242no—
324D32—242no—
325D32—242no—
326C32×C4—22no—
327C32×C4C23×D311yescharacteristic
328C3⋊Dic3—31no—
329C3×Dic3—31no—
330C3×Dic3D3×D411yescharacteristic
331C22×C9—31no—
332C22×C9—62no—
333C2×D9—22no—
334C2×D9—31no—
335C2×D9—61no—
336C2×D9—62no—
337C2×D9—122no—
338C2×D9—242no—
339C2×D9—242no—
340C2×D9—242no—
341C2×D9D3×D411yescharacteristic
342C4×C9—22no—
343C4×C9—21no—
344C4×C9—31no—
345C4×C9—42no—
346C4×C9C23×D311yescharacteristic
347Dic9—31no—
348Dic9—61no—
349Dic9D3×D411yescharacteristic
350H(5,𝔽2)—272no—
351AGL(1,ℤ/8)—272no—
352AGL(1,ℤ/8)—272no—
353AGL(1,ℤ/8)—272no—
354#135—31no—
355#135—91no—
356#135—271no—
357C2×D8—271no—
358C2×D8—271no—
359C2×D8—271no—
360#131—271no—
361C3×C9#57611yescharacteristic, complemented, Sylow
362C22×D3—182no—
363C22×D3—182no—
364C22×D3—182no—
365C22×D3—182no—
366C22×D3—182no—
367C22×D3—182no—
368C3×Q8—22no—
369C3×Q8C22×D912yes—
370C3×Q8C2×D3212yes—
371C3×D4—32no—
372C3×D4—62no—
373C3×D4—92no—
374C3×D4—182no—
375C2×C3×C4—31no—
376C2×C3×C4—62no—
377C2×C3×C4—91no—
378C2×C3×C4—182no—
379#67—91no—
380#67—91no—
381#67—91no—
382#67—91no—
383#67—182no—
384#67—182no—
385#67—182no—
386#67—182no—
387D12—22no—
388D12—32no—
389D12—62no—
390D12—91no—
391D12—91no—
392D12—92no—
393D12—92no—
394D12—182no—
395D12—182no—
396D12—182no—
397D12—182no—
398D12—182no—
399D12—362no—
400D12C22×D912yes—
401C4×D3—22no—
402C4×D3—31no—
403C4×D3—62no—
404C4×D3—91no—
405C4×D3—91no—
406C4×D3—181no—
407C4×D3—182no—
408C4×D3—182no—
409C4×D3—362no—
410C4×D3C22×D911yescharacteristic
411Dic6—91no—
412Dic6—91no—
413C3×C8—21no—
414C3×C8—31no—
415C3×C8—91no—
416C3×C8C22×D911yescharacteristic
417C3×C8C2×D3211yescharacteristic
418C3⋊C8—31no—
419C3⋊C8—91no—
420C3⋊C8—91no—
421C3⋊C8—181no—
422C3⋊C8C22×D911yescharacteristic
423C2×C32C2×D3×D411yescharacteristic
424C3⋊D3—61no—
425C3⋊D3—122no—
426C3×D3—21no—
427C3×D3—42no—
428C3×D3—61no—
429C3×D3—122no—
430C2×C9—21no—
431C2×C9—61no—
432C2×C9—122no—
433C2×C9C2×D3×D411yescharacteristic
434D9—21no—
435D9—42no—
436D9—61no—
437D9—121no—
438D9—122no—
439D9—242no—
440#41—32no—
441#41—92no—
442#41—271no—
443#41—272no—
444#41—542no—
445C2×D4—272no—
446C2×D4—272no—
447C2×D4—272no—
448C2×D4—542no—
449C2×D4—542no—
450C2×D4—542no—
451Q16D3×D911yescharacteristic, complemented
452SD16—32no—
453SD16—92no—
454SD16—272no—
455D8—31no—
456D8—91no—
457D8—272no—
458D8—272no—
459D8—272no—
460D8—271no—
461M16—271no—
462M16—271no—
463M16—271no—
464C2×C8—31no—
465C2×C8—91no—
466C2×C8—271no—
467C22×C3—31no—
468C22×C3—62no—
469C22×C3—91no—
470C22×C3—182no—
471C2×D3—22no—
472C2×D3—31no—
473C2×D3—62no—
474C2×D3—91no—
475C2×D3—91no—
476C2×D3—181no—
477C2×D3—182no—
478C2×D3—182no—
479C2×D3—362no—
480C2×D3—722no—
481C2×D3—722no—
482C2×D3—722no—
483C2×D3—722no—
484C2×D3—722no—
485C2×D3—722no—
486C2×D3D4×D911yescharacteristic
487C3×C4—22no—
488C3×C4—22no—
489C3×C4—21no—
490C3×C4—31no—
491C3×C4—42no—
492C3×C4—91no—
493C3×C4C23×D911yescharacteristic
494C3×C4C22×D3211yescharacteristic
495Dic3—31no—
496Dic3—91no—
497Dic3—91no—
498Dic3—181no—
499Dic3D4×D911yescharacteristic
500C32#576911yescharacteristic
501C9—21no—
502C9#576911yescharacteristic, complemented
503C23—542no—
504C23—542no—
505C23—542no—
506Q8—271no—
507Q8C2×D3×D912yes—
508D4—32no—
509D4—62no—
510D4—92no—
511D4—182no—
512D4—271no—
513D4—271no—
514D4—271no—
515D4—272no—
516D4—542no—
517D4—542no—
518D4—542no—
519D4—542no—
520C2×C4—31no—
521C2×C4—62no—
522C2×C4—91no—
523C2×C4—182no—
524C2×C4—271no—
525C2×C4—542no—
526C8—31no—
527C8—91no—
528C8—271no—
529C8C2×D3×D911yescharacteristic
530C2×C3—21no—
531C2×C3—61no—
532C2×C3—122no—
533C2×C3—181no—
534C2×C3—362no—
535C2×C3C2×D4×D911yescharacteristic
536C2×C3D32×D411yescharacteristic
537D3—21no—
538D3—42no—
539D3—61no—
540D3—122no—
541D3—181no—
542D3—181no—
543D3—361no—
544D3—362no—
545D3—362no—
546D3—722no—
547C22—31no—
548C22—62no—
549C22—91no—
550C22—182no—
551C22—271no—
552C22—542no—
553C22—2162no—
554C22—2162no—
555C22—2162no—
556C4—22no—
557C4—31no—
558C4—91no—
559C4—271no—
560C4C22×D3×D911yescharacteristic
561C3—21no—
562C3#1058987111yescharacteristic, complemented
563C3#1059477111yescharacteristic
564C2—61no—
565C2—122no—
566C2—181no—
567C2—362no—
568C2—541no—
569C2—1082no—
570C2D3×D4×D911yescharacteristic
subgroupG/H
centerC2D3×D4×D9
commutator, cosocleC3×C4×C9C24
FrattiniC3×C4C22×D32
FittingC3×C9×Q16C22
socleC2×C32C2×D3×D4
hypercenterQ16D3×D9
hypocenterC3×C9#576
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C4 ◁ Q16

Complements

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

NH ≅ G/Ntype
1. #11722515C2semidirect
2. #11721805C2semidirect
3. C9×#919C2semidirect
4. C3×#63481C2semidirect
5. #11721213C2semidirect
6. #11721211C2semidirect
7. #11721205C2semidirect
8. #11720818C2semidirect
9. D9×D24C2semidirect
10. #11720806C2semidirect
11. D3×D72C2semidirect
14. #88746C22semidirect
15. C9×#239C22semidirect
16. C3×#3701C22semidirect
18. #88646C22semidirect
20. D9×D12C22semidirect
22. D3×D36C22semidirect
25. C3×C9×Q16C22semidirect
26. #88548C22semidirect
27. #88525C22semidirect
28. C8×C3⋊D9C22semidirect
29. C9×#215C22semidirect
30. C3×#3675C22semidirect
31. C9×D24C22semidirect
32. D3×C8×C9C22semidirect
33. C3×D72C22semidirect
34. C3×C8×D9C22semidirect
35. #88422C22semidirect
36. #88421C22semidirect
37. #88420C22semidirect
38. #88417C22semidirect
39. #88414C22semidirect
40. #88413C22semidirect
57. #63481D3semidirect
123. #3701C2×D3semidirect
133. C9×Q16C2×D3semidirect
139. #3675C2×D3semidirect
146. D72C2×D3semidirect
151. C8×D9C2×D3semidirect
176. #919D9semidirect
264. #239C2×D9semidirect
282. C3×Q16C2×D9semidirect
294. #215C2×D9semidirect
307. D24C2×D9semidirect
316. D3×C8C2×D9semidirect
361. C3×C9#576semidirect
451. Q16D3×D9semidirect
502. C9#5769semidirect
562. C3#10589871semidirect

Maximal subgroups

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

subgroupindex
#117225152
#117218052
C9×#9192
C3×#634812
#117212132
#117212112
#117212052
#117208182
D9×D242
#117208062
D3×D722
#105947713
#105898713