Finite Group Databasebeta

← #13149904

Subgroups — #13149904

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#11722499C212yesmaximal, complemented
2#11721791C211yesmaximal, characteristic, complemented
3D3×C9×D8C211yesmaximal, characteristic, complemented
4C3×D8×D9C211yesmaximal, characteristic, complemented
5#11721176C212yesmaximal, complemented
6#11721175C212yesmaximal, complemented
7#11721168C212yesmaximal, complemented
8#11720822C211yesmaximal, characteristic, complemented
9#11720818C211yesmaximal, characteristic, complemented
10#11720811C211yesmaximal, characteristic, complemented
11#11720806C211yesmaximal, characteristic, complemented
12#10594731—31nomaximal
13#10589859—31nomaximal
14#88742C2212yescomplemented
15D3×D4×C9C2212yescomplemented
16C3×D4×D9C2212yescomplemented
17D9×#67—22no—
18D3×#611—22no—
19C2×#6194—22no—
20#88663—22no—
21#88646C2212yescomplemented
22#88641C2212yescomplemented
23#88640C2211yescharacteristic
24#88638C2212yescomplemented
25C3×C9×D8C2211yescharacteristic, complemented
26#88543C2212yescomplemented
27C8×C3⋊D9C2211yescharacteristic, complemented
28#88521C2211yescharacteristic, complemented
29C9×#213C2212yescomplemented
30C3×#3673C2212yescomplemented
31C9×D24C2211yescharacteristic, complemented
32D3×C8×C9C2211yescharacteristic, complemented
33C3×D72C2211yescharacteristic, complemented
34C3×C8×D9C2211yescharacteristic, complemented
35#88424C2212yescomplemented
36#88421C2212yescomplemented
37#88420C2211yescharacteristic, complemented
38#88415C2212yescomplemented
39#88414C2211yescharacteristic, complemented
40#88413C2211yescharacteristic, complemented
41#64313—32no—
42#64121—31no—
43C3×D3×D8—31no—
44C3×D3×D8D311yescharacteristic
45#63932—32no—
46#63932—32no—
47#63927—32no—
48#63807—31no—
49#63807—31no—
50#63803—31no—
51#63797—31no—
52#63710—32no—
53C2×C9×D8—31no—
54#63495—32no—
55#63474—31no—
56#63474—31no—
57#63474—61no—
58#63473—32no—
59#63473—32no—
60D8×D9—31no—
61D8×D9D311yescharacteristic, complemented
62#63467—31no—
63#63464—31no—
64C22×D3×C9—22no—
65C22×C3×D9—22no—
66C2×D3×D9—22no—
67C3×D4×C9C2312yes—
68#6233—22no—
69C2×C3⋊Dic9—22no—
70C4×C3⋊D9C2311yescharacteristic
71#6226C2312yes—
72C9×#67—22no—
73C3×#611—22no—
74C9×D12C2312yes—
75C4×D3×C9C2311yescharacteristic
76C3×D36C2312yes—
77C3×C4×D9C2311yescharacteristic
78#6195D411yescharacteristic
79#6194—22no—
80#6194—22no—
81#6194—22no—
82#6194D411yescharacteristic
83D3×Dic9—22no—
84#6192D411yescharacteristic
85Dic3×D9—22no—
86#6189D411yescharacteristic
87C3×C8×C9C2311yescharacteristic
88C3⋊(C9⋊C8)C2311yescharacteristic
89C9×C3⋊C8C2311yescharacteristic
90C3×C9⋊C8C2311yescharacteristic
91#5757—91no—
92#5757—91no—
93#3830—32no—
94C3×D3×D4—32no—
95C3×D3×D4C2×D312yes—
96D3×#67—62no—
97D3×#67—62no—
98C2×#625—62no—
99#3805—62no—
100#3802—32no—
101#3798—31no—
102#3795—32no—
103#3795—32no—
104C32×D8C2×D311yescharacteristic
105#3754—32no—
106#3746—31no—
107C8×C3⋊D3—31no—
108C3×#213—32no—
109C3×#213C2×D312yes—
110C3×D24—31no—
111C3×D24C2×D311yescharacteristic
112C3×D3×C8—31no—
113C3×D3×C8C2×D311yescharacteristic
114#3716—32no—
115#3716—32no—
116#3713—32no—
117#3712—31no—
118#3711—31no—
119#3711—31no—
120C2×D4×C9—32no—
121C2×#611—62no—
122C2×#611—62no—
123#3699—32no—
124#3699—32no—
125#3699—62no—
126#3699—62no—
127D4×D9—32no—
128D4×D9—62no—
129D4×D9C2×D312yescomplemented
130#3697—31no—
131#3697—32no—
132C9×D8—21no—
133C9×D8—32no—
134C9×D8—31no—
135C9×D8C2×D311yescharacteristic, complemented
136C2×C8×C9—31no—
137#3673—32no—
138#3673C2×D312yescomplemented
139#3672—32no—
140#3672—32no—
141#3672—62no—
142#3667—31no—
143D72C2×D311yescharacteristic, complemented
144#3664—32no—
145#3663—31no—
146#3663—31no—
147C8×D9—31no—
148C8×D9—61no—
149C8×D9C2×D311yescharacteristic, complemented
150Dic36—31no—
151Dic36—61no—
152C22×C3×C9—22no—
153C2×C3⋊D9C2×D411yescharacteristic
154C2×D3×C9—22no—
155C2×D3×C9—82no—
156C2×D3×C9C2×D411yescharacteristic
157C2×C3×D9—22no—
158C2×C3×D9—82no—
159C2×C3×D9C2×D411yescharacteristic
160D3×D9—82no—
161C3×C4×C9C2411yescharacteristic
162C3⋊Dic9—22no—
163C3⋊Dic9C2×D411yescharacteristic
164C9×Dic3C2×D411yescharacteristic
165C3×Dic9C2×D411yescharacteristic
166#1004—92no—
167#1004—92no—
168C2×C3×D8—31no—
169C2×C3×D8—91no—
170#932—92no—
171#932—92no—
172#912—91no—
173#912—91no—
174#912—91no—
175#912—91no—
176#912—181no—
177#911—92no—
178#911—92no—
179#911—92no—
180#911—92no—
181D3×D8—31no—
182D3×D8—91no—
183D3×D8—91no—
184D3×D8D911yescharacteristic, complemented
185#904—91no—
186#904—91no—
187#901—91no—
188#901—91no—
189C22×C3×D3—22no—
190C22×C3×D3—62no—
191C2×D32—62no—
192C32×D4C22×D312yes—
193#638—62no—
194C2×C3⋊Dic3—62no—
195C4×C3⋊D3—31no—
196#634—32no—
197C3×#67—22no—
198C3×#67—62no—
199C3×D12—32no—
200C3×D12C22×D312yes—
201C3×C4×D3—31no—
202C3×C4×D3C22×D311yescharacteristic
203#627—31no—
204#626—31no—
205#626—31no—
206#625—31no—
207#625—62no—
208#625—62no—
209#625—62no—
210D3×Dic3—62no—
211D3×Dic3—62no—
212C23×C9—62no—
213C22×D9—22no—
214C22×D9—62no—
215C32×C8C22×D311yescharacteristic
216C3⋊(C3⋊C8)—31no—
217C3×C3⋊C8—31no—
218C3×C3⋊C8C22×D311yescharacteristic
219D4×C9—22no—
220D4×C9—32no—
221D4×C9—62no—
222D4×C9C22×D312yes—
223C2×C4×C9—31no—
224#611—22no—
225#611—31no—
226#611—31no—
227#611—62no—
228#611—62no—
229#611—62no—
230#611—62no—
231#611—122no—
232C2×Dic9—62no—
233C2×Dic9—62no—
234C2×Dic9—122no—
235D36—31no—
236D36C22×D312yes—
237C4×D9—31no—
238C4×D9—61no—
239C4×D9—62no—
240C4×D9C22×D311yescharacteristic
241Dic18—31no—
242Dic18—32no—
243Dic18—62no—
244C8×C9—21no—
245C8×C9—31no—
246C8×C9C22×D311yescharacteristic
247C9⋊C8—31no—
248C9⋊C8—61no—
249C9⋊C8C22×D311yescharacteristic
250#576—271noSylow
251C2×C3×C9—42no—
252C2×C3×C9C22×D411yescharacteristic
253C3⋊D9—21no—
254D3×C9—21no—
255D3×C9—42no—
256C3×D9—21no—
257C3×D9—42no—
258C2×C3×D4—32no—
259C2×C3×D4—92no—
260C2×#67—182no—
261C2×#67—182no—
262C2×#67—182no—
263C2×#67—182no—
264#237—92no—
265#237—92no—
266#237—92no—
267#237—92no—
268#237—182no—
269#237—182no—
270#237—182no—
271D3×D4—32no—
272D3×D4—92no—
273D3×D4—92no—
274D3×D4—182no—
275D3×D4—182no—
276D3×D4C2×D912yescomplemented
277#235—91no—
278#235—91no—
279#235—92no—
280#235—92no—
281C3×D8—21no—
282C3×D8—32no—
283C3×D8—31no—
284C3×D8—92no—
285C3×D8—91no—
286C3×D8C2×D911yescharacteristic, complemented
287C3×D8D3211yescharacteristic
288C2×C3×C8—31no—
289C2×C3×C8—91no—
290#214—92no—
291#214—92no—
292#214—92no—
293#214—92no—
294#214—182no—
295#213—32no—
296#213—92no—
297#213—92no—
298#213C2×D912yescomplemented
299#208—91no—
300#208—91no—
301Dic12—91no—
302Dic12—91no—
303Dic12—181no—
304D24—31no—
305D24C2×D911yescharacteristic, complemented
306#204—92no—
307#204—92no—
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
317C22×C32—22no—
318C2×C3⋊D3—31no—
319C2×C3×D3—22no—
320C2×C3×D3—31no—
321C2×C3×D3—62no—
322C2×C3×D3—82no—
323C2×C3×D3—242no—
324C2×C3×D3D3×D411yescharacteristic
325D32—242no—
326C32×C4C23×D311yescharacteristic
327C3⋊Dic3—31no—
328C3⋊Dic3—62no—
329C3×Dic3—31no—
330C3×Dic3D3×D411yescharacteristic
331C22×C9—22no—
332C22×C9—31no—
333C22×C9—42no—
334C22×C9—62no—
335C22×C9—242no—
336C2×D9—22no—
337C2×D9—31no—
338C2×D9—61no—
339C2×D9—82no—
340C2×D9—242no—
341C2×D9D3×D411yescharacteristic
342C4×C9—21no—
343C4×C9—31no—
344C4×C9C23×D311yescharacteristic
345Dic9—31no—
346Dic9—61no—
347Dic9—62no—
348Dic9—122no—
349Dic9D3×D411yescharacteristic
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—
361C3×C9#57611yescharacteristic, complemented, Sylow
362C23×C3—62no—
363C23×C3—182no—
364C22×D3—22no—
365C22×D3—62no—
366C22×D3—182no—
367C22×D3—182no—
368C3×D4—22no—
369C3×D4—32no—
370C3×D4—62no—
371C3×D4—92no—
372C3×D4—182no—
373C3×D4C22×D912yes—
374C3×D4C2×D3212yes—
375C2×C3×C4—31no—
376C2×C3×C4—91no—
377#67—22no—
378#67—62no—
379#67—91no—
380#67—91no—
381#67—91no—
382#67—91no—
383#67—182no—
384#67—182no—
385#67—182no—
386#67—182no—
387#67—182no—
388#67—182no—
389#67—182no—
390#67—182no—
391#67—362no—
392C2×Dic3—182no—
393C2×Dic3—182no—
394C2×Dic3—182no—
395C2×Dic3—182no—
396C2×Dic3—362no—
397D12—32no—
398D12—91no—
399D12—91no—
400D12C22×D912yes—
401C4×D3—31no—
402C4×D3—91no—
403C4×D3—91no—
404C4×D3—181no—
405C4×D3—182no—
406C4×D3—182no—
407C4×D3C22×D911yescharacteristic
408Dic6—91no—
409Dic6—91no—
410Dic6—92no—
411Dic6—92no—
412Dic6—182no—
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×C32—42no—
424C2×C32C2×D3×D411yescharacteristic
425C3⋊D3—61no—
426C3×D3—21no—
427C3×D3—42no—
428C3×D3—61no—
429C3×D3—122no—
430C2×C9—21no—
431C2×C9—42no—
432C2×C9—61no—
433C2×C9—82no—
434C2×C9—122no—
435C2×C9C2×D3×D411yescharacteristic
436D9—21no—
437D9—42no—
438D9—61no—
439D9—121no—
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×D911yescharacteristic, complemented
461M16—271no—
462M16—271no—
463M16—271no—
464C2×C8—31no—
465C2×C8—91no—
466C2×C8—271no—
467C22×C3—22no—
468C22×C3—22no—
469C22×C3—31no—
470C22×C3—42no—
471C22×C3—62no—
472C22×C3—91no—
473C22×C3—182no—
474C22×C3—242no—
475C22×C3—722no—
476C2×D3—22no—
477C2×D3—31no—
478C2×D3—62no—
479C2×D3—82no—
480C2×D3—91no—
481C2×D3—91no—
482C2×D3—181no—
483C2×D3—242no—
484C2×D3—722no—
485C2×D3—722no—
486C2×D3D4×D911yescharacteristic
487C3×C4—21no—
488C3×C4—31no—
489C3×C4—91no—
490C3×C4C23×D911yescharacteristic
491C3×C4C22×D3211yescharacteristic
492Dic3—31no—
493Dic3—91no—
494Dic3—91no—
495Dic3—181no—
496Dic3—182no—
497Dic3—182no—
498Dic3—362no—
499Dic3D4×D911yescharacteristic
500C32#575711yescharacteristic
501C9—21no—
502C9#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×D912yes—
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×D911yescharacteristic
530C2×C3—21no—
531C2×C3—42no—
532C2×C3—42no—
533C2×C3—61no—
534C2×C3—82no—
535C2×C3—122no—
536C2×C3—181no—
537C2×C3—362no—
538C2×C3C2×D4×D911yescharacteristic
539C2×C3D32×D411yescharacteristic
540D3—21no—
541D3—42no—
542D3—61no—
543D3—122no—
544D3—181no—
545D3—181no—
546D3—361no—
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×D911yescharacteristic
561C3—21no—
562C3#1058985911yescharacteristic, complemented
563C3#1059473111yescharacteristic
564C2—42no—
565C2—61no—
566C2—122no—
567C2—181no—
568C2—362no—
569C2—541no—
570C2D3×D4×D911yescharacteristic
subgroupG/H
centerC2D3×D4×D9
commutator, cosocleC3×C4×C9C24
FrattiniC3×C4C22×D32
FittingC3×C9×D8C22
socleC2×C32C2×D3×D4
hypercenterD8D3×D9
hypocenterC3×C9#576
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C4 ◁ D8

Complements

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

NH ≅ G/Ntype
1. #11722499C2semidirect
2. #11721791C2semidirect
3. D3×C9×D8C2semidirect
4. C3×D8×D9C2semidirect
5. #11721176C2semidirect
6. #11721175C2semidirect
7. #11721168C2semidirect
8. #11720822C2semidirect
9. #11720818C2semidirect
10. #11720811C2semidirect
11. #11720806C2semidirect
14. #88742C22semidirect
15. D3×D4×C9C22semidirect
16. C3×D4×D9C22semidirect
21. #88646C22semidirect
22. #88641C22semidirect
24. #88638C22semidirect
25. C3×C9×D8C22semidirect
26. #88543C22semidirect
27. C8×C3⋊D9C22semidirect
28. #88521C22semidirect
29. C9×#213C22semidirect
30. C3×#3673C22semidirect
31. C9×D24C22semidirect
32. D3×C8×C9C22semidirect
33. C3×D72C22semidirect
34. C3×C8×D9C22semidirect
35. #88424C22semidirect
36. #88421C22semidirect
37. #88420C22semidirect
38. #88415C22semidirect
39. #88414C22semidirect
40. #88413C22semidirect
61. D8×D9D3semidirect
129. D4×D9C2×D3semidirect
135. C9×D8C2×D3semidirect
138. #3673C2×D3semidirect
143. D72C2×D3semidirect
149. C8×D9C2×D3semidirect
184. D3×D8D9semidirect
276. D3×D4C2×D9semidirect
286. C3×D8C2×D9semidirect
298. #213C2×D9semidirect
305. D24C2×D9semidirect
316. D3×C8C2×D9semidirect
361. C3×C9#576semidirect
460. D8D3×D9semidirect
502. C9#5757semidirect
562. C3#10589859semidirect

Maximal subgroups

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

subgroupindex
#117224992
#117217912
D3×C9×D82
C3×D8×D92
#117211762
#117211752
#117211682
#117208222
#117208182
#117208112
#117208062
#105947313
#105898593