Finite Group Databasebeta

← #13151897

Subgroups — #13151897

Order 1728.

836 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63716C211yesmaximal, characteristic, complemented
2#11722476C211yesmaximal, characteristic, complemented
3#11721868C211yesmaximal, characteristic, complemented
4C9×#949C211yesmaximal, characteristic, complemented
5#11721214C212yesmaximal, complemented
6D9×#215C212yesmaximal, complemented
7#11721177C212yesmaximal, complemented
8#11721173C212yesmaximal, complemented
9C2×#88423C211yesmaximal, characteristic, complemented
10#11720871C211yesmaximal, characteristic, complemented
11#11720859C211yesmaximal, characteristic, complemented
12#10595455—31nomaximal
13#10589877—31nomaximal
14C3×C9×#41C2211yescharacteristic, complemented
15#88736C2211yescharacteristic
16C3×Q8×D9C2212yescomplemented
17C3×#3699—22no—
18C3×#3699C2212yescomplemented
19C2×C9×D12C2211yescharacteristic
20C3×#3697—21no—
21C3×#3697C2211yescharacteristic
22C2×C3×Dic18—21no—
23C2×C3×Dic18C2211yescharacteristic, complemented
24D3×#611—22no—
25#88646—21no—
26D9×D12C2212yes—
27#88639—21no—
28#88638C2212yes—
29#88548C2211yescharacteristic, complemented
30#88547C2211yescharacteristic, complemented
31#88544C2211yescharacteristic, complemented
32#88543C2211yescharacteristic, complemented
33C2×C3⋊(C9⋊C8)C2211yescharacteristic, complemented
34C9×#215C2212yescomplemented
35C9×#213C2212yescomplemented
36C9×#208C2211yescharacteristic, complemented
37#88423C2212yescomplemented
38#88423C2212yescomplemented
39#88418C2212yescomplemented
40#88416C2212yescomplemented
41#88413C2212yescomplemented
42D9×C3⋊C8C2212yescomplemented
43C3×#1010—31no—
44#64307—31no—
45#64159—31no—
46C3×#949D311yescharacteristic
47#63946—32no—
48D3×#215—32no—
49#63933—32no—
50#63925—32no—
51#63833—31no—
52C2×#3715—31no—
53#63815—31no—
54#63716D311yescharacteristic, complemented
55#63710—31no—
56C9×AGL(1,ℤ/8)—31no—
57#63513—31no—
58#63513—61no—
59C2×#3672—31no—
60#63477—32no—
61SD16×D9—32no—
62#63474—32no—
63#63473—32no—
64#63471—31no—
65#63469—31no—
66C22×D3×C9—21no—
67C2×D3×D9—22no—
68C3×Q8×C9C2311yescharacteristic
69C3×D4×C9—21no—
70C3×D4×C9C2311yescharacteristic
71C2×C3×C4×C9—21no—
72C2×C3×C4×C9C2311yescharacteristic
73#6233—21no—
74C4⋊(C3⋊D9)C2311yescharacteristic
75C4×C3⋊D9—21no—
76#6226C2311yescharacteristic
77C3×#611—22no—
78C3×#611D412yes—
79C2×C3×Dic9—21no—
80C2×C3×Dic9—22no—
81C9×D12—21no—
82C9×D12C2312yes—
83C3×D36D411yescharacteristic
84C3×C4×D9—22no—
85C3×C4×D9C2312yes—
86C3×Dic18—21no—
87C3×Dic18—22no—
88C3×Dic18D411yescharacteristic
89C3×Dic18C2312yes—
90#6195—22no—
91#6194—22no—
92D3×Dic9—22no—
93C3⋊(C9⋊C8)C2311yescharacteristic
94C3⋊(C9⋊C8)C2311yescharacteristic
95C9×C3⋊C8C2312yes—
96#5837—91no—
97#5775—91no—
98C32×#41C2×D311yescharacteristic
99#3828—31no—
100C3×D3×Q8—32no—
101C3×#237—32no—
102C3×#237—62no—
103C3×#235—31no—
104C3×#235—61no—
105C2×C3×D12C2×D311yescharacteristic
106C2×C3×Dic6—31no—
107C2×C3×Dic6—61no—
108D3×#67—62no—
109#3802—61no—
110D3×D12—32no—
111#3796—61no—
112#3795—32no—
113#3756—31no—
114#3755—31no—
115#3754—31no—
116#3753—31no—
117C2×C3⋊(C3⋊C8)—31no—
118C3×#215C2×D312yes—
119C3×#213C2×D312yes—
120C3×#208C2×D311yescharacteristic
121#3719—32no—
122#3717—32no—
123#3715—32no—
124#3715—32no—
125#3711—32no—
126D3×C3⋊C8—32no—
127C9×#41—21no—
128C9×#41C2×D311yescharacteristic, complemented
129C2×D4×C9—31no—
130C2×#611—62no—
131Q8×D9C2×D312yescomplemented
132#3699—22no—
133#3699—32no—
134#3699—61no—
135#3699C2×D312yescomplemented
136D4×D9—32no—
137D4×D9—61no—
138#3697—21no—
139#3697—31no—
140#3697—61no—
141#3697C2×D311yescharacteristic
142C2×Dic18—21no—
143C2×Dic18C2×D311yescharacteristic, complemented
144C9×SD16—32no—
145C9×D8—32no—
146C9×M16—31no—
147#3675—31no—
148#3675—61no—
149#3674—31no—
150#3674—61no—
151#3673—31no—
152#3673—61no—
153#3672—31no—
154#3672—32no—
155#3672—32no—
156#3672—61no—
157C2×C9⋊C8—31no—
158C2×C9⋊C8—61no—
159#3664—32no—
160#3663—32no—
161C8×D9—32no—
162Dic36—32no—
163C22×C3×C9—21no—
164C22×C3×C9C2×D411yescharacteristic
165C2×C3⋊D9—21no—
166C2×D3×C9—22no—
167C2×D3×C9—81no—
168C2×C3×D9C2×D412yes—
169D3×D9—82no—
170C3×C4×C9—21no—
171C3×C4×C9C2×D411yescharacteristic
172C3×C4×C9C2411yescharacteristic
173C3⋊Dic9—21no—
174C3×Dic9—22no—
175C3×Dic9C2×D412yes—
176C3×2−1+4—91no—
177#1010—31no—
178#1009—91no—
179#1004—91no—
180C3×AGL(1,ℤ/8)—31no—
181#950—91no—
182#950—91no—
183#950—92no—
184#950—181no—
185#949—92no—
186#949D911yescharacteristic, complemented
187#948—91no—
188#942—92no—
189#942—91no—
190C2×#215—91no—
191C2×#215—92no—
192C2×#214—91no—
193#915—92no—
194D3×SD16—92no—
195#912—92no—
196#911—92no—
197#909—91no—
198#907—91no—
199C22×C3×D3—21no—
200C2×D32—62no—
201C32×Q8C22×D311yescharacteristic
202C32×D4—21no—
203C32×D4C22×D311yescharacteristic
204C2×C32×C4—21no—
205C2×C32×C4C22×D311yescharacteristic
206#638—61no—
207C4⋊(C3⋊D3)—31no—
208C4×C3⋊D3—61no—
209#634—31no—
210C3×#67—32no—
211C3×#67—62no—
212C2×C3×Dic3—62no—
213C2×C3×Dic3—61no—
214C3×D12—21no—
215C3×D12—31no—
216C3×D12C22×D312yes—
217C3×C4×D3—32no—
218C3×C4×D3—62no—
219C3×Dic6—31no—
220C3×Dic6—32no—
221C3×Dic6—62no—
222C3×Dic6—61no—
223#626—62no—
224#625—62no—
225D3×Dic3—62no—
226C23×C9—61no—
227C22×D9—62no—
228C3⋊(C3⋊C8)—31no—
229C3⋊(C3⋊C8)—31no—
230C3×C3⋊C8C22×D312yes—
231Q8×C9—21no—
232Q8×C9C22×D311yescharacteristic
233D4×C9—21no—
234D4×C9—21no—
235D4×C9—32no—
236D4×C9—41no—
237D4×C9—61no—
238D4×C9C22×D311yescharacteristic
239C2×C4×C9—21no—
240C2×C4×C9—21no—
241C2×C4×C9—41no—
242C2×C4×C9C22×D311yescharacteristic
243#611—22no—
244#611—61no—
245#611—62no—
246#611—62no—
247#611—121no—
248#611#6712yes—
249C2×Dic9—21no—
250C2×Dic9—22no—
251C2×Dic9—62no—
252D36—31no—
253D36—61no—
254D36#6711yescharacteristic
255C4×D9—22no—
256C4×D9—61no—
257C4×D9—121no—
258C4×D9C22×D312yes—
259Dic18—21no—
260Dic18—22no—
261Dic18—31no—
262Dic18—61no—
263Dic18#6711yescharacteristic
264Dic18C22×D312yes—
265C8×C9—32no—
266C9⋊C8—31no—
267C9⋊C8—31no—
268C9⋊C8—61no—
269C9⋊C8—61no—
270#577—271noSylow
271C2×C3×C9—21no—
272C2×C3×C9—41no—
273C2×C3×C9C22×D411yescharacteristic
274C3⋊D9—41no—
275D3×C9—42no—
276C3×D9—22no—
277C3×#41—21no—
278C3×#41—91no—
279C3×#41—92no—
280C3×#41—181no—
281C3×#41—182no—
282C3×#41C2×D911yescharacteristic, complemented
283C3×#41D3211yescharacteristic
284C2×C3×Q8—92no—
285C2×C3×Q8—91no—
286C2×C3×Q8—181no—
287C2×C3×D4—31no—
288C2×#67—182no—
289#239—181no—
290D3×Q8—32no—
291#237—32no—
292#237—62no—
293#237—92no—
294#237—181no—
295D3×D4—92no—
296D3×D4—181no—
297D3×D4—181no—
298D3×D4—182no—
299#235—31no—
300#235—61no—
301#235—91no—
302#235—91no—
303#235—92no—
304#235—181no—
305C2×D12—92no—
306C2×D12C2×D911yescharacteristic
307C2×Dic6—31no—
308C2×Dic6—61no—
309C3×SD16—32no—
310C3×D8—32no—
311C3×M16—31no—
312#216—91no—
313#216—91no—
314#216—92no—
315#216—181no—
316#215—91no—
317#215—91no—
318#215—92no—
319#215—92no—
320#215—92no—
321#215—181no—
322#215C2×D912yescomplemented
323#214—92no—
324#214—92no—
325#214—91no—
326#214—91no—
327#214—181no—
328#213—91no—
329#213—91no—
330#213—181no—
331#213C2×D912yescomplemented
332#208—92no—
333#208C2×D911yescharacteristic, complemented
334C2×C3⋊C8—91no—
335C2×C3⋊C8—91no—
336C2×C3⋊C8—92no—
337C2×C3⋊C8—181no—
338Dic12—92no—
339#204—92no—
340ΓL(1,𝔽25)—92no—
341D3×C8—92no—
342C22×C32—21no—
343C22×C32D3×D411yescharacteristic
344C2×C3⋊D3—61no—
345C2×C3×D3—22no—
346C2×C3×D3—32no—
347C2×C3×D3—81no—
348D32—242no—
349C32×C4—21no—
350C32×C4D3×D411yescharacteristic
351C32×C4C23×D311yescharacteristic
352C3⋊Dic3—61no—
353C3×Dic3—32no—
354C3×Dic3—62no—
355C22×C9—21no—
356C22×C9—21no—
357C22×C9—41no—
358C22×C9—62no—
359C22×C9—241no—
360C22×C9C2×#6711yescharacteristic
361C2×D9—61no—
362C2×D9—121no—
363C2×D9—242no—
364C2×D9C2×#6712yes—
365C4×C9—21no—
366C4×C9—21no—
367C4×C9—21no—
368C4×C9—41no—
369C4×C9C2×#6711yescharacteristic
370C4×C9C23×D311yescharacteristic
371Dic9—22no—
372Dic9—61no—
373Dic9—121no—
374Dic9C2×#6712yes—
3752−1+4—91no—
376H(5,𝔽2)—271no—
377#137—271no—
378#137—272no—
379AGL(1,ℤ/8)—31no—
380AGL(1,ℤ/8)—272no—
381#135—272no—
382#135—271no—
383C2×SD16—272no—
384C2×SD16—271no—
385#131—271no—
386C3×C9#57711yescharacteristic, complemented, Sylow
387C23×C3—61no—
388C22×D3—21no—
389C22×D3—182no—
390C22×D3—182no—
391C3×Q8—21no—
392C3×Q8—91no—
393C3×Q8—92no—
394C3×Q8—182no—
395C3×Q8—181no—
396C3×Q8C22×D911yescharacteristic
397C3×Q8C2×D3211yescharacteristic
398C3×D4—21no—
399C3×D4—21no—
400C3×D4—21no—
401C3×D4—32no—
402C3×D4—41no—
403C3×D4—61no—
404C3×D4—92no—
405C3×D4—91no—
406C3×D4—182no—
407C3×D4C22×D911yescharacteristic
408C3×D4C2×D3211yescharacteristic
409C2×C3×C4—21no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—41no—
413C2×C3×C4—92no—
414C2×C3×C4—182no—
415C2×C3×C4—182no—
416C2×C3×C4—181no—
417C2×C3×C4C22×D911yescharacteristic
418C2×C3×C4C2×D3211yescharacteristic
419#67—32no—
420#67—62no—
421#67—182no—
422#67—182no—
423#67—182no—
424#67—181no—
425#67—181no—
426#67—361no—
427C2×Dic3—62no—
428C2×Dic3—61no—
429C2×Dic3—182no—
430D12—21no—
431D12—31no—
432D12—91no—
433D12—91no—
434D12—181no—
435D12—182no—
436D12C22×D912yes—
437C4×D3—32no—
438C4×D3—62no—
439C4×D3—182no—
440C4×D3—181no—
441C4×D3—181no—
442C4×D3—361no—
443Dic6—31no—
444Dic6—32no—
445Dic6—62no—
446Dic6—61no—
447Dic6—91no—
448Dic6—91no—
449Dic6—181no—
450C3×C8—32no—
451C3⋊C8—91no—
452C3⋊C8—91no—
453C3⋊C8—91no—
454C3⋊C8—91no—
455C3⋊C8—181no—
456C3⋊C8—181no—
457C3⋊C8C22×D912yes—
458C2×C32—21no—
459C2×C32—41no—
460C2×C32C2×D3×D411yescharacteristic
461C3⋊D3—121no—
462C3×D3—42no—
463C3×D3—62no—
464C2×C9—21no—
465C2×C9—21no—
466C2×C9—41no—
467C2×C9—41no—
468C2×C9—81no—
469C2×C9—122no—
470C2×C9C22×#6711yescharacteristic
471D9—22no—
472D9—121no—
473D9—241no—
474#41—91no—
475#41—92no—
476#41—181no—
477#41—182no—
478#41—272no—
479#41—271no—
480#41—541no—
481#41D3×D911yescharacteristic, complemented
482C2×Q8—92no—
483C2×Q8—91no—
484C2×Q8—181no—
485C2×D4—31no—
486C2×D4—272no—
487C2×D4—541no—
488C2×D4—542no—
489Q16—272no—
490Q16—271no—
491SD16—32no—
492SD16—272no—
493SD16—272no—
494SD16—272no—
495SD16—271no—
496SD16—271no—
497D8—32no—
498D8—271no—
499M16—31no—
500M16—272no—
501C2×C8—272no—
502C2×C8—271no—
503C22×C3—21no—
504C22×C3—21no—
505C22×C3—21no—
506C22×C3—41no—
507C22×C3—62no—
508C22×C3—92no—
509C22×C3—241no—
510C22×C3D4×D911yescharacteristic
511C22×C3D3×#6711yescharacteristic
512C2×D3—22no—
513C2×D3—32no—
514C2×D3—81no—
515C2×D3—181no—
516C2×D3—181no—
517C2×D3—361no—
518C2×D3—722no—
519C2×D3—722no—
520C3×C4—21no—
521C3×C4—21no—
522C3×C4—21no—
523C3×C4—21no—
524C3×C4—41no—
525C3×C4—92no—
526C3×C4—182no—
527C3×C4D4×D911yescharacteristic
528C3×C4C23×D911yescharacteristic
529C3×C4D3×#6711yescharacteristic
530C3×C4C22×D3211yescharacteristic
531Dic3—32no—
532Dic3—62no—
533Dic3—181no—
534Dic3—181no—
535Dic3—361no—
536C32#577511yescharacteristic
537C9—21no—
538C9#583711yescharacteristic, complemented
539C23—61no—
540C23—542no—
541Q8—91no—
542Q8—92no—
543Q8—182no—
544Q8—181no—
545Q8—271no—
546Q8C2×D3×D911yescharacteristic
547D4—21no—
548D4—32no—
549D4—61no—
550D4—92no—
551D4—91no—
552D4—182no—
553D4—271no—
554D4—542no—
555D4—542no—
556D4—541no—
557D4C2×D3×D911yescharacteristic
558C2×C4—21no—
559C2×C4—92no—
560C2×C4—182no—
561C2×C4—182no—
562C2×C4—181no—
563C2×C4—542no—
564C2×C4—541no—
565C2×C4C2×D3×D911yescharacteristic
566C8—32no—
567C8—271no—
568C8—271no—
569C2×C3—21no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—41no—
573C2×C3—41no—
574C2×C3—41no—
575C2×C3—81no—
576C2×C3—122no—
577C2×C3—182no—
578C2×C3C2×D4×D911yescharacteristic
579C2×C3C2×D3×#6711yescharacteristic
580D3—42no—
581D3—62no—
582D3—361no—
583D3—361no—
584D3—721no—
585C22—21no—
586C22—62no—
587C22—92no—
588C22—241no—
589C22—541no—
590C22—2162no—
591C22D9×#6711yescharacteristic
592C4—21no—
593C4—92no—
594C4—182no—
595C4—541no—
596C4D9×#6711yescharacteristic
597C4C22×D3×D911yescharacteristic
598C3—21no—
599C3#1058987711yescharacteristic, complemented
600C3#1059545511yescharacteristic
601C2—21no—
602C2—41no—
603C2—122no—
604C2—182no—
605C2—1081no—
606C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC2C2×D9×#67
commutator, cosocleC3×C4×C9C24
FrattiniC3×C4C22×D32
FittingC3×C9×#41C22
socleC2×C32C2×D3×D4
hypercenter#41D3×D9
hypocenterC3×C9#577
LCSG ▷ C3×C4×C9 ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. C3×#63716C2semidirect
2. #11722476C2semidirect
3. #11721868C2semidirect
4. C9×#949C2semidirect
5. #11721214C2semidirect
6. D9×#215C2semidirect
7. #11721177C2semidirect
8. #11721173C2semidirect
9. C2×#88423C2semidirect
10. #11720871C2semidirect
11. #11720859C2semidirect
14. C3×C9×#41C22semidirect
16. C3×Q8×D9C22semidirect
18. C3×#3699C22semidirect
23. C2×C3×Dic18C22semidirect
29. #88548C22semidirect
30. #88547C22semidirect
31. #88544C22semidirect
32. #88543C22semidirect
33. C2×C3⋊(C9⋊C8)C22semidirect
34. C9×#215C22semidirect
35. C9×#213C22semidirect
36. C9×#208C22semidirect
37. #88423C22semidirect
38. #88423C22semidirect
39. #88418C22semidirect
40. #88416C22semidirect
41. #88413C22semidirect
42. D9×C3⋊C8C22semidirect
54. #63716D3semidirect
128. C9×#41C2×D3semidirect
131. Q8×D9C2×D3semidirect
135. #3699C2×D3semidirect
143. C2×Dic18C2×D3semidirect
186. #949D9semidirect
282. C3×#41C2×D9semidirect
322. #215C2×D9semidirect
331. #213C2×D9semidirect
333. #208C2×D9semidirect
386. C3×C9#577semidirect
481. #41D3×D9semidirect
538. C9#5837semidirect
599. C3#10589877semidirect

Maximal subgroups

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

subgroupindex
C3×#637162
#117224762
#117218682
C9×#9492
#117212142
D9×#2152
#117211772
#117211732
C2×#884232
#117208712
#117208592
#105954553
#105898773