Finite Group Databasebeta

← #13151917

Subgroups — #13151917

Order 1728.

854 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#41×D9C211yesmaximal, characteristic, complemented
2C2×#88635C211yesmaximal, characteristic, complemented
3#11721867C211yesmaximal, characteristic, complemented
4C9×#951C211yesmaximal, characteristic, complemented
5#11721223C212yesmaximal, complemented
6#11721222C212yesmaximal, complemented
7#11721185C212yesmaximal, complemented
8#11721184C212yesmaximal, complemented
9#11720893C211yesmaximal, characteristic, complemented
10#11720873C211yesmaximal, characteristic, complemented
11D9×#208C211yesmaximal, characteristic, complemented
12#10595470—31nomaximal
13#10589879—31nomaximal
14C3×C9×#41C2211yescharacteristic
15C2×C4⋊(C3⋊D9)C2211yescharacteristic, complemented
16C3×#3701C2211yescharacteristic, complemented
17C3×Q8×D9C2211yescharacteristic, complemented
18C3×#3699—21no—
19C3×#3699C2211yescharacteristic, complemented
20C3×D4×D9—21no—
21C3×D4×D9C2211yescharacteristic, complemented
22C3×#3697—21no—
23C3×#3697C2211yescharacteristic
24C2×C3×C4×D9—21no—
25C2×C3×C4×D9C2211yescharacteristic
26C2×C9×Dic6C2211yescharacteristic, complemented
27C2×#6192—21no—
28C2×(C3⋊D9)⋊C4—21no—
29#88635—21no—
30#88635—21no—
31#88635C2212yes—
32#88635C2212yescomplemented
33#88548C2212yescomplemented
34#88544C2212yescomplemented
35#88532C2211yescharacteristic, complemented
36C9×#216C2212yescomplemented
37C9×#214C2212yescomplemented
38C9×#208C2211yescharacteristic, complemented
39#88427C2212yescomplemented
40#88425C2212yescomplemented
41#88417C2212yescomplemented
42#88416C2212yescomplemented
43#88413C2212yescomplemented
44D9×C3⋊C8C2212yescomplemented
45C3×D3×#41—31no—
46C2×#3799—31no—
47#64158—31no—
48C3×#951D311yescharacteristic
49#63949—32no—
50#63948—32no—
51#63935—32no—
52#63934—32no—
53#63827—31no—
54#63825—31no—
55D3×#208—31no—
56#41×D9D311yescharacteristic, complemented
57C2×#3701—31no—
58C9×#137—31no—
59#63512—31no—
60#63512—61no—
61#63505—31no—
62#63481—32no—
63#63480—32no—
64#63478—32no—
65#63476—32no—
66#63470—31no—
67M16×D9—31no—
68C22×C3⋊D9—21no—
69C22×C3×D9—21no—
70C22×C3×D9D411yescharacteristic
71C3×Q8×C9C2311yescharacteristic
72C3×D4×C9—21no—
73C3×D4×C9C2311yescharacteristic
74C2×C3×C4×C9—21no—
75C2×C3×C4×C9C2311yescharacteristic
76C4⋊(C3⋊D9)—21no—
77C4⋊(C3⋊D9)C2312yes—
78C3×#611—21no—
79C3×#611—21no—
80C3×#611—21no—
81C2×C9×Dic3—21no—
82C2×C3×Dic9—21no—
83C2×C3×Dic9D411yescharacteristic
84C3×D36—21no—
85C3×D36C2311yescharacteristic
86C3×C4×D9—21no—
87C3×C4×D9—21no—
88C3×C4×D9—21no—
89C3×C4×D9D411yescharacteristic
90C3×C4×D9D411yescharacteristic
91C3×C4×D9C2311yescharacteristic
92C3×C4×D9C2311yescharacteristic
93C9×Dic6—21no—
94C9×Dic6C2312yes—
95C3×Dic18—21no—
96C3×Dic18C2311yescharacteristic
97#6192—22no—
98#6192—22no—
99(C3⋊D9)⋊C4—22no—
100(C3⋊D9)⋊C4—22no—
101C3⋊(C9⋊C8)C2312yes—
102C9×C3⋊C8C2312yes—
103#5822—91no—
104#5777—91no—
105C32×#41C2×D311yescharacteristic
106C2×C4⋊(C3⋊D3)—31no—
107C3×#239—31no—
108C3×D3×Q8—31no—
109C3×#237—31no—
110C3×#237—61no—
111C3×D3×D4—31no—
112C3×D3×D4—61no—
113C3×#235—31no—
114C3×#235—61no—
115C2×C3×C4×D3—31no—
116C2×C3×C4×D3—61no—
117C2×C3×Dic6C2×D311yescharacteristic
118C2×#626—61no—
119C2×#624—61no—
120#3799—32no—
121#3799—32no—
122#3799—61no—
123#3799—61no—
124#3755—32no—
125#3753—32no—
126#3748—31no—
127C3×#216C2×D312yes—
128C3×#214C2×D312yes—
129C3×#208C2×D311yescharacteristic
130#3718—32no—
131#3717—32no—
132#3715—32no—
133#3714—32no—
134#3711—32no—
135D3×C3⋊C8—32no—
136C9×#41—21no—
137C9×#41C2×D311yescharacteristic
138C2×Q8×C9—31no—
139#3701—32no—
140#3701—32no—
141#3701—61no—
142#3701—61no—
143#3701C2×D311yescharacteristic, complemented
144Q8×D9C2×D311yescharacteristic, complemented
145#3699—21no—
146#3699C2×D311yescharacteristic, complemented
147D4×D9—21no—
148D4×D9C2×D311yescharacteristic, complemented
149#3697—21no—
150#3697C2×D311yescharacteristic
151C2×D36—31no—
152C2×D36—61no—
153C2×D36—61no—
154C2×C4×D9—21no—
155C2×C4×D9—61no—
156C2×C4×D9C2×D311yescharacteristic
157C9×Q16—32no—
158C9×SD16—32no—
159C9×M16—31no—
160#3675—32no—
161#3675—32no—
162#3675—62no—
163#3674—32no—
164#3673—32no—
165#3673—62no—
166#3667—31no—
167#3667—61no—
168D72—32no—
169#3664—32no—
170#3663—32no—
171C8×D9—32no—
172C22×C3×C9—21no—
173C22×C3×C9C2×D411yescharacteristic
174C2×C3⋊D9—22no—
175C2×C3⋊D9—42no—
176C2×C3×D9—21no—
177C2×C3×D9—41no—
178C2×C3×D9—42no—
179C2×C3×D9C2×D411yescharacteristic
180C2×C3×D9C2×D411yescharacteristic
181C3×C4×C9—21no—
182C3×C4×C9C2×D411yescharacteristic
183C3×C4×C9C2411yescharacteristic
184C9×Dic3—22no—
185C3×Dic9—21no—
186C3×Dic9C2×D411yescharacteristic
187C3×Dic9C2×D411yescharacteristic
188C2×C3×#41—91no—
189D3×#41—31no—
190C2×#239—91no—
191C2×#235—91no—
192C3×#137—31no—
193#951—91no—
194#951D911yescharacteristic, complemented
195#950—92no—
196#950—92no—
197#949—91no—
198#949—91no—
199#949—91no—
200#949—181no—
201#942—91no—
202#942—92no—
203#932—92no—
204C2×#208—91no—
205#919—92no—
206#918—92no—
207#916—92no—
208#914—92no—
209#908—91no—
210D3×M16—91no—
211C22×C3⋊D3—61no—
212C22×C3×D3—31no—
213C22×C3×D3—61no—
214C32×Q8C22×D311yescharacteristic
215C32×D4—21no—
216C32×D4C22×D311yescharacteristic
217C2×C32×C4—21no—
218C2×C32×C4C22×D311yescharacteristic
219C4⋊(C3⋊D3)—32no—
220C4⋊(C3⋊D3)—61no—
221C3×#67—61no—
222C3×#67—61no—
223C3×#67—61no—
224C2×C3×Dic3—21no—
225C2×C3×Dic3—31no—
226C2×C3×Dic3—61no—
227C3×D12—31no—
228C3×D12—61no—
229C3×C4×D3—31no—
230C3×C4×D3—31no—
231C3×C4×D3—31no—
232C3×C4×D3—31no—
233C3×C4×D3—61no—
234C3×C4×D3—61no—
235C3×C4×D3—61no—
236C3×Dic6—21no—
237C3×Dic6—31no—
238C3×Dic6—61no—
239C3×Dic6C22×D312yes—
240#626—62no—
241#626—62no—
242#624—62no—
243#624—62no—
244C22×D9—21no—
245C22×D9—61no—
246C22×D9—121no—
247C22×D9#6711yescharacteristic
248C3⋊(C3⋊C8)—32no—
249C3×C3⋊C8C22×D312yes—
250Q8×C9—21no—
251Q8×C9—32no—
252Q8×C9—61no—
253Q8×C9C22×D311yescharacteristic
254D4×C9—21no—
255D4×C9—21no—
256D4×C9—41no—
257D4×C9C22×D311yescharacteristic
258C2×C4×C9—21no—
259C2×C4×C9—21no—
260C2×C4×C9—41no—
261C2×C4×C9—61no—
262C2×C4×C9C22×D311yescharacteristic
263#611—21no—
264#611—21no—
265#611—21no—
266C2×Dic9—21no—
267C2×Dic9#6711yescharacteristic
268D36—21no—
269D36—32no—
270D36—62no—
271D36—62no—
272D36—62no—
273D36—61no—
274D36—121no—
275D36C22×D311yescharacteristic
276C4×D9—21no—
277C4×D9—21no—
278C4×D9—21no—
279C4×D9—62no—
280C4×D9—62no—
281C4×D9#6711yescharacteristic
282C4×D9#6711yescharacteristic
283C4×D9C22×D311yescharacteristic
284C4×D9C22×D311yescharacteristic
285Dic18—21no—
286Dic18C22×D311yescharacteristic
287C8×C9—32no—
288C9⋊C8—32no—
289C9⋊C8—62no—
290#575—271noSylow
291C2×C3×C9—21no—
292C2×C3×C9—41no—
293C2×C3×C9C22×D411yescharacteristic
294C3⋊D9—42no—
295C3×D9—21no—
296C3×D9—21no—
297C3×D9—41no—
298C3×#41—21no—
299C3×#41—91no—
300C3×#41—91no—
301C3×#41—91no—
302C3×#41—181no—
303C3×#41—181no—
304C3×#41C2×D911yescharacteristic
305C3×#41D3211yescharacteristic
306C2×C3×Q8—31no—
307C2×C3×Q8—91no—
308C2×C3×D4—91no—
309C2×C3×D4—181no—
310C22×C3×C4—91no—
311C22×C3×C4—181no—
312C2×#67—181no—
313#239—31no—
314#239—92no—
315#239—92no—
316#239—181no—
317#239—181no—
318D3×Q8—31no—
319#237—31no—
320#237—61no—
321D3×D4—31no—
322D3×D4—61no—
323#235—31no—
324#235—61no—
325#235—92no—
326#235—92no—
327#235—181no—
328#235—181no—
329C2×D12—91no—
330C2×D12—91no—
331C2×D12—181no—
332C2×D12—181no—
333C2×C4×D3—31no—
334C2×C4×D3—61no—
335C2×C4×D3—181no—
336C2×C4×D3—181no—
337C2×Dic6C2×D911yescharacteristic, complemented
338C3×Q16—32no—
339C3×SD16—32no—
340C3×M16—31no—
341#216—92no—
342#216—92no—
343#216C2×D912yescomplemented
344#215—92no—
345#215—92no—
346#215—92no—
347#215—92no—
348#215—182no—
349#214—92no—
350#214C2×D912yescomplemented
351#213—92no—
352#213—92no—
353#213—92no—
354#213—182no—
355#208—91no—
356#208—91no—
357#208—92no—
358#208—181no—
359#208C2×D911yescharacteristic, complemented
360C2×C3⋊C8—92no—
361D24—92no—
362#204—92no—
363ΓL(1,𝔽25)—92no—
364D3×C8—92no—
365C22×C32—21no—
366C22×C32D3×D411yescharacteristic
367C2×C3⋊D3—62no—
368C2×C3⋊D3—122no—
369C2×C3×D3—31no—
370C2×C3×D3—31no—
371C2×C3×D3—61no—
372C2×C3×D3—121no—
373C2×C3×D3—122no—
374C32×C4—21no—
375C32×C4D3×D411yescharacteristic
376C32×C4C23×D311yescharacteristic
377C3×Dic3—22no—
378C3×Dic3—31no—
379C3×Dic3—31no—
380C3×Dic3—61no—
381C22×C9—21no—
382C22×C9—21no—
383C22×C9—41no—
384C22×C9C2×#6711yescharacteristic
385C2×D9—21no—
386C2×D9—41no—
387C2×D9—42no—
388C2×D9—62no—
389C2×D9—122no—
390C2×D9—122no—
391C2×D9—242no—
392C2×D9C2×#6711yescharacteristic
393C2×D9C2×#6711yescharacteristic
394C4×C9—21no—
395C4×C9—21no—
396C4×C9—21no—
397C4×C9—41no—
398C4×C9—62no—
399C4×C9C2×#6711yescharacteristic
400C4×C9C23×D311yescharacteristic
401Dic9—21no—
402Dic9C2×#6711yescharacteristic
403Dic9C2×#6711yescharacteristic
404C2×#41—91no—
405C2×#41—271no—
406#137—31no—
407#137—272no—
408#137—271no—
409AGL(1,ℤ/8)—271no—
410AGL(1,ℤ/8)—272no—
411AGL(1,ℤ/8)—271no—
412#135—272no—
413#135—272no—
414C2×M16—271no—
415C3×C9#57511yescharacteristic, complemented, Sylow
416C23×C3—91no—
417C23×C3—181no—
418C22×D3—31no—
419C22×D3—61no—
420C22×D3—181no—
421C22×D3—181no—
422C22×D3—361no—
423C3×Q8—21no—
424C3×Q8—32no—
425C3×Q8—61no—
426C3×Q8—91no—
427C3×Q8—181no—
428C3×Q8C22×D911yescharacteristic
429C3×Q8C2×D3211yescharacteristic
430C3×D4—21no—
431C3×D4—21no—
432C3×D4—21no—
433C3×D4—41no—
434C3×D4—91no—
435C3×D4—181no—
436C3×D4—181no—
437C3×D4—181no—
438C3×D4—181no—
439C3×D4C22×D911yescharacteristic
440C3×D4C2×D3211yescharacteristic
441C2×C3×C4—21no—
442C2×C3×C4—21no—
443C2×C3×C4—21no—
444C2×C3×C4—41no—
445C2×C3×C4—61no—
446C2×C3×C4—91no—
447C2×C3×C4—91no—
448C2×C3×C4—91no—
449C2×C3×C4—91no—
450C2×C3×C4—91no—
451C2×C3×C4—181no—
452C2×C3×C4—181no—
453C2×C3×C4—181no—
454C2×C3×C4—181no—
455C2×C3×C4C22×D911yescharacteristic
456C2×C3×C4C2×D3211yescharacteristic
457#67—61no—
458#67—61no—
459#67—61no—
460#67—182no—
461#67—182no—
462C2×Dic3—21no—
463C2×Dic3—31no—
464C2×Dic3—61no—
465D12—31no—
466D12—61no—
467D12—92no—
468D12—92no—
469D12—182no—
470D12—182no—
471D12—182no—
472D12—181no—
473D12—181no—
474D12—361no—
475C4×D3—31no—
476C4×D3—31no—
477C4×D3—31no—
478C4×D3—31no—
479C4×D3—61no—
480C4×D3—61no—
481C4×D3—61no—
482C4×D3—182no—
483C4×D3—182no—
484C4×D3—182no—
485C4×D3—182no—
486Dic6—21no—
487Dic6—31no—
488Dic6—61no—
489Dic6C22×D912yes—
490C3×C8—32no—
491C3⋊C8—92no—
492C3⋊C8—92no—
493C3⋊C8—182no—
494C3⋊C8C22×D912yes—
495C2×C32—21no—
496C2×C32—41no—
497C2×C32C2×D3×D411yescharacteristic
498C3⋊D3—122no—
499C3×D3—61no—
500C3×D3—61no—
501C3×D3—121no—
502C2×C9—21no—
503C2×C9—21no—
504C2×C9—41no—
505C2×C9—41no—
506C2×C9—81no—
507C2×C9C22×#6711yescharacteristic
508D9—21no—
509D9—21no—
510D9—41no—
511D9—122no—
512D9—242no—
513#41—91no—
514#41—91no—
515#41—91no—
516#41—181no—
517#41—181no—
518#41—272no—
519#41—272no—
520#41—541no—
521#41—541no—
522#41D3×D911yescharacteristic
523C2×Q8—31no—
524C2×Q8—91no—
525C2×D4—91no—
526C2×D4—181no—
527C2×D4—271no—
528C2×D4—541no—
529C22×C4—91no—
530C22×C4—181no—
531C22×C4—541no—
532Q16—32no—
533Q16—272no—
534SD16—32no—
535SD16—272no—
536SD16—272no—
537SD16—272no—
538D8—272no—
539D8—272no—
540M16—31no—
541M16—272no—
542M16—271no—
543C2×C8—272no—
544C22×C3—21no—
545C22×C3—21no—
546C22×C3—21no—
547C22×C3—41no—
548C22×C3—91no—
549C22×C3—91no—
550C22×C3—181no—
551C22×C3—361no—
552C22×C3—362no—
553C22×C3D4×D911yescharacteristic
554C22×C3D3×#6711yescharacteristic
555C2×D3—31no—
556C2×D3—31no—
557C2×D3—61no—
558C2×D3—121no—
559C2×D3—122no—
560C2×D3—182no—
561C2×D3—182no—
562C2×D3—362no—
563C2×D3—362no—
564C2×D3—362no—
565C2×D3—722no—
566C3×C4—21no—
567C3×C4—21no—
568C3×C4—21no—
569C3×C4—21no—
570C3×C4—41no—
571C3×C4—62no—
572C3×C4—91no—
573C3×C4—91no—
574C3×C4—181no—
575C3×C4D4×D911yescharacteristic
576C3×C4C23×D911yescharacteristic
577C3×C4D3×#6711yescharacteristic
578C3×C4C22×D3211yescharacteristic
579Dic3—22no—
580Dic3—31no—
581Dic3—31no—
582Dic3—61no—
583C32#577711yescharacteristic
584C9—21no—
585C9#582211yescharacteristic, complemented
586C23—91no—
587C23—181no—
588C23—541no—
589Q8—32no—
590Q8—61no—
591Q8—91no—
592Q8—181no—
593Q8C2×D3×D911yescharacteristic
594D4—21no—
595D4—91no—
596D4—181no—
597D4—181no—
598D4—181no—
599D4—181no—
600D4—272no—
601D4—542no—
602D4—542no—
603D4—541no—
604D4C2×D3×D911yescharacteristic
605C2×C4—21no—
606C2×C4—61no—
607C2×C4—91no—
608C2×C4—91no—
609C2×C4—91no—
610C2×C4—91no—
611C2×C4—91no—
612C2×C4—181no—
613C2×C4—181no—
614C2×C4—181no—
615C2×C4—181no—
616C2×C4—542no—
617C2×C4—542no—
618C2×C4C2×D3×D911yescharacteristic
619C8—32no—
620C8—272no—
621C2×C3—21no—
622C2×C3—21no—
623C2×C3—21no—
624C2×C3—41no—
625C2×C3—41no—
626C2×C3—41no—
627C2×C3—81no—
628C2×C3—181no—
629C2×C3—181no—
630C2×C3—361no—
631C2×C3C2×D4×D911yescharacteristic
632C2×C3C2×D3×#6711yescharacteristic
633D3—61no—
634D3—61no—
635D3—121no—
636D3—362no—
637D3—362no—
638D3—722no—
639C22—21no—
640C22—91no—
641C22—91no—
642C22—181no—
643C22—361no—
644C22—362no—
645C22—542no—
646C22—1082no—
647C22D9×#6711yescharacteristic
648C4—21no—
649C4—62no—
650C4—91no—
651C4—91no—
652C4—181no—
653C4D9×#6711yescharacteristic
654C4C22×D3×D911yescharacteristic
655C3—21no—
656C3#1058987911yescharacteristic, complemented
657C3#1059547011yescharacteristic
658C2—21no—
659C2—41no—
660C2—181no—
661C2—181no—
662C2—361no—
663C2—1082no—
664C2C2×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#575
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×#41×D9C2semidirect
2. C2×#88635C2semidirect
3. #11721867C2semidirect
4. C9×#951C2semidirect
5. #11721223C2semidirect
6. #11721222C2semidirect
7. #11721185C2semidirect
8. #11721184C2semidirect
9. #11720893C2semidirect
10. #11720873C2semidirect
11. D9×#208C2semidirect
15. C2×C4⋊(C3⋊D9)C22semidirect
16. C3×#3701C22semidirect
17. C3×Q8×D9C22semidirect
19. C3×#3699C22semidirect
21. C3×D4×D9C22semidirect
26. C2×C9×Dic6C22semidirect
32. #88635C22semidirect
33. #88548C22semidirect
34. #88544C22semidirect
35. #88532C22semidirect
36. C9×#216C22semidirect
37. C9×#214C22semidirect
38. C9×#208C22semidirect
39. #88427C22semidirect
40. #88425C22semidirect
41. #88417C22semidirect
42. #88416C22semidirect
43. #88413C22semidirect
44. D9×C3⋊C8C22semidirect
56. #41×D9D3semidirect
143. #3701C2×D3semidirect
144. Q8×D9C2×D3semidirect
146. #3699C2×D3semidirect
148. D4×D9C2×D3semidirect
194. #951D9semidirect
337. C2×Dic6C2×D9semidirect
343. #216C2×D9semidirect
350. #214C2×D9semidirect
359. #208C2×D9semidirect
415. C3×C9#575semidirect
585. C9#5822semidirect
656. C3#10589879semidirect

Maximal subgroups

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

subgroupindex
C3×#41×D92
C2×#886352
#117218672
C9×#9512
#117212232
#117212222
#117211852
#117211842
#117208932
#117208732
D9×#2082
#105954703
#105898793