Finite Group Databasebeta

← #13151907

Subgroups — #13151907

Order 1728.

846 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#41×D9C211yesmaximal, characteristic, complemented
2C2×#88638C211yesmaximal, characteristic, complemented
3#11721866C211yesmaximal, characteristic, complemented
4C9×#949C211yesmaximal, characteristic, complemented
5#11721215C212yesmaximal, complemented
6#11721214C212yesmaximal, complemented
7#11721177C212yesmaximal, complemented
8#11721176C212yesmaximal, complemented
9#11720883C211yesmaximal, characteristic, complemented
10#11720869C211yesmaximal, characteristic, complemented
11D9×#208C211yesmaximal, characteristic, complemented
12#10595469—31nomaximal
13#10589875—31nomaximal
14C3×C9×#41C2211yescharacteristic
15C2×#6226C2211yescharacteristic, complemented
16C3×#3701C2211yescharacteristic, complemented
17C3×Q8×D9C2211yescharacteristic, complemented
18C3×#3699—21no—
19C3×#3699C2211yescharacteristic, complemented
20C3×D4×D9—21no—
21C3×D4×D9C2211yescharacteristic, complemented
22C2×C9×D12C2211yescharacteristic, complemented
23C3×#3697—21no—
24C3×#3697C2211yescharacteristic
25C2×C3×C4×D9—21no—
26C2×C3×C4×D9C2211yescharacteristic
27C2×#6194—21no—
28C2×D3×Dic9—21no—
29#88638—21no—
30#88638—21no—
31#88638C2212yes—
32#88638C2212yescomplemented
33#88547C2212yescomplemented
34#88543C2212yescomplemented
35#88532C2211yescharacteristic, complemented
36C9×#215C2212yescomplemented
37C9×#213C2212yescomplemented
38C9×#208C2211yescharacteristic, complemented
39#88423C2212yescomplemented
40#88421C2212yescomplemented
41#88418C2212yescomplemented
42#88415C2212yescomplemented
43#88413C2212yescomplemented
44D9×C3⋊C8C2212yescomplemented
45C3×D3×#41—31no—
46C2×#3795—31no—
47#64160—31no—
48C3×#949D311yescharacteristic
49#63947—32no—
50#63946—32no—
51#63933—32no—
52#63932—32no—
53#63831—31no—
54#63823—31no—
55D3×#208—31no—
56#41×D9D311yescharacteristic, complemented
57C2×#3699—31no—
58C9×AGL(1,ℤ/8)—31no—
59#63511—31no—
60#63511—61no—
61#63495—31no—
62#63478—32no—
63#63477—32no—
64#63474—32no—
65#63473—32no—
66#63471—31no—
67M16×D9—31no—
68C22×D3×C9—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
76C2×C3⋊Dic9—21no—
77#6226—21no—
78#6226C2312yes—
79C3×#611—21no—
80C3×#611—21no—
81C3×#611—21no—
82C2×C3×Dic9—21no—
83C2×C3×Dic9D411yescharacteristic
84C9×D12—21no—
85C9×D12C2312yes—
86C3×D36—21no—
87C3×D36C2311yescharacteristic
88C3×C4×D9—21no—
89C3×C4×D9—21no—
90C3×C4×D9—21no—
91C3×C4×D9D411yescharacteristic
92C3×C4×D9D411yescharacteristic
93C3×C4×D9C2311yescharacteristic
94C3×C4×D9C2311yescharacteristic
95C3×Dic18—21no—
96C3×Dic18C2311yescharacteristic
97#6194—22no—
98#6194—22no—
99D3×Dic9—22no—
100D3×Dic9—22no—
101C3⋊(C9⋊C8)C2312yes—
102C9×C3⋊C8C2312yes—
103#5822—91no—
104#5773—91no—
105C32×#41C2×D311yescharacteristic
106C2×#634—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×D12C2×D311yescharacteristic
116C2×C3×C4×D3—31no—
117C2×C3×C4×D3—61no—
118C2×#625—61no—
119C2×D3×Dic3—61no—
120#3795—32no—
121#3795—32no—
122#3795—61no—
123#3795—61no—
124#3756—32no—
125#3754—32no—
126#3748—31no—
127C3×#215C2×D312yes—
128C3×#213C2×D312yes—
129C3×#208C2×D311yescharacteristic
130#3719—32no—
131#3716—32no—
132#3715—32no—
133#3713—32no—
134#3711—32no—
135D3×C3⋊C8—32no—
136C9×#41—21no—
137C9×#41C2×D311yescharacteristic
138C2×D4×C9—31no—
139C2×#611—61no—
140C22×Dic9—61no—
141#3701C2×D311yescharacteristic, complemented
142Q8×D9C2×D311yescharacteristic, complemented
143#3699—21no—
144#3699—32no—
145#3699—32no—
146#3699—61no—
147#3699—61no—
148#3699C2×D311yescharacteristic, complemented
149D4×D9—21no—
150D4×D9C2×D311yescharacteristic, complemented
151#3697—21no—
152#3697C2×D311yescharacteristic
153C2×C4×D9—21no—
154C2×C4×D9C2×D311yescharacteristic
155C2×Dic18—31no—
156C2×Dic18—61no—
157C9×SD16—32no—
158C9×D8—32no—
159C9×M16—31no—
160#3674—32no—
161#3674—62no—
162#3673—32no—
163#3672—32no—
164#3672—32no—
165#3672—62no—
166#3667—31no—
167#3667—61no—
168#3664—32no—
169#3663—32no—
170C8×D9—32no—
171Dic36—32no—
172C22×C3×C9—21no—
173C22×C3×C9C2×D411yescharacteristic
174C2×D3×C9—22no—
175C2×D3×C9—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
184C3⋊Dic9—22no—
185C3×Dic9—21no—
186C3×Dic9C2×D411yescharacteristic
187C3×Dic9C2×D411yescharacteristic
188C2×C3×#41—91no—
189D3×#41—31no—
190C2×#237—91no—
191C2×#235—91no—
192C3×AGL(1,ℤ/8)—31no—
193#951—91no—
194#951—91no—
195#951—91no—
196#951—181no—
197#950—92no—
198#950—92no—
199#949—91no—
200#949D911yescharacteristic, complemented
201#942—92no—
202#932—91no—
203#932—92no—
204C2×#208—91no—
205#916—92no—
206#915—92no—
207#912—92no—
208#911—92no—
209#909—91no—
210D3×M16—91no—
211C22×C3×D3—21no—
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
219C2×C3⋊Dic3—61no—
220#634—32no—
221#634—61no—
222C3×#67—61no—
223C3×#67—61no—
224C3×#67—61no—
225C2×C3×Dic3—31no—
226C2×C3×Dic3—61no—
227C3×D12—21no—
228C3×D12—31no—
229C3×D12—61no—
230C3×D12C22×D312yes—
231C3×C4×D3—31no—
232C3×C4×D3—31no—
233C3×C4×D3—31no—
234C3×C4×D3—31no—
235C3×C4×D3—61no—
236C3×C4×D3—61no—
237C3×C4×D3—61no—
238C3×Dic6—31no—
239C3×Dic6—61no—
240#625—62no—
241#625—62no—
242D3×Dic3—62no—
243D3×Dic3—62no—
244C23×C9—61no—
245C22×D9—21no—
246C22×D9#6711yescharacteristic
247C3⋊(C3⋊C8)—32no—
248C3×C3⋊C8C22×D312yes—
249Q8×C9—21no—
250Q8×C9C22×D311yescharacteristic
251D4×C9—21no—
252D4×C9—21no—
253D4×C9—32no—
254D4×C9—41no—
255D4×C9—61no—
256D4×C9C22×D311yescharacteristic
257C2×C4×C9—21no—
258C2×C4×C9—21no—
259C2×C4×C9—41no—
260C2×C4×C9C22×D311yescharacteristic
261#611—21no—
262#611—21no—
263#611—21no—
264#611—62no—
265#611—62no—
266C2×Dic9—21no—
267C2×Dic9—61no—
268C2×Dic9—62no—
269C2×Dic9—62no—
270C2×Dic9—121no—
271C2×Dic9#6711yescharacteristic
272D36—21no—
273D36C22×D311yescharacteristic
274C4×D9—21no—
275C4×D9—21no—
276C4×D9—21no—
277C4×D9#6711yescharacteristic
278C4×D9#6711yescharacteristic
279C4×D9C22×D311yescharacteristic
280C4×D9C22×D311yescharacteristic
281Dic18—21no—
282Dic18—32no—
283Dic18—62no—
284Dic18—61no—
285Dic18—121no—
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
294D3×C9—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—91no—
307C2×C3×D4—31no—
308C2×C3×D4—91no—
309C2×C3×D4—181no—
310C22×C3×C4—91no—
311C22×C3×C4—181no—
312C2×#67—181no—
313C2×#67—181no—
314C22×Dic3—181no—
315#239—31no—
316D3×Q8—31no—
317#237—31no—
318#237—61no—
319#237—92no—
320#237—92no—
321#237—181no—
322#237—181no—
323D3×D4—31no—
324D3×D4—61no—
325#235—31no—
326#235—61no—
327#235—92no—
328#235—92no—
329#235—181no—
330#235—181no—
331C2×D12C2×D911yescharacteristic, complemented
332C2×C4×D3—31no—
333C2×C4×D3—61no—
334C2×C4×D3—181no—
335C2×Dic6—91no—
336C2×Dic6—91no—
337C2×Dic6—181no—
338C3×SD16—32no—
339C3×D8—32no—
340C3×M16—31no—
341#216—92no—
342#216—92no—
343#216—92no—
344#216—182no—
345#215—92no—
346#215C2×D912yescomplemented
347#214—92no—
348#214—92no—
349#214—92no—
350#214—92no—
351#214—182no—
352#213—92no—
353#213—92no—
354#213C2×D912yescomplemented
355#208—91no—
356#208—91no—
357#208—92no—
358#208—181no—
359#208C2×D911yescharacteristic, complemented
360C2×C3⋊C8—92no—
361Dic12—92no—
362#204—92no—
363ΓL(1,𝔽25)—92no—
364D3×C8—92no—
365C22×C32—21no—
366C22×C32D3×D411yescharacteristic
367C2×C3×D3—22no—
368C2×C3×D3—31no—
369C2×C3×D3—31no—
370C2×C3×D3—42no—
371C2×C3×D3—61no—
372C2×C3×D3—121no—
373C2×C3×D3—122no—
374C32×C4—21no—
375C32×C4D3×D411yescharacteristic
376C32×C4C23×D311yescharacteristic
377C3⋊Dic3—62no—
378C3×Dic3—31no—
379C3×Dic3—31no—
380C3×Dic3—61no—
381C22×C9—21no—
382C22×C9—21no—
383C22×C9—41no—
384C22×C9—62no—
385C22×C9—122no—
386C22×C9C2×#6711yescharacteristic
387C2×D9—21no—
388C2×D9—41no—
389C2×D9—42no—
390C2×D9C2×#6711yescharacteristic
391C2×D9C2×#6711yescharacteristic
392C4×C9—21no—
393C4×C9—21no—
394C4×C9—21no—
395C4×C9—41no—
396C4×C9C2×#6711yescharacteristic
397C4×C9C23×D311yescharacteristic
398Dic9—21no—
399Dic9—62no—
400Dic9—122no—
401Dic9C2×#6711yescharacteristic
402Dic9C2×#6711yescharacteristic
403C2×#41—91no—
404C2×#41—271no—
405#137—271no—
406#137—272no—
407#137—271no—
408AGL(1,ℤ/8)—31no—
409AGL(1,ℤ/8)—272no—
410AGL(1,ℤ/8)—271no—
411#135—272no—
412#135—272no—
413C2×M16—271no—
414C3×C9#57511yescharacteristic, complemented, Sylow
415C23×C3—61no—
416C23×C3—91no—
417C23×C3—181no—
418C22×D3—21no—
419C22×D3—31no—
420C22×D3—61no—
421C3×Q8—21no—
422C3×Q8—91no—
423C3×Q8—181no—
424C3×Q8C22×D911yescharacteristic
425C3×Q8C2×D3211yescharacteristic
426C3×D4—21no—
427C3×D4—21no—
428C3×D4—21no—
429C3×D4—32no—
430C3×D4—41no—
431C3×D4—61no—
432C3×D4—91no—
433C3×D4—181no—
434C3×D4—181no—
435C3×D4—181no—
436C3×D4—181no—
437C3×D4C22×D911yescharacteristic
438C3×D4C2×D3211yescharacteristic
439C2×C3×C4—21no—
440C2×C3×C4—21no—
441C2×C3×C4—21no—
442C2×C3×C4—41no—
443C2×C3×C4—91no—
444C2×C3×C4—91no—
445C2×C3×C4—91no—
446C2×C3×C4—91no—
447C2×C3×C4—91no—
448C2×C3×C4—181no—
449C2×C3×C4—181no—
450C2×C3×C4—181no—
451C2×C3×C4—181no—
452C2×C3×C4C22×D911yescharacteristic
453C2×C3×C4C2×D3211yescharacteristic
454#67—61no—
455#67—61no—
456#67—61no—
457#67—182no—
458#67—182no—
459#67—182no—
460#67—182no—
461C2×Dic3—31no—
462C2×Dic3—61no—
463C2×Dic3—182no—
464C2×Dic3—182no—
465C2×Dic3—181no—
466C2×Dic3—181no—
467C2×Dic3—361no—
468D12—21no—
469D12—31no—
470D12—61no—
471D12C22×D912yes—
472C4×D3—31no—
473C4×D3—31no—
474C4×D3—31no—
475C4×D3—31no—
476C4×D3—61no—
477C4×D3—61no—
478C4×D3—61no—
479C4×D3—182no—
480C4×D3—182no—
481Dic6—31no—
482Dic6—61no—
483Dic6—92no—
484Dic6—92no—
485Dic6—182no—
486Dic6—181no—
487Dic6—181no—
488Dic6—361no—
489C3×C8—32no—
490C3⋊C8—92no—
491C3⋊C8—92no—
492C3⋊C8—182no—
493C3⋊C8C22×D912yes—
494C2×C32—21no—
495C2×C32—41no—
496C2×C32C2×D3×D411yescharacteristic
497C3×D3—42no—
498C3×D3—61no—
499C3×D3—61no—
500C3×D3—121no—
501C2×C9—21no—
502C2×C9—21no—
503C2×C9—41no—
504C2×C9—41no—
505C2×C9—81no—
506C2×C9—122no—
507C2×C9C22×#6711yescharacteristic
508D9—21no—
509D9—21no—
510D9—41no—
511#41—91no—
512#41—91no—
513#41—91no—
514#41—181no—
515#41—181no—
516#41—272no—
517#41—272no—
518#41—541no—
519#41—541no—
520#41D3×D911yescharacteristic
521C2×Q8—91no—
522C2×Q8—271no—
523C2×D4—31no—
524C2×D4—91no—
525C2×D4—181no—
526C2×D4—541no—
527C22×C4—91no—
528C22×C4—181no—
529C22×C4—541no—
530Q16—272no—
531Q16—272no—
532SD16—32no—
533SD16—272no—
534SD16—272no—
535SD16—272no—
536D8—32no—
537D8—272no—
538M16—31no—
539M16—272no—
540M16—271no—
541C2×C8—272no—
542C22×C3—21no—
543C22×C3—21no—
544C22×C3—21no—
545C22×C3—41no—
546C22×C3—62no—
547C22×C3—91no—
548C22×C3—91no—
549C22×C3—122no—
550C22×C3—181no—
551C22×C3—361no—
552C22×C3—362no—
553C22×C3D4×D911yescharacteristic
554C22×C3D3×#6711yescharacteristic
555C2×D3—22no—
556C2×D3—31no—
557C2×D3—31no—
558C2×D3—42no—
559C2×D3—61no—
560C2×D3—121no—
561C2×D3—122no—
562C3×C4—21no—
563C3×C4—21no—
564C3×C4—21no—
565C3×C4—21no—
566C3×C4—41no—
567C3×C4—91no—
568C3×C4—91no—
569C3×C4—181no—
570C3×C4D4×D911yescharacteristic
571C3×C4C23×D911yescharacteristic
572C3×C4D3×#6711yescharacteristic
573C3×C4C22×D3211yescharacteristic
574Dic3—31no—
575Dic3—31no—
576Dic3—61no—
577Dic3—182no—
578Dic3—182no—
579Dic3—362no—
580C32#577311yescharacteristic
581C9—21no—
582C9#582211yescharacteristic, complemented
583C23—61no—
584C23—91no—
585C23—181no—
586Q8—91no—
587Q8—181no—
588Q8—272no—
589Q8—541no—
590Q8C2×D3×D911yescharacteristic
591D4—21no—
592D4—32no—
593D4—61no—
594D4—91no—
595D4—181no—
596D4—181no—
597D4—181no—
598D4—181no—
599D4—542no—
600D4—542no—
601D4C2×D3×D911yescharacteristic
602C2×C4—21no—
603C2×C4—91no—
604C2×C4—91no—
605C2×C4—91no—
606C2×C4—91no—
607C2×C4—91no—
608C2×C4—181no—
609C2×C4—181no—
610C2×C4—181no—
611C2×C4—181no—
612C2×C4—542no—
613C2×C4—542no—
614C2×C4—541no—
615C2×C4C2×D3×D911yescharacteristic
616C8—32no—
617C8—272no—
618C2×C3—21no—
619C2×C3—21no—
620C2×C3—21no—
621C2×C3—41no—
622C2×C3—41no—
623C2×C3—41no—
624C2×C3—81no—
625C2×C3—122no—
626C2×C3—181no—
627C2×C3—181no—
628C2×C3—361no—
629C2×C3C2×D4×D911yescharacteristic
630C2×C3C2×D3×#6711yescharacteristic
631D3—42no—
632D3—61no—
633D3—61no—
634D3—121no—
635C22—21no—
636C22—62no—
637C22—91no—
638C22—91no—
639C22—122no—
640C22—181no—
641C22—361no—
642C22—362no—
643C22D9×#6711yescharacteristic
644C4—21no—
645C4—91no—
646C4—91no—
647C4—181no—
648C4—542no—
649C4D9×#6711yescharacteristic
650C4C22×D3×D911yescharacteristic
651C3—21no—
652C3#1058987511yescharacteristic, complemented
653C3#1059546911yescharacteristic
654C2—21no—
655C2—41no—
656C2—122no—
657C2—181no—
658C2—181no—
659C2—361no—
660C2C2×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×#88638C2semidirect
3. #11721866C2semidirect
4. C9×#949C2semidirect
5. #11721215C2semidirect
6. #11721214C2semidirect
7. #11721177C2semidirect
8. #11721176C2semidirect
9. #11720883C2semidirect
10. #11720869C2semidirect
11. D9×#208C2semidirect
15. C2×#6226C22semidirect
16. C3×#3701C22semidirect
17. C3×Q8×D9C22semidirect
19. C3×#3699C22semidirect
21. C3×D4×D9C22semidirect
22. C2×C9×D12C22semidirect
32. #88638C22semidirect
33. #88547C22semidirect
34. #88543C22semidirect
35. #88532C22semidirect
36. C9×#215C22semidirect
37. C9×#213C22semidirect
38. C9×#208C22semidirect
39. #88423C22semidirect
40. #88421C22semidirect
41. #88418C22semidirect
42. #88415C22semidirect
43. #88413C22semidirect
44. D9×C3⋊C8C22semidirect
56. #41×D9D3semidirect
141. #3701C2×D3semidirect
142. Q8×D9C2×D3semidirect
148. #3699C2×D3semidirect
150. D4×D9C2×D3semidirect
200. #949D9semidirect
331. C2×D12C2×D9semidirect
346. #215C2×D9semidirect
354. #213C2×D9semidirect
359. #208C2×D9semidirect
414. C3×C9#575semidirect
582. C9#5822semidirect
652. C3#10589875semidirect

Maximal subgroups

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

subgroupindex
C3×#41×D92
C2×#886382
#117218662
C9×#9492
#117212152
#117212142
#117211772
#117211762
#117208832
#117208692
D9×#2082
#105954693
#105898753