Finite Group Databasebeta

← D9×#1010 #13164778

Subgroups — D9×#1010 #13164778

Order 1728.

2528 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11722673C211yesmaximal, characteristic, complemented
2C9×#1010C211yesmaximal, characteristic, complemented
3C3×#41×D9C211yesmaximal, characteristic, complemented
4D3×Q8×D9C212yesmaximal, complemented
5#11722510C212yesmaximal, complemented
6D9×#237C216yesmaximal, complemented
7#11722494C216yesmaximal, complemented
8D9×#235C213yesmaximal, complemented
9#11722465C213yesmaximal, complemented
10#11722458C213yesmaximal, complemented
11C2×D9×Dic6C213yesmaximal, complemented
12D3×#1010—31nomaximal
13D9×2−1+4—31nomaximal
14C3×C9×#41C2211yescharacteristic, complemented
15Q8×C3⋊D9C2212yescomplemented
16#88742C2216yescomplemented
17#88736C2213yescomplemented
18C2×#6226C2213yescomplemented
19D3×Q8×C9C2212yescomplemented
20C3×#3701C2211yescharacteristic, complemented
21C3×Q8×D9C2211yescharacteristic
22C9×#237C2216yescomplemented
23C3×#3699C2213yescomplemented
24C3×D4×D9C2213yes—
25C9×#235C2213yescomplemented
26C3×#3697C2213yescomplemented
27C2×C3×C4×D9C2213yes—
28C2×C9×Dic6C2213yescomplemented
29D9×#67C2216yes—
30#88663C22112yescomplemented
31#88662C2216yescomplemented
32#88658C22112yescomplemented
33C2×Dic3×D9C2216yes—
34C2×#6189C2216yescomplemented
35D9×D12C2211yescharacteristic
36C4×D3×D9C2216yes—
37#88641C2216yescomplemented
38#88640C2216yescomplemented
39#88638C2213yescomplemented
40D3×Dic18C2216yescomplemented
41(C3⋊D9)⋊Q8C2216yescomplemented
42#88635C2213yescomplemented
43#88634C2216yescomplemented
44D9×Dic6C2216yes—
45D9×Dic6C2213yes—
46D9×Dic6C2216yescomplemented
47D9×Dic6C2216yescomplemented
48#64367—31no—
49C3×#1010D311yescharacteristic
50C3×D3×#41—31no—
51D32×Q8—32no—
52#64315—32no—
53D3×#237—36no—
54#64309—36no—
55D3×#235—33no—
56#64298—33no—
57#64297—33no—
58C2×D3×Dic6—33no—
59C9×2−1+4—31no—
60#63716—31no—
61#63716—36no—
62#63716—33no—
63#63716—61no—
64#41×D9—36no—
65#41×D9—33no—
66#41×D9D311yescharacteristic, complemented
67#63713—32no—
68#63713—33no—
69C2×Q8×D9—32no—
70C2×Q8×D9—33no—
71C22×C3×D9C2313yes—
72C2×D3×D9C2312yes—
73C3×Q8×C9C2311yescharacteristic
74C3×D4×C9C2313yes—
75C2×C3×C4×C9C2313yes—
76#6233C2316yes—
77C2×C3⋊Dic9C2316yes—
78C4⋊(C3⋊D9)C2311yescharacteristic
79C4×C3⋊D9C2316yes—
80#6226C2313yes—
81#6226C2316yes—
82C9×#67C2316yes—
83C3×#611C2316yes—
84C2×C9×Dic3C2316yes—
85C2×C3×Dic9C2313yes—
86C9×D12C2311yescharacteristic
87C4×D3×C9C2316yes—
88C3×D36C2313yes—
89C3×C4×D9C2311yescharacteristic
90C3×C4×D9C2313yes—
91C3×C4×D9C2313yes—
92C3×C4×D9C2313yes—
93C9×Dic6C2313yes—
94C9×Dic6C2316yes—
95C3×Dic18C2313yes—
96#6195C2312yes—
97#6194C2316yes—
98D3×Dic9C2316yes—
99#6192C2316yes—
100(C3⋊D9)⋊C4C2316yes—
101Dic3×D9C2316yes—
102Dic3×D9C23112yes—
103#6189C2316yes—
104#6189C23112yes—
105D3×2−1+4—91no—
106C2×#1010—91no—
107C32×#41C2×D311yescharacteristic
108Q8×C3⋊D3—32no—
109#3830—36no—
110#3828—33no—
111C2×#634—33no—
112C3×#239—31no—
113C3×D3×Q8—31no—
114C3×D3×Q8C2×D312yes—
115C3×#237—33no—
116C3×#237C2×D316yes—
117C3×D3×D4—33no—
118C3×#235—33no—
119C3×#235C2×D313yes—
120C2×C3×C4×D3—33no—
121C2×C3×Dic6C2×D313yes—
122D3×#67—36no—
123C2×#627—36no—
124#3805—312no—
125#3804—36no—
126#3804—312no—
127C2×D3×Dic3—36no—
128D3×D12—31no—
129C4×D32—36no—
130#3799—33no—
131#3798—36no—
132#3797—36no—
133#3796—36no—
134#3795—33no—
135#3795—36no—
136D3×Dic6—36no—
137D3×Dic6—33no—
138D3×Dic6—36no—
139D3×Dic6—36no—
140D3×Dic6—36no—
141C9×#41—21no—
142C9×#41—36no—
143C9×#41—33no—
144C9×#41C2×D311yescharacteristic, complemented
145C2×Q8×C9—32no—
146C2×Q8×C9—33no—
147#3701—36no—
148#3701—33no—
149#3701C2×D311yescharacteristic, complemented
150Q8×D9—32no—
151Q8×D9—36no—
152Q8×D9—33no—
153Q8×D9—36no—
154Q8×D9—36no—
155Q8×D9—36no—
156Q8×D9—62no—
157Q8×D9C2×D311yescharacteristic
158#3699—36no—
159#3699—312no—
160#3699—36no—
161#3699—33no—
162#3699—66no—
163#3699C2×D313yescomplemented
164D4×D9—36no—
165D4×D9—31no—
166D4×D9C2×D313yes—
167#3697—33no—
168#3697—312no—
169#3697—36no—
170#3697—36no—
171#3697—63no—
172#3697C2×D313yescomplemented
173C2×C4×D9—36no—
174C2×C4×D9—36no—
175C2×C4×D9C2×D313yes—
176C2×Dic18—33no—
177C2×Dic18—36no—
178C2×Dic18—36no—
179C2×Dic18—63no—
180C22×C3×C9C2413yes—
181C2×C3⋊D9C2412yes—
182C2×D3×C9C2412yes—
183C2×C3×D9—26no—
184C2×C3×D9C2413yes—
185C2×C3×D9C2411yescharacteristic
186D3×D9—24no—
187C3×C4×C9C2413yes—
188C3×C4×C9C2411yescharacteristic
189C3⋊Dic9C2416yes—
190C9×Dic3C2416yes—
191C3×Dic9C2411yescharacteristic
192C3×Dic9C2413yes—
193C3×2−1+4—31no—
194C2×C3×#41—91no—
195#1010—93no—
196#1010—96no—
197#1010—91no—
198#1010—91no—
199#1010—92no—
200#1010—96no—
201#1010—93no—
202#1010—93no—
203#1010—181no—
204#1010D911yescharacteristic, direct product factor
205D3×#41—31no—
206D3×#41—93no—
207D3×#41—96no—
208#1007—93no—
209#1007—92no—
210C2×D3×Q8—93no—
211C2×D3×Q8—92no—
212C2×D3×Q8—92no—
213C2×#237—96no—
214C2×#235—93no—
215C22×Dic6—93no—
216C22×C3×D3—33no—
217C2×D32—32no—
218C32×Q8C22×D311yescharacteristic
219C32×D4C22×D313yes—
220C2×C32×C4C22×D313yes—
221#638—36no—
222C2×C3⋊Dic3—36no—
223C4⋊(C3⋊D3)—31no—
224C4×C3⋊D3—36no—
225#634—33no—
226#634—36no—
227C3×#67—36no—
228C3×#67C22×D316yes—
229C2×C3×Dic3—33no—
230C2×C3×Dic3C22×D316yes—
231C3×D12—33no—
232C3×D12C22×D311yescharacteristic
233C3×C4×D3—31no—
234C3×C4×D3—33no—
235C3×C4×D3—33no—
236C3×C4×D3—33no—
237C3×C4×D3C22×D316yes—
238C3×Dic6—33no—
239C3×Dic6C22×D313yes—
240C3×Dic6C22×D316yes—
241#627—312no—
242#627—36no—
243#626—32no—
244#626—36no—
245#625—36no—
246#624—36no—
247D3×Dic3—36no—
248D3×Dic3—36no—
249D3×Dic3—312no—
250C22×D9—32no—
251C22×D9C22×D313yes—
252Q8×C9—21no—
253Q8×C9—33no—
254Q8×C9—36no—
255Q8×C9C22×D311yescharacteristic
256D4×C9—23no—
257D4×C9—36no—
258D4×C9—31no—
259D4×C9C22×D313yes—
260C2×C4×C9—23no—
261C2×C4×C9—36no—
262C2×C4×C9—36no—
263C2×C4×C9C22×D313yes—
264#611—36no—
265#611—36no—
266#611—32no—
267#611—66no—
268#611C22×D316yes—
269C2×Dic9—36no—
270C2×Dic9—36no—
271C2×Dic9—66no—
272C2×Dic9C22×D313yes—
273D36—36no—
274D36—31no—
275D36—61no—
276D36C22×D313yes—
277C4×D9—36no—
278C4×D9—312no—
279C4×D9—36no—
280C4×D9—36no—
281C4×D9—66no—
282C4×D9C22×D313yes—
283C4×D9C22×D311yescharacteristic
284C4×D9C22×D313yes—
285C4×D9C22×D313yes—
286Dic18—33no—
287Dic18—36no—
288Dic18—36no—
289Dic18—312no—
290Dic18—63no—
291Dic18—66no—
292Dic18C22×D313yes—
293C2×2−1+4—271noSylow
294C2×C3×C9—23no—
295C2×C3×C9C2511yescharacteristic
296C3⋊D9—22no—
297D3×C9—22no—
298C3×D9—23no—
299C3×D92−1+412yescomplemented
300C3×#41—21no—
301C3×#41—36no—
302C3×#41—33no—
303C3×#41—91no—
304C3×#41—93no—
305C3×#41—93no—
306C3×#41C2×D911yescharacteristic, complemented
307C3×#41D3211yescharacteristic
308C2×C3×Q8—32no—
309C2×C3×Q8—33no—
310C2×C3×Q8—91no—
311C2×C3×D4—93no—
312C22×C3×C4—93no—
313C2×#67—96no—
314C22×Dic3—96no—
315#239—31no—
316#239—93no—
317#239—96no—
318D3×Q8—31no—
319D3×Q8—96no—
320D3×Q8—93no—
321D3×Q8—96no—
322D3×Q8—96no—
323D3×Q8—96no—
324D3×Q8—92no—
325D3×Q8—92no—
326D3×Q8—96no—
327D3×Q8—96no—
328D3×Q8—182no—
329D3×Q8C2×D912yescomplemented
330#237—33no—
331#237—93no—
332#237—96no—
333#237—912no—
334#237—96no—
335#237—96no—
336#237—912no—
337#237—912no—
338#237—96no—
339#237—96no—
340#237—186no—
341#237C2×D916yescomplemented
342D3×D4—33no—
343D3×D4—91no—
344D3×D4—96no—
345#235—33no—
346#235—96no—
347#235—96no—
348#235—912no—
349#235—93no—
350#235—93no—
351#235—96no—
352#235—93no—
353#235—96no—
354#235—93no—
355#235—183no—
356#235C2×D913yescomplemented
357C2×D12—91no—
358C2×C4×D3—33no—
359C2×C4×D3—96no—
360C2×C4×D3—96no—
361C2×C4×D3—96no—
362C2×Dic6—96no—
363C2×Dic6—96no—
364C2×Dic6—93no—
365C2×Dic6—93no—
366C2×Dic6—96no—
367C2×Dic6—93no—
368C2×Dic6—96no—
369C2×Dic6—96no—
370C2×Dic6—96no—
371C2×Dic6—183no—
372C2×Dic6C2×D913yescomplemented
373C22×C32C23×D313yes—
374C2×C3⋊D3—32no—
375C2×C3×D3—31no—
376C2×C3×D3—33no—
377C2×C3×D3—66no—
378C2×C3×D3C23×D312yes—
379D32—64no—
380C32×C4C23×D313yes—
381C32×C4C23×D311yescharacteristic
382C3⋊Dic3—36no—
383C3×Dic3—33no—
384C3×Dic3—31no—
385C3×Dic3C23×D316yes—
386C22×C9—23no—
387C22×C9—32no—
388C22×C9C23×D313yes—
389C2×D9—26no—
390C2×D9—32no—
391C2×D9—62no—
392C2×D9—64no—
393C2×D9C23×D313yes—
394C2×D9C23×D311yescharacteristic
395C4×C9—21no—
396C4×C9—23no—
397C4×C9—36no—
398C4×C9C23×D311yescharacteristic
399C4×C9C23×D313yes—
400Dic9—36no—
401Dic9—66no—
402Dic9C23×D311yescharacteristic
403Dic9C23×D313yes—
4042−1+4—31no—
4052−1+4—272no—
4062−1+4—273no—
4072−1+4—276no—
4082−1+4—273no—
4092−1+4—271no—
410C2×#41—91no—
411C2×#41—273no—
412C2×#41—276no—
413C22×Q8—273no—
414C22×Q8—272no—
415C3×C9C2×2−1+411yescharacteristic, complemented, Sylow
416C23×C3—93no—
417C22×D3—33no—
418C22×D3—92no—
419C22×D3—92no—
420C3×Q8—21no—
421C3×Q8—36no—
422C3×Q8—33no—
423C3×Q8—93no—
424C3×Q8C22×D911yescharacteristic
425C3×Q8C2×D3211yescharacteristic
426C3×D4—23no—
427C3×D4—36no—
428C3×D4—31no—
429C3×D4—96no—
430C3×D4—93no—
431C3×D4C22×D913yes—
432C3×D4C2×D3213yes—
433C2×C3×C4—23no—
434C2×C3×C4—36no—
435C2×C3×C4—36no—
436C2×C3×C4—93no—
437C2×C3×C4—91no—
438C2×C3×C4—93no—
439C2×C3×C4—93no—
440C2×C3×C4—93no—
441C2×C3×C4C22×D913yes—
442C2×C3×C4C2×D3213yes—
443#67—36no—
444#67—92no—
445#67—96no—
446#67—96no—
447#67—96no—
448#67—96no—
449#67—96no—
450#67—186no—
451#67C22×D916yes—
452C2×Dic3—33no—
453C2×Dic3—96no—
454C2×Dic3—96no—
455C2×Dic3—96no—
456C2×Dic3—96no—
457C2×Dic3—912no—
458C2×Dic3—186no—
459C2×Dic3C22×D916yes—
460D12—33no—
461D12—96no—
462D12—92no—
463D12—91no—
464D12—91no—
465D12—181no—
466D12C22×D911yescharacteristic
467C4×D3—31no—
468C4×D3—33no—
469C4×D3—33no—
470C4×D3—33no—
471C4×D3—96no—
472C4×D3—96no—
473C4×D3—912no—
474C4×D3—96no—
475C4×D3—96no—
476C4×D3—96no—
477C4×D3—96no—
478C4×D3—186no—
479C4×D3C22×D916yes—
480Dic6—33no—
481Dic6—912no—
482Dic6—96no—
483Dic6—96no—
484Dic6—93no—
485Dic6—93no—
486Dic6—96no—
487Dic6—96no—
488Dic6—912no—
489Dic6—183no—
490Dic6—186no—
491Dic6C22×D913yes—
492Dic6C22×D916yes—
493C2×C32—23no—
494C2×C32C24×D311yescharacteristic
495C3⋊D3—62no—
496C3×D3—22no—
497C3×D3—32no—
498C3×D3—63no—
499C2×C9—23no—
500C2×C9—21no—
501C2×C9—43no—
502C2×C9—62no—
503C2×C9C24×D311yescharacteristic
504D9—23no—
505D9—62no—
506D9—122no—
507D9#101012yesdirect product factor
508#41—36no—
509#41—33no—
510#41—91no—
511#41—93no—
512#41—93no—
513#41—276no—
514#41—2712no—
515#41—276no—
516#41—276no—
517#41—273no—
518#41—273no—
519#41—2712no—
520#41—276no—
521#41—276no—
522#41—273no—
523#41D3×D911yescharacteristic, complemented
524C2×Q8—32no—
525C2×Q8—33no—
526C2×Q8—91no—
527C2×Q8—276no—
528C2×Q8—276no—
529C2×Q8—273no—
530C2×Q8—276no—
531C2×Q8—276no—
532C2×Q8—276no—
533C2×Q8—276no—
534C2×Q8—272no—
535C2×Q8—273no—
536C2×D4—93no—
537C2×D4—271no—
538C2×D4—276no—
539C22×C4—93no—
540C22×C4—276no—
541C22×C4—276no—
542C22×C3—23no—
543C22×C3—32no—
544C22×C3—91no—
545C22×C3—93no—
546C22×C3—186no—
547C22×C3C23×D913yes—
548C22×C3C22×D3213yes—
549C2×D3—31no—
550C2×D3—33no—
551C2×D3—66no—
552C2×D3—92no—
553C2×D3—92no—
554C2×D3—182no—
555C2×D3—184no—
556C2×D3—184no—
557C2×D3C23×D912yes—
558C3×C4—21no—
559C3×C4—23no—
560C3×C4—36no—
561C3×C4—91no—
562C3×C4—93no—
563C3×C4C23×D911yescharacteristic
564C3×C4C23×D913yes—
565C3×C4C22×D3213yes—
566C3×C4C22×D3211yescharacteristic
567Dic3—33no—
568Dic3—31no—
569Dic3—96no—
570Dic3—96no—
571Dic3—186no—
572Dic3C23×D916yes—
573C32D3×2−1+411yescharacteristic
574C9—21no—
575C9C2×#101011yescharacteristic, complemented
576C23—93no—
577C23—272no—
578Q8—36no—
579Q8—33no—
580Q8—93no—
581Q8—276no—
582Q8—2712no—
583Q8—276no—
584Q8—273no—
585Q8C2×D3×D911yescharacteristic
586D4—36no—
587D4—31no—
588D4—96no—
589D4—93no—
590D4—276no—
591D4—272no—
592D4—276no—
593D4—276no—
594D4—271no—
595D4C2×D3×D913yes—
596C2×C4—36no—
597C2×C4—36no—
598C2×C4—93no—
599C2×C4—91no—
600C2×C4—93no—
601C2×C4—93no—
602C2×C4—93no—
603C2×C4—276no—
604C2×C4—276no—
605C2×C4—2712no—
606C2×C4—276no—
607C2×C4—276no—
608C2×C4—276no—
609C2×C4C2×D3×D913yes—
610C2×C3—23no—
611C2×C3—23no—
612C2×C3—21no—
613C2×C3—43no—
614C2×C3—62no—
615C2×C3—92no—
616C2×C3—183no—
617C2×C3C24×D911yescharacteristic
618C2×C3C23×D3211yescharacteristic
619D3—22no—
620D3—32no—
621D3—63no—
622D3—182no—
623D3—182no—
624D3—362no—
625C22—32no—
626C22—91no—
627C22—93no—
628C22—186no—
629C22—272no—
630C22—544no—
631C22C22×D3×D913yes—
632C4—36no—
633C4—91no—
634C4—93no—
635C4—276no—
636C4C22×D3×D913yes—
637C4C22×D3×D911yescharacteristic
638C3—21no—
639C3D9×2−1+411yescharacteristic, complemented
640C3D3×#101011yescharacteristic
641C2—23no—
642C2—62no—
643C2—92no—
644C2—183no—
645C2—542no—
646C2C23×D3×D911yescharacteristic
subgroupG/H
centerC2C23×D3×D9
commutator, cosocleC2×C3×C9C25
FrattiniC2×C3C23×D32
FittingC3×C9×#41C22
socleC2×C32C24×D3
hypercenter#41D3×D9
hypocenterC3×C9C2×2−1+4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #11722673C2semidirect
2. C9×#1010C2semidirect
3. C3×#41×D9C2semidirect
4. D3×Q8×D9C2semidirect
5. #11722510C2semidirect
6. D9×#237C2semidirect
7. #11722494C2semidirect
8. D9×#235C2semidirect
9. #11722465C2semidirect
10. #11722458C2semidirect
11. C2×D9×Dic6C2semidirect
14. C3×C9×#41C22semidirect
15. Q8×C3⋊D9C22semidirect
16. #88742C22semidirect
17. #88736C22semidirect
18. C2×#6226C22semidirect
19. D3×Q8×C9C22semidirect
20. C3×#3701C22semidirect
22. C9×#237C22semidirect
23. C3×#3699C22semidirect
25. C9×#235C22semidirect
26. C3×#3697C22semidirect
28. C2×C9×Dic6C22semidirect
30. #88663C22semidirect
31. #88662C22semidirect
32. #88658C22semidirect
34. C2×#6189C22semidirect
37. #88641C22semidirect
38. #88640C22semidirect
39. #88638C22semidirect
40. D3×Dic18C22semidirect
41. (C3⋊D9)⋊Q8C22semidirect
42. #88635C22semidirect
43. #88634C22semidirect
46. D9×Dic6C22semidirect
47. D9×Dic6C22semidirect
66. #41×D9D3semidirect
144. C9×#41C2×D3semidirect
149. #3701C2×D3semidirect
163. #3699C2×D3semidirect
172. #3697C2×D3semidirect
204. #1010D9direct (N × H)
299. C3×D92−1+4semidirect
306. C3×#41C2×D9semidirect
329. D3×Q8C2×D9semidirect
341. #237C2×D9semidirect
356. #235C2×D9semidirect
372. C2×Dic6C2×D9semidirect
415. C3×C9C2×2−1+4semidirect
507. D9#1010direct (N × H)
523. #41D3×D9semidirect
575. C9C2×#1010semidirect
639. C3D9×2−1+4semidirect

Maximal subgroups

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

subgroupindex
#117226732
C9×#10102
C3×#41×D92
D3×Q8×D92
#117225102
D9×#2372
#117224942
D9×#2352
#117224652
#117224582
C2×D9×Dic62
D3×#10103
D9×2−1+43