Finite Group Databasebeta

← #13164783

Subgroups — #13164783

Order 1728.

2576 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11722673C211yesmaximal, characteristic, complemented
2D3×C9×#41C211yesmaximal, characteristic, complemented
3C3×#63715C211yesmaximal, characteristic, complemented
4#11722514C212yesmaximal, complemented
5#11722510C212yesmaximal, complemented
6#11722499C216yesmaximal, complemented
7D9×#237C216yesmaximal, complemented
8#11722477C213yesmaximal, complemented
9D3×#3697C213yesmaximal, complemented
10C2×#88641C213yesmaximal, complemented
11#11722465C213yesmaximal, complemented
12#10596696—31nomaximal
13#10593139—31nomaximal
14C3×C9×#41C2211yescharacteristic, complemented
15Q8×C3⋊D9C2212yescomplemented
16#88742C2216yescomplemented
17#88736C2213yescomplemented
18C2×#6226C2213yescomplemented
19C9×#239C2211yescharacteristic, complemented
20D3×Q8×C9C2211yescharacteristic, complemented
21C3×#3701C2212yescomplemented
22C9×#237C2213yescomplemented
23D3×D4×C9C2213yescomplemented
24C3×D4×D9C2216yescomplemented
25C9×#235C2213yescomplemented
26C2×C4×D3×C9C2213yescomplemented
27C3×#3697C2213yescomplemented
28C2×C3×D36C2213yescomplemented
29D9×#67C22112yescomplemented
30D3×#611C2216yescomplemented
31C2×#6194C2216yescomplemented
32#88663C22112yescomplemented
33#88658C2216yescomplemented
34C2×Dic3×D9C2216yescomplemented
35#88646C2216yescomplemented
36D3×D36C2213yescomplemented
37C4×D3×D9C2216yescomplemented
38#88642C2211yescharacteristic, complemented
39#88641C2216yescomplemented
40#88641C2216yescomplemented
41#88641C2216yescomplemented
42#88641C2213yescomplemented
43#88640C2216yescomplemented
44#88638C2216yescomplemented
45D3×Dic18C2213yescomplemented
46#88634C2216yescomplemented
47D9×Dic6C2216yescomplemented
48#64367—31no—
49C3×#1009—31no—
50C3×D3×#41D311yescharacteristic
51#64319—32no—
52#64315—32no—
53#64313—36no—
54D3×#237—36no—
55#64307—33no—
56D3×#235—33no—
57#64298—33no—
58C2×#3795—33no—
59C2×C9×#41—31no—
60#63716—31no—
61#63716—33no—
62#63716—61no—
63#63715—33no—
64#63715D311yescharacteristic, complemented
65#41×D9—32no—
66#41×D9—36no—
67#63713—32no—
68#63710—36no—
69C2×#3697—33no—
70C2×#3697—33no—
71C22×D3×C9C2313yes—
72C22×C3×D9C2316yes—
73C2×D3×D9C2316yes—
74C3×Q8×C9C2311yescharacteristic
75C3×D4×C9C2313yes—
76C2×C3×C4×C9C2313yes—
77#6233C2316yes—
78C2×C3⋊Dic9C2316yes—
79C4⋊(C3⋊D9)C2311yescharacteristic
80C4×C3⋊D9C2316yes—
81#6226C2316yes—
82#6226C2313yes—
83C9×#67C2316yes—
84C3×#611C2316yes—
85C2×C9×Dic3C2313yes—
86C9×D12C2313yes—
87C4×D3×C9C2313yes—
88C4×D3×C9C2313yes—
89C4×D3×C9C2311yescharacteristic
90C4×D3×C9C2313yes—
91C3×D36C2316yes—
92C3×D36C2313yes—
93C3×C4×D9C2316yes—
94C9×Dic6C2313yes—
95C3×Dic18C2311yescharacteristic
96#6195C2312yes—
97#6194C23112yes—
98#6194C2316yes—
99D3×Dic9C2316yes—
100#6192C2316yes—
101(C3⋊D9)⋊C4C2312yes—
102Dic3×D9C23112yes—
103Dic3×D9C2316yes—
104#6189C2316yes—
105#5966—91no—
106#5964—91no—
107C32×#41C2×D311yescharacteristic
108Q8×C3⋊D3—32no—
109#3830—36no—
110#3828—33no—
111C2×#634—33no—
112C3×#239—32no—
113C3×#239C2×D311yescharacteristic
114C3×D3×Q8C2×D311yescharacteristic
115C3×#237C2×D313yes—
116C3×D3×D4—36no—
117C3×D3×D4C2×D313yes—
118C3×#235—33no—
119C3×#235C2×D313yes—
120C2×C3×D12—33no—
121C2×C3×C4×D3C2×D313yes—
122D3×#67—36no—
123D3×#67—312no—
124C2×#625—36no—
125#3805—312no—
126#3804—36no—
127C2×D3×Dic3—36no—
128#3802—36no—
129D3×D12—33no—
130C4×D32—36no—
131#3799—31no—
132#3798—36no—
133#3796—36no—
134#3795—36no—
135#3795—33no—
136#3795—36no—
137#3795—36no—
138#3795—36no—
139D3×Dic6—33no—
140D3×Dic6—36no—
141C9×#41—21no—
142C9×#41—31no—
143C9×#41—33no—
144C9×#41—33no—
145C9×#41C2×D311yescharacteristic, complemented
146C2×Q8×C9—31no—
147C2×D4×C9—33no—
148C22×C4×C9—33no—
149C2×#611—36no—
150C2×#611—36no—
151#3701—36no—
152#3701C2×D312yescomplemented
153Q8×D9—32no—
154Q8×D9—36no—
155Q8×D9—62no—
156#3699—36no—
157#3699—312no—
158#3699—36no—
159#3699—66no—
160D4×D9—312no—
161D4×D9—36no—
162D4×D9C2×D316yescomplemented
163#3697—33no—
164#3697—36no—
165#3697—33no—
166#3697—36no—
167#3697—36no—
168#3697—36no—
169#3697—36no—
170#3697—31no—
171#3697—63no—
172#3697C2×D313yescomplemented
173C2×D36—33no—
174C2×D36C2×D313yescomplemented
175C2×C4×D9—36no—
176C2×C4×D9—36no—
177C2×Dic18—33no—
178C2×Dic18—33no—
179C2×Dic18—63no—
180C22×C3×C9C2413yes—
181C2×C3⋊D9C2412yes—
182C2×D3×C9—43no—
183C2×D3×C9C2413yes—
184C2×D3×C9C2411yescharacteristic
185C2×C3×D9—46no—
186C2×C3×D9C2416yes—
187D3×D9—46no—
188C3×C4×C9C2411yescharacteristic
189C3×C4×C9C2413yes—
190C3⋊Dic9C2416yes—
191C9×Dic3C2413yes—
192C9×Dic3C2411yescharacteristic
193C3×Dic9C2412yes—
194C3×H(5,𝔽2)—91no—
195C2×C3×#41—31no—
196#1010—93no—
197#1010—91no—
198#1010—91no—
199#1010—92no—
200#1010—93no—
201#1010—181no—
202#1009—31no—
203#1009—93no—
204D3×#41—96no—
205D3×#41—92no—
206D3×#41—92no—
207D3×#41—93no—
208D3×#41D911yescharacteristic, complemented
209#1007—92no—
210#1004—96no—
211#1004—96no—
212#1004—93no—
213C2×#237—96no—
214C2×#237—93no—
215C2×#235—93no—
216C2×#235—93no—
217C22×C3×D3—36no—
218C22×C3×D3C22×D313yes—
219C2×D32—36no—
220C32×Q8C22×D311yescharacteristic
221C32×D4C22×D313yes—
222C2×C32×C4C22×D313yes—
223#638—36no—
224C2×C3⋊Dic3—36no—
225C4⋊(C3⋊D3)—31no—
226C4×C3⋊D3—36no—
227#634—33no—
228#634—36no—
229C3×#67—36no—
230C3×#67C22×D316yes—
231C2×C3×Dic3C22×D313yes—
232C3×D12—33no—
233C3×D12—36no—
234C3×D12C22×D313yes—
235C3×C4×D3—36no—
236C3×C4×D3C22×D313yes—
237C3×C4×D3C22×D311yescharacteristic
238C3×C4×D3C22×D313yes—
239C3×C4×D3C22×D313yes—
240C3×Dic6—31no—
241C3×Dic6C22×D313yes—
242#627—36no—
243#626—32no—
244#626—36no—
245#625—36no—
246#625—312no—
247#624—32no—
248D3×Dic3—36no—
249D3×Dic3—312no—
250D3×Dic3—36no—
251C23×C9—33no—
252C22×D9—36no—
253C22×D9C22×D316yes—
254Q8×C9—21no—
255Q8×C9—33no—
256Q8×C9C22×D311yescharacteristic
257D4×C9—23no—
258D4×C9—36no—
259D4×C9—33no—
260D4×C9C22×D313yes—
261C2×C4×C9—23no—
262C2×C4×C9—33no—
263C2×C4×C9—33no—
264C2×C4×C9—33no—
265C2×C4×C9—31no—
266C2×C4×C9—33no—
267C2×C4×C9C22×D313yes—
268#611—36no—
269#611—36no—
270#611—312no—
271#611—32no—
272#611—66no—
273#611C22×D316yes—
274C2×Dic9—36no—
275C2×Dic9—36no—
276C2×Dic9—66no—
277D36—36no—
278D36—31no—
279D36—61no—
280D36C22×D313yes—
281D36C22×D316yes—
282C4×D9—36no—
283C4×D9—312no—
284C4×D9—36no—
285C4×D9—32no—
286C4×D9—66no—
287C4×D9C22×D316yes—
288Dic18—33no—
289Dic18—36no—
290Dic18—36no—
291Dic18—63no—
292Dic18—66no—
293Dic18C22×D311yescharacteristic
294#585—271noSylow
295C2×C3×C9—23no—
296C2×C3×C9C2511yescharacteristic
297C3⋊D9—22no—
298D3×C9—23no—
299D3×C9—21no—
300C3×D9—26no—
301C3×#41—21no—
302C3×#41—31no—
303C3×#41—33no—
304C3×#41—33no—
305C3×#41—92no—
306C3×#41—93no—
307C3×#41C2×D911yescharacteristic, complemented
308C3×#41D3211yescharacteristic
309C2×C3×Q8—31no—
310C2×C3×D4—33no—
311C2×C3×D4—96no—
312C2×C3×D4—93no—
313C22×C3×C4—33no—
314C2×#67—96no—
315C2×#67—96no—
316C2×#67—96no—
317C2×#67—912no—
318C22×Dic3—96no—
319#239—32no—
320#239—96no—
321#239—91no—
322#239C2×D911yescharacteristic, complemented
323D3×Q8—96no—
324D3×Q8—92no—
325D3×Q8—92no—
326D3×Q8—93no—
327D3×Q8—182no—
328D3×Q8C2×D911yescharacteristic, complemented
329#237—96no—
330#237—912no—
331#237—96no—
332#237—96no—
333#237—912no—
334#237—96no—
335#237—96no—
336#237—93no—
337#237—96no—
338#237—96no—
339#237—96no—
340#237—186no—
341#237C2×D913yescomplemented
342D3×D4—36no—
343D3×D4—96no—
344D3×D4—912no—
345D3×D4—96no—
346D3×D4—96no—
347D3×D4—93no—
348D3×D4C2×D913yescomplemented
349#235—33no—
350#235—91no—
351#235—96no—
352#235—93no—
353#235—96no—
354#235—96no—
355#235—96no—
356#235—96no—
357#235—93no—
358#235—93no—
359#235—96no—
360#235—96no—
361#235—183no—
362#235C2×D913yescomplemented
363C2×D12—33no—
364C2×D12—93no—
365C2×C4×D3—96no—
366C2×C4×D3—96no—
367C2×C4×D3—96no—
368C2×C4×D3C2×D913yescomplemented
369C2×Dic6—93no—
370C2×Dic6—93no—
371C2×Dic6—93no—
372C2×Dic6—96no—
373C2×Dic6—183no—
374C22×C32C23×D313yes—
375C2×C3⋊D3—32no—
376C2×C3×D3—36no—
377C2×C3×D3—43no—
378C2×C3×D3—126no—
379C2×C3×D3C23×D313yes—
380C2×C3×D3C23×D311yescharacteristic
381D32—126no—
382C32×C4C23×D311yescharacteristic
383C32×C4C23×D313yes—
384C3⋊Dic3—36no—
385C3×Dic3—32no—
386C3×Dic3C23×D311yescharacteristic
387C3×Dic3C23×D313yes—
388C22×C9—23no—
389C22×C9—33no—
390C22×C9—31no—
391C22×C9—123no—
392C22×C9C23×D313yes—
393C2×D9—32no—
394C2×D9—46no—
395C2×D9—62no—
396C2×D9—126no—
397C2×D9C23×D316yes—
398C4×C9—21no—
399C4×C9—23no—
400C4×C9—33no—
401C4×C9—31no—
402C4×C9C23×D311yescharacteristic
403C4×C9C23×D313yes—
404Dic9—36no—
405Dic9—66no—
406Dic9C23×D312yes—
4072−1+4—272no—
4082−1+4—273no—
4092−1+4—271no—
410H(5,𝔽2)—91no—
411H(5,𝔽2)—276no—
412H(5,𝔽2)—273no—
413C2×#41—31no—
414C2×#41—273no—
415C2×#41—273no—
416C2×#41—276no—
417C2×#41—272no—
418C3×C9#58511yescharacteristic, complemented, Sylow
419C23×C3—33no—
420C23×C3—96no—
421C22×D3—36no—
422C22×D3—96no—
423C22×D3—96no—
424C22×D3C22×D913yes—
425C3×Q8—21no—
426C3×Q8—33no—
427C3×Q8—91no—
428C3×Q8C22×D911yescharacteristic
429C3×Q8C2×D3211yescharacteristic
430C3×D4—23no—
431C3×D4—36no—
432C3×D4—33no—
433C3×D4—96no—
434C3×D4—93no—
435C3×D4—96no—
436C3×D4C22×D913yes—
437C3×D4C2×D3213yes—
438C2×C3×C4—23no—
439C2×C3×C4—33no—
440C2×C3×C4—33no—
441C2×C3×C4—33no—
442C2×C3×C4—31no—
443C2×C3×C4—33no—
444C2×C3×C4—96no—
445C2×C3×C4C22×D913yes—
446C2×C3×C4C2×D3213yes—
447#67—36no—
448#67—92no—
449#67—96no—
450#67—912no—
451#67—96no—
452#67—96no—
453#67—912no—
454#67—96no—
455#67—96no—
456#67—186no—
457#67C22×D916yes—
458C2×Dic3—96no—
459C2×Dic3—96no—
460C2×Dic3—96no—
461C2×Dic3—912no—
462C2×Dic3—96no—
463C2×Dic3—186no—
464C2×Dic3C22×D913yes—
465D12—33no—
466D12—36no—
467D12—96no—
468D12—92no—
469D12—91no—
470D12—91no—
471D12—181no—
472D12C22×D913yes—
473C4×D3—36no—
474C4×D3—92no—
475C4×D3—96no—
476C4×D3—912no—
477C4×D3—96no—
478C4×D3—96no—
479C4×D3—96no—
480C4×D3—92no—
481C4×D3—186no—
482C4×D3C22×D913yes—
483C4×D3C22×D911yescharacteristic
484C4×D3C22×D913yes—
485C4×D3C22×D913yes—
486Dic6—31no—
487Dic6—96no—
488Dic6—93no—
489Dic6—96no—
490Dic6—93no—
491Dic6—96no—
492Dic6—96no—
493Dic6—183no—
494Dic6—186no—
495Dic6C22×D913yes—
496C2×C32—23no—
497C2×C32C24×D311yescharacteristic
498C3⋊D3—62no—
499C3×D3—23no—
500C3×D3—21no—
501C3×D3—66no—
502C2×C9—23no—
503C2×C9—21no—
504C2×C9—43no—
505C2×C9—63no—
506C2×C9—61no—
507C2×C9C24×D311yescharacteristic
508D9—26no—
509D9—62no—
510D9—122no—
511#41—31no—
512#41—33no—
513#41—33no—
514#41—92no—
515#41—93no—
516#41—276no—
517#41—276no—
518#41—273no—
519#41—276no—
520#41—276no—
521#41—276no—
522#41—271no—
523#41—276no—
524#41—2712no—
525#41—276no—
526#41—276no—
527#41—273no—
528#41D3×D911yescharacteristic, complemented
529C2×Q8—31no—
530C2×Q8—273no—
531C2×Q8—276no—
532C2×Q8—272no—
533C2×Q8—273no—
534C2×D4—33no—
535C2×D4—96no—
536C2×D4—93no—
537C2×D4—273no—
538C2×D4—276no—
539C2×D4—2712no—
540C2×D4—276no—
541C2×D4—276no—
542C22×C4—33no—
543C22×C4—276no—
544C22×C4—276no—
545C22×C3—23no—
546C22×C3—33no—
547C22×C3—31no—
548C22×C3—96no—
549C22×C3—123no—
550C22×C3—366no—
551C22×C3C23×D913yes—
552C22×C3C22×D3213yes—
553C2×D3—36no—
554C2×D3—43no—
555C2×D3—92no—
556C2×D3—92no—
557C2×D3—126no—
558C2×D3—182no—
559C2×D3—366no—
560C2×D3—366no—
561C2×D3C23×D913yes—
562C2×D3C23×D911yescharacteristic
563C3×C4—21no—
564C3×C4—23no—
565C3×C4—33no—
566C3×C4—31no—
567C3×C4—92no—
568C3×C4C23×D911yescharacteristic
569C3×C4C23×D913yes—
570C3×C4C22×D3211yescharacteristic
571C3×C4C22×D3213yes—
572Dic3—32no—
573Dic3—96no—
574Dic3—96no—
575Dic3—186no—
576Dic3C23×D913yes—
577Dic3C23×D911yescharacteristic
578C32#596411yescharacteristic
579C9—21no—
580C9#596611yescharacteristic, complemented
581C23—33no—
582C23—96no—
583C23—276no—
584Q8—33no—
585Q8—91no—
586Q8—276no—
587Q8—273no—
588Q8—276no—
589Q8C2×D3×D911yescharacteristic
590D4—36no—
591D4—33no—
592D4—96no—
593D4—93no—
594D4—96no—
595D4—276no—
596D4—272no—
597D4—276no—
598D4—2712no—
599D4—276no—
600D4—271no—
601D4C2×D3×D913yes—
602C2×C4—33no—
603C2×C4—31no—
604C2×C4—33no—
605C2×C4—33no—
606C2×C4—33no—
607C2×C4—96no—
608C2×C4—272no—
609C2×C4—276no—
610C2×C4—2712no—
611C2×C4—276no—
612C2×C4—276no—
613C2×C4—276no—
614C2×C4C2×D3×D913yes—
615C2×C3—23no—
616C2×C3—23no—
617C2×C3—21no—
618C2×C3—43no—
619C2×C3—63no—
620C2×C3—61no—
621C2×C3—186no—
622C2×C3C24×D911yescharacteristic
623C2×C3C23×D3211yescharacteristic
624D3—21no—
625D3—23no—
626D3—66no—
627D3—182no—
628D3—182no—
629D3—362no—
630C22—31no—
631C22—33no—
632C22—96no—
633C22—123no—
634C22—272no—
635C22—366no—
636C22—1086no—
637C22C22×D3×D913yes—
638C4—31no—
639C4—33no—
640C4—92no—
641C4—276no—
642C4C22×D3×D911yescharacteristic
643C4C22×D3×D913yes—
644C3—21no—
645C3#1059313911yescharacteristic, complemented
646C3#1059669611yescharacteristic
647C2—23no—
648C2—61no—
649C2—63no—
650C2—186no—
651C2—542no—
652C2C23×D3×D911yescharacteristic
subgroupG/H
centerC2C23×D3×D9
commutator, cosocleC2×C3×C9C25
FrattiniC2×C3C23×D32
FittingC3×C9×#41C22
socleC2×C32C24×D3
hypercenter#41D3×D9
hypocenterC3×C9#585
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. D3×C9×#41C2semidirect
3. C3×#63715C2semidirect
4. #11722514C2semidirect
5. #11722510C2semidirect
6. #11722499C2semidirect
7. D9×#237C2semidirect
8. #11722477C2semidirect
9. D3×#3697C2semidirect
10. C2×#88641C2semidirect
11. #11722465C2semidirect
14. C3×C9×#41C22semidirect
15. Q8×C3⋊D9C22semidirect
16. #88742C22semidirect
17. #88736C22semidirect
18. C2×#6226C22semidirect
19. C9×#239C22semidirect
20. D3×Q8×C9C22semidirect
21. C3×#3701C22semidirect
22. C9×#237C22semidirect
23. D3×D4×C9C22semidirect
24. C3×D4×D9C22semidirect
25. C9×#235C22semidirect
26. C2×C4×D3×C9C22semidirect
27. C3×#3697C22semidirect
28. C2×C3×D36C22semidirect
29. D9×#67C22semidirect
30. D3×#611C22semidirect
31. C2×#6194C22semidirect
32. #88663C22semidirect
33. #88658C22semidirect
34. C2×Dic3×D9C22semidirect
35. #88646C22semidirect
36. D3×D36C22semidirect
37. C4×D3×D9C22semidirect
38. #88642C22semidirect
39. #88641C22semidirect
40. #88641C22semidirect
41. #88641C22semidirect
42. #88641C22semidirect
43. #88640C22semidirect
44. #88638C22semidirect
45. D3×Dic18C22semidirect
46. #88634C22semidirect
47. D9×Dic6C22semidirect
64. #63715D3semidirect
145. C9×#41C2×D3semidirect
152. #3701C2×D3semidirect
162. D4×D9C2×D3semidirect
172. #3697C2×D3semidirect
174. C2×D36C2×D3semidirect
208. D3×#41D9semidirect
307. C3×#41C2×D9semidirect
322. #239C2×D9semidirect
328. D3×Q8C2×D9semidirect
341. #237C2×D9semidirect
348. D3×D4C2×D9semidirect
362. #235C2×D9semidirect
368. C2×C4×D3C2×D9semidirect
418. C3×C9#585semidirect
528. #41D3×D9semidirect
580. C9#5966semidirect
645. C3#10593139semidirect

Maximal subgroups

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

subgroupindex
#117226732
D3×C9×#412
C3×#637152
#117225142
#117225102
#117224992
D9×#2372
#117224772
D3×#36972
C2×#886412
#117224652
#105966963
#105931393