Finite Group Databasebeta

← D3×#63716 #13164779

Subgroups — D3×#63716 #13164779

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
2D3×C9×#41C211yesmaximal, characteristic, complemented
3C3×#63716C211yesmaximal, characteristic, complemented
4D3×Q8×D9C212yesmaximal, complemented
5#11722508C212yesmaximal, complemented
6D3×#3699C216yesmaximal, complemented
7#11722494C216yesmaximal, complemented
8D3×#3697C213yesmaximal, complemented
9#11722465C213yesmaximal, complemented
10#11722464C213yesmaximal, complemented
11C2×D3×Dic18C213yesmaximal, complemented
12D3×#1010—31nomaximal
13C2×#63716—31nomaximal
14C3×C9×#41C2211yescharacteristic, complemented
15Q8×C3⋊D9C2212yescomplemented
16#88742C2216yescomplemented
17#88736C2213yescomplemented
18C2×#6226C2213yescomplemented
19C9×#239C2211yescharacteristic, complemented
20D3×Q8×C9C2211yescharacteristic
21C3×Q8×D9C2212yescomplemented
22C9×#237C2213yescomplemented
23D3×D4×C9C2213yes—
24C3×#3699C2216yescomplemented
25C9×#235C2213yescomplemented
26C2×C4×D3×C9C2213yes—
27C3×#3697C2213yescomplemented
28C2×C3×Dic18C2213yescomplemented
29D3×#611C2216yes—
30#88663C22112yescomplemented
31#88662C22112yescomplemented
32C2×D3×Dic9C2216yes—
33#88658C2216yescomplemented
34C2×#6189C2216yescomplemented
35D3×D36C2211yescharacteristic
36C4×D3×D9C2216yes—
37#88642C2213yescomplemented
38#88641C2213yescomplemented
39#88640C2216yescomplemented
40#88639C2216yescomplemented
41#88638C2216yescomplemented
42D3×Dic18C2216yes—
43D3×Dic18C2213yes—
44D3×Dic18C2216yescomplemented
45D3×Dic18C2216yescomplemented
46(C3⋊D9)⋊Q8C2216yescomplemented
47D9×Dic6C2216yescomplemented
48#64367—31no—
49C3×#1010—31no—
50C3×D3×#41D311yescharacteristic
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—
59C2×C9×#41—31no—
60#63716—31no—
61#63716—32no—
62#63716—36no—
63#63716—33no—
64#63716—33no—
65#63716—61no—
66#63716D311yescharacteristic, direct product factor
67C2×Q8×D9—32no—
68C2×#3699—36no—
69C2×#3697—33no—
70C22×Dic18—33no—
71C22×D3×C9C2313yes—
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#6226C2316yes—
81#6226C2313yes—
82C9×#67C2316yes—
83C3×#611C2316yes—
84C2×C9×Dic3C2313yes—
85C2×C3×Dic9C2316yes—
86C9×D12C2313yes—
87C4×D3×C9C2311yescharacteristic
88C4×D3×C9C2313yes—
89C4×D3×C9C2313yes—
90C4×D3×C9C2313yes—
91C3×D36C2311yescharacteristic
92C3×C4×D9C2316yes—
93C9×Dic6C2313yes—
94C3×Dic18C2316yes—
95C3×Dic18C2313yes—
96#6195C2316yes—
97#6194C2316yes—
98D3×Dic9C2316yes—
99D3×Dic9C23112yes—
100#6192C2312yes—
101(C3⋊D9)⋊C4C2316yes—
102Dic3×D9C2316yes—
103#6189C23112yes—
104#6189C2316yes—
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×#239C2×D311yescharacteristic
113C3×D3×Q8—32no—
114C3×D3×Q8C2×D311yescharacteristic
115C3×#237—36no—
116C3×#237C2×D313yes—
117C3×D3×D4C2×D313yes—
118C3×#235—33no—
119C3×#235C2×D313yes—
120C2×C3×C4×D3C2×D313yes—
121C2×C3×Dic6—33no—
122D3×#67—36no—
123C2×#627—36no—
124#3805—312no—
125#3804—312no—
126#3804—36no—
127C2×D3×Dic3—36no—
128D3×D12—31no—
129C4×D32—36no—
130#3799—33no—
131#3798—36no—
132#3797—36no—
133#3796—36no—
134#3795—36no—
135#3795—33no—
136D3×Dic6—33no—
137D3×Dic6—36no—
138D3×Dic6—36no—
139D3×Dic6—36no—
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—
150C22×Dic9—36no—
151Q8×D9—32no—
152Q8×D9—36no—
153Q8×D9—36no—
154Q8×D9—62no—
155Q8×D9C2×D312yescomplemented
156#3699—36no—
157#3699—312no—
158#3699—312no—
159#3699—36no—
160#3699—36no—
161#3699—66no—
162#3699C2×D316yescomplemented
163#3697—33no—
164#3697—36no—
165#3697—33no—
166#3697—36no—
167#3697—33no—
168#3697—63no—
169#3697C2×D313yescomplemented
170C2×D36—31no—
171C2×C4×D9—36no—
172C2×Dic18—33no—
173C2×Dic18—36no—
174C2×Dic18—33no—
175C2×Dic18—36no—
176C2×Dic18—36no—
177C2×Dic18—36no—
178C2×Dic18—63no—
179C2×Dic18C2×D313yescomplemented
180C22×C3×C9C2413yes—
181C2×C3⋊D9C2412yes—
182C2×D3×C9—26no—
183C2×D3×C9C2413yes—
184C2×D3×C9C2411yescharacteristic
185C2×C3×D9C2412yes—
186D3×D9—24no—
187C3×C4×C9C2411yescharacteristic
188C3×C4×C9C2413yes—
189C3⋊Dic9C2416yes—
190C9×Dic3C2411yescharacteristic
191C9×Dic3C2413yes—
192C3×Dic9C2416yes—
193C3×2−1+4—91no—
194C2×C3×#41—31no—
195#1010—31no—
196#1010—93no—
197#1010—93no—
198#1010—96no—
199#1010—92no—
200#1010—91no—
201#1010—91no—
202#1010—96no—
203#1010—93no—
204#1010—181no—
205D3×#41—96no—
206D3×#41—93no—
207D3×#41D911yescharacteristic, complemented
208#1007—92no—
209#1007—93no—
210C2×D3×Q8—92no—
211C2×D3×Q8—92no—
212C2×D3×Q8—93no—
213C2×#237—96no—
214C2×#235—93no—
215C22×Dic6—93no—
216C22×C3×D3C22×D313yes—
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—36no—
230C2×C3×Dic3C22×D313yes—
231C3×D12—31no—
232C3×D12C22×D313yes—
233C3×C4×D3—36no—
234C3×C4×D3C22×D311yescharacteristic
235C3×C4×D3C22×D313yes—
236C3×C4×D3C22×D313yes—
237C3×C4×D3C22×D313yes—
238C3×Dic6—33no—
239C3×Dic6—36no—
240C3×Dic6C22×D313yes—
241#627—312no—
242#627—36no—
243#626—36no—
244#626—32no—
245#625—36no—
246#624—36no—
247D3×Dic3—36no—
248D3×Dic3—312no—
249D3×Dic3—36no—
250C23×C9—33no—
251C22×D9—32no—
252Q8×C9—21no—
253Q8×C9—33no—
254Q8×C9C22×D311yescharacteristic
255D4×C9—23no—
256D4×C9—36no—
257D4×C9—33no—
258D4×C9C22×D313yes—
259C2×C4×C9—23no—
260C2×C4×C9—33no—
261C2×C4×C9—31no—
262C2×C4×C9—33no—
263C2×C4×C9—33no—
264C2×C4×C9—33no—
265C2×C4×C9C22×D313yes—
266#611—36no—
267#611—36no—
268#611—36no—
269#611—66no—
270#611C22×D316yes—
271C2×Dic9—36no—
272C2×Dic9—36no—
273C2×Dic9—312no—
274C2×Dic9—66no—
275C2×Dic9C22×D316yes—
276D36—32no—
277D36—31no—
278D36—61no—
279D36C22×D311yescharacteristic
280C4×D9—36no—
281C4×D9—36no—
282C4×D9—36no—
283C4×D9—66no—
284C4×D9C22×D316yes—
285Dic18—33no—
286Dic18—36no—
287Dic18—36no—
288Dic18—312no—
289Dic18—63no—
290Dic18—66no—
291Dic18C22×D313yes—
292Dic18C22×D316yes—
293C2×2−1+4—271noSylow
294C2×C3×C9—23no—
295C2×C3×C9C2511yescharacteristic
296C3⋊D9—22no—
297D3×C9—23no—
298D3×C92−1+412yescomplemented
299C3×D9—22no—
300C3×#41—21no—
301C3×#41—31no—
302C3×#41—33no—
303C3×#41—33no—
304C3×#41—96no—
305C3×#41—93no—
306C3×#41C2×D911yescharacteristic, complemented
307C3×#41D3211yescharacteristic
308C2×C3×Q8—31no—
309C2×C3×Q8—92no—
310C2×C3×Q8—93no—
311C2×C3×D4—33no—
312C22×C3×C4—33no—
313C2×#67—96no—
314C22×Dic3—96no—
315#239—96no—
316#239—93no—
317#239C2×D911yescharacteristic, complemented
318D3×Q8—32no—
319D3×Q8—96no—
320D3×Q8—96no—
321D3×Q8—92no—
322D3×Q8—92no—
323D3×Q8—96no—
324D3×Q8—93no—
325D3×Q8—96no—
326D3×Q8—96no—
327D3×Q8—96no—
328D3×Q8—182no—
329D3×Q8C2×D911yescharacteristic
330#237—36no—
331#237—96no—
332#237—96no—
333#237—912no—
334#237—912no—
335#237—96no—
336#237—96no—
337#237—912no—
338#237—96no—
339#237—93no—
340#237—186no—
341#237C2×D913yescomplemented
342D3×D4—96no—
343D3×D4—91no—
344D3×D4C2×D913yes—
345#235—33no—
346#235—93no—
347#235—96no—
348#235—93no—
349#235—96no—
350#235—93no—
351#235—93no—
352#235—912no—
353#235—96no—
354#235—96no—
355#235—183no—
356#235C2×D913yescomplemented
357C2×D12—91no—
358C2×C4×D3—96no—
359C2×C4×D3—96no—
360C2×C4×D3—96no—
361C2×C4×D3C2×D913yes—
362C2×Dic6—33no—
363C2×Dic6—93no—
364C2×Dic6—96no—
365C2×Dic6—96no—
366C2×Dic6—96no—
367C2×Dic6—96no—
368C2×Dic6—93no—
369C2×Dic6—93no—
370C2×Dic6—96no—
371C2×Dic6—96no—
372C2×Dic6—183no—
373C22×C32C23×D313yes—
374C2×C3⋊D3—32no—
375C2×C3×D3—26no—
376C2×C3×D3—32no—
377C2×C3×D3C23×D313yes—
378C2×C3×D3C23×D311yescharacteristic
379D32—64no—
380C32×C4C23×D311yescharacteristic
381C32×C4C23×D313yes—
382C3⋊Dic3—36no—
383C3×Dic3—36no—
384C3×Dic3C23×D313yes—
385C3×Dic3C23×D311yescharacteristic
386C22×C9—23no—
387C22×C9—31no—
388C22×C9—33no—
389C22×C9—66no—
390C22×C9C23×D313yes—
391C2×D9—32no—
392C2×D9—62no—
393C2×D9—64no—
394C2×D9C23×D312yes—
395C4×C9—21no—
396C4×C9—23no—
397C4×C9—31no—
398C4×C9—33no—
399C4×C9C23×D311yescharacteristic
400C4×C9C23×D313yes—
401Dic9—36no—
402Dic9—66no—
403Dic9C23×D316yes—
4042−1+4—91no—
4052−1+4—272no—
4062−1+4—273no—
4072−1+4—276no—
4082−1+4—273no—
4092−1+4—271no—
410C2×#41—31no—
411C2×#41—273no—
412C2×#41—276no—
413C22×Q8—273no—
414C22×Q8—272no—
415C3×C9C2×2−1+411yescharacteristic, complemented, Sylow
416C23×C3—33no—
417C22×D3—92no—
418C22×D3—92no—
419C22×D3C22×D913yes—
420C3×Q8—21no—
421C3×Q8—33no—
422C3×Q8—93no—
423C3×Q8—96no—
424C3×Q8C22×D911yescharacteristic
425C3×Q8C2×D3211yescharacteristic
426C3×D4—23no—
427C3×D4—36no—
428C3×D4—33no—
429C3×D4—96no—
430C3×D4—91no—
431C3×D4C22×D913yes—
432C3×D4C2×D3213yes—
433C2×C3×C4—23no—
434C2×C3×C4—33no—
435C2×C3×C4—31no—
436C2×C3×C4—33no—
437C2×C3×C4—33no—
438C2×C3×C4—33no—
439C2×C3×C4—96no—
440C2×C3×C4—96no—
441C2×C3×C4C22×D913yes—
442C2×C3×C4C2×D3213yes—
443#67—36no—
444#67—96no—
445#67—96no—
446#67—96no—
447#67—96no—
448#67—96no—
449#67—92no—
450#67—186no—
451#67C22×D916yes—
452C2×Dic3—36no—
453C2×Dic3—96no—
454C2×Dic3—912no—
455C2×Dic3—96no—
456C2×Dic3—96no—
457C2×Dic3—96no—
458C2×Dic3—186no—
459C2×Dic3C22×D913yes—
460D12—31no—
461D12—92no—
462D12—96no—
463D12—91no—
464D12—91no—
465D12—181no—
466D12C22×D913yes—
467C4×D3—36no—
468C4×D3—96no—
469C4×D3—96no—
470C4×D3—96no—
471C4×D3—912no—
472C4×D3—96no—
473C4×D3—96no—
474C4×D3—96no—
475C4×D3—186no—
476C4×D3C22×D913yes—
477C4×D3C22×D911yescharacteristic
478C4×D3C22×D913yes—
479C4×D3C22×D913yes—
480Dic6—33no—
481Dic6—36no—
482Dic6—96no—
483Dic6—912no—
484Dic6—93no—
485Dic6—96no—
486Dic6—93no—
487Dic6—96no—
488Dic6—96no—
489Dic6—912no—
490Dic6—183no—
491Dic6—186no—
492Dic6C22×D913yes—
493C2×C32—23no—
494C2×C32C24×D311yescharacteristic
495C3⋊D3—62no—
496C3×D3—23no—
497C3×D3—62no—
498C3×D3#101012yes—
499C2×C9—23no—
500C2×C9—21no—
501C2×C9—32no—
502C2×C9—43no—
503C2×C9—63no—
504C2×C9C24×D311yescharacteristic
505D9—22no—
506D9—62no—
507D9—122no—
508#41—31no—
509#41—33no—
510#41—33no—
511#41—96no—
512#41—93no—
513#41—276no—
514#41—276no—
515#41—273no—
516#41—273no—
517#41—2712no—
518#41—276no—
519#41—2712no—
520#41—276no—
521#41—276no—
522#41—273no—
523#41D3×D911yescharacteristic, complemented
524C2×Q8—31no—
525C2×Q8—92no—
526C2×Q8—93no—
527C2×Q8—276no—
528C2×Q8—273no—
529C2×Q8—276no—
530C2×Q8—276no—
531C2×Q8—276no—
532C2×Q8—276no—
533C2×Q8—276no—
534C2×Q8—272no—
535C2×Q8—273no—
536C2×D4—33no—
537C2×D4—271no—
538C2×D4—276no—
539C22×C4—33no—
540C22×C4—276no—
541C22×C4—276no—
542C22×C3—23no—
543C22×C3—31no—
544C22×C3—33no—
545C22×C3—66no—
546C22×C3—92no—
547C22×C3C23×D913yes—
548C22×C3C22×D3213yes—
549C2×D3—26no—
550C2×D3—32no—
551C2×D3—92no—
552C2×D3—92no—
553C2×D3—182no—
554C2×D3—184no—
555C2×D3—184no—
556C2×D3C23×D913yes—
557C2×D3C23×D911yescharacteristic
558C3×C4—21no—
559C3×C4—23no—
560C3×C4—31no—
561C3×C4—33no—
562C3×C4—96no—
563C3×C4C23×D911yescharacteristic
564C3×C4C23×D913yes—
565C3×C4C22×D3211yescharacteristic
566C3×C4C22×D3213yes—
567Dic3—36no—
568Dic3—96no—
569Dic3—96no—
570Dic3—186no—
571Dic3C23×D911yescharacteristic
572Dic3C23×D913yes—
573C32C2×#101011yescharacteristic
574C9—21no—
575C9D3×2−1+411yescharacteristic, complemented
576C23—33no—
577C23—272no—
578Q8—33no—
579Q8—93no—
580Q8—96no—
581Q8—276no—
582Q8—2712no—
583Q8—273no—
584Q8—276no—
585Q8C2×D3×D911yescharacteristic
586D4—36no—
587D4—33no—
588D4—96no—
589D4—91no—
590D4—272no—
591D4—276no—
592D4—276no—
593D4—276no—
594D4—271no—
595D4C2×D3×D913yes—
596C2×C4—33no—
597C2×C4—31no—
598C2×C4—33no—
599C2×C4—33no—
600C2×C4—33no—
601C2×C4—96no—
602C2×C4—96no—
603C2×C4—276no—
604C2×C4—276no—
605C2×C4—276no—
606C2×C4—2712no—
607C2×C4—276no—
608C2×C4—276no—
609C2×C4C2×D3×D913yes—
610C2×C3—23no—
611C2×C3—23no—
612C2×C3—21no—
613C2×C3—32no—
614C2×C3—43no—
615C2×C3—63no—
616C2×C3—182no—
617C2×C3C24×D911yescharacteristic
618C2×C3C23×D3211yescharacteristic
619D3—23no—
620D3—62no—
621D3—182no—
622D3—182no—
623D3—362no—
624D3#6371612yesdirect product factor
625C22—31no—
626C22—33no—
627C22—66no—
628C22—92no—
629C22—272no—
630C22—544no—
631C22C22×D3×D913yes—
632C4—31no—
633C4—33no—
634C4—96no—
635C4—276no—
636C4C22×D3×D911yescharacteristic
637C4C22×D3×D913yes—
638C3—21no—
639C3C2×#6371611yescharacteristic, complemented
640C3D3×#101011yescharacteristic
641C2—23no—
642C2—32no—
643C2—63no—
644C2—182no—
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. D3×C9×#41C2semidirect
3. C3×#63716C2semidirect
4. D3×Q8×D9C2semidirect
5. #11722508C2semidirect
6. D3×#3699C2semidirect
7. #11722494C2semidirect
8. D3×#3697C2semidirect
9. #11722465C2semidirect
10. #11722464C2semidirect
11. C2×D3×Dic18C2semidirect
14. C3×C9×#41C22semidirect
15. Q8×C3⋊D9C22semidirect
16. #88742C22semidirect
17. #88736C22semidirect
18. C2×#6226C22semidirect
19. C9×#239C22semidirect
21. C3×Q8×D9C22semidirect
22. C9×#237C22semidirect
24. C3×#3699C22semidirect
25. C9×#235C22semidirect
27. C3×#3697C22semidirect
28. C2×C3×Dic18C22semidirect
30. #88663C22semidirect
31. #88662C22semidirect
33. #88658C22semidirect
34. C2×#6189C22semidirect
37. #88642C22semidirect
38. #88641C22semidirect
39. #88640C22semidirect
40. #88639C22semidirect
41. #88638C22semidirect
44. D3×Dic18C22semidirect
45. D3×Dic18C22semidirect
46. (C3⋊D9)⋊Q8C22semidirect
47. D9×Dic6C22semidirect
66. #63716D3direct (N × H)
145. C9×#41C2×D3semidirect
155. Q8×D9C2×D3semidirect
162. #3699C2×D3semidirect
169. #3697C2×D3semidirect
179. C2×Dic18C2×D3semidirect
207. D3×#41D9semidirect
298. D3×C92−1+4semidirect
306. C3×#41C2×D9semidirect
317. #239C2×D9semidirect
341. #237C2×D9semidirect
356. #235C2×D9semidirect
415. C3×C9C2×2−1+4semidirect
523. #41D3×D9semidirect
575. C9D3×2−1+4semidirect
624. D3#63716direct (N × H)
639. C3C2×#63716semidirect

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×#637162
D3×Q8×D92
#117225082
D3×#36992
#117224942
D3×#36972
#117224652
#117224642
C2×D3×Dic182
D3×#10103
C2×#637163