Finite Group Databasebeta

← D3×#64367 #13181641

Subgroups — D3×#64367 #13181641

Order 1728.

5348 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#11724685C211yesmaximal, characteristic, complemented
2C3×#64367C211yesmaximal, characteristic, complemented
3C32×D3×#41C211yesmaximal, characteristic, complemented
4D3×Q8×C3⋊D3C212yesmaximal, complemented
5#11724590C212yesmaximal, complemented
6D3×#3830C216yesmaximal, complemented
7#11724582C216yesmaximal, complemented
8D3×#3828C213yesmaximal, complemented
9#11724563C213yesmaximal, complemented
10#11724562C213yesmaximal, complemented
11C2×D3×#634C213yesmaximal, complemented
12C2×#64367—31nomaximal
13D3×#1010—34nomaximal
14C33×#41C2211yescharacteristic, complemented
15Q8×C3⋊(C3⋊D3)C2212yescomplemented
16#89078C2216yescomplemented
17#89076C2213yescomplemented
18C2×#6308C2213yescomplemented
19C3×Q8×C3⋊D3C2212yescomplemented
20C3×#3830C2216yescomplemented
21C3×#3828C2213yescomplemented
22C2×C3×#634C2213yescomplemented
23C32×#239C2211yescharacteristic, complemented
24C32×D3×Q8C2211yescharacteristic
25C32×#237C2213yescomplemented
26C32×D3×D4C2213yes—
27C32×#235C2213yescomplemented
28C2×C32×C4×D3C2213yes—
29D3×#638C2216yes—
30C2×#6293C2216yescomplemented
31#89031C2216yescomplemented
32#89029C22112yescomplemented
33#89028C22112yescomplemented
34C2×D3×C3⋊Dic3C2216yes—
35D3×C4⋊(C3⋊D3)C2211yescharacteristic
36C4×D3×C3⋊D3C2216yes—
37#89022C2213yescomplemented
38#89021C2213yescomplemented
39#89020C2216yescomplemented
40#89019C2216yescomplemented
41C3⋊D3×Dic6C2216yescomplemented
42#89015C2216yescomplemented
43#89014C2216yescomplemented
44D3×#634C2213yes—
45D3×#634C2216yes—
46D3×#634C2216yescomplemented
47D3×#634C2216yescomplemented
48C2×C32×#41—31no—
49#64367—31no—
50#64367—32no—
51#64367—36no—
52#64367—33no—
53#64367—34no—
54#64367—33no—
55#64367—64no—
56#64367D311yescharacteristic, direct product factor
57C2×Q8×C3⋊D3—32no—
58C2×#3830—36no—
59C2×#3828—33no—
60C22×#634—33no—
61C3×#1010—34no—
62C3×D3×#41D314yescomplemented
63D32×Q8—38no—
64#64315—38no—
65D3×#237—324no—
66#64309—324no—
67D3×#235—312no—
68#64298—312no—
69#64297—312no—
70C2×D3×Dic6—312no—
71C22×C32×D3C2313yes—
72C2×D3×C3⋊D3C2312yes—
73C33×Q8C2311yescharacteristic
74C33×D4C2313yes—
75C2×C33×C4C2313yes—
76#6312C2316yes—
77C2×C3⋊(C3⋊Dic3)C2316yes—
78C4⋊(C3⋊(C3⋊D3))C2311yescharacteristic
79C4×C3⋊(C3⋊D3)C2316yes—
80#6308C2313yes—
81#6308C2316yes—
82C3×#638C2316yes—
83C2×C3×C3⋊Dic3C2316yes—
84C3×C4⋊(C3⋊D3)C2311yescharacteristic
85C3×C4×C3⋊D3C2316yes—
86C3×#634C2313yes—
87C3×#634C2316yes—
88C32×#67C2316yes—
89C2×C32×Dic3C2313yes—
90C32×D12C2313yes—
91C32×C4×D3C2311yescharacteristic
92C32×C4×D3C2313yes—
93C32×C4×D3C2313yes—
94C32×C4×D3C2313yes—
95C32×Dic6C2313yes—
96#6293C2316yes—
97#6293C23112yes—
98#6292C2312yes—
99#6291C2316yes—
100#6290C2316yes—
101#6289C2316yes—
102Dic3×C3⋊D3C2316yes—
103D3×C3⋊Dic3C2316yes—
104D3×C3⋊Dic3C23112yes—
105D3×2−1+4—91no—
106C2×#1010—94no—
107C32×#41—24no—
108C32×#41—31no—
109C32×#41—33no—
110C32×#41—33no—
111C32×#41C2×D314yescomplemented
112C32×#41C2×D311yescharacteristic, complemented
113C2×C32×Q8—31no—
114C2×C32×D4—33no—
115C22×C32×C4—33no—
116C2×#638—36no—
117C22×C3⋊Dic3—36no—
118Q8×C3⋊D3—32no—
119Q8×C3⋊D3—38no—
120Q8×C3⋊D3—36no—
121Q8×C3⋊D3—36no—
122Q8×C3⋊D3—68no—
123Q8×C3⋊D3C2×D312yescomplemented
124#3830—36no—
125#3830—324no—
126#3830—312no—
127#3830—312no—
128#3830—36no—
129#3830—36no—
130#3830—624no—
131#3830C2×D316yescomplemented
132#3828—33no—
133#3828—312no—
134#3828—36no—
135#3828—33no—
136#3828—36no—
137#3828—33no—
138#3828—612no—
139#3828C2×D313yescomplemented
140C2×C4⋊(C3⋊D3)—31no—
141C2×C4×C3⋊D3—36no—
142C2×#634—33no—
143C2×#634—312no—
144C2×#634—36no—
145C2×#634—33no—
146C2×#634—36no—
147C2×#634—36no—
148C2×#634—36no—
149C2×#634—612no—
150C2×#634C2×D313yescomplemented
151C3×#239C2×D314yescomplemented
152C3×D3×Q8—38no—
153C3×D3×Q8C2×D314yes—
154C3×#237—324no—
155C3×#237C2×D3112yescomplemented
156C3×D3×D4C2×D3112yes—
157C3×#235—312no—
158C3×#235C2×D3112yescomplemented
159C2×C3×C4×D3C2×D3112yes—
160C2×C3×Dic6—312no—
161D3×#67—324no—
162C2×#627—324no—
163#3805—348no—
164#3804—348no—
165#3804—324no—
166C2×D3×Dic3—324no—
167D3×D12—34no—
168C4×D32—324no—
169#3799—312no—
170#3798—324no—
171#3797—324no—
172#3796—324no—
173#3795—324no—
174#3795—312no—
175D3×Dic6—312no—
176D3×Dic6—324no—
177D3×Dic6—324no—
178D3×Dic6—324no—
179D3×Dic6—324no—
180C22×C33C2413yes—
181C2×C3⋊(C3⋊D3)C2412yes—
182C2×C3×C3⋊D3C2412yes—
183C2×C32×D3—26no—
184C2×C32×D3C2413yes—
185C2×C32×D3C2411yescharacteristic
186D3×C3⋊D3—24no—
187C33×C4C2413yes—
188C33×C4C2411yescharacteristic
189C3⋊(C3⋊Dic3)C2416yes—
190C3×C3⋊Dic3C2416yes—
191C32×Dic3C2413yes—
192C32×Dic3C2411yescharacteristic
193C3×2−1+4—91no—
194C2×C3×#41—34no—
195#1010—34no—
196#1010—912no—
197#1010—912no—
198#1010—924no—
199#1010—98no—
200#1010—94no—
201#1010—91no—
202#1010—96no—
203#1010—93no—
204#1010—184no—
205D3×#41—96no—
206D3×#41—93no—
207D3×#41C3⋊D311yescharacteristic, complemented
208#1007—92no—
209#1007—93no—
210C2×D3×Q8—98no—
211C2×D3×Q8—92no—
212C2×D3×Q8—93no—
213C2×#237—924no—
214C2×#235—912no—
215C22×Dic6—912no—
216C23×C32—33no—
217C22×C3⋊D3—32no—
218C22×C3×D3C22×D3112yes—
219C2×D32—38no—
220C32×Q8—24no—
221C32×Q8—33no—
222C32×Q8C22×D314yes—
223C32×Q8C22×D311yescharacteristic
224C32×D4—212no—
225C32×D4—36no—
226C32×D4—33no—
227C32×D4C22×D3112yes—
228C32×D4C22×D313yes—
229C2×C32×C4—212no—
230C2×C32×C4—33no—
231C2×C32×C4—31no—
232C2×C32×C4—33no—
233C2×C32×C4—33no—
234C2×C32×C4—33no—
235C2×C32×C4C22×D3112yes—
236C2×C32×C4C22×D313yes—
237#638—36no—
238#638—36no—
239#638—36no—
240#638—324no—
241#638—624no—
242#638C22×D316yes—
243C2×C3⋊Dic3—36no—
244C2×C3⋊Dic3—36no—
245C2×C3⋊Dic3—312no—
246C2×C3⋊Dic3—324no—
247C2×C3⋊Dic3—624no—
248C2×C3⋊Dic3C22×D316yes—
249C4⋊(C3⋊D3)—32no—
250C4⋊(C3⋊D3)—31no—
251C4⋊(C3⋊D3)—34no—
252C4⋊(C3⋊D3)—64no—
253C4⋊(C3⋊D3)C22×D311yescharacteristic
254C4×C3⋊D3—36no—
255C4×C3⋊D3—36no—
256C4×C3⋊D3—36no—
257C4×C3⋊D3—324no—
258C4×C3⋊D3—624no—
259C4×C3⋊D3C22×D316yes—
260#634—33no—
261#634—36no—
262#634—36no—
263#634—312no—
264#634—312no—
265#634—324no—
266#634—612no—
267#634—624no—
268#634C22×D313yes—
269#634C22×D316yes—
270C3×#67—324no—
271C3×#67C22×D3124yes—
272C2×C3×Dic3—324no—
273C2×C3×Dic3C22×D3112yes—
274C3×D12—34no—
275C3×D12C22×D3112yes—
276C3×C4×D3—324no—
277C3×C4×D3C22×D314yes—
278C3×C4×D3C22×D3112yes—
279C3×C4×D3C22×D3112yes—
280C3×C4×D3C22×D3112yes—
281C3×Dic6—312no—
282C3×Dic6—324no—
283C3×Dic6C22×D3112yes—
284#627—348no—
285#627—324no—
286#626—324no—
287#626—38no—
288#625—324no—
289#624—324no—
290D3×Dic3—324no—
291D3×Dic3—348no—
292D3×Dic3—324no—
293C2×2−1+4—271noSylow
294C2×C33—23no—
295C2×C33C2511yescharacteristic
296C3⋊(C3⋊D3)—22no—
297C3×C3⋊D3—22no—
298C32×D3—23no—
299C32×D32−1+412yescomplemented
300C3×#41—24no—
301C3×#41—34no—
302C3×#41—312no—
303C3×#41—312no—
304C3×#41—96no—
305C3×#41—93no—
306C3×#41D3214yescomplemented
307C3×#41C2×C3⋊D311yescharacteristic, complemented
308C2×C3×Q8—34no—
309C2×C3×Q8—92no—
310C2×C3×Q8—93no—
311C2×C3×D4—312no—
312C22×C3×C4—312no—
313C2×#67—924no—
314C22×Dic3—924no—
315#239—96no—
316#239—93no—
317#239C2×C3⋊D311yescharacteristic, complemented
318D3×Q8—38no—
319D3×Q8—924no—
320D3×Q8—924no—
321D3×Q8—98no—
322D3×Q8—92no—
323D3×Q8—96no—
324D3×Q8—93no—
325D3×Q8—96no—
326D3×Q8—96no—
327D3×Q8—96no—
328D3×Q8—188no—
329D3×Q8C2×C3⋊D311yescharacteristic
330#237—324no—
331#237—924no—
332#237—924no—
333#237—948no—
334#237—948no—
335#237—924no—
336#237—96no—
337#237—912no—
338#237—96no—
339#237—93no—
340#237—1824no—
341#237C2×C3⋊D313yescomplemented
342D3×D4—96no—
343D3×D4—91no—
344D3×D4C2×C3⋊D313yes—
345#235—312no—
346#235—912no—
347#235—924no—
348#235—912no—
349#235—924no—
350#235—912no—
351#235—93no—
352#235—912no—
353#235—96no—
354#235—96no—
355#235—1812no—
356#235C2×C3⋊D313yescomplemented
357C2×D12—94no—
358C2×C4×D3—924no—
359C2×C4×D3—96no—
360C2×C4×D3—96no—
361C2×C4×D3C2×C3⋊D313yes—
362C2×Dic6—312no—
363C2×Dic6—912no—
364C2×Dic6—924no—
365C2×Dic6—924no—
366C2×Dic6—924no—
367C2×Dic6—924no—
368C2×Dic6—912no—
369C2×Dic6—93no—
370C2×Dic6—96no—
371C2×Dic6—96no—
372C2×Dic6—1812no—
373C22×C32—212no—
374C22×C32—31no—
375C22×C32—33no—
376C22×C32—66no—
377C22×C32C23×D3112yes—
378C22×C32C23×D313yes—
379C2×C3⋊D3—32no—
380C2×C3⋊D3—38no—
381C2×C3⋊D3—68no—
382C2×C3⋊D3—64no—
383C2×C3⋊D3C23×D312yes—
384C2×C3×D3—224no—
385C2×C3×D3—38no—
386C2×C3×D3C23×D3112yes—
387C2×C3×D3C23×D314yes—
388D32—616no—
389C32×C4—24no—
390C32×C4—212no—
391C32×C4—31no—
392C32×C4—33no—
393C32×C4C23×D314yes—
394C32×C4C23×D3112yes—
395C32×C4C23×D311yescharacteristic
396C32×C4C23×D313yes—
397C3⋊Dic3—36no—
398C3⋊Dic3—324no—
399C3⋊Dic3—624no—
400C3⋊Dic3C23×D316yes—
401C3×Dic3—324no—
402C3×Dic3C23×D3112yes—
403C3×Dic3C23×D314yes—
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—
415C33C2×2−1+411yescharacteristic, complemented, Sylow
416C23×C3—312no—
417C22×D3—98no—
418C22×D3—92no—
419C22×D3C22×C3⋊D313yes—
420C3×Q8—24no—
421C3×Q8—312no—
422C3×Q8—93no—
423C3×Q8—96no—
424C3×Q8C2×D3214yes—
425C3×Q8C22×C3⋊D311yescharacteristic
426C3×D4—212no—
427C3×D4—324no—
428C3×D4—312no—
429C3×D4—96no—
430C3×D4—91no—
431C3×D4C2×D32112yes—
432C3×D4C22×C3⋊D313yes—
433C2×C3×C4—212no—
434C2×C3×C4—312no—
435C2×C3×C4—34no—
436C2×C3×C4—312no—
437C2×C3×C4—312no—
438C2×C3×C4—312no—
439C2×C3×C4—96no—
440C2×C3×C4—96no—
441C2×C3×C4C2×D32112yes—
442C2×C3×C4C22×C3⋊D313yes—
443#67—324no—
444#67—924no—
445#67—924no—
446#67—924no—
447#67—96no—
448#67—96no—
449#67—92no—
450#67—1824no—
451#67C22×C3⋊D316yes—
452C2×Dic3—324no—
453C2×Dic3—924no—
454C2×Dic3—948no—
455C2×Dic3—924no—
456C2×Dic3—96no—
457C2×Dic3—96no—
458C2×Dic3—1824no—
459C2×Dic3C22×C3⋊D313yes—
460D12—34no—
461D12—98no—
462D12—96no—
463D12—94no—
464D12—91no—
465D12—184no—
466D12C22×C3⋊D313yes—
467C4×D3—324no—
468C4×D3—924no—
469C4×D3—924no—
470C4×D3—96no—
471C4×D3—912no—
472C4×D3—924no—
473C4×D3—96no—
474C4×D3—96no—
475C4×D3—1824no—
476C4×D3C22×C3⋊D313yes—
477C4×D3C22×C3⋊D311yescharacteristic
478C4×D3C22×C3⋊D313yes—
479C4×D3C22×C3⋊D313yes—
480Dic6—312no—
481Dic6—324no—
482Dic6—924no—
483Dic6—948no—
484Dic6—912no—
485Dic6—924no—
486Dic6—93no—
487Dic6—96no—
488Dic6—96no—
489Dic6—912no—
490Dic6—1812no—
491Dic6—1824no—
492Dic6C22×C3⋊D313yes—
493C2×C32—23no—
494C2×C32—24no—
495C2×C32—212no—
496C2×C32—32no—
497C2×C32—412no—
498C2×C32—63no—
499C2×C32C24×D314yes—
500C2×C32C24×D311yescharacteristic
501C3⋊D3—22no—
502C3⋊D3—62no—
503C3⋊D3—68no—
504C3⋊D3—128no—
505C3×D3—212no—
506C3×D3—68no—
507C3×D3#101018yescomplemented
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×C3⋊D311yescharacteristic, 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—212no—
543C22×C3—34no—
544C22×C3—312no—
545C22×C3—624no—
546C22×C3—92no—
547C22×C3C22×D32112yes—
548C22×C3C23×C3⋊D313yes—
549C2×D3—26no—
550C2×D3—38no—
551C2×D3—98no—
552C2×D3—92no—
553C2×D3—188no—
554C2×D3—1816no—
555C2×D3—184no—
556C2×D3C23×C3⋊D313yes—
557C2×D3C23×C3⋊D311yescharacteristic
558C3×C4—24no—
559C3×C4—212no—
560C3×C4—34no—
561C3×C4—312no—
562C3×C4—96no—
563C3×C4C22×D3214yes—
564C3×C4C22×D32112yes—
565C3×C4C23×C3⋊D311yescharacteristic
566C3×C4C23×C3⋊D313yes—
567Dic3—324no—
568Dic3—924no—
569Dic3—96no—
570Dic3—1824no—
571Dic3C23×C3⋊D311yescharacteristic
572Dic3C23×C3⋊D313yes—
573C32—24no—
574C32C2×#101014yescomplemented
575C32D3×2−1+411yescharacteristic, complemented
576C23—33no—
577C23—272no—
578Q8—33no—
579Q8—93no—
580Q8—96no—
581Q8—276no—
582Q8—2712no—
583Q8—273no—
584Q8—276no—
585Q8C2×D3×C3⋊D311yescharacteristic
586D4—36no—
587D4—33no—
588D4—96no—
589D4—91no—
590D4—272no—
591D4—276no—
592D4—276no—
593D4—276no—
594D4—271no—
595D4C2×D3×C3⋊D313yes—
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×C3⋊D313yes—
610C2×C3—212no—
611C2×C3—23no—
612C2×C3—24no—
613C2×C3—38no—
614C2×C3—412no—
615C2×C3—612no—
616C2×C3—182no—
617C2×C3C23×D3214yes—
618C2×C3C24×C3⋊D311yescharacteristic
619D3—23no—
620D3—68no—
621D3—188no—
622D3—182no—
623D3—368no—
624D3#6436712yesdirect product factor
625C22—31no—
626C22—33no—
627C22—66no—
628C22—92no—
629C22—272no—
630C22—544no—
631C22C22×D3×C3⋊D313yes—
632C4—31no—
633C4—33no—
634C4—96no—
635C4—276no—
636C4C22×D3×C3⋊D311yescharacteristic
637C4C22×D3×C3⋊D313yes—
638C3—24no—
639C3D3×#101014yescomplemented
640C3C2×#6436711yescharacteristic, complemented
641C2—23no—
642C2—32no—
643C2—63no—
644C2—182no—
645C2—542no—
646C2C23×D3×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C23×D3×C3⋊D3
commutator, socle, cosocleC2×C33C25
FittingC33×#41C22
hypercenter#41D3×C3⋊D3
hypocenterC33C2×2−1+4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #11724685C2semidirect
2. C3×#64367C2semidirect
3. C32×D3×#41C2semidirect
4. D3×Q8×C3⋊D3C2semidirect
5. #11724590C2semidirect
6. D3×#3830C2semidirect
7. #11724582C2semidirect
8. D3×#3828C2semidirect
9. #11724563C2semidirect
10. #11724562C2semidirect
11. C2×D3×#634C2semidirect
14. C33×#41C22semidirect
15. Q8×C3⋊(C3⋊D3)C22semidirect
16. #89078C22semidirect
17. #89076C22semidirect
18. C2×#6308C22semidirect
19. C3×Q8×C3⋊D3C22semidirect
20. C3×#3830C22semidirect
21. C3×#3828C22semidirect
22. C2×C3×#634C22semidirect
23. C32×#239C22semidirect
25. C32×#237C22semidirect
27. C32×#235C22semidirect
30. C2×#6293C22semidirect
31. #89031C22semidirect
32. #89029C22semidirect
33. #89028C22semidirect
37. #89022C22semidirect
38. #89021C22semidirect
39. #89020C22semidirect
40. #89019C22semidirect
41. C3⋊D3×Dic6C22semidirect
42. #89015C22semidirect
43. #89014C22semidirect
46. D3×#634C22semidirect
47. D3×#634C22semidirect
56. #64367D3direct (N × H)
62. C3×D3×#41D3semidirect
111. C32×#41C2×D3semidirect
112. C32×#41C2×D3semidirect
123. Q8×C3⋊D3C2×D3semidirect
131. #3830C2×D3semidirect
139. #3828C2×D3semidirect
150. C2×#634C2×D3semidirect
151. C3×#239C2×D3semidirect
155. C3×#237C2×D3semidirect
158. C3×#235C2×D3semidirect
207. D3×#41C3⋊D3semidirect
299. C32×D32−1+4semidirect
306. C3×#41D32semidirect
307. C3×#41C2×C3⋊D3semidirect
317. #239C2×C3⋊D3semidirect
341. #237C2×C3⋊D3semidirect
356. #235C2×C3⋊D3semidirect
415. C33C2×2−1+4semidirect
507. C3×D3#1010semidirect
523. #41D3×C3⋊D3semidirect
574. C32C2×#1010semidirect
575. C32D3×2−1+4semidirect
624. D3#64367direct (N × H)
639. C3D3×#1010semidirect
640. C3C2×#64367semidirect

Maximal subgroups

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

subgroupindex
#117246852
C3×#643672
C32×D3×#412
D3×Q8×C3⋊D32
#117245902
D3×#38302
#117245822
D3×#38282
#117245632
#117245622
C2×D3×#6342
C2×#643673
D3×#10103