Finite Group Databasebeta

← D3×#2852 #11001060

Subgroups — D3×#2852 #11001060

Order 768.

3354 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#126C211yesmaximal, characteristic, complemented
2C22×C42×D3C211yesmaximal, characteristic, complemented
3C3×#2852C211yesmaximal, characteristic, complemented
4#83120C211yesmaximal, characteristic, complemented
5#82078C211yesmaximal, characteristic, complemented
6#75958C213yesmaximal, complemented
7D3×#388C213yesmaximal, complemented
8#75446C213yesmaximal, complemented
9D3×#387C213yesmaximal, complemented
10D3×#386C213yesmaximal
11#74870C213yesmaximal, complemented
12D3×#385C213yesmaximal, complemented
13#74587C213yesmaximal, complemented
14#74317C211yesmaximal, characteristic, complemented
15D3×#383C211yesmaximal, characteristic, complemented
16C2×#2852—31nomaximal, Sylow
17C22×D3×D4C2211yescharacteristic
18C23×C4×D3C2213yes—
19C2×C3×#126C2211yescharacteristic, complemented
20C22×C3×C42C2211yescharacteristic, complemented
21C22×C4×Dic3C2213yescomplemented
22D3×#126—28no—
23C2×#898C2213yescomplemented
24C2×D3×C4⋊C4C2213yes—
25C2×D3×C4⋊C4C2213yes—
26C2×D3×C22⋊C4C2213yes—
27C2×D3×C22⋊C4C2213yes—
28C2×D3×C22⋊C4C2213yes—
29C2×#879C2213yescomplemented
30C2×#876C2211yescharacteristic, complemented
31C2×C42×D3—28no—
32C2×C42×D3C2213yes—
33C2×C42×D3C2211yescharacteristic
34C2×C42×D3C2211yescharacteristic, complemented
35C2×C42×D3C2213yescomplemented
36C3×#388C2213yescomplemented
37C3×#387C2213yescomplemented
38C3×#386C2213yes—
39C3×#385C2213yescomplemented
40C3×#383C2211yescharacteristic, complemented
41#5220C2213yes—
42#5213C2213yes—
43#5210C2213yescomplemented
44#4990C2213yescomplemented
45#4989C2213yescomplemented
46#4989C2213yescomplemented
47#4980C2213yescomplemented
48#4978C2213yescomplemented
49#4958C2213yes—
50#4956C2213yes—
51#4956C2213yes—
52#4951C2213yescomplemented
53#4950C2213yescomplemented
54#4946C2213yescomplemented
55#4945C2213yescomplemented
56#4944C2213yescomplemented
57#4940C2213yescomplemented
58#4937C2211yescharacteristic, complemented
59#4933C2213yescomplemented
60#4670C2213yescomplemented
61#4669C2213yescomplemented
62#4668C2213yescomplemented
63#4668C2213yescomplemented
64#4665C2213yes—
65#4665C2213yes—
66D3×#95C2213yes—
67D3×#95C2213yes—
68D3×#95C2213yes—
69D3×#95C2213yes—
70#4655C2213yescomplemented
71#4654C2213yescomplemented
72#4653C2213yescomplemented
73#4652C2213yescomplemented
74#4651C2213yescomplemented
75#4651C2213yescomplemented
76C22×#126—31no—
77C23×C42—31no—
78#2852—31no—
79#2852—33no—
80#2852—33no—
81#2852—33no—
82#2852—33no—
83#2852—31no—
84#2852—31no—
85#2852D311yescharacteristic, direct product factor
86C2×#388—33no—
87C2×#387—33no—
88C2×#386—33no—
89C2×#385—33no—
90C2×#383—31no—
91C24×D3C2311yescharacteristic
92C24×D3C2311yescharacteristic
93C22×C3×D4C2311yescharacteristic
94C23×C3×C4C2313yes—
95C22×#67C2311yescharacteristic
96C22×#67C2311yescharacteristic
97C23×Dic3C2313yes—
98C2×D3×D4—212no—
99C22×D12C2311yescharacteristic
100C22×C4×D3—212no—
101C22×C4×D3—212no—
102C22×C4×D3—212no—
103C22×C4×D3C2311yescharacteristic
104C22×C4×D3C2313yes—
105C22×C4×D3C2313yes—
106C22×C4×D3C2313yes—
107C22×C4×D3C2313yes—
108C22×C4×D3C2313yes—
109C22×C4×D3C2313yes—
110C3×#126—24no—
111C2×C3×C4⋊C4C2313yes—
112C2×C3×C4⋊C4C2313yes—
113C2×C3×C22⋊C4C2313yes—
114C2×C3×C22⋊C4C2313yes—
115C2×C3×C22⋊C4C2313yes—
116C2×C3×C42—24no—
117C2×C3×C42C2313yes—
118C2×C3×C42C2311yescharacteristic
119C2×#217C2313yes—
120C2×#217C2313yes—
121C2×#217C2313yes—
122C2×(C2×D3)⋊C4C2313yes—
123C2×(C2×D3)⋊C4C2313yes—
124C2×(C2×D3)⋊C4C2313yes—
125C2×(C2×D3)⋊C4C2313yes—
126C2×(C2×D3)⋊C4C2313yes—
127C2×(C2×D3)⋊C4C2313yes—
128C2×C4⋊Dic3C2313yes—
129C2×C4⋊Dic3C2313yes—
130C2×Dic3⋊C4C2313yes—
131C2×Dic3⋊C4C2313yes—
132C2×Dic3⋊C4C2313yes—
133C2×Dic3⋊C4C2313yes—
134C2×C4×Dic3—212no—
135C2×C4×Dic3C2313yes—
136C2×C4×Dic3C2313yes—
137C2×C4×Dic3C2313yes—
138C2×C4×Dic3C2313yes—
139#898—212no—
140D3×C4⋊C4—212no—
141D3×C4⋊C4—212no—
142D3×C22⋊C4—212no—
143D3×C22⋊C4—212no—
144D3×C22⋊C4—212no—
145#879—212no—
146#876—24no—
147C42×D3—212no—
148C42×D3—24no—
149C42×D3—212no—
150C42×D3D418yescomplemented
151C3×#95C2313yes—
152C3×#95C2313yes—
153C3×#95C2313yes—
154C3×#95C2313yes—
155C3⋊#95C2313yes—
156C3⋊#95C2313yes—
157C3⋊#95C2313yes—
158C3⋊#95C2313yes—
159C3⋊#95C2313yes—
160C3⋊#95C2313yes—
161C3⋊#95C2313yes—
162C3⋊#95C2313yes—
163C3⋊#95C2313yes—
164C3⋊#95C2313yes—
165C3⋊#95C2313yes—
166C3⋊#95C2313yes—
167C23×D4—31no—
168C24×C4—33no—
169C2×#126—33no—
170C2×#126—33no—
171C2×#126—31no—
172C2×#126—68no—
173C2×#126C2×D311yescharacteristic, complemented
174C22×C4⋊C4—33no—
175C22×C4⋊C4—33no—
176C22×C22⋊C4—33no—
177C22×C22⋊C4—33no—
178C22×C22⋊C4—33no—
179C22×C42—31no—
180C22×C42—33no—
181C22×C42—33no—
182C22×C42—31no—
183C22×C42—33no—
184C22×C42—68no—
185C22×C42C2×D311yescharacteristic, complemented
186#388—33no—
187#388—33no—
188#388—33no—
189#388—33no—
190#388—33no—
191#388—33no—
192#388—33no—
193#388C2×D313yescomplemented
194#387—33no—
195#387—33no—
196#387—33no—
197#387—33no—
198#387—33no—
199#387—33no—
200#387—33no—
201#387C2×D313yescomplemented
202#386—33no—
203#386—33no—
204#386—33no—
205#386—33no—
206#386—33no—
207#386—33no—
208#386—33no—
209#386C2×D313yes—
210#385—33no—
211#385—33no—
212#385—33no—
213#385—33no—
214#385—33no—
215#385—33no—
216#385—33no—
217#385C2×D313yescomplemented
218#383—31no—
219#383—33no—
220#383—33no—
221#383C2×D311yescharacteristic, complemented
222C2×#95—33no—
223C2×#95—33no—
224C2×#95—33no—
225C2×#95—33no—
226C24×C3C2411yescharacteristic
227C24×C3C2411yescharacteristic
228C23×D3—22no—
229C23×D3—26no—
230C23×D3—22no—
231C23×D3—26no—
232C23×D3—26no—
233C23×D3#41112yes—
234C23×D3C2411yescharacteristic
235C23×D3C2411yescharacteristic
236C23×D3C2411yescharacteristic
237C2×C3×D4—26no—
238C22×C3×C4—26no—
239C22×C3×C4—26no—
240C22×C3×C4—26no—
241C22×C3×C4C2411yescharacteristic
242C22×C3×C4C2413yes—
243C22×C3×C4C2413yes—
244C22×C3×C4C2413yes—
245C22×C3×C4C2413yes—
246C2×#67—26no—
247C2×#67—26no—
248C22×Dic3—26no—
249C22×Dic3—26no—
250C22×Dic3—26no—
251C22×Dic3C2411yescharacteristic
252C22×Dic3C2413yes—
253C22×Dic3C2413yes—
254C22×Dic3C2413yes—
255C22×Dic3C2413yes—
256D3×D4—416no—
257C2×D12—26no—
258C2×C4×D3—296no—
259C2×C4×D3—26no—
260C2×C4×D3—26no—
261C2×C4×D3—26no—
262C2×C4×D3—26no—
263C2×C4×D3—26no—
264C2×C4×D3—26no—
265C2×C4×D3—26no—
266C2×C4×D3—26no—
267C2×C4×D3—26no—
268C2×C4×D3—26no—
269C2×C4×D3—26no—
270C2×C4×D3—26no—
271C2×C4×D3—26no—
272C2×C4×D3—26no—
273C2×C4×D3—26no—
274C2×C4×D3—26no—
275C2×C4×D3—26no—
276C2×C4×D3C2×D4112yes—
277C2×C4×D3#41112yes—
278C3×C4⋊C4—26no—
279C3×C4⋊C4—26no—
280C3×C22⋊C4—26no—
281C3×C22⋊C4—26no—
282C3×C22⋊C4—26no—
283C3×C42—26no—
284C3×C42C2×D414yescomplemented
285#217—26no—
286#217—26no—
287#217—26no—
288(C2×D3)⋊C4—26no—
289(C2×D3)⋊C4—26no—
290(C2×D3)⋊C4—26no—
291(C2×D3)⋊C4—26no—
292(C2×D3)⋊C4—26no—
293(C2×D3)⋊C4—26no—
294C4⋊Dic3—26no—
295C4⋊Dic3—26no—
296Dic3⋊C4—26no—
297Dic3⋊C4—26no—
298Dic3⋊C4—26no—
299Dic3⋊C4—26no—
300C4×Dic3—26no—
301C4×Dic3—26no—
302C4×Dic3—26no—
303C4×Dic3C2×D4112yescomplemented
304C25—31no—
305C25—31no—
306C22×D4—31no—
307C22×D4—31no—
308C22×D4—31no—
309C22×D4—612no—
310C22×D4C22×D311yescharacteristic
311C23×C4—33no—
312C23×C4—33no—
313C23×C4—33no—
314C23×C4—33no—
315C23×C4—33no—
316C23×C4—33no—
317C23×C4—33no—
318C23×C4—31no—
319C23×C4—612no—
320C23×C4—612no—
321C23×C4—612no—
322C23×C4C22×D313yes—
323#126—24no—
324#126—612no—
325#126—612no—
326#126—64no—
327C2×C4⋊C4—33no—
328C2×C4⋊C4—33no—
329C2×C4⋊C4—33no—
330C2×C4⋊C4—33no—
331C2×C4⋊C4—33no—
332C2×C4⋊C4—33no—
333C2×C4⋊C4—612no—
334C2×C4⋊C4—612no—
335C2×C4⋊C4C22×D313yes—
336C2×C4⋊C4C22×D313yes—
337C2×C22⋊C4—33no—
338C2×C22⋊C4—33no—
339C2×C22⋊C4—33no—
340C2×C22⋊C4—33no—
341C2×C22⋊C4—33no—
342C2×C22⋊C4—33no—
343C2×C22⋊C4—33no—
344C2×C22⋊C4—33no—
345C2×C22⋊C4—33no—
346C2×C22⋊C4—612no—
347C2×C22⋊C4—612no—
348C2×C22⋊C4—612no—
349C2×C22⋊C4C22×D313yes—
350C2×C22⋊C4C22×D313yes—
351C2×C22⋊C4C22×D313yes—
352C2×C42—24no—
353C2×C42—33no—
354C2×C42—38no—
355C2×C42—33no—
356C2×C42—33no—
357C2×C42—33no—
358C2×C42—612no—
359C2×C42—612no—
360C2×C42—612no—
361C2×C42—64no—
362C2×C42C22×D313yes—
363C2×C42C22×D311yescharacteristic
364#95—33no—
365#95—33no—
366#95—33no—
367#95—33no—
368#95—33no—
369#95—33no—
370#95—33no—
371#95—33no—
372#95—33no—
373#95—33no—
374#95—33no—
375#95—33no—
376#95C22×D313yes—
377#95C22×D313yes—
378#95C22×D313yes—
379#95C22×D313yes—
380C23×C3—21no—
381C23×C3—23no—
382C23×C3—21no—
383C23×C3—23no—
384C23×C3—23no—
385C23×C3C2×#4116yes—
386C23×C3C2511yescharacteristic
387C22×D3—21no—
388C22×D3—23no—
389C22×D3—224no—
390C22×D3—224no—
391C22×D3—21no—
392C22×D3—23no—
393C22×D3—23no—
394C22×D3—412no—
395C22×D3—44no—
396C22×D3—412no—
397C22×D3#126116yescomplemented
398C22×D3C22×D412yes—
399C22×D3C2×#4116yes—
400C22×D3C2×#4116yes—
401C22×D3C2×#4116yes—
402C3×D4—44no—
403C2×C3×C4—224no—
404C2×C3×C4—23no—
405C2×C3×C4—23no—
406C2×C3×C4—23no—
407C2×C3×C4—23no—
408C2×C3×C4—23no—
409C2×C3×C4—23no—
410C2×C3×C4—23no—
411C2×C3×C4—23no—
412C2×C3×C4—23no—
413C2×C3×C4—23no—
414C2×C3×C4—23no—
415C2×C3×C4C22×D416yes—
416C2×C3×C4C2×#4116yes—
417#67—44no—
418#67—44no—
419C2×Dic3—224no—
420C2×Dic3—23no—
421C2×Dic3—23no—
422C2×Dic3—23no—
423C2×Dic3—23no—
424C2×Dic3—23no—
425C2×Dic3—23no—
426C2×Dic3—23no—
427C2×Dic3—23no—
428C2×Dic3—23no—
429C2×Dic3—23no—
430C2×Dic3—23no—
431C2×Dic3C22×D416yes—
432C2×Dic3C2×#4116yes—
433D12—44no—
434C4×D3—224no—
435C4×D3—224no—
436C4×D3—224no—
437C4×D3—224no—
438C4×D3—412no—
439C4×D3—44no—
440C4×D3—412no—
441C24—31no—
442C24—312no—
443C24—31no—
444C24—31no—
445C24—62no—
446C24—66no—
447C24—62no—
448C24—66no—
449C24—66no—
450C24C23×D311yescharacteristic
451C24C23×D311yescharacteristic
452C2×D4—26no—
453C2×D4—66no—
454C2×D4—66no—
455C2×D4—66no—
456C2×D4—1216no—
457C22×C4—26no—
458C22×C4—26no—
459C22×C4—26no—
460C22×C4—33no—
461C22×C4—33no—
462C22×C4—312no—
463C22×C4—312no—
464C22×C4—31no—
465C22×C4—33no—
466C22×C4—33no—
467C22×C4—66no—
468C22×C4—66no—
469C22×C4—66no—
470C22×C4—66no—
471C22×C4—66no—
472C22×C4—66no—
473C22×C4—66no—
474C22×C4—66no—
475C22×C4—696no—
476C22×C4—66no—
477C22×C4—66no—
478C22×C4—66no—
479C22×C4—66no—
480C22×C4—66no—
481C22×C4—66no—
482C22×C4—66no—
483C22×C4—66no—
484C22×C4—66no—
485C22×C4—66no—
486C22×C4—66no—
487C22×C4—66no—
488C22×C4C23×D311yescharacteristic
489C22×C4C23×D313yes—
490C22×C4C23×D313yes—
491C22×C4C23×D313yes—
492C22×C4C23×D313yes—
493C4⋊C4—26no—
494C4⋊C4—26no—
495C4⋊C4—66no—
496C4⋊C4—66no—
497C4⋊C4—66no—
498C4⋊C4—66no—
499C4⋊C4—66no—
500C4⋊C4—66no—
501C22⋊C4—26no—
502C22⋊C4—26no—
503C22⋊C4—26no—
504C22⋊C4—66no—
505C22⋊C4—66no—
506C22⋊C4—66no—
507C22⋊C4—66no—
508C22⋊C4—66no—
509C22⋊C4—66no—
510C22⋊C4—66no—
511C22⋊C4—66no—
512C22⋊C4—66no—
513C42—26no—
514C42—312no—
515C42—66no—
516C42—66no—
517C42—66no—
518C42D3×D414yescomplemented
519C22×C3—26no—
520C22×C3—26no—
521C22×C3—43no—
522C22×C3—41no—
523C22×C3—43no—
524C22×C3C2×#12614yescomplemented
525C22×C3C23×D411yescharacteristic
526C22×C3C22×#4113yes—
527C22×C3C22×#4113yes—
528C2×D3—26no—
529C2×D3—26no—
530C2×D3—232no—
531C2×D3—43no—
532C2×D3—41no—
533C2×D3—43no—
534C2×D3—88no—
535C2×D3#525112yes—
536C2×D3C2×#12614yes—
537C2×D3C2×#12614yescomplemented
538C2×D3#529112yes—
539C3×C4—26no—
540C3×C4—26no—
541C3×C4—43no—
542C3×C4—41no—
543C3×C4—43no—
544Dic3—26no—
545Dic3—26no—
546Dic3—43no—
547Dic3—41no—
548Dic3—43no—
549C23—21no—
550C23—23no—
551C23—21no—
552C23—23no—
553C23—23no—
554C23—32no—
555C23—36no—
556C23—36no—
557C23—316no—
558C23—36no—
559C23—624no—
560C23—624no—
561C23—61no—
562C23—63no—
563C23—61no—
564C23—63no—
565C23—63no—
566C23—124no—
567C23—1212no—
568C23—1212no—
569C23D3×#4116yes—
570C23C24×D311yescharacteristic
571D4—44no—
572D4—124no—
573D4—124no—
574D4—124no—
575C2×C4—224no—
576C2×C4—23no—
577C2×C4—23no—
578C2×C4—23no—
579C2×C4—23no—
580C2×C4—23no—
581C2×C4—23no—
582C2×C4—23no—
583C2×C4—23no—
584C2×C4—23no—
585C2×C4—23no—
586C2×C4—23no—
587C2×C4—36no—
588C2×C4—36no—
589C2×C4—624no—
590C2×C4—624no—
591C2×C4—624no—
592C2×C4—624no—
593C2×C4—624no—
594C2×C4—63no—
595C2×C4—63no—
596C2×C4—63no—
597C2×C4—63no—
598C2×C4—63no—
599C2×C4—63no—
600C2×C4—63no—
601C2×C4—63no—
602C2×C4—63no—
603C2×C4—63no—
604C2×C4—63no—
605C2×C4—124no—
606C2×C4—1212no—
607C2×C4—1212no—
608C2×C4C2×D3×D416yes—
609C2×C4D3×#4116yes—
610C2×C3—24no—
611C2×C3—81no—
612C2×C3C2×#52513yes—
613C2×C3C22×#12611yescharacteristic
614C2×C3C2×#52913yes—
615D3—24no—
616D3—81no—
617D3#285218yesdirect product factor
618C22—26no—
619C22—26no—
620C22—312no—
621C22—34no—
622C22—312no—
623C22—34no—
624C22—43no—
625C22—41no—
626C22—43no—
627C22—632no—
628C22—66no—
629C22—66no—
630C22—123no—
631C22—121no—
632C22—123no—
633C22—248no—
634C22D3×#12614yescomplemented
635C22C22×D3×D411yescharacteristic
636C22C2×D3×#4113yes—
637C22C2×D3×#4113yes—
638C4—26no—
639C4—26no—
640C4—43no—
641C4—41no—
642C4—43no—
643C4—66no—
644C4—66no—
645C4—123no—
646C4—121no—
647C4—123no—
648C3C2×#285211yescharacteristic, complemented, Sylow
649C2—24no—
650C2—38no—
651C2—64no—
652C2—81no—
653C2—241no—
654C2D3×#52513yes—
655C2C2×D3×#12611yescharacteristic
656C2D3×#52913yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2852C2
prime radicalC22×D3×D4C22
hypercenter#2852D3
hypocenterC3C2×#2852
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2852

Complements

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

NH ≅ G/Ntype
1. C2×D3×#126C2semidirect
2. C22×C42×D3C2semidirect
3. C3×#2852C2semidirect
4. #83120C2semidirect
5. #82078C2semidirect
6. #75958C2semidirect
7. D3×#388C2semidirect
8. #75446C2semidirect
9. D3×#387C2semidirect
11. #74870C2semidirect
12. D3×#385C2semidirect
13. #74587C2semidirect
14. #74317C2semidirect
15. D3×#383C2semidirect
19. C2×C3×#126C22semidirect
20. C22×C3×C42C22semidirect
21. C22×C4×Dic3C22semidirect
23. C2×#898C22semidirect
29. C2×#879C22semidirect
30. C2×#876C22semidirect
34. C2×C42×D3C22semidirect
35. C2×C42×D3C22semidirect
36. C3×#388C22semidirect
37. C3×#387C22semidirect
39. C3×#385C22semidirect
40. C3×#383C22semidirect
43. #5210C22semidirect
44. #4990C22semidirect
45. #4989C22semidirect
46. #4989C22semidirect
47. #4980C22semidirect
48. #4978C22semidirect
52. #4951C22semidirect
53. #4950C22semidirect
54. #4946C22semidirect
55. #4945C22semidirect
56. #4944C22semidirect
57. #4940C22semidirect
58. #4937C22semidirect
59. #4933C22semidirect
60. #4670C22semidirect
61. #4669C22semidirect
62. #4668C22semidirect
63. #4668C22semidirect
70. #4655C22semidirect
71. #4654C22semidirect
72. #4653C22semidirect
73. #4652C22semidirect
74. #4651C22semidirect
75. #4651C22semidirect
85. #2852D3direct (N × H)
150. C42×D3D4semidirect
173. C2×#126C2×D3semidirect
185. C22×C42C2×D3semidirect
193. #388C2×D3semidirect
201. #387C2×D3semidirect
217. #385C2×D3semidirect
221. #383C2×D3semidirect
284. C3×C42C2×D4semidirect
303. C4×Dic3C2×D4semidirect
397. C22×D3#126semidirect
518. C42D3×D4semidirect
524. C22×C3C2×#126semidirect
537. C2×D3C2×#126semidirect
617. D3#2852direct (N × H)
634. C22D3×#126semidirect
648. C3C2×#2852semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#1262
C22×C42×D32
C3×#28522
#831202
#820782
#759582
D3×#3882
#754462
D3×#3872
D3×#3862
#748702
D3×#3852
#745872
#743172
D3×#3832
C2×#28523