Finite Group Databasebeta

← D3×#2537 #10966922

Subgroups — D3×#2537 #10966922

Order 768.

4782 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#121C214yesmaximal, complemented
2C2×C4×D3×D4C212yesmaximal, complemented
3C3×#2537C211yesmaximal, characteristic, complemented
4#82667C214yesmaximal, complemented
5#82417C212yesmaximal, complemented
6D3×#394C214yesmaximal, complemented
7#76944C214yesmaximal, complemented
8#74368C211yesmaximal, characteristic, complemented
9D3×#384C211yesmaximal, characteristic, complemented
10D3×#380C214yesmaximal, complemented
11#73935C214yesmaximal, complemented
12C2×#2537—31nomaximal, Sylow
13C22×D3×D4C2214yes—
14C22×D3×D4C2212yes—
15C23×C4×D3C2214yes—
16C2×C3×#121C2214yescomplemented
17C2×C3×C4×D4C2212yescomplemented
18C2×#939C2218yescomplemented
19C2×#938C2214yescomplemented
20C2×D4×Dic3C2212yescomplemented
21C2×#930C2214yescomplemented
22C2×C4×#67C2214yescomplemented
23D3×#121—232no—
24C4×D3×D4—216no—
25C2×#895C2214yescomplemented
26C2×#893C2212yescomplemented
27C2×D3×C4⋊C4C2212yes—
28C2×D3×C4⋊C4C2214yes—
29C2×D3×C4⋊C4C2212yes—
30C2×#884C2218yescomplemented
31C2×#67⋊C4C2214yescomplemented
32C2×D3×C22⋊C4C2214yes—
33C2×D3×C22⋊C4C2218yes—
34C2×C4×D12C2212yescomplemented
35C2×C42×D3C2211yescharacteristic
36C3×#394C2214yescomplemented
37C3×#384C2211yescharacteristic, complemented
38C3×#380C2214yescomplemented
39#5222C2214yescomplemented
40#5212C2214yescomplemented
41#4991C2214yescomplemented
42#4987C2214yescomplemented
43#4987C2214yescomplemented
44#4975C2212yescomplemented
45#4975C2214yescomplemented
46#4961C2218yescomplemented
47#4959C2218yescomplemented
48#4942C2218yescomplemented
49#4935C2211yescharacteristic, complemented
50#4672C2214yescomplemented
51#4667C2218yescomplemented
52D3×#95C2214yes—
53C22×#121—34no—
54C22×C4×D4—32no—
55#2537—34no—
56#2537—31no—
57#2537—34no—
58#2537—34no—
59#2537—32no—
60#2537D311yescharacteristic, direct product factor
61C2×#394—34no—
62C2×#384—31no—
63C2×#380—34no—
64C24×D3C2314yes—
65C22×C3×D4C2314yes—
66C22×C3×D4C2312yes—
67C23×C3×C4C2314yes—
68C22×#67C2318yes—
69C22×#67C2314yes—
70C23×Dic3C2314yes—
71C2×D3×D4—232no—
72C2×D3×D4—28no—
73C2×D3×D4—28no—
74C2×D3×D4—48no—
75C2×D3×D4D4116yescomplemented
76C22×D12C2314yes—
77C22×D12C2312yes—
78C22×C4×D3—216no—
79C22×C4×D3—216no—
80C22×C4×D3—216no—
81C22×C4×D3C2314yes—
82C22×C4×D3C2314yes—
83C22×C4×D3C2314yes—
84C22×C4×D3C2314yes—
85C22×C4×D3C2311yescharacteristic
86C22×C4×D3C2312yes—
87C3×#121—216no—
88C3×C4×D4—28no—
89C2×C3×C4⋊C4C2314yes—
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C4⋊C4C2312yes—
92C2×C3×C22⋊C4C2318yes—
93C2×C3×C22⋊C4C2314yes—
94C2×C3×C42C2311yescharacteristic
95C2×#217C2318yes—
96C2×#217C2314yes—
97#939—232no—
98#938—216no—
99D4×Dic3—28no—
100#930—216no—
101C4×#67—216no—
102C2×(C2×D3)⋊C4C2318yes—
103C2×(C2×D3)⋊C4C2318yes—
104C2×(C2×D3)⋊C4C2314yes—
105C2×(C2×D3)⋊C4C2314yes—
106C2×C4⋊Dic3C2314yes—
107C2×C4⋊Dic3C2312yes—
108C2×C4⋊Dic3C2312yes—
109C2×Dic3⋊C4C2314yes—
110C2×Dic3⋊C4C2318yes—
111C2×Dic3⋊C4C2314yes—
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2312yes—
114#895—216no—
115#893—28no—
116D3×C4⋊C4—28no—
117D3×C4⋊C4—216no—
118D3×C4⋊C4—44no—
119#884—232no—
120#67⋊C4—216no—
121D3×C22⋊C4—232no—
122D3×C22⋊C4—216no—
123C4×D12—28no—
124C42×D3D418yescomplemented
125C3×#95C2314yes—
126C3⋊#95C2318yes—
127C3⋊#95C2314yes—
128C23×D4—32no—
129C23×D4—34no—
130C24×C4—34no—
131C2×#121—34no—
132C2×#121—38no—
133C2×#121—34no—
134C2×#121—34no—
135C2×#121—38no—
136C2×#121—632no—
137C2×#121C2×D314yescomplemented
138C2×C4×D4—32no—
139C2×C4×D4—34no—
140C2×C4×D4—32no—
141C2×C4×D4—34no—
142C2×C4×D4—32no—
143C2×C4×D4—616no—
144C2×C4×D4C2×D312yescomplemented
145C22×C4⋊C4—32no—
146C22×C4⋊C4—34no—
147C22×C4⋊C4—32no—
148C22×C22⋊C4—34no—
149C22×C22⋊C4—38no—
150C22×C42—31no—
151#394—34no—
152#394—34no—
153#394—38no—
154#394—38no—
155#394—34no—
156#394C2×D314yescomplemented
157#384—32no—
158#384—34no—
159#384—31no—
160#384C2×D311yescharacteristic, complemented
161#380—34no—
162#380—34no—
163#380—34no—
164#380—38no—
165#380—38no—
166#380C2×D314yescomplemented
167C2×#95—34no—
168C24×C3C2414yes—
169C23×D3—28no—
170C23×D3—28no—
171C23×D3—28no—
172C23×D3—28no—
173C23×D3—28no—
174C23×D3—28no—
175C23×D3C2×D4116yes—
176C23×D3C2414yes—
177C23×D3C2411yescharacteristic
178C2×C3×D4—216no—
179C2×C3×D4—24no—
180C2×C3×D4—24no—
181C2×C3×D4—44no—
182C2×C3×D4C2×D418yescomplemented
183C22×C3×C4—28no—
184C22×C3×C4—28no—
185C22×C3×C4—28no—
186C22×C3×C4C2414yes—
187C22×C3×C4C2414yes—
188C22×C3×C4C2411yescharacteristic
189C22×C3×C4C2412yes—
190C2×#67—216no—
191C2×#67—216no—
192C2×#67—28no—
193C2×#67—28no—
194C2×#67—48no—
195C2×#67C2×D4116yescomplemented
196C22×Dic3—28no—
197C22×Dic3—28no—
198C22×Dic3—28no—
199C22×Dic3C2414yes—
200C22×Dic3C2414yes—
201C22×Dic3C2411yescharacteristic
202C22×Dic3C2412yes—
203D3×D4—264no—
204D3×D4—464no—
205C2×D12—24no—
206C2×D12—24no—
207C2×D12—216no—
208C2×D12—44no—
209C2×D12C2×D418yescomplemented
210C2×C4×D3—22no—
211C2×C4×D3—22no—
212C2×C4×D3—28no—
213C2×C4×D3—28no—
214C2×C4×D3—216no—
215C2×C4×D3—28no—
216C2×C4×D3—28no—
217C2×C4×D3—28no—
218C2×C4×D3—28no—
219C2×C4×D3—28no—
220C2×C4×D3—28no—
221C2×C4×D3—28no—
222C2×C4×D3—28no—
223C2×C4×D3—464no—
224C2×C4×D3C2×D414yes—
225C2×C4×D3C2×D418yes—
226C2×C4×D3C2×D418yes—
227C2×C4×D3#4118yes—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—28no—
230C3×C4⋊C4—42no—
231C3×C22⋊C4—216no—
232C3×C22⋊C4—28no—
233C3×C42C2×D414yescomplemented
234#217—216no—
235#217—28no—
236(C2×D3)⋊C4—216no—
237(C2×D3)⋊C4—216no—
238(C2×D3)⋊C4—28no—
239(C2×D3)⋊C4—28no—
240C4⋊Dic3—24no—
241C4⋊Dic3—28no—
242C4⋊Dic3—42no—
243Dic3⋊C4—28no—
244Dic3⋊C4—216no—
245Dic3⋊C4—44no—
246C4×Dic3C2×D414yescomplemented
247C4×Dic3C2×D418yescomplemented
248C25—34no—
249C22×D4—34no—
250C22×D4—316no—
251C22×D4—38no—
252C22×D4—32no—
253C22×D4—34no—
254C22×D4—68no—
255C22×D4—68no—
256C22×D4—632no—
257C22×D4—128no—
258C22×D4C22×D314yes—
259C22×D4C22×D312yes—
260C23×C4—32no—
261C23×C4—31no—
262C23×C4—34no—
263C23×C4—34no—
264C23×C4—34no—
265C23×C4—34no—
266C23×C4—34no—
267C23×C4—616no—
268C23×C4—616no—
269C23×C4—616no—
270C23×C4C22×D314yes—
271#121—216no—
272#121—632no—
273#121—616no—
274#121—616no—
275#121—632no—
276#121—616no—
277C4×D4—28no—
278C4×D4—616no—
279C4×D4—68no—
280C4×D4—616no—
281C4×D4—68no—
282C4×D4—68no—
283C2×C4⋊C4—34no—
284C2×C4⋊C4—32no—
285C2×C4⋊C4—38no—
286C2×C4⋊C4—34no—
287C2×C4⋊C4—34no—
288C2×C4⋊C4—32no—
289C2×C4⋊C4—68no—
290C2×C4⋊C4—616no—
291C2×C4⋊C4—124no—
292C2×C4⋊C4C22×D312yes—
293C2×C4⋊C4C22×D314yes—
294C2×C4⋊C4C22×D312yes—
295C2×C22⋊C4—34no—
296C2×C22⋊C4—38no—
297C2×C22⋊C4—34no—
298C2×C22⋊C4—34no—
299C2×C22⋊C4—38no—
300C2×C22⋊C4—38no—
301C2×C22⋊C4—616no—
302C2×C22⋊C4—632no—
303C2×C22⋊C4C22×D314yes—
304C2×C22⋊C4C22×D318yes—
305C2×C42—31no—
306C2×C42—32no—
307C2×C42—38no—
308C2×C42C22×D311yescharacteristic
309#95—34no—
310#95—38no—
311#95C22×D314yes—
312C23×C3—24no—
313C23×C3—24no—
314C23×C3—24no—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3—24no—
318C23×C3C22×D418yes—
319C23×C3C2511yescharacteristic
320C22×D3—24no—
321C22×D3—24no—
322C22×D3—24no—
323C22×D3—24no—
324C22×D3—24no—
325C22×D3—24no—
326C22×D3—232no—
327C22×D3—232no—
328C22×D3—232no—
329C22×D3—416no—
330C22×D3—416no—
331C22×D3—416no—
332C22×D3—416no—
333C22×D3C22×D418yes—
334C22×D3C22×D412yes—
335C22×D3C22×D414yes—
336C22×D3C2×#4114yes—
337C22×D3H(5,𝔽2)12yes—
338C22×D32−1+412yes—
339C3×D4—216no—
340C3×D4—416no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—24no—
344C2×C3×C4—28no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—416no—
349C2×C3×C4C22×D412yes—
350C2×C3×C4C22×D414yes—
351C2×C3×C4C22×D414yes—
352C2×C3×C4C2×#4114yes—
353#67—232no—
354#67—432no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—24no—
358C2×Dic3—28no—
359C2×Dic3—24no—
360C2×Dic3—24no—
361C2×Dic3—24no—
362C2×Dic3—416no—
363C2×Dic3C22×D412yes—
364C2×Dic3C22×D414yes—
365C2×Dic3C22×D414yes—
366C2×Dic3C2×#4114yes—
367D12—216no—
368D12—416no—
369C4×D3—28no—
370C4×D3—416no—
371C4×D3—416no—
372C4×D3—416no—
373C4×D3—416no—
374C4×D3#121132yescomplemented
375C24—31no—
376C24—316no—
377C24—34no—
378C24—68no—
379C24—68no—
380C24—68no—
381C24—68no—
382C24—68no—
383C24—68no—
384C24C23×D314yes—
385C2×D4—216no—
386C2×D4—24no—
387C2×D4—24no—
388C2×D4—316no—
389C2×D4—38no—
390C2×D4—44no—
391C2×D4—616no—
392C2×D4—664no—
393C2×D4—68no—
394C2×D4—68no—
395C2×D4—616no—
396C2×D4—616no—
397C2×D4—64no—
398C2×D4—64no—
399C2×D4—124no—
400C2×D4—128no—
401C2×D4—1264no—
402C2×D4D3×D418yescomplemented
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—32no—
407C22×C4—38no—
408C22×C4—38no—
409C22×C4—38no—
410C22×C4—31no—
411C22×C4—34no—
412C22×C4—34no—
413C22×C4—34no—
414C22×C4—68no—
415C22×C4—68no—
416C22×C4—68no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—68no—
420C22×C4—68no—
421C22×C4—616no—
422C22×C4—68no—
423C22×C4—68no—
424C22×C4—68no—
425C22×C4—68no—
426C22×C4—68no—
427C22×C4—68no—
428C22×C4—68no—
429C22×C4—68no—
430C22×C4—1264no—
431C22×C4C23×D314yes—
432C22×C4C23×D314yes—
433C22×C4C23×D311yescharacteristic
434C22×C4C23×D312yes—
435C4⋊C4—24no—
436C4⋊C4—28no—
437C4⋊C4—42no—
438C4⋊C4—68no—
439C4⋊C4—616no—
440C4⋊C4—64no—
441C4⋊C4—68no—
442C4⋊C4—124no—
443C4⋊C4—122no—
444C22⋊C4—216no—
445C22⋊C4—28no—
446C22⋊C4—68no—
447C22⋊C4—616no—
448C22⋊C4—616no—
449C22⋊C4—68no—
450C22⋊C4—616no—
451C22⋊C4—68no—
452C42—34no—
453C42—38no—
454C42D3×D414yescomplemented
455C22×C3—28no—
456C22×C3—28no—
457C22×C3—28no—
458C22×C3—44no—
459C22×C3—44no—
460C22×C3—44no—
461C22×C3—44no—
462C22×C3C23×D411yescharacteristic
463C22×C3C23×D412yes—
464C22×C3C22×#4112yes—
465C22×C3C2×H(5,𝔽2)11yescharacteristic
466C22×C3C2×2−1+411yescharacteristic
467C2×D3—28no—
468C2×D3—28no—
469C2×D3—28no—
470C2×D3—44no—
471C2×D3—44no—
472C2×D3—44no—
473C2×D3—44no—
474C2×D3—464no—
475C2×D3C2×#12118yes—
476C2×D3#53714yes—
477C2×D3D4214yes—
478C2×D3#54718yes—
479C2×D3#55514yes—
480C3×C4—22no—
481C3×C4—44no—
482C3×C4—44no—
483C3×C4C2×#12118yescomplemented
484Dic3—22no—
485Dic3—44no—
486Dic3—44no—
487Dic3C2×#12118yescomplemented
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—32no—
495C23—34no—
496C23—34no—
497C23—32no—
498C23—32no—
499C23—38no—
500C23—632no—
501C23—632no—
502C23—632no—
503C23—64no—
504C23—64no—
505C23—64no—
506C23—64no—
507C23—64no—
508C23—64no—
509C23—1216no—
510C23—1216no—
511C23—1216no—
512C23—1216no—
513C23C2×D3×D418yes—
514C23C24×D311yescharacteristic
515D4—216no—
516D4—416no—
517D4—632no—
518D4—616no—
519D4—1216no—
520D4—1232no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—34no—
529C2×C4—34no—
530C2×C4—34no—
531C2×C4—332no—
532C2×C4—32no—
533C2×C4—416no—
534C2×C4—68no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—64no—
538C2×C4—64no—
539C2×C4—68no—
540C2×C4—64no—
541C2×C4—64no—
542C2×C4—1216no—
543C2×C4—1216no—
544C2×C4—1216no—
545C2×C4—1216no—
546C2×C4—1216no—
547C2×C4C2×D3×D412yes—
548C2×C4C2×D3×D414yes—
549C2×C4C2×D3×D414yes—
550C2×C4D3×#4114yes—
551C2×C3—48no—
552C2×C3C22×#12112yes—
553C2×C3C2×#53711yescharacteristic
554C2×C3C2×D4211yescharacteristic
555C2×C3C2×#54712yes—
556C2×C3C2×#55511yescharacteristic
557D3—48no—
558D3#253718yesdirect product factor
559C22—28no—
560C22—28no—
561C22—28no—
562C22—34no—
563C22—34no—
564C22—38no—
565C22—38no—
566C22—34no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—68no—
572C22—68no—
573C22—68no—
574C22—1264no—
575C22—124no—
576C22—124no—
577C22—124no—
578C22—124no—
579C22C22×D3×D412yes—
580C22C22×D3×D411yescharacteristic
581C22C2×D3×#4112yes—
582C22D3×H(5,𝔽2)11yescharacteristic
583C22D3×2−1+411yescharacteristic
584C4—22no—
585C4—38no—
586C4—44no—
587C4—44no—
588C4—62no—
589C4—124no—
590C4—124no—
591C4D3×#12118yescomplemented
592C3C2×#253711yescharacteristic, complemented, Sylow
593C2—38no—
594C2—48no—
595C2—128no—
596C2C2×D3×#12112yes—
597C2D3×#53711yescharacteristic
598C2D3×D4211yescharacteristic
599C2D3×#54712yes—
600C2D3×#55511yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2537C2
hypercenter#2537D3
hypocenterC3C2×#2537
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2537

Complements

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

NH ≅ G/Ntype
1. C2×D3×#121C2semidirect
2. C2×C4×D3×D4C2semidirect
3. C3×#2537C2semidirect
4. #82667C2semidirect
5. #82417C2semidirect
6. D3×#394C2semidirect
7. #76944C2semidirect
8. #74368C2semidirect
9. D3×#384C2semidirect
10. D3×#380C2semidirect
11. #73935C2semidirect
16. C2×C3×#121C22semidirect
17. C2×C3×C4×D4C22semidirect
18. C2×#939C22semidirect
19. C2×#938C22semidirect
20. C2×D4×Dic3C22semidirect
21. C2×#930C22semidirect
22. C2×C4×#67C22semidirect
25. C2×#895C22semidirect
26. C2×#893C22semidirect
30. C2×#884C22semidirect
31. C2×#67⋊C4C22semidirect
34. C2×C4×D12C22semidirect
36. C3×#394C22semidirect
37. C3×#384C22semidirect
38. C3×#380C22semidirect
39. #5222C22semidirect
40. #5212C22semidirect
41. #4991C22semidirect
42. #4987C22semidirect
43. #4987C22semidirect
44. #4975C22semidirect
45. #4975C22semidirect
46. #4961C22semidirect
47. #4959C22semidirect
48. #4942C22semidirect
49. #4935C22semidirect
50. #4672C22semidirect
51. #4667C22semidirect
60. #2537D3direct (N × H)
75. C2×D3×D4D4semidirect
124. C42×D3D4semidirect
137. C2×#121C2×D3semidirect
144. C2×C4×D4C2×D3semidirect
156. #394C2×D3semidirect
160. #384C2×D3semidirect
166. #380C2×D3semidirect
182. C2×C3×D4C2×D4semidirect
195. C2×#67C2×D4semidirect
209. C2×D12C2×D4semidirect
233. C3×C42C2×D4semidirect
246. C4×Dic3C2×D4semidirect
247. C4×Dic3C2×D4semidirect
374. C4×D3#121semidirect
402. C2×D4D3×D4semidirect
454. C42D3×D4semidirect
483. C3×C4C2×#121semidirect
487. Dic3C2×#121semidirect
558. D3#2537direct (N × H)
591. C4D3×#121semidirect
592. C3C2×#2537semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#1212
C2×C4×D3×D42
C3×#25372
#826672
#824172
D3×#3942
#769442
#743682
D3×#3842
D3×#3802
#739352
C2×#25373