Finite Group Databasebeta

← C2×#76745 #11001703

Subgroups — C2×#76745 #11001703

Order 768.

3342 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#936C211yesmaximal, characteristic, complemented
2C22×#885C211yesmaximal, characteristic, complemented
3C2×C3×#392C211yesmaximal, characteristic, complemented
4C2×#5223C212yesmaximal, complemented
5C2×#5223C211yesmaximal, characteristic, complemented
6C2×#5222C212yesmaximal, complemented
7C2×#4959C212yesmaximal, complemented
8C2×#4947C211yesmaximal, characteristic, complemented
9C2×#4947C212yesmaximal, complemented
10#76745C2116yesmaximal, direct product factor
11C2×#4673C211yesmaximal, characteristic, complemented
12C2×#4650C211yesmaximal, characteristic, complemented
13C2×#2819—31nomaximal, Sylow
14C23×C3×D4C2211yescharacteristic
15C23×C3×D4C2212yes—
16C23×#67C2211yescharacteristic
17C23×#67C2212yes—
18C23×Dic6C2211yescharacteristic
19C23×Dic6C2211yescharacteristic
20C22×C3×C22⋊C4C2211yescharacteristic
21C22×C3×C22⋊C4C2212yes—
22C22×#217C2211yescharacteristic
23C22×#217C2212yes—
24C22×#217C2212yes—
25C22×#217C2212yes—
26C22×#217C2212yes—
27C2×#936—28no—
28C2×#936C2218yescomplemented
29C22×(C2×D3)⋊C4C2211yescharacteristic
30C22×(C2×D3)⋊C4C2211yescharacteristic
31C22×(C2×D3)⋊C4C2212yes—
32C22×Dic3⋊C4C2212yes—
33C22×C4×Dic3C2211yescharacteristic
34C2×#885—28no—
35C2×#885C2218yescomplemented
36C3×#392C2218yescomplemented
37C2×C3×#95C2211yescharacteristic
38#5223C2218yescomplemented
39#5223C22116yescomplemented
40#5222C22116yescomplemented
41C2×C3⋊#95C2212yes—
42C2×C3⋊#95C2211yescharacteristic
43C2×C3⋊#95C2212yes—
44C2×C3⋊#95C2212yes—
45#4959C22116yescomplemented
46#4947C22116yescomplemented
47#4947C2218yescomplemented
48#4673C2218yescomplemented
49#4650C2218yescomplemented
50C22×#124—31no—
51C22×#124—31no—
52#2819—316no—
53C2×#394—32no—
54C2×#394—32no—
55C2×#393—31no—
56C2×#393—32no—
57C2×#393—31no—
58C2×#392—32no—
59C2×#392—31no—
60C2×#392D311yescharacteristic, complemented
61C2×#382—31no—
62C25×C3C2312yes—
63C25×C3C2311yescharacteristic
64C24×D3C2311yescharacteristic
65C22×C3×D4—24no—
66C22×C3×D4—44no—
67C22×C3×D4—44no—
68C22×C3×D4—44no—
69C22×C3×D4—44no—
70C22×C3×D4C2314yes—
71C22×C3×D4C2318yes—
72C23×C3×C4C2311yescharacteristic
73C23×C3×C4C2312yes—
74C23×C3×C4C2311yescharacteristic
75C22×#67—24no—
76C22×#67—44no—
77C22×#67—44no—
78C22×#67—44no—
79C22×#67—44no—
80C22×#67C2314yes—
81C22×#67C2318yes—
82C23×Dic3C2311yescharacteristic
83C23×Dic3C2312yes—
84C23×Dic3C2312yes—
85C23×Dic3C2311yescharacteristic
86C23×Dic3C2311yescharacteristic
87C22×Dic6—24no—
88C22×Dic6—24no—
89C22×Dic6—44no—
90C22×Dic6—44no—
91C22×Dic6C2314yes—
92C22×Dic6C2314yes—
93C2×C3×C22⋊C4—24no—
94C2×C3×C22⋊C4—44no—
95C2×C3×C22⋊C4C2314yes—
96C2×C3×C22⋊C4C2318yes—
97C2×#217—24no—
98C2×#217—28no—
99C2×#217—28no—
100C2×#217—44no—
101C2×#217—44no—
102C2×#217C2318yes—
103C2×#217C2318yes—
104C2×#217C2314yes—
105C2×#217C2318yes—
106C2×#217C2318yes—
107#936—232no—
108C2×(C2×D3)⋊C4—24no—
109C2×(C2×D3)⋊C4—24no—
110C2×(C2×D3)⋊C4—44no—
111C2×(C2×D3)⋊C4C2314yes—
112C2×(C2×D3)⋊C4C2314yes—
113C2×(C2×D3)⋊C4C2318yes—
114C2×Dic3⋊C4—44no—
115C2×Dic3⋊C4C2318yes—
116C2×C4×Dic3D418yescomplemented
117C2×C4×Dic3C2314yes—
118#885—232no—
119C3×#95C2314yes—
120C3⋊#95C2314yes—
121C3⋊#95C2318yes—
122C3⋊#95C2318yes—
123C3⋊#95C2318yes—
124C23×Q8—31no—
125C23×Q8—31no—
126C23×D4—31no—
127C23×D4—32no—
128C23×D4C2×D311yescharacteristic
129C23×D4C2×D312yes—
130C2×#124—38no—
131C2×#124—38no—
132C2×#124—68no—
133C2×#124—68no—
134C22×C4⋊C4—32no—
135C22×C22⋊C4—32no—
136C22×C22⋊C4—31no—
137C22×C22⋊C4—32no—
138C22×C22⋊C4—31no—
139C22×C22⋊C4—31no—
140C22×C22⋊C4—32no—
141C22×C22⋊C4—32no—
142C22×C22⋊C4—32no—
143C22×C22⋊C4C2×D311yescharacteristic
144C22×C22⋊C4C2×D312yes—
145C22×C42—31no—
146#394—316no—
147#394—316no—
148#393—316no—
149#393—38no—
150#393—38no—
151#392—316no—
152#392—38no—
153#392C2×D318yescomplemented
154#382—38no—
155C2×#95—32no—
156C2×#95—31no—
157C2×#95—32no—
158C2×#95—32no—
159C2×#95C2×D311yescharacteristic
160C24×C3—24no—
161C24×C3—22no—
162C24×C3—24no—
163C24×C3—24no—
164C24×C3—24no—
165C24×C3—24no—
166C24×C3—24no—
167C24×C3—24no—
168C24×C3—24no—
169C24×C3—22no—
170C24×C3—24no—
171C24×C3—22no—
172C24×C3C2414yes—
173C24×C3C2412yes—
174C24×C3C2411yescharacteristic
175C23×D3—22no—
176C23×D3—24no—
177C23×D3—24no—
178C23×D3—22no—
179C23×D3—22no—
180C23×D3C2412yes—
181C2×C3×D4—28no—
182C2×C3×D4—48no—
183C2×C3×D4—48no—
184C2×C3×D4—48no—
185C2×C3×D4—48no—
186C2×C3×D4—416no—
187C2×C3×D4—432no—
188C22×C3×C4—24no—
189C22×C3×C4—22no—
190C22×C3×C4—24no—
191C22×C3×C4—24no—
192C22×C3×C4—24no—
193C22×C3×C4—24no—
194C22×C3×C4C2×D414yes—
195C22×C3×C4C2414yes—
196C22×C3×C4C2412yes—
197C22×C3×C4C2412yes—
198C2×#67—28no—
199C2×#67—48no—
200C2×#67—48no—
201C2×#67—48no—
202C2×#67—48no—
203C2×#67—416no—
204C2×#67—432no—
205C22×Dic3—22no—
206C22×Dic3—24no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—24no—
210C22×Dic3—24no—
211C22×Dic3—24no—
212C22×Dic3—24no—
213C22×Dic3—24no—
214C22×Dic3—24no—
215C22×Dic3—24no—
216C22×Dic3C2×D414yes—
217C22×Dic3C2×D414yes—
218C22×Dic3C2414yes—
219C22×Dic3C2412yes—
220C22×Dic3C2414yes—
221C22×Dic3C2412yes—
222C22×Dic3C2412yes—
223C2×Dic6—28no—
224C2×Dic6—28no—
225C2×Dic6—48no—
226C2×Dic6—48no—
227C2×Dic6—416no—
228C2×Dic6—416no—
229C3×C22⋊C4—28no—
230C3×C22⋊C4—48no—
231#217—28no—
232#217—216no—
233#217—216no—
234#217—48no—
235#217—48no—
236(C2×D3)⋊C4—28no—
237(C2×D3)⋊C4—28no—
238(C2×D3)⋊C4—48no—
239Dic3⋊C4—48no—
240C4×Dic3C2×D4116yescomplemented
241C25—31no—
242C25C22×D311yescharacteristic
243C25C22×D312yes—
244C22×Q8—34no—
245C22×Q8—34no—
246C22×Q8—64no—
247C22×Q8—64no—
248C22×Q8—124no—
249C22×Q8—124no—
250C22×D4—24no—
251C22×D4—34no—
252C22×D4—38no—
253C22×D4—44no—
254C22×D4—44no—
255C22×D4—44no—
256C22×D4—44no—
257C22×D4—64no—
258C22×D4—124no—
259C22×D4—124no—
260C22×D4—124no—
261C22×D4—124no—
262C22×D4C22×D314yes—
263C22×D4C22×D318yes—
264C23×C4—31no—
265C23×C4—31no—
266C23×C4—32no—
267C23×C4—31no—
268C23×C4—32no—
269C23×C4C22×D311yescharacteristic
270C23×C4C22×D312yes—
271C23×C4C22×D311yescharacteristic
272#124—632no—
273#124—632no—
274C2×C4⋊C4—38no—
275C2×C4⋊C4—124no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—38no—
278C2×C22⋊C4—38no—
279C2×C22⋊C4—34no—
280C2×C22⋊C4—34no—
281C2×C22⋊C4—34no—
282C2×C22⋊C4—38no—
283C2×C22⋊C4—38no—
284C2×C22⋊C4—38no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—64no—
287C2×C22⋊C4—64no—
288C2×C22⋊C4—68no—
289C2×C22⋊C4—68no—
290C2×C22⋊C4—64no—
291C2×C22⋊C4—124no—
292C2×C22⋊C4—124no—
293C2×C22⋊C4—124no—
294C2×C22⋊C4C22×D314yes—
295C2×C22⋊C4C22×D318yes—
296C2×C42—38no—
297C2×C42—34no—
298#95—38no—
299#95—34no—
300#95—38no—
301#95—38no—
302#95C22×D314yes—
303C23×C3—24no—
304C23×C3—22no—
305C23×C3—24no—
306C23×C3—24no—
307C23×C3—24no—
308C23×C3—24no—
309C23×C3—24no—
310C23×C3—24no—
311C23×C3—24no—
312C23×C3—22no—
313C23×C3—24no—
314C23×C3—22no—
315C23×C3—44no—
316C23×C3—48no—
317C23×C3—48no—
318C23×C3—44no—
319C23×C3—48no—
320C23×C3—44no—
321C23×C3—48no—
322C23×C3—48no—
323C23×C3—48no—
324C23×C3—48no—
325C23×C3—48no—
326C23×C3—48no—
327C23×C3C22×D412yes—
328C23×C3H(5,𝔽2)14yes—
329C23×C3H(5,𝔽2)14yes—
330C23×C3H(5,𝔽2)12yes—
331C23×C32−1+412yes—
332C23×C3C2511yescharacteristic
333C22×D3—22no—
334C22×D3—24no—
335C22×D3—24no—
336C22×D3—22no—
337C22×D3—22no—
338C22×D3—44no—
339C22×D3—44no—
340C22×D3—44no—
341C22×D3—48no—
342C22×D3—48no—
343C3×D4—416no—
344C3×D4—432no—
345C2×C3×C4—28no—
346C2×C3×C4—24no—
347C2×C3×C4—22no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—44no—
353C2×C3×C4—48no—
354C2×C3×C4C22×D414yes—
355#67—416no—
356#67—432no—
357C2×Dic3—28no—
358C2×Dic3—28no—
359C2×Dic3—22no—
360C2×Dic3—24no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—24no—
364C2×Dic3—24no—
365C2×Dic3—24no—
366C2×Dic3—24no—
367C2×Dic3—24no—
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—44no—
371C2×Dic3—48no—
372C2×Dic3—48no—
373C2×Dic3C22×D414yes—
374C2×Dic3C22×D414yes—
375Dic6—416no—
376Dic6—416no—
377C24—22no—
378C24—24no—
379C24—24no—
380C24—24no—
381C24—24no—
382C24—24no—
383C24—22no—
384C24—24no—
385C24—24no—
386C24—24no—
387C24—22no—
388C24—24no—
389C24—32no—
390C24—62no—
391C24—64no—
392C24—64no—
393C24—62no—
394C24—62no—
395C24C23×D312yes—
396C24C23×D314yes—
397C24C23×D311yescharacteristic
398C2×Q8—68no—
399C2×Q8—68no—
400C2×Q8—128no—
401C2×Q8—128no—
402C2×Q8—1216no—
403C2×Q8—1216no—
404C2×D4—28no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—416no—
410C2×D4—432no—
411C2×D4—68no—
412C2×D4—128no—
413C2×D4—128no—
414C2×D4—128no—
415C2×D4—128no—
416C2×D4—1216no—
417C2×D4—1232no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—34no—
425C22×C4—34no—
426C22×C4—32no—
427C22×C4—32no—
428C22×C4—34no—
429C22×C4—34no—
430C22×C4—32no—
431C22×C4—62no—
432C22×C4—64no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—64no—
436C22×C4—64no—
437C22×C4—64no—
438C22×C4—64no—
439C22×C4—64no—
440C22×C4—64no—
441C22×C4—64no—
442C22×C4D3×D414yes—
443C22×C4C23×D314yes—
444C22×C4C23×D312yes—
445C22×C4C23×D312yes—
446C4⋊C4—128no—
447C22⋊C4—28no—
448C22⋊C4—48no—
449C22⋊C4—68no—
450C22⋊C4—68no—
451C22⋊C4—616no—
452C22⋊C4—616no—
453C22⋊C4—68no—
454C22⋊C4—128no—
455C22⋊C4—128no—
456C22⋊C4—128no—
457C42—316no—
458C22×C3—42no—
459C22×C3—44no—
460C22×C3—42no—
461C22×C3—44no—
462C22×C3—44no—
463C22×C3—42no—
464C22×C3—44no—
465C22×C3—44no—
466C22×C3—44no—
467C22×C3—44no—
468C22×C3—44no—
469C22×C3—44no—
470C22×C3—88no—
471C22×C3—816no—
472C22×C3#53514yes—
473C22×C3#53514yes—
474C22×C3#53714yes—
475C22×C3H(3,𝔽4)18yes—
476C22×C3#56218yes—
477C22×C3C23×D411yescharacteristic
478C22×C3C2×H(5,𝔽2)12yes—
479C22×C3C2×H(5,𝔽2)12yes—
480C22×C3C2×H(5,𝔽2)11yescharacteristic
481C22×C3C2×2−1+411yescharacteristic
482C2×D3—42no—
483C2×D3—42no—
484C2×D3—42no—
485C2×D3—44no—
486C2×D3—44no—
487C2×D3—88no—
488C3×C4—24no—
489C3×C4—42no—
490C3×C4—44no—
491Dic3—24no—
492Dic3—24no—
493Dic3—44no—
494Dic3—42no—
495Dic3—44no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—22no—
507C23—24no—
508C23—44no—
509C23—48no—
510C23—48no—
511C23—44no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—44no—
516C23—48no—
517C23—48no—
518C23—48no—
519C23—48no—
520C23—62no—
521C23—64no—
522C23—62no—
523C23—64no—
524C23—62no—
525C23—124no—
526C23—124no—
527C23—128no—
528C23—124no—
529C23—128no—
530C23C2×D3×D412yes—
531C23#100414yes—
532C23#100414yes—
533C23#100412yes—
534C23#101012yes—
535C23C24×D311yescharacteristic
536Q8—1216no—
537Q8—1216no—
538D4—416no—
539D4—432no—
540D4—1216no—
541D4—1232no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—34no—
550C2×C4—34no—
551C2×C4—44no—
552C2×C4—48no—
553C2×C4—62no—
554C2×C4—64no—
555C2×C4—62no—
556C2×C4—68no—
557C2×C4—68no—
558C2×C4—64no—
559C2×C4—62no—
560C2×C4—64no—
561C2×C4—64no—
562C2×C4—64no—
563C2×C4—64no—
564C2×C4—64no—
565C2×C4—64no—
566C2×C4—124no—
567C2×C4—128no—
568C2×C4—128no—
569C2×C4C2×D3×D414yes—
570C2×C3—82no—
571C2×C3—84no—
572C2×C3#281918yescomplemented
573C2×C3C2×#53511yescharacteristic
574C2×C3C2×#53511yescharacteristic
575C2×C3C2×#53711yescharacteristic
576C2×C3C2×H(3,𝔽4)12yes—
577C2×C3C2×#56212yes—
578D3—82no—
579C22—44no—
580C22—42no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—42no—
585C22—44no—
586C22—42no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—88no—
592C22—816no—
593C22—122no—
594C22—124no—
595C22—122no—
596C22—122no—
597C22—124no—
598C22—248no—
599C22#559214yes—
600C22#559418yes—
601C22#560814yes—
602C22#562118yes—
603C22#571514yes—
604C22C22×D3×D411yescharacteristic
605C22C2×#100412yes—
606C22C2×#100411yescharacteristic
607C22C2×#100412yes—
608C22C2×#101011yescharacteristic
609C4—24no—
610C4—44no—
611C4—42no—
612C4—64no—
613C4—64no—
614C4—124no—
615C4—124no—
616C4—122no—
617C3C2×#281911yescharacteristic, complemented, Sylow
618C2—84no—
619C2—82no—
620C2—242no—
621C2#7674518yesdirect product factor
622C2C2×#559211yescharacteristic
623C2C2×#559412yes—
624C2C2×#560811yescharacteristic
625C2C2×#562112yes—
626C2C2×#571511yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC2×C3×#392C2
socleC24×C3C24
hypercenterC2×#392D3
hypocenterC3C2×#2819
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#392

Complements

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

NH ≅ G/Ntype
1. C22×#936C2semidirect
2. C22×#885C2semidirect
3. C2×C3×#392C2semidirect
4. C2×#5223C2semidirect
5. C2×#5223C2semidirect
6. C2×#5222C2semidirect
7. C2×#4959C2semidirect
8. C2×#4947C2semidirect
9. C2×#4947C2semidirect
10. #76745C2direct (N × H)
11. C2×#4673C2semidirect
12. C2×#4650C2semidirect
28. C2×#936C22semidirect
35. C2×#885C22semidirect
36. C3×#392C22semidirect
38. #5223C22semidirect
39. #5223C22semidirect
40. #5222C22semidirect
45. #4959C22semidirect
46. #4947C22semidirect
47. #4947C22semidirect
48. #4673C22semidirect
49. #4650C22semidirect
60. C2×#392D3semidirect
116. C2×C4×Dic3D4semidirect
153. #392C2×D3semidirect
240. C4×Dic3C2×D4semidirect
572. C2×C3#2819semidirect
617. C3C2×#2819semidirect
621. C2#76745direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9362
C22×#8852
C2×C3×#3922
C2×#52232
C2×#52222
C2×#49592
C2×#49472
#767452
C2×#46732
C2×#46502
C2×#28193