Finite Group Databasebeta

← C2×#83811 #11659497

Subgroups — C2×#83811 #11659497

Order 768.

2674 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#568C211yesmaximal, characteristic, complemented
2C2×#5822C211yesmaximal, characteristic
3C2×#5745C211yesmaximal, characteristic, complemented
4C22×#837C211yesmaximal, characteristic, complemented
5#83811C2116yesmaximal, direct product factor
6C2×#5145C214yesmaximal, complemented
7C2×#5144C214yesmaximal, complemented
8C22×#825C211yesmaximal, characteristic, complemented
9C2×#5124C211yesmaximal, characteristic, complemented
10C2×Dic3×M16C211yesmaximal, characteristic, complemented
11C2×#2956—31nomaximal, Sylow
12C22×C3×#41C2211yescharacteristic
13C22×#235C2211yescharacteristic
14C3×#568—24no—
15C3×#568C2218yescomplemented
16C2×C3×#131—22no—
17C2×C3×#131C2214yescomplemented
18C22×C3×M16—22no—
19C22×C3×M16C2211yescharacteristic
20C22×C3×M16C2211yescharacteristic
21C2×#951C412yescomplemented
22#5822C418yescomplemented
23#5822C2218yes—
24C2×#950C2214yes—
25C2×#949C412yescomplemented
26C2×#942C412yescomplemented
27C2×#932C412yescomplemented
28C22×C4×Dic3C2211yescharacteristic
29C22×#208C2211yescharacteristic
30C22×#208C2211yescharacteristic
31#5745—24no—
32#5745C2218yescomplemented
33C2×#907—22no—
34C2×D3×M16—22no—
35C2×#901C2214yescomplemented
36C2×#837C2212yescomplemented
37C2×#837C2212yescomplemented
38C2×#837C2218yescomplemented
39#5145C22132yescomplemented
40#5144C22132yescomplemented
41C2×#825C2212yescomplemented
42C2×#825C2212yescomplemented
43C2×#825C2218yescomplemented
44#5124C2218yescomplemented
45C2×#822C2214yescomplemented
46Dic3×M16C2218yescomplemented
47C2×#815C2212yescomplemented
48C2×C8×Dic3C2212yescomplemented
49C2×#575—31no—
50C2×#568—31no—
51C2×#568D311yescharacteristic, complemented
52#2956—316no—
53C2×#444—34no—
54C2×#443—34no—
55C2×#430—31no—
56C22×#104—31no—
57C22×#104—31no—
58C2×C4×M16—31no—
59C2×C3×#41—21no—
60C2×C3×#41—22no—
61C2×C3×#41—21no—
62C2×C3×#41—22no—
63C2×C3×#41C2×C412yes—
64C2×C3×#41C2×C414yes—
65C2×C3×#41C2312yes—
66C2×C3×#41C2314yes—
67C22×C3×Q8C2×C411yescharacteristic
68C22×C3×D4—21no—
69C22×C3×D4C2×C411yescharacteristic
70C23×C3×C4—21no—
71C23×C3×C4C2311yescharacteristic
72C22×#67—21no—
73C23×Dic3—21no—
74C2×#235—21no—
75C2×#235—22no—
76C2×#235—21no—
77C2×#235—22no—
78C2×#235C2×C412yes—
79C2×#235C2×C414yes—
80C2×#235C2312yes—
81C2×#235C2314yes—
82C22×D12C2×C411yescharacteristic
83C22×C4×D3—21no—
84C22×Dic6C2×C411yescharacteristic
85C3×#131—216no—
86C3×#131—28no—
87C3×#131C23116yescomplemented
88C2×C3×M16—21no—
89C2×C3×M16—22no—
90C2×C3×M16—216no—
91C2×C3×M16—24no—
92C2×C3×M16D412yes—
93C2×C3×M16D412yes—
94C2×C3×M16D414yes—
95C2×C3×M16C2312yes—
96C2×C3×M16C2314yes—
97C2×C3×M16C2314yes—
98C22×C3×C8—21no—
99C22×C3×C8—24no—
100C22×C3×C8C2312yes—
101#951C2×C4116yescomplemented
102#950C2×C4116yescomplemented
103#950C23116yes—
104#949C2×C4116yescomplemented
105C2×#216C2×C414yescomplemented
106#942C2×C4116yescomplemented
107C2×#215C2×C414yescomplemented
108C2×#214C2×C414yescomplemented
109#932C2×C4116yescomplemented
110C2×#213C2×C414yescomplemented
111C2×C4×Dic3—21no—
112C2×C4×Dic3—22no—
113C2×C4×Dic3C2311yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C2×C4×Dic3C2314yes—
116C2×#208—21no—
117C2×#208—22no—
118C2×#208C2×C412yes—
119C2×#208C2×C412yes—
120C2×#208C2×C414yes—
121C2×#208C2312yes—
122C2×#208C2314yes—
123C2×#208C2314yes—
124C22×C3⋊C8—21no—
125C22×C3⋊C8C2312yes—
126#907—216no—
127D3×M16—216no—
128#901—28no—
129#901C23116yescomplemented
130C2×ΓL(1,𝔽25)—24no—
131C2×D3×C8—24no—
132#837C2318yescomplemented
133#837C2318yescomplemented
134#825C2318yescomplemented
135#825C2318yescomplemented
136#822C23116yescomplemented
137#815C2318yescomplemented
138C8×Dic3C2318yescomplemented
139C22×#41—31no—
140C22×#41C2×D311yescharacteristic
141#575—38no—
142#575—38no—
143C2×#137—32no—
144C2×#137—32no—
145C2×AGL(1,ℤ/8)—32no—
146C2×AGL(1,ℤ/8)—32no—
147C2×#135—34no—
148#568—24no—
149#568—38no—
150#568—64no—
151#568C2×D318yescomplemented
152C2×#131—22no—
153C2×#131—34no—
154C2×#131—62no—
155C2×#131C2×D314yescomplemented
156C22×M16—22no—
157C22×M16—31no—
158C22×M16—31no—
159C22×M16—62no—
160C22×M16C2×D311yescharacteristic
161C22×M16C2×D311yescharacteristic
162C22×C42—31no—
163#444—332no—
164#443—332no—
165#430—38no—
166C2×#108—34no—
167C2×#104—32no—
168C2×#104—32no—
169C2×#104—38no—
170C2×#104—32no—
171C2×#104—32no—
172C2×#104—38no—
173C4×M16—38no—
174C2×#97—32no—
175C2×C4×C8—32no—
176C24×C3—21no—
177C24×C3#4111yescharacteristic
178C23×D3—21no—
179C3×#41—24no—
180C3×#41—24no—
181C3×#41—24no—
182C3×#41—24no—
183C3×#41—22no—
184C3×#41—22no—
185C3×#41—24no—
186C3×#41C22×C418yes—
187C3×#41C22×C414yes—
188C3×#41C2414yes—
189C2×C3×Q8—21no—
190C2×C3×Q8—22no—
191C2×C3×Q8C22×C412yes—
192C2×C3×Q8C22×C414yes—
193C2×C3×D4—22no—
194C2×C3×D4—24no—
195C2×C3×D4—22no—
196C2×C3×D4—24no—
197C2×C3×D4—21no—
198C2×C3×D4—22no—
199C2×C3×D4C22×C412yes—
200C2×C3×D4C22×C414yes—
201C22×C3×C4—21no—
202C22×C3×C4—22no—
203C22×C3×C4—24no—
204C22×C3×C4—22no—
205C22×C3×C4—24no—
206C22×C3×C4C22×C411yescharacteristic
207C22×C3×C4C22×C411yescharacteristic
208C22×C3×C4C22×C412yes—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D412yes—
212C22×C3×C4#4111yescharacteristic
213C22×C3×C4#4112yes—
214C22×C3×C4C2411yescharacteristic
215C22×C3×C4C2412yes—
216C2×#67—22no—
217C2×#67—24no—
218C2×#67—22no—
219C2×#67—24no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—24no—
223C22×Dic3—21no—
224C22×Dic3—44no—
225#235—24no—
226#235—24no—
227#235—24no—
228#235—22no—
229#235—24no—
230#235—22no—
231#235—24no—
232#235C22×C418yes—
233#235C22×C414yes—
234#235C2414yes—
235C2×D12—21no—
236C2×D12—22no—
237C2×D12C22×C412yes—
238C2×D12C22×C414yes—
239C2×C4×D3—22no—
240C2×C4×D3—24no—
241C2×C4×D3—22no—
242C2×C4×D3—24no—
243C2×Dic6—21no—
244C2×Dic6—22no—
245C2×Dic6C22×C412yes—
246C2×Dic6C22×C414yes—
247C3×M16—24no—
248C3×M16—22no—
249C3×M16—216no—
250C3×M16C2×D418yes—
251C3×M16C2×D418yes—
252C3×M16C2414yes—
253C2×C3×C8—22no—
254C2×C3×C8—24no—
255C2×C3×C8—216no—
256C2×C3×C8C2×D414yes—
257C2×C3×C8C2×D414yes—
258C2×C3×C8C2414yes—
259#216C22×C4116yescomplemented
260#215C22×C4116yescomplemented
261#214C22×C4116yescomplemented
262#213C22×C4116yescomplemented
263C4×Dic3—21no—
264C4×Dic3—24no—
265C4×Dic3—21no—
266C4×Dic3C2412yes—
267C4×Dic3C2412yes—
268#208—22no—
269#208—24no—
270#208C22×C418yes—
271#208C22×C418yes—
272#208C2414yes—
273C2×C3⋊C8—22no—
274C2×C3⋊C8—24no—
275C2×C3⋊C8C22×C414yes—
276C2×C3⋊C8C22×C414yes—
277C2×C3⋊C8C2414yes—
278ΓL(1,𝔽25)—216no—
279D3×C8—216no—
280C2×#41—21no—
281C2×#41—22no—
282C2×#41—21no—
283C2×#41—22no—
284C2×#41—32no—
285C2×#41—34no—
286C2×#41—32no—
287C2×#41—34no—
288C2×#41—61no—
289C2×#41—62no—
290C2×#41—61no—
291C2×#41—62no—
292C2×#41C4×D312yes—
293C2×#41C4×D314yes—
294C2×#41C22×D312yes—
295C2×#41C22×D314yes—
296C22×Q8—31no—
297C22×Q8C4×D311yescharacteristic
298C22×D4—21no—
299C22×D4—31no—
300C22×D4—61no—
301C22×D4C4×D311yescharacteristic
302C23×C4—21no—
303C23×C4—61no—
304C23×C4—61no—
305C23×C4C22×D311yescharacteristic
306#137—316no—
307#137—316no—
308AGL(1,ℤ/8)—316no—
309AGL(1,ℤ/8)—316no—
310#135—316no—
311#135—316no—
312C2×Q16—34no—
313C2×SD16—34no—
314C2×SD16—34no—
315C2×D8—34no—
316#131—216no—
317#131—28no—
318#131—316no—
319#131—616no—
320#131—68no—
321#131C22×D3116yescomplemented
322C2×M16—21no—
323C2×M16—22no—
324C2×M16—216no—
325C2×M16—24no—
326C2×M16—32no—
327C2×M16—32no—
328C2×M16—34no—
329C2×M16—34no—
330C2×M16—32no—
331C2×M16—34no—
332C2×M16—61no—
333C2×M16—62no—
334C2×M16—616no—
335C2×M16—64no—
336C2×M16#6712yes—
337C2×M16#6712yes—
338C2×M16#6714yes—
339C2×M16C22×D312yes—
340C2×M16C22×D314yes—
341C2×M16C22×D314yes—
342C22×C8—21no—
343C22×C8—24no—
344C22×C8—32no—
345C22×C8—61no—
346C22×C8—64no—
347C22×C8C22×D312yes—
348C2×C42—31no—
349C2×C42—31no—
350C2×C42—34no—
351C2×C42—61no—
352C2×C42—62no—
353#108—316no—
354#104—38no—
355#104—38no—
356#104—38no—
357#104—38no—
358#97—38no—
359C4×C8—38no—
360C23×C3—22no—
361C23×C3—24no—
362C23×C3—42no—
363C23×C3—82no—
364C23×C3C4×D411yescharacteristic
365C23×C3C4×D411yescharacteristic
366C23×C3C4×D412yes—
367C23×C3C2×#4111yescharacteristic
368C23×C3C2×#4112yes—
369C22×D3—22no—
370C22×D3—24no—
371C22×D3—82no—
372C3×Q8—22no—
373C3×Q8—24no—
374C3×Q8C23×C414yes—
375C3×D4—28no—
376C3×D4—24no—
377C3×D4—24no—
378C3×D4—22no—
379C3×D4—24no—
380C3×D4C23×C414yes—
381C2×C3×C4—24no—
382C2×C3×C4—22no—
383C2×C3×C4—28no—
384C2×C3×C4—24no—
385C2×C3×C4—24no—
386C2×C3×C4C4×D411yescharacteristic
387C2×C3×C4C4×D412yes—
388C2×C3×C4C4×D411yescharacteristic
389C2×C3×C4C4×D412yes—
390C2×C3×C4C4×D412yes—
391C2×C3×C4C4×D412yes—
392C2×C3×C4C4×D412yes—
393C2×C3×C4C23×C411yescharacteristic
394C2×C3×C4C23×C411yescharacteristic
395C2×C3×C4C23×C412yes—
396C2×C3×C4C23×C412yes—
397C2×C3×C4C22×D412yes—
398C2×C3×C4C22×D412yes—
399C2×C3×C4C22×D411yescharacteristic
400C2×C3×C4C22×D411yescharacteristic
401C2×C3×C4C2×#4112yes—
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2511yescharacteristic
404#67—28no—
405#67—24no—
406#67—24no—
407C2×Dic3—24no—
408C2×Dic3—24no—
409C2×Dic3—22no—
410C2×Dic3—22no—
411C2×Dic3—22no—
412C2×Dic3—48no—
413C2×Dic3—42no—
414D12—24no—
415D12—22no—
416D12C23×C414yes—
417C4×D3—24no—
418C4×D3—28no—
419C4×D3—24no—
420Dic6—24no—
421Dic6—22no—
422Dic6C23×C414yes—
423C3×C8—24no—
424C3×C8C22×D418yes—
425C3⋊C8—24no—
426C3⋊C8C23×C418yes—
427C24—21no—
428C24—61no—
429C24#23511yescharacteristic
430#41—24no—
431#41—24no—
432#41—24no—
433#41—22no—
434#41—24no—
435#41—24no—
436#41—22no—
437#41—34no—
438#41—38no—
439#41—34no—
440#41—64no—
441#41—64no—
442#41—64no—
443#41—62no—
444#41—64no—
445#41—64no—
446#41—62no—
447#41C2×C4×D314yes—
448#41C2×C4×D318yes—
449#41C23×D314yes—
450C2×Q8—21no—
451C2×Q8—22no—
452C2×Q8—32no—
453C2×Q8—34no—
454C2×Q8—61no—
455C2×Q8—62no—
456C2×Q8C2×C4×D312yes—
457C2×Q8C2×C4×D314yes—
458C2×D4—22no—
459C2×D4—24no—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—21no—
463C2×D4—22no—
464C2×D4—32no—
465C2×D4—34no—
466C2×D4—62no—
467C2×D4—64no—
468C2×D4—62no—
469C2×D4—64no—
470C2×D4—61no—
471C2×D4—62no—
472C2×D4C2×C4×D312yes—
473C2×D4C2×C4×D314yes—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—62no—
482C22×C4—64no—
483C22×C4—64no—
484C22×C4—62no—
485C22×C4—64no—
486C22×C4—61no—
487C22×C4—124no—
488C22×C4C2×C4×D311yescharacteristic
489C22×C4C2×C4×D311yescharacteristic
490C22×C4C2×C4×D312yes—
491C22×C4#23511yescharacteristic
492C22×C4#23512yes—
493C22×C4C2×#6712yes—
494C22×C4C2×#6711yescharacteristic
495C22×C4C2×#6711yescharacteristic
496C22×C4C23×D312yes—
497C22×C4C23×D311yescharacteristic
498Q16—316no—
499SD16—316no—
500SD16—316no—
501D8—316no—
502M16—24no—
503M16—22no—
504M16—216no—
505M16—38no—
506M16—38no—
507M16—34no—
508M16—64no—
509M16—62no—
510M16—616no—
511M16C2×#6718yes—
512M16C2×#6718yes—
513M16C23×D314yes—
514C2×C8—24no—
515C2×C8—22no—
516C2×C8—216no—
517C2×C8—34no—
518C2×C8—34no—
519C2×C8—34no—
520C2×C8—64no—
521C2×C8—62no—
522C2×C8—616no—
523C2×C8C2×#6714yes—
524C2×C8C2×#6714yes—
525C2×C8C23×D314yes—
526C42—32no—
527C42—32no—
528C42—61no—
529C42—64no—
530C42—61no—
531C22×C3—22no—
532C22×C3—22no—
533C22×C3—22no—
534C22×C3—24no—
535C22×C3—42no—
536C22×C3—42no—
537C22×C3—44no—
538C22×C3—84no—
539C22×C3C2×C4×D412yes—
540C22×C3C2×C4×D412yes—
541C22×C3C2×C4×D411yescharacteristic
542C22×C3C2×C4×D411yescharacteristic
543C22×C3C22×#4111yescharacteristic
544C2×D3—24no—
545C2×D3—44no—
546C2×D3—84no—
547C3×C4—24no—
548C3×C4C2×C4×D412yes—
549C3×C4C2×C4×D411yescharacteristic
550C3×C4C2×C4×D411yescharacteristic
551C3×C4C2×C4×D412yes—
552C3×C4C24×C411yescharacteristic
553C3×C4C23×D411yescharacteristic
554Dic3—24no—
555Dic3—24no—
556Dic3—42no—
557C23—22no—
558C23—24no—
559C23—42no—
560C23—62no—
561C23—64no—
562C23—82no—
563C23—242no—
564C23C4×#6711yescharacteristic
565C23C4×#6711yescharacteristic
566C23C4×#6712yes—
567C23C2×#23511yescharacteristic
568C23C2×#23512yes—
569Q8—24no—
570Q8—22no—
571Q8—34no—
572Q8—64no—
573Q8—62no—
574Q8C22×C4×D314yes—
575D4—24no—
576D4—28no—
577D4—24no—
578D4—24no—
579D4—22no—
580D4—34no—
581D4—64no—
582D4—68no—
583D4—64no—
584D4—64no—
585D4—62no—
586D4C22×C4×D314yes—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—64no—
593C2×C4—62no—
594C2×C4—64no—
595C2×C4—62no—
596C2×C4—64no—
597C2×C4—64no—
598C2×C4—68no—
599C2×C4—62no—
600C2×C4—128no—
601C2×C4—122no—
602C2×C4C4×#6711yescharacteristic
603C2×C4C4×#6712yes—
604C2×C4C4×#6711yescharacteristic
605C2×C4C4×#6712yes—
606C2×C4C4×#6712yes—
607C2×C4C4×#6712yes—
608C2×C4C4×#6712yes—
609C2×C4C22×C4×D311yescharacteristic
610C2×C4C22×C4×D311yescharacteristic
611C2×C4C22×C4×D312yes—
612C2×C4C22×C4×D312yes—
613C2×C4C2×#23511yescharacteristic
614C2×C4C2×#23512yes—
615C2×C4C22×#6712yes—
616C2×C4C22×#6712yes—
617C2×C4C22×#6711yescharacteristic
618C2×C4C22×#6711yescharacteristic
619C2×C4C24×D311yescharacteristic
620C8—24no—
621C8—38no—
622C8—64no—
623C8C22×#6718yes—
624C2×C3—21no—
625C2×C3—22no—
626C2×C3—22no—
627C2×C3—21no—
628C2×C3—44no—
629C2×C3#295612yescomplemented
630C2×C3C22×C4×D411yescharacteristic
631D3—44no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—24no—
636C22—42no—
637C22—42no—
638C22—44no—
639C22—64no—
640C22—84no—
641C22—124no—
642C22—244no—
643C22C2×C4×#6712yes—
644C22C2×C4×#6712yes—
645C22C2×C4×#6711yescharacteristic
646C22C2×C4×#6711yescharacteristic
647C22C22×#23511yescharacteristic
648C4—24no—
649C4—64no—
650C4—64no—
651C4—122no—
652C4C2×C4×#6711yescharacteristic
653C4C2×C4×#6712yes—
654C4C2×C4×#6712yes—
655C4C2×C4×#6711yescharacteristic
656C4C23×C4×D311yescharacteristic
657C4C23×#6711yescharacteristic
658C3C2×#295611yescharacteristic, complemented, Sylow
659C2—21no—
660C2—22no—
661C2—22no—
662C2—21no—
663C2—44no—
664C2—124no—
665C2#8381112yesdirect product factor
666C2C22×C4×#6711yescharacteristic
subgroupG/H
centerC2×C4C22×#67
commutatorC3×C4C24×C4
FrattiniC2×C4C24×D3
FittingC2×C3×#568C2
socleC22×C3C2×C4×D4
prime radicalC2×#5822C2
cosocleC2×C3×C4C25
hypercenterC2×#568D3
hypocenterC3C2×#2956
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×M16 ◁ C2×#568

Complements

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

NH ≅ G/Ntype
1. C2×C3×#568C2semidirect
3. C2×#5745C2semidirect
4. C22×#837C2semidirect
5. #83811C2direct (N × H)
6. C2×#5145C2semidirect
7. C2×#5144C2semidirect
8. C22×#825C2semidirect
9. C2×#5124C2semidirect
10. C2×Dic3×M16C2semidirect
15. C3×#568C22semidirect
17. C2×C3×#131C22semidirect
21. C2×#951C4semidirect
22. #5822C4semidirect
25. C2×#949C4semidirect
26. C2×#942C4semidirect
27. C2×#932C4semidirect
32. #5745C22semidirect
35. C2×#901C22semidirect
36. C2×#837C22semidirect
37. C2×#837C22semidirect
38. C2×#837C22semidirect
39. #5145C22semidirect
40. #5144C22semidirect
41. C2×#825C22semidirect
42. C2×#825C22semidirect
43. C2×#825C22semidirect
44. #5124C22semidirect
45. C2×#822C22semidirect
46. Dic3×M16C22semidirect
47. C2×#815C22semidirect
48. C2×C8×Dic3C22semidirect
51. C2×#568D3semidirect
87. C3×#131C23semidirect
101. #951C2×C4semidirect
102. #950C2×C4semidirect
104. #949C2×C4semidirect
105. C2×#216C2×C4semidirect
106. #942C2×C4semidirect
107. C2×#215C2×C4semidirect
108. C2×#214C2×C4semidirect
109. #932C2×C4semidirect
110. C2×#213C2×C4semidirect
129. #901C23semidirect
132. #837C23semidirect
133. #837C23semidirect
134. #825C23semidirect
135. #825C23semidirect
136. #822C23semidirect
137. #815C23semidirect
138. C8×Dic3C23semidirect
151. #568C2×D3semidirect
155. C2×#131C2×D3semidirect
259. #216C22×C4semidirect
260. #215C22×C4semidirect
261. #214C22×C4semidirect
262. #213C22×C4semidirect
321. #131C22×D3semidirect
629. C2×C3#2956semidirect
658. C3C2×#2956semidirect
665. C2#83811direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5682
C2×#58222
C2×#57452
C22×#8372
#838112
C2×#51452
C2×#51442
C22×#8252
C2×#51242
C2×Dic3×M162
C2×#29563