Finite Group Databasebeta

← C2×#83529 #11670329

Subgroups — C2×#83529 #11670329

Order 768.

2742 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#547C211yesmaximal, characteristic
2C22×#951C212yesmaximal, complemented
3C2×D4×Dic6C211yesmaximal, characteristic, complemented
4C2×#5239C212yesmaximal, complemented
5#83529C2116yesmaximal, direct product factor
6C2×#5090C211yesmaximal, characteristic, complemented
7C2×#5092C211yesmaximal, characteristic, complemented
8C2×#5050C212yesmaximal, complemented
9C2×#5052C212yesmaximal, complemented
10C2×#5019C211yesmaximal, characteristic, complemented
11C2×C4×#214C211yesmaximal, characteristic, complemented
12C2×#5012C211yesmaximal, characteristic, complemented
13C2×#3276—31nomaximal, Sylow
14C22×C3×#41C2212yes—
15C23×Dic6C2212yes—
16C3×#547—24no—
17C3×#547C2218yes—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#123—22no—
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×#121—21no—
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic
24C22×C3×C4⋊C4—21no—
25C2×#951—216no—
26C2×#951C22116yescomplemented
27C22×#216C2212yes—
28C22×#216C2212yes—
29C22×#216C2212yes—
30C2×D4×Dic3—21no—
31C22×#214C2212yes—
32C22×#214C2211yescharacteristic
33C2×#924C2212yes—
34C22×#208C2212yes—
35D4×Dic6—24no—
36D4×Dic6C2218yescomplemented
37C2×#888—21no—
38C2×#878—22no—
39C2×#869C2211yescharacteristic
40C2×C4×Dic6C2211yescharacteristic
41#5239C22116yescomplemented
42C2×#835C2212yes—
43C2×#832C2211yescharacteristic
44C2×#830C2212yes—
45#5092C2218yescomplemented
46#5090C2218yescomplemented
47#5052C22116yescomplemented
48#5050C22116yescomplemented
49#5019C2218yescomplemented
50C4×#214C2218yescomplemented
51#5012C2218yescomplemented
52C2×#810C2212yes—
53C2×#810C2211yescharacteristic
54C2×#808C2211yescharacteristic
55C2×C4⋊(C3⋊C8)C2211yescharacteristic
56C2×C4×C3⋊C8C2211yescharacteristic
57C22×#137—32no—
58C2×D4×Q8—31no—
59C2×#547D311yescharacteristic
60#3276—316no—
61C2×#494—31no—
62C2×#470—32no—
63C2×#462—31no—
64C2×#461—31no—
65C2×#452—32no—
66C2×#451—32no—
67C2×C4×SD16—31no—
68C2×#436—31no—
69C2×C3×#41—24no—
70C2×C3×#41—28no—
71C2×C3×#41—44no—
72C2×C3×#41C2318yes—
73C22×C3×Q8C2312yes—
74C22×C3×D4—22no—
75C22×C3×D4D411yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—22no—
78C23×C3×C4—21no—
79C23×C3×C4C2312yes—
80C23×Dic3—22no—
81C22×Dic6—28no—
82C22×Dic6—22no—
83C22×Dic6—24no—
84C22×Dic6—44no—
85C22×Dic6C2312yes—
86C22×Dic6C2318yes—
87C22×Dic6C2311yescharacteristic
88C3×C4⋊Q8—22no—
89C3×C4⋊Q8C2314yes—
90C3×#123—216no—
91C3×C22⋊Q8—24no—
92C3×C22⋊Q8C2318yes—
93C3×#121—28no—
94C3×C4×D4—22no—
95C3×C4×D4D414yes—
96C3×C4×D4C2314yes—
97C3×C4×D4C2314yes—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—22no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—28no—
102C2×C3×C4⋊C4D411yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4D412yes—
107C2×C3×C42C2311yescharacteristic
108C2×#217—22no—
109#951—264no—
110C2×#216—28no—
111C2×#216—28no—
112C2×#216—44no—
113C2×#216C2318yes—
114C2×#216C2318yes—
115C2×#216C2318yes—
116D4×Dic3—28no—
117C2×#214—28no—
118C2×#214D418yescomplemented
119C2×#214C2318yes—
120C2×#214C2314yes—
121C2×C4⋊Dic3—21no—
122C2×C4⋊Dic3—21no—
123C2×C4⋊Dic3C2311yescharacteristic
124#924—24no—
125#924C2318yes—
126C2×Dic3⋊C4—21no—
127C2×Dic3⋊C4—22no—
128C2×C4×Dic3—21no—
129C2×#208—28no—
130C2×#208C2318yes—
131C22×C3⋊C8C2311yescharacteristic
132C22×C3⋊C8C2312yes—
133C22×C3⋊C8C2311yescharacteristic
134#888—28no—
135#878—216no—
136#869—22no—
137#869C2314yes—
138C4×Dic6—22no—
139C4×Dic6C2314yes—
140#835C2318yes—
141#832C2314yes—
142#830C2318yes—
143#810C2314yes—
144#810C2318yes—
145#808C2314yes—
146C4⋊(C3⋊C8)C2314yes—
147C4×C3⋊C8C2314yes—
148C22×#41C2×D312yes—
149C23×Q8—32no—
150C2×#137—316no—
151C2×#137—616no—
152C22×Q16—32no—
153C22×Q16—32no—
154C22×Q16—32no—
155C22×SD16—31no—
156C22×SD16—32no—
157C22×M16—32no—
158D4×Q8—38no—
159D4×Q8—64no—
160#547—24no—
161#547C2×D318yes—
162C2×C4⋊Q8—31no—
163C2×C4⋊Q8—61no—
164C2×C4⋊Q8C2×D311yescharacteristic
165C2×#123—22no—
166C2×C22⋊Q8—32no—
167C2×C22⋊Q8—62no—
168C2×C22⋊Q8C2×D312yes—
169C2×#121—21no—
170C2×C4×Q8—31no—
171C2×C4×D4—61no—
172C2×C4×D4C2×D311yescharacteristic
173C2×C4×D4C2×D311yescharacteristic
174C22×C4⋊C4—21no—
175#494—38no—
176#470—316no—
177#462—38no—
178#461—38no—
179#452—316no—
180#451—316no—
181C4×SD16—38no—
182#436—38no—
183C2×#106—31no—
184C2×C4⋊C8—31no—
185C2×#103—32no—
186C2×#103—31no—
187C2×#103—32no—
188C2×#102—31no—
189C2×C22⋊C8—32no—
190C2×C4×C8—31no—
191C24×C3—21no—
192C24×C3C2×D412yes—
193C3×#41—216no—
194C3×#41—48no—
195C3×#41—416no—
196C2×C3×Q8—22no—
197C2×C3×Q8—24no—
198C2×C3×Q8—24no—
199C2×C3×Q8C2414yes—
200C2×C3×D4—28no—
201C2×C3×D4—24no—
202C2×C3×D4—21no—
203C2×C3×D4—22no—
204C2×C3×D4—48no—
205C2×C3×D4C2×D414yes—
206C2×C3×D4C2×D414yes—
207C2×C3×D4C2412yes—
208C22×C3×C4—24no—
209C22×C3×C4—24no—
210C22×C3×C4—21no—
211C22×C3×C4—22no—
212C22×C3×C4—24no—
213C22×C3×C4—28no—
214C22×C3×C4—48no—
215C22×C3×C4—44no—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D412yes—
218C22×C3×C4C2×D414yes—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2414yes—
221C22×C3×C4C2411yescharacteristic
222C22×Dic3—21no—
223C22×Dic3—28no—
224C22×Dic3—21no—
225C22×Dic3—22no—
226C22×Dic3—48no—
227C2×Dic6—21no—
228C2×Dic6—22no—
229C2×Dic6—24no—
230C2×Dic6—24no—
231C2×Dic6—216no—
232C2×Dic6—22no—
233C2×Dic6—28no—
234C2×Dic6—48no—
235C2×Dic6—416no—
236C2×Dic6—48no—
237C2×Dic6C2×D414yes—
238C2×Dic6C2414yes—
239C2×Dic6C2412yes—
240C3×C4⋊C4—24no—
241C3×C4⋊C4—28no—
242C3×C4⋊C4—24no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4C2×D414yes—
245C3×C4⋊C4C2412yes—
246C3×C22⋊C4—28no—
247C3×C22⋊C4—24no—
248C3×C22⋊C4C2×D418yes—
249C3×C42C2×D412yes—
250C3×C42C2412yes—
251#217—28no—
252#216—216no—
253#216—216no—
254#216—48no—
255#214—216no—
256#214C2×D4116yescomplemented
257C4⋊Dic3—21no—
258C4⋊Dic3—24no—
259C4⋊Dic3—24no—
260C4⋊Dic3C2412yes—
261Dic3⋊C4—24no—
262Dic3⋊C4—28no—
263C4×Dic3—24no—
264#208—216no—
265C2×C3⋊C8—24no—
266C2×C3⋊C8—22no—
267C2×C3⋊C8C2×D414yes—
268C2×C3⋊C8C2412yes—
269C2×C3⋊C8C2414yes—
270C2×C3⋊C8C2412yes—
271C2×#41—24no—
272C2×#41—28no—
273C2×#41—44no—
274C2×#41C22×D318yes—
275C22×Q8—31no—
276C22×Q8—32no—
277C22×Q8—38no—
278C22×Q8—68no—
279C22×Q8—62no—
280C22×Q8—64no—
281C22×Q8—124no—
282C22×Q8C22×D312yes—
283C22×D4—22no—
284C22×D4#6711yescharacteristic
285C22×D4C22×D311yescharacteristic
286C23×C4—21no—
287C23×C4—22no—
288C23×C4—62no—
289C23×C4C22×D312yes—
290#137—664no—
291C2×Q16—38no—
292C2×Q16—38no—
293C2×Q16—38no—
294C2×Q16—68no—
295C2×Q16—68no—
296C2×Q16—124no—
297C2×SD16—38no—
298C2×SD16—34no—
299C2×SD16—38no—
300C2×SD16—68no—
301C2×M16—38no—
302C2×M16—68no—
303C22×C8—31no—
304C22×C8—32no—
305C22×C8—31no—
306C4⋊Q8—22no—
307C4⋊Q8—34no—
308C4⋊Q8—68no—
309C4⋊Q8—62no—
310C4⋊Q8C22×D314yes—
311#123—216no—
312C22⋊Q8—24no—
313C22⋊Q8—38no—
314C22⋊Q8—64no—
315C22⋊Q8—616no—
316C22⋊Q8C22×D318yes—
317#121—28no—
318C4×Q8—34no—
319C4×Q8—62no—
320C4×D4—22no—
321C4×D4—68no—
322C4×D4#6714yes—
323C4×D4C22×D314yes—
324C4×D4C22×D314yes—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—28no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—62no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4#6711yescharacteristic
335C2×C4⋊C4C22×D311yescharacteristic
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—62no—
339C2×C22⋊C4#6712yes—
340C2×C42—61no—
341C2×C42C22×D311yescharacteristic
342#106—34no—
343C4⋊C8—34no—
344#103—38no—
345#103—34no—
346#103—38no—
347#102—34no—
348C22⋊C8—38no—
349C4×C8—34no—
350C23×C3—24no—
351C23×C3—24no—
352C23×C3—24no—
353C23×C3—24no—
354C23×C3—22no—
355C23×C3—42no—
356C23×C3—42no—
357C23×C3C22×D414yes—
358C23×C3C22×D411yescharacteristic
359C3×Q8—24no—
360C3×Q8—24no—
361C3×Q8—44no—
362C3×D4—24no—
363C3×D4—28no—
364C3×D4—22no—
365C3×D4—416no—
366C3×D4C22×D414yes—
367C2×C3×C4—24no—
368C2×C3×C4—22no—
369C2×C3×C4—28no—
370C2×C3×C4—24no—
371C2×C3×C4—24no—
372C2×C3×C4—22no—
373C2×C3×C4—24no—
374C2×C3×C4—416no—
375C2×C3×C4—44no—
376C2×C3×C4—42no—
377C2×C3×C4—48no—
378C2×C3×C4#13714yescomplemented
379C2×C3×C4C22×D412yes—
380C2×C3×C4C22×D412yes—
381C2×C3×C4C22×D414yes—
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C22×D412yes—
384C2×C3×C4H(5,𝔽2)12yes—
385C2×C3×C4C2511yescharacteristic
386C2×Dic3—24no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—24no—
390C2×Dic3—416no—
391C2×Dic3—44no—
392C2×Dic3—42no—
393Dic6—24no—
394Dic6—22no—
395Dic6—24no—
396Dic6—24no—
397Dic6—416no—
398Dic6—44no—
399Dic6C22×D414yes—
400C3⋊C8—24no—
401C3⋊C8—22no—
402C3⋊C8C22×D414yes—
403C24—21no—
404C24C2×#6712yes—
405#41—216no—
406#41—48no—
407#41—416no—
408C2×Q8—22no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—34no—
412C2×Q8—32no—
413C2×Q8—34no—
414C2×Q8—616no—
415C2×Q8—62no—
416C2×Q8—61no—
417C2×Q8—68no—
418C2×Q8—62no—
419C2×Q8—64no—
420C2×Q8—64no—
421C2×Q8—128no—
422C2×Q8—128no—
423C2×Q8—1216no—
424C2×Q8C23×D314yes—
425C2×D4—28no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—21no—
429C2×D4—48no—
430C2×D4D3×D414yes—
431C2×D4C2×#6714yes—
432C2×D4C23×D312yes—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—28no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—61no—
442C22×C4—68no—
443C22×C4—61no—
444C22×C4—62no—
445C22×C4—128no—
446C22×C4C2×#6714yes—
447C22×C4C2×#6711yescharacteristic
448C22×C4C2×#6712yes—
449C22×C4C2×#6711yescharacteristic
450C22×C4C23×D314yes—
451C22×C4C23×D311yescharacteristic
452Q16—616no—
453Q16—616no—
454Q16—128no—
455SD16—316no—
456SD16—616no—
457M16—616no—
458C2×C8—34no—
459C2×C8—32no—
460C2×C8—32no—
461C2×C8—34no—
462C2×C8—64no—
463C2×C8—62no—
464C4⋊C4—24no—
465C4⋊C4—28no—
466C4⋊C4—24no—
467C4⋊C4—21no—
468C4⋊C4—32no—
469C4⋊C4—61no—
470C4⋊C4—64no—
471C4⋊C4—64no—
472C4⋊C4—68no—
473C4⋊C4—64no—
474C4⋊C4C2×#6714yes—
475C4⋊C4C23×D312yes—
476C22⋊C4—28no—
477C22⋊C4—24no—
478C22⋊C4—68no—
479C22⋊C4C2×#6718yes—
480C42—64no—
481C42C2×#6712yes—
482C42C23×D312yes—
483C22×C3—24no—
484C22×C3—24no—
485C22×C3—24no—
486C22×C3—24no—
487C22×C3—48no—
488C22×C3—48no—
489C22×C3—42no—
490C22×C3—42no—
491C22×C3D4212yes—
492C22×C3C2×#13712yes—
493C22×C3#57812yes—
494C22×C3C23×D411yescharacteristic
495C3×C4—24no—
496C3×C4—22no—
497C3×C4—44no—
498C3×C4—42no—
499C3×C4D4212yes—
500C3×C4C2×#13714yescomplemented
501C3×C4C23×D411yescharacteristic
502C3×C4C2×H(5,𝔽2)11yescharacteristic
503Dic3—24no—
504Dic3—44no—
505Dic3—42no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—42no—
512C23—42no—
513C23C22×#6714yes—
514C23C22×#6711yescharacteristic
515Q8—24no—
516Q8—24no—
517Q8—34no—
518Q8—44no—
519Q8—62no—
520Q8—64no—
521Q8—64no—
522Q8—64no—
523Q8—1216no—
524Q8—124no—
525D4—22no—
526D4—24no—
527D4—28no—
528D4—416no—
529D4C2×D3×D414yes—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—416no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—48no—
541C2×C4—62no—
542C2×C4—64no—
543C2×C4—62no—
544C2×C4—64no—
545C2×C4—1216no—
546C2×C4—124no—
547C2×C4—122no—
548C2×C4#95114yescomplemented
549C2×C4C2×D3×D412yes—
550C2×C4#100412yes—
551C2×C4C22×#6712yes—
552C2×C4C22×#6714yes—
553C2×C4C22×#6711yescharacteristic
554C2×C4C22×#6712yes—
555C2×C4C24×D311yescharacteristic
556C8—34no—
557C8—64no—
558C8—62no—
559C2×C3—44no—
560C2×C3—44no—
561C2×C3#327614yescomplemented
562C2×C3C2×D4211yescharacteristic
563C2×C3C22×#13711yescharacteristic
564C2×C3C2×#57811yescharacteristic
565C22—24no—
566C22—24no—
567C22—24no—
568C22—24no—
569C22—48no—
570C22—48no—
571C22—42no—
572C22—42no—
573C22D4×#6712yes—
574C22C2×#95112yes—
575C22#583812yes—
576C22C23×#6711yescharacteristic
577C4—24no—
578C4—22no—
579C4—44no—
580C4—42no—
581C4—64no—
582C4—124no—
583C4—122no—
584C4D4×#6712yes—
585C4C2×#95114yescomplemented
586C4C22×D3×D411yescharacteristic
587C4C2×#100411yescharacteristic
588C3C2×#327611yescharacteristic, complemented, Sylow
589C2—44no—
590C2—44no—
591C2#8352914yesdirect product factor
592C2C2×D4×#6711yescharacteristic
593C2C22×#95111yescharacteristic
594C2C2×#583811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
Fitting, prime radicalC2×C3×#547C2
socleC23×C3C22×D4
hypercenterC2×#547D3
hypocenterC3C2×#3276
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
2. C22×#951C2semidirect
3. C2×D4×Dic6C2semidirect
4. C2×#5239C2semidirect
5. #83529C2direct (N × H)
6. C2×#5090C2semidirect
7. C2×#5092C2semidirect
8. C2×#5050C2semidirect
9. C2×#5052C2semidirect
10. C2×#5019C2semidirect
11. C2×C4×#214C2semidirect
12. C2×#5012C2semidirect
26. C2×#951C22semidirect
36. D4×Dic6C22semidirect
41. #5239C22semidirect
45. #5092C22semidirect
46. #5090C22semidirect
47. #5052C22semidirect
48. #5050C22semidirect
49. #5019C22semidirect
50. C4×#214C22semidirect
51. #5012C22semidirect
118. C2×#214D4semidirect
256. #214C2×D4semidirect
378. C2×C3×C4#137semidirect
500. C3×C4C2×#137semidirect
548. C2×C4#951semidirect
561. C2×C3#3276semidirect
585. C4C2×#951semidirect
588. C3C2×#3276semidirect
591. C2#83529direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5472
C22×#9512
C2×D4×Dic62
C2×#52392
#835292
C2×#50902
C2×#50922
C2×#50502
C2×#50522
C2×#50192
C2×C4×#2142
C2×#50122
C2×#32763