Finite Group Databasebeta

← C2×#83560 #11670333

Subgroups — C2×#83560 #11670333

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×D4×Q8C211yesmaximal, characteristic, complemented
2C22×#942C212yesmaximal, complemented
3C2×#5551C211yesmaximal, characteristic
4C2×#5228C212yesmaximal, complemented
5#83560C2116yesmaximal, direct product factor
6C2×#5083C211yesmaximal, characteristic, complemented
7C2×#5092C211yesmaximal, characteristic, complemented
8C2×#5047C212yesmaximal, complemented
9C2×#5052C212yesmaximal, complemented
10C2×#5031C211yesmaximal, characteristic, complemented
11C2×C4×#215C211yesmaximal, characteristic, complemented
12C2×#5012C211yesmaximal, characteristic, complemented
13C2×#3276—31nomaximal, Sylow
14C23×C3×Q8C2212yes—
15C22×#235C2212yes—
16C3×D4×Q8—24no—
17C3×D4×Q8C2218yescomplemented
18C2×C3×C4⋊Q8—21no—
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×C22⋊Q8—22no—
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4C2211yescharacteristic
25C22×#216C2212yes—
26C22×#216C2212yes—
27C22×#216C2212yes—
28C2×#942—216no—
29C2×#942C22116yescomplemented
30C22×#215C2212yes—
31C22×#215C2211yescharacteristic
32C2×#938—21no—
33C2×#924C2212yes—
34C22×#208C2212yes—
35#5551—24no—
36#5551C2218yes—
37C2×D3×C4⋊C4—21no—
38C2×#883—22no—
39C2×C4×D12C2211yescharacteristic
40C2×#869C2211yescharacteristic
41#5228C22116yescomplemented
42C2×#835C2212yes—
43C2×#835C2211yescharacteristic
44C2×#830C2212yes—
45#5092C2218yescomplemented
46#5083C2218yescomplemented
47#5052C22116yescomplemented
48#5047C22116yescomplemented
49#5031C2218yescomplemented
50C4×#215C2218yescomplemented
51#5012C2218yescomplemented
52C2×#810C2212yes—
53C2×#809C2211yescharacteristic
54C2×#808C2211yescharacteristic
55C2×C4⋊(C3⋊C8)C2211yescharacteristic
56C2×C4×C3⋊C8C2211yescharacteristic
57C22×#137—32no—
58C2×D4×Q8D311yescharacteristic, complemented
59C2×#547—31no—
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—
69C22×C3×Q8—22no—
70C22×C3×Q8—24no—
71C22×C3×Q8—28no—
72C22×C3×Q8—44no—
73C22×C3×Q8C2312yes—
74C22×C3×Q8C2318yes—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4D411yescharacteristic
77C23×C3×C4—22no—
78C23×C3×C4C2312yes—
79C22×#67—22no—
80C2×#235—24no—
81C2×#235—28no—
82C2×#235—44no—
83C2×#235C2318yes—
84C22×D12C2311yescharacteristic
85C22×C4×D3—22no—
86C22×C4×D3—21no—
87C22×Dic6C2312yes—
88C3×C4⋊Q8—28no—
89C3×C4⋊Q8—22no—
90C3×C4⋊Q8C2314yes—
91C3×C22⋊Q8—24no—
92C3×C22⋊Q8—216no—
93C3×C22⋊Q8C2318yes—
94C3×C4×Q8—22no—
95C3×C4×Q8C2314yes—
96C3×C4×D4—28no—
97C3×C4×D4D414yes—
98C3×C4×D4C2314yes—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—22no—
103C2×C3×C4⋊C4D411yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4D412yes—
107C2×C3×C42—21no—
108C2×C3×C42C2311yescharacteristic
109C2×#217—22no—
110C2×#216—28no—
111C2×#216—28no—
112C2×#216—44no—
113C2×#216C2318yes—
114C2×#216C2318yes—
115C2×#216C2318yes—
116#942—264no—
117C2×#215—28no—
118C2×#215D418yescomplemented
119C2×#215C2318yes—
120C2×#215C2314yes—
121#938—28no—
122C2×(C2×D3)⋊C4—21no—
123C2×C4⋊Dic3—21no—
124C2×C4⋊Dic3—21no—
125C2×C4⋊Dic3C2311yescharacteristic
126#924—24no—
127#924C2318yes—
128C2×Dic3⋊C4—22no—
129C2×#208—28no—
130C2×#208C2318yes—
131C22×C3⋊C8C2311yescharacteristic
132C22×C3⋊C8C2312yes—
133C22×C3⋊C8C2311yescharacteristic
134D3×C4⋊C4—28no—
135#883—216no—
136C4×D12—22no—
137C4×D12C2314yes—
138#869—22no—
139#869C2314yes—
140#835C2314yes—
141#835C2318yes—
142#830C2318yes—
143#810C2318yes—
144#809C2314yes—
145#808C2314yes—
146C4⋊(C3⋊C8)C2314yes—
147C4×C3⋊C8C2314yes—
148C22×#41—32no—
149C23×Q8C2×D312yes—
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—24no—
159D4×Q8C2×D318yescomplemented
160#547—38no—
161#547—64no—
162C2×C4⋊Q8—21no—
163C2×C4⋊Q8—31no—
164C2×C4⋊Q8C2×D311yescharacteristic
165C2×#123—62no—
166C2×C22⋊Q8—22no—
167C2×C22⋊Q8—32no—
168C2×C22⋊Q8C2×D312yes—
169C2×#121—61no—
170C2×C4×Q8C2×D311yescharacteristic
171C2×C4×D4—21no—
172C2×C4×D4—31no—
173C2×C4×D4C2×D311yescharacteristic
174C22×C4⋊C4—61no—
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—31no—
186C2×#103—32no—
187C2×#103—32no—
188C2×#102—31no—
189C2×C22⋊C8—32no—
190C2×C4×C8—31no—
191C24×C3C2×D412yes—
192C23×D3—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—28no—
195C2×C3×Q8—22no—
196C2×C3×Q8—22no—
197C2×C3×Q8—216no—
198C2×C3×Q8—24no—
199C2×C3×Q8—24no—
200C2×C3×Q8—48no—
201C2×C3×Q8—48no—
202C2×C3×Q8—416no—
203C2×C3×Q8C2×D414yes—
204C2×C3×Q8C2414yes—
205C2×C3×Q8C2412yes—
206C2×C3×D4—24no—
207C2×C3×D4C2×D414yes—
208C22×C3×C4—21no—
209C22×C3×C4—24no—
210C22×C3×C4—24no—
211C22×C3×C4—21no—
212C22×C3×C4—22no—
213C22×C3×C4—28no—
214C22×C3×C4—48no—
215C22×C3×C4C2×D412yes—
216C22×C3×C4C2×D414yes—
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4C2414yes—
220C22×C3×C4C2411yescharacteristic
221C2×#67—28no—
222C2×#67—48no—
223C22×Dic3—21no—
224C22×Dic3—22no—
225#235—216no—
226#235—48no—
227#235—416no—
228C2×D12—21no—
229C2×D12—22no—
230C2×D12C2×D414yes—
231C2×D12C2412yes—
232C2×C4×D3—24no—
233C2×C4×D3—28no—
234C2×C4×D3—48no—
235C2×C4×D3—44no—
236C2×Dic6—24no—
237C2×Dic6—24no—
238C2×Dic6—22no—
239C2×Dic6C2414yes—
240C3×C4⋊C4—24no—
241C3×C4⋊C4—24no—
242C3×C4⋊C4—24no—
243C3×C4⋊C4—28no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4C2×D414yes—
246C3×C4⋊C4C2412yes—
247C3×C22⋊C4—28no—
248C3×C22⋊C4C2×D418yes—
249C3×C42—24no—
250C3×C42C2×D412yes—
251C3×C42C2412yes—
252#217—28no—
253#216—216no—
254#216—216no—
255#216—48no—
256#215—216no—
257#215C2×D4116yescomplemented
258(C2×D3)⋊C4—24no—
259C4⋊Dic3—24no—
260C4⋊Dic3—24no—
261C4⋊Dic3—21no—
262C4⋊Dic3C2412yes—
263Dic3⋊C4—28no—
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—38no—
272C2×#41—68no—
273C2×#41—64no—
274C2×#41—124no—
275C22×Q8—22no—
276C22×Q8—24no—
277C22×Q8—28no—
278C22×Q8—32no—
279C22×Q8—44no—
280C22×Q8C22×D312yes—
281C22×Q8C22×D318yes—
282C22×Q8C22×D311yescharacteristic
283C22×D4—31no—
284C22×D4—62no—
285C22×D4#6711yescharacteristic
286C23×C4—22no—
287C23×C4—61no—
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—28no—
307C4⋊Q8—22no—
308C4⋊Q8—34no—
309C4⋊Q8—62no—
310C4⋊Q8C22×D314yes—
311#123—616no—
312C22⋊Q8—24no—
313C22⋊Q8—216no—
314C22⋊Q8—38no—
315C22⋊Q8—64no—
316C22⋊Q8C22×D318yes—
317#121—68no—
318C4×Q8—22no—
319C4×Q8C22×D314yes—
320C4×D4—28no—
321C4×D4—34no—
322C4×D4—62no—
323C4×D4#6714yes—
324C4×D4C22×D314yes—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—62no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—68no—
334C2×C4⋊C4#6711yescharacteristic
335C2×C4⋊C4C22×D311yescharacteristic
336C2×C22⋊C4—22no—
337C2×C22⋊C4—62no—
338C2×C22⋊C4—61no—
339C2×C22⋊C4#6712yes—
340C2×C42—21no—
341C2×C42C22×D311yescharacteristic
342#106—34no—
343C4⋊C8—34no—
344#103—34no—
345#103—38no—
346#103—38no—
347#102—34no—
348C22⋊C8—38no—
349C4×C8—34no—
350C23×C3—24no—
351C23×C3—24no—
352C23×C3—24no—
353C23×C3C22×D414yes—
354C23×C3C22×D411yescharacteristic
355C22×D3—24no—
356C22×D3—22no—
357C22×D3—42no—
358C22×D3—42no—
359C3×Q8—24no—
360C3×Q8—22no—
361C3×Q8—24no—
362C3×Q8—24no—
363C3×Q8—416no—
364C3×Q8—44no—
365C3×Q8C22×D414yes—
366C3×D4—28no—
367C2×C3×C4—24no—
368C2×C3×C4—24no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—28no—
372C2×C3×C4—24no—
373C2×C3×C4—24no—
374C2×C3×C4—22no—
375C2×C3×C4—24no—
376C2×C3×C4—416no—
377C2×C3×C4—44no—
378C2×C3×C4—42no—
379C2×C3×C4#13714yescomplemented
380C2×C3×C4C22×D412yes—
381C2×C3×C4C22×D414yes—
382C2×C3×C4C22×D412yes—
383C2×C3×C4C22×D411yescharacteristic
384C2×C3×C4C22×D412yes—
385C2×C3×C4H(5,𝔽2)12yes—
386C2×C3×C4C2511yescharacteristic
387#67—416no—
388C2×Dic3—22no—
389C2×Dic3—24no—
390C2×Dic3—44no—
391C2×Dic3—42no—
392D12—24no—
393D12—22no—
394D12C22×D414yes—
395C4×D3—416no—
396C4×D3—48no—
397Dic6—24no—
398Dic6—24no—
399Dic6—44no—
400C3⋊C8—24no—
401C3⋊C8—22no—
402C3⋊C8C22×D414yes—
403C24—61no—
404C24C2×#6712yes—
405#41—616no—
406#41—128no—
407#41—1216no—
408C2×Q8—22no—
409C2×Q8—21no—
410C2×Q8—28no—
411C2×Q8—22no—
412C2×Q8—216no—
413C2×Q8—24no—
414C2×Q8—24no—
415C2×Q8—34no—
416C2×Q8—48no—
417C2×Q8—48no—
418C2×Q8—416no—
419C2×Q8—62no—
420C2×Q8—64no—
421C2×Q8—64no—
422C2×Q8D3×D414yes—
423C2×Q8C23×D314yes—
424C2×Q8C23×D312yes—
425C2×D4—24no—
426C2×D4—34no—
427C2×D4—32no—
428C2×D4—68no—
429C2×D4—62no—
430C2×D4—61no—
431C2×D4—128no—
432C2×D4C2×#6714yes—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—28no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—48no—
440C22×C4—61no—
441C22×C4—62no—
442C22×C4—64no—
443C22×C4—68no—
444C22×C4—128no—
445C22×C4—124no—
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—24no—
466C4⋊C4—24no—
467C4⋊C4—28no—
468C4⋊C4—21no—
469C4⋊C4—32no—
470C4⋊C4—61no—
471C4⋊C4—64no—
472C4⋊C4—68no—
473C4⋊C4—64no—
474C4⋊C4C2×#6714yes—
475C4⋊C4C23×D312yes—
476C22⋊C4—28no—
477C22⋊C4—68no—
478C22⋊C4—64no—
479C22⋊C4C2×#6718yes—
480C42—24no—
481C42C2×#6712yes—
482C42C23×D312yes—
483C22×C3—24no—
484C22×C3—24no—
485C22×C3—24no—
486C22×C3—48no—
487C22×C3D4212yes—
488C22×C3C2×#13712yes—
489C22×C3#57812yes—
490C22×C3C23×D411yescharacteristic
491C2×D3—24no—
492C2×D3—48no—
493C2×D3—42no—
494C2×D3—42no—
495C3×C4—24no—
496C3×C4—24no—
497C3×C4—22no—
498C3×C4—44no—
499C3×C4—42no—
500C3×C4D4212yes—
501C3×C4C2×#13714yescomplemented
502C3×C4C23×D411yescharacteristic
503C3×C4C2×H(5,𝔽2)11yescharacteristic
504Dic3—44no—
505Dic3—42no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—62no—
510C23—64no—
511C23—122no—
512C23—122no—
513C23C22×#6714yes—
514C23C22×#6711yescharacteristic
515Q8—22no—
516Q8—24no—
517Q8—24no—
518Q8—24no—
519Q8—416no—
520Q8—44no—
521Q8—64no—
522Q8—64no—
523Q8—124no—
524Q8C2×D3×D414yes—
525D4—28no—
526D4—34no—
527D4—62no—
528D4—64no—
529D4—1216no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—416no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—62no—
543C2×C4—64no—
544C2×C4—1216no—
545C2×C4—124no—
546C2×C4—122no—
547C2×C4—128no—
548C2×C4#94214yescomplemented
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#327614yescomplemented
561C2×C3C2×D4211yescharacteristic
562C2×C3C22×#13711yescharacteristic
563C2×C3C2×#57811yescharacteristic
564D3—44no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—48no—
569C22—64no—
570C22—128no—
571C22—122no—
572C22—122no—
573C22D4×#6712yes—
574C22C2×#94212yes—
575C22#583812yes—
576C22C23×#6711yescharacteristic
577C4—24no—
578C4—24no—
579C4—22no—
580C4—44no—
581C4—42no—
582C4—124no—
583C4—122no—
584C4D4×#6712yes—
585C4C2×#94214yescomplemented
586C4C22×D3×D411yescharacteristic
587C4C2×#100411yescharacteristic
588C3C2×#327611yescharacteristic, complemented, Sylow
589C2—44no—
590C2—124no—
591C2#8356014yesdirect product factor
592C2C2×D4×#6711yescharacteristic
593C2C22×#94211yescharacteristic
594C2C2×#583811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×D4×Q8C2
socleC23×C3C22×D4
prime radicalC2×#5551C2
hypercenterC2×D4×Q8D3
hypocenterC3C2×#3276
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×Q8C2semidirect
2. C22×#942C2semidirect
4. C2×#5228C2semidirect
5. #83560C2direct (N × H)
6. C2×#5083C2semidirect
7. C2×#5092C2semidirect
8. C2×#5047C2semidirect
9. C2×#5052C2semidirect
10. C2×#5031C2semidirect
11. C2×C4×#215C2semidirect
12. C2×#5012C2semidirect
17. C3×D4×Q8C22semidirect
29. C2×#942C22semidirect
41. #5228C22semidirect
45. #5092C22semidirect
46. #5083C22semidirect
47. #5052C22semidirect
48. #5047C22semidirect
49. #5031C22semidirect
50. C4×#215C22semidirect
51. #5012C22semidirect
58. C2×D4×Q8D3semidirect
118. C2×#215D4semidirect
159. D4×Q8C2×D3semidirect
257. #215C2×D4semidirect
379. C2×C3×C4#137semidirect
501. C3×C4C2×#137semidirect
548. C2×C4#942semidirect
560. C2×C3#3276semidirect
585. C4C2×#942semidirect
588. C3C2×#3276semidirect
591. C2#83560direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×Q82
C22×#9422
C2×#55512
C2×#52282
#835602
C2×#50832
C2×#50922
C2×#50472
C2×#50522
C2×#50312
C2×C4×#2152
C2×#50122
C2×#32763