Finite Group Databasebeta

← C2×#80351 #11670443

Subgroups — C2×#80351 #11670443

Order 768.

2858 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#912C212yesmaximal, complemented
2C2×#5718C211yesmaximal, characteristic, complemented
3C2×#5555C211yesmaximal, characteristic, complemented
4C2×C3×#459C211yesmaximal, characteristic, complemented
5C2×#5158C212yesmaximal, complemented
6C2×#5076C211yesmaximal, characteristic, complemented
7C2×C4×#214C211yesmaximal, characteristic, complemented
8#80351C2116yesmaximal, direct product factor
9C2×#4849C211yesmaximal, characteristic, complemented
10C2×#4834C211yesmaximal, characteristic, complemented
11C2×#4778C212yesmaximal, complemented
12C2×#4783C212yesmaximal, complemented
13C2×#3284—31nomaximal, Sylow
14C22×#237C2212yes—
15C22×#237C2212yes—
16C22×C3×D8C2212yes—
17C2×C3×#127C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×#940—21no—
20C2×#939—22no—
21C2×#938C2212yes—
22C2×D4×Dic3—21no—
23C22×#214C2212yes—
24C22×#214C2212yes—
25C22×#214C2211yescharacteristic
26C2×#930—21no—
27C22×C4⋊Dic3—21no—
28C2×#912—216no—
29C2×#912C22116yescomplemented
30C22×Dic12C2212yes—
31C22×D3×C8C2212yes—
32#5718—24no—
33#5718C2218yescomplemented
34#5555—24no—
35#5555C2218yescomplemented
36C2×#897C2212yes—
37C2×#886—22no—
38C2×C4×D12C2211yescharacteristic
39C2×#869C2211yescharacteristic
40C2×C4×Dic6C2211yescharacteristic
41C3×#459C2218yescomplemented
42C2×C3×C4⋊C8C2211yescharacteristic
43C2×C3×#102C2212yes—
44C2×#832C2211yescharacteristic
45#5158C22116yescomplemented
46C2×(C2×D3)⋊C8C2212yes—
47C2×#816C2212yes—
48#5076C2218yescomplemented
49C4×#214C2218yescomplemented
50C2×#810C2211yescharacteristic
51C2×#808C2211yescharacteristic
52C2×C4×C3⋊C8C2211yescharacteristic
53#4849C2218yescomplemented
54#4834C2218yescomplemented
55#4783C22116yescomplemented
56#4778C22116yescomplemented
57C22×#135—32no—
58C2×#550—31no—
59C2×#547—31no—
60#3284—316no—
61C2×#492—31no—
62C2×#465—32no—
63C2×#462—31no—
64C2×#459D311yescharacteristic, complemented
65C2×#452—32no—
66C2×#449—32no—
67C2×C4×SD16—31no—
68C2×C8×D4—31no—
69C22×C3×D4—21no—
70C22×C3×D4—21no—
71C22×C3×D4C2312yes—
72C22×C3×D4C2311yescharacteristic
73C23×C3×C4—21no—
74C22×#67—22no—
75C22×#67—22no—
76C22×#67—22no—
77C23×Dic3—22no—
78C23×Dic3—22no—
79C2×#237—28no—
80C2×#237—24no—
81C2×#237—24no—
82C2×#237—28no—
83C2×#237—44no—
84C2×#237—44no—
85C2×#237C2318yes—
86C2×#237C2318yes—
87C22×D12D411yescharacteristic
88C22×C4×D3C2312yes—
89C22×Dic6C2312yes—
90C22×Dic6C2311yescharacteristic
91C2×C3×D8—28no—
92C2×C3×D8C2318yes—
93C22×C3×C8C2312yes—
94C3×#127—22no—
95C3×#127C2314yes—
96C3×C4×D4—22no—
97C3×C4×D4C2314yes—
98C2×C3×C4⋊C4C2311yescharacteristic
99C2×C3×C22⋊C4—21no—
100C2×C3×C42C2311yescharacteristic
101C2×#217—22no—
102#940—28no—
103#939—216no—
104#938—24no—
105#938C2318yes—
106D4×Dic3—28no—
107C2×#214—28no—
108C2×#214—44no—
109C2×#214D418yescomplemented
110C2×#214C2318yes—
111C2×#214C2318yes—
112C2×#214C2314yes—
113#930—28no—
114C2×(C2×D3)⋊C4—22no—
115C2×(C2×D3)⋊C4D412yes—
116C2×C4⋊Dic3—21no—
117C2×C4⋊Dic3—21no—
118C2×C4⋊Dic3—22no—
119C2×C4⋊Dic3—28no—
120C2×C4⋊Dic3D411yescharacteristic
121C2×C4⋊Dic3C2311yescharacteristic
122C2×Dic3⋊C4—21no—
123C2×C4×Dic3—21no—
124C22×C3⋊C8C2311yescharacteristic
125C22×C3⋊C8C2311yescharacteristic
126#912—264no—
127C2×Dic12—28no—
128C2×Dic12C2318yes—
129C2×D3×C8—28no—
130C2×D3×C8C2318yes—
131#897—24no—
132#897C2318yes—
133#886—216no—
134C4×D12D414yes—
135C4×D12C2314yes—
136#869—22no—
137#869C2314yes—
138C4×Dic6—22no—
139C4×Dic6C2314yes—
140C3×C4⋊C8C2314yes—
141C3×#102C2318yes—
142#832C2314yes—
143(C2×D3)⋊C8C2318yes—
144#816C2318yes—
145#810C2314yes—
146#808C2314yes—
147C4×C3⋊C8C2314yes—
148C22×#41—32no—
149C22×#41—32no—
150C2×#135—316no—
151C2×#135—616no—
152C22×Q16—32no—
153C22×SD16—31no—
154C22×SD16—32no—
155C22×SD16—32no—
156C22×D8C2×D312yes—
157C23×C8—32no—
158#550—38no—
159#550—64no—
160#547—38no—
161#547—64no—
162C2×C4⋊Q8—31no—
163C2×#127—61no—
164C2×#127C2×D311yescharacteristic
165C2×#123—62no—
166C2×C22⋊Q8—32no—
167C2×#121—32no—
168C2×#121—62no—
169C2×#121—61no—
170C2×C4×Q8—31no—
171C2×C4×D4—31no—
172C2×C4×D4—61no—
173C2×C4×D4C2×D311yescharacteristic
174C22×C4⋊C4—61no—
175#492—38no—
176#465—316no—
177#462—38no—
178#459C2×D318yescomplemented
179#452—316no—
180#449—316no—
181C4×SD16—38no—
182C8×D4—38no—
183C2×#106—31no—
184C2×C4⋊C8C2×D311yescharacteristic
185C2×#103—32no—
186C2×#103—31no—
187C2×#102—31no—
188C2×#102C2×D312yes—
189C2×C22⋊C8—32no—
190C2×C4×C8—31no—
191C24×C3—21no—
192C24×C3—22no—
193C23×D3C2×D412yes—
194C2×C3×D4—21no—
195C2×C3×D4—24no—
196C2×C3×D4—24no—
197C2×C3×D4—22no—
198C2×C3×D4—24no—
199C2×C3×D4—22no—
200C2×C3×D4—24no—
201C2×C3×D4—44no—
202C2×C3×D4—82no—
203C2×C3×D4C2×D414yes—
204C2×C3×D4C2414yes—
205C2×C3×D4C2412yes—
206C22×C3×C4—21no—
207C22×C3×C4—24no—
208C22×C3×C4—44no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C2×#67—28no—
213C2×#67—28no—
214C2×#67—28no—
215C2×#67—48no—
216C2×#67—48no—
217C2×#67—84no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C22×Dic3—22no—
221C22×Dic3—28no—
222C22×Dic3—28no—
223C22×Dic3—48no—
224C22×Dic3—48no—
225C22×Dic3C2×D412yes—
226#237—216no—
227#237—216no—
228#237—48no—
229#237—48no—
230#237—416no—
231#237—416no—
232C2×D12—24no—
233C2×D12C2×D414yes—
234C2×C4×D3—24no—
235C2×C4×D3—24no—
236C2×C4×D3C2×D414yes—
237C2×C4×D3C2414yes—
238C2×Dic6—21no—
239C2×Dic6—22no—
240C2×Dic6—24no—
241C2×Dic6—24no—
242C2×Dic6—22no—
243C2×Dic6C2×D414yes—
244C2×Dic6C2414yes—
245C2×Dic6C2412yes—
246C3×D8—216no—
247C2×C3×C8—24no—
248C2×C3×C8C2414yes—
249C3×C4⋊C4—21no—
250C3×C4⋊C4C2412yes—
251C3×C22⋊C4—24no—
252C3×C42C2×D412yes—
253C3×C42C2412yes—
254#217—28no—
255#214—216no—
256#214—48no—
257#214C2×D4116yescomplemented
258(C2×D3)⋊C4—28no—
259(C2×D3)⋊C4C2×D418yes—
260C4⋊Dic3—21no—
261C4⋊Dic3—24no—
262C4⋊Dic3—24no—
263C4⋊Dic3—28no—
264C4⋊Dic3C2×D414yes—
265C4⋊Dic3C2412yes—
266Dic3⋊C4—24no—
267C4×Dic3—24no—
268C2×C3⋊C8—22no—
269C2×C3⋊C8C2×D414yes—
270C2×C3⋊C8C2412yes—
271C2×C3⋊C8C2412yes—
272Dic12—216no—
273D3×C8—216no—
274C2×#41—38no—
275C2×#41—38no—
276C2×#41—64no—
277C2×#41—64no—
278C2×#41—68no—
279C2×#41—68no—
280C2×#41—124no—
281C2×#41—124no—
282C22×Q8—31no—
283C22×Q8—32no—
284C22×D4—21no—
285C22×D4—21no—
286C22×D4—31no—
287C22×D4—62no—
288C22×D4—62no—
289C22×D4—62no—
290C22×D4C22×D312yes—
291C22×D4C22×D311yescharacteristic
292C23×C4—21no—
293C23×C4—32no—
294C23×C4—62no—
295C23×C4—62no—
296#135—664no—
297C2×Q16—38no—
298C2×Q16—68no—
299C2×SD16—38no—
300C2×SD16—34no—
301C2×SD16—38no—
302C2×SD16—38no—
303C2×SD16—68no—
304C2×SD16—124no—
305C2×D8—28no—
306C2×D8C22×D318yes—
307C22×C8—31no—
308C22×C8—31no—
309C22×C8—38no—
310C22×C8—68no—
311C22×C8C22×D312yes—
312C4⋊Q8—34no—
313C4⋊Q8—62no—
314#127—22no—
315#127—68no—
316#127C22×D314yes—
317#123—616no—
318C22⋊Q8—38no—
319C22⋊Q8—64no—
320#121—38no—
321#121—616no—
322#121—68no—
323#121—64no—
324C4×Q8—34no—
325C4×Q8—62no—
326C4×D4—22no—
327C4×D4—34no—
328C4×D4—34no—
329C4×D4—68no—
330C4×D4C22×D314yes—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—68no—
334C2×C4⋊C4—62no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4C22×D311yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—32no—
341C2×C22⋊C4—62no—
342C2×C22⋊C4—62no—
343C2×C42—61no—
344C2×C42C22×D311yescharacteristic
345#106—34no—
346C4⋊C8C22×D314yes—
347#103—38no—
348#103—34no—
349#102—34no—
350#102C22×D318yes—
351C22⋊C8—38no—
352C4×C8—34no—
353C23×C3—24no—
354C23×C3—22no—
355C23×C3—24no—
356C23×C3—42no—
357C23×C3—42no—
358C23×C3—44no—
359C23×C3—44no—
360C23×C3—44no—
361C23×C3C22×D411yescharacteristic
362C22×D3—24no—
363C22×D3—24no—
364C22×D3—24no—
365C22×D3C22×D414yes—
366C3×D4—24no—
367C3×D4—24no—
368C3×D4—22no—
369C3×D4—24no—
370C3×D4—44no—
371C3×D4—48no—
372C3×D4—84no—
373C3×D4C22×D414yes—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—42no—
377C2×C3×C4—48no—
378C2×C3×C4#13514yescomplemented
379C2×C3×C4C22×D412yes—
380C2×C3×C4C22×D411yescharacteristic
381C2×C3×C4C22×D412yes—
382C2×C3×C4C22×D412yes—
383C2×C3×C4H(5,𝔽2)12yes—
384C2×C3×C4C2511yescharacteristic
385#67—416no—
386#67—416no—
387#67—88no—
388C2×Dic3—24no—
389C2×Dic3—24no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—24no—
393C2×Dic3—416no—
394C2×Dic3—416no—
395C2×Dic3—44no—
396C2×Dic3—42no—
397C2×Dic3C22×D414yes—
398D12—28no—
399C4×D3—28no—
400C4×D3—24no—
401C4×D3—24no—
402Dic6—24no—
403Dic6—22no—
404Dic6—24no—
405Dic6—24no—
406Dic6—44no—
407Dic6C22×D414yes—
408C3×C8—24no—
409C3⋊C8—22no—
410C3⋊C8C22×D414yes—
411C24—21no—
412C24—22no—
413C24—32no—
414#41—616no—
415#41—616no—
416#41—128no—
417#41—128no—
418#41—1216no—
419#41—1216no—
420C2×Q8—34no—
421C2×Q8—32no—
422C2×Q8—34no—
423C2×Q8—61no—
424C2×Q8—62no—
425C2×Q8—62no—
426C2×Q8—64no—
427C2×Q8—64no—
428C2×D4—21no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—22no—
432C2×D4—24no—
433C2×D4—22no—
434C2×D4—24no—
435C2×D4—34no—
436C2×D4—44no—
437C2×D4—68no—
438C2×D4—68no—
439C2×D4—68no—
440C2×D4—64no—
441C2×D4—82no—
442C2×D4—128no—
443C2×D4—128no—
444C2×D4—244no—
445C2×D4D3×D414yes—
446C2×D4C23×D314yes—
447C2×D4C23×D312yes—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—32no—
451C22×C4—34no—
452C22×C4—34no—
453C22×C4—44no—
454C22×C4—68no—
455C22×C4—68no—
456C22×C4—61no—
457C22×C4—64no—
458C22×C4—64no—
459C22×C4—61no—
460C22×C4—62no—
461C22×C4—128no—
462C22×C4—128no—
463C22×C4D3×D411yescharacteristic
464C22×C4D3×D411yescharacteristic
465C22×C4C23×D311yescharacteristic
466Q16—616no—
467SD16—316no—
468SD16—616no—
469SD16—128no—
470D8—216no—
471C2×C8—24no—
472C2×C8—34no—
473C2×C8—32no—
474C2×C8—32no—
475C2×C8—616no—
476C2×C8—62no—
477C2×C8C23×D314yes—
478C4⋊C4—21no—
479C4⋊C4—34no—
480C4⋊C4—32no—
481C4⋊C4—68no—
482C4⋊C4—61no—
483C4⋊C4—64no—
484C4⋊C4—64no—
485C4⋊C4—64no—
486C4⋊C4C23×D312yes—
487C22⋊C4—24no—
488C22⋊C4—38no—
489C22⋊C4—68no—
490C22⋊C4—68no—
491C42—64no—
492C42D3×D412yes—
493C42C23×D312yes—
494C22×C3—24no—
495C22×C3—48no—
496C22×C3—42no—
497C22×C3—42no—
498C22×C3—44no—
499C22×C3—44no—
500C22×C3—44no—
501C22×C3—88no—
502C22×C3D4212yes—
503C22×C3C2×#13512yes—
504C22×C3#57712yes—
505C22×C3C23×D411yescharacteristic
506C2×D3—24no—
507C2×D3—24no—
508C2×D3—24no—
509C2×D3—48no—
510C3×C4—22no—
511C3×C4—42no—
512C3×C4D4212yes—
513C3×C4C2×#13514yescomplemented
514C3×C4C23×D411yescharacteristic
515C3×C4C2×H(5,𝔽2)11yescharacteristic
516Dic3—24no—
517Dic3—24no—
518Dic3—44no—
519Dic3—42no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—34no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—64no—
530C23—64no—
531C23—64no—
532C23C2×D3×D411yescharacteristic
533Q8—34no—
534Q8—62no—
535Q8—64no—
536Q8—64no—
537Q8—64no—
538Q8—124no—
539D4—24no—
540D4—22no—
541D4—24no—
542D4—24no—
543D4—48no—
544D4—44no—
545D4—68no—
546D4—84no—
547D4—1216no—
548D4—1216no—
549D4—248no—
550D4C2×D3×D414yes—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—34no—
554C2×C4—42no—
555C2×C4—48no—
556C2×C4—62no—
557C2×C4—64no—
558C2×C4—64no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—68no—
562C2×C4—62no—
563C2×C4—64no—
564C2×C4—1216no—
565C2×C4—1216no—
566C2×C4—122no—
567C2×C4—124no—
568C2×C4#91214yescomplemented
569C2×C4C2×D3×D412yes—
570C2×C4C2×D3×D412yes—
571C2×C4C2×D3×D412yes—
572C2×C4C2×D3×D411yescharacteristic
573C2×C4#100912yes—
574C2×C4C24×D311yescharacteristic
575C8—24no—
576C8—34no—
577C8—62no—
578C2×C3—44no—
579C2×C3—84no—
580C2×C3#328414yescomplemented
581C2×C3C2×D4211yescharacteristic
582C2×C3C22×#13511yescharacteristic
583C2×C3C2×#57711yescharacteristic
584D3—44no—
585C22—24no—
586C22—42no—
587C22—42no—
588C22—48no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—64no—
593C22—64no—
594C22—64no—
595C22—88no—
596C22—128no—
597C22#560912yes—
598C22C2×#91212yes—
599C22#577512yes—
600C22C22×D3×D411yescharacteristic
601C4—22no—
602C4—42no—
603C4—64no—
604C4—64no—
605C4—122no—
606C4—124no—
607C4#560912yes—
608C4C2×#91214yescomplemented
609C4C22×D3×D411yescharacteristic
610C4C2×#100911yescharacteristic
611C3C2×#328411yescharacteristic, complemented, Sylow
612C2—44no—
613C2—84no—
614C2—124no—
615C2#8035114yesdirect product factor
616C2C2×#560911yescharacteristic
617C2C22×#91211yescharacteristic
618C2C2×#577511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#459C2
socleC23×C3C22×D4
hypercenterC2×#459D3
hypocenterC3C2×#3284
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. C22×#912C2semidirect
2. C2×#5718C2semidirect
3. C2×#5555C2semidirect
4. C2×C3×#459C2semidirect
5. C2×#5158C2semidirect
6. C2×#5076C2semidirect
7. C2×C4×#214C2semidirect
8. #80351C2direct (N × H)
9. C2×#4849C2semidirect
10. C2×#4834C2semidirect
11. C2×#4778C2semidirect
12. C2×#4783C2semidirect
29. C2×#912C22semidirect
33. #5718C22semidirect
35. #5555C22semidirect
41. C3×#459C22semidirect
45. #5158C22semidirect
48. #5076C22semidirect
49. C4×#214C22semidirect
53. #4849C22semidirect
54. #4834C22semidirect
55. #4783C22semidirect
56. #4778C22semidirect
64. C2×#459D3semidirect
109. C2×#214D4semidirect
178. #459C2×D3semidirect
257. #214C2×D4semidirect
378. C2×C3×C4#135semidirect
513. C3×C4C2×#135semidirect
568. C2×C4#912semidirect
580. C2×C3#3284semidirect
608. C4C2×#912semidirect
611. C3C2×#3284semidirect
615. C2#80351direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9122
C2×#57182
C2×#55552
C2×C3×#4592
C2×#51582
C2×#50762
C2×C4×#2142
#803512
C2×#48492
C2×#48342
C2×#47782
C2×#47832
C2×#32843