Finite Group Databasebeta

← C2×#80502 #11670435

Subgroups — C2×#80502 #11670435

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×#919C212yesmaximal, complemented
2C2×#5726C211yesmaximal, characteristic, complemented
3C2×#5577C211yesmaximal, characteristic, complemented
4C2×C3×#462C211yesmaximal, characteristic, complemented
5C2×#5187C212yesmaximal, complemented
6C2×#5085C211yesmaximal, characteristic, complemented
7C2×C4×#215C211yesmaximal, characteristic, complemented
8#80502C2116yesmaximal, direct product factor
9C2×#4843C211yesmaximal, characteristic, complemented
10C2×#4834C211yesmaximal, characteristic, complemented
11C2×#4807C212yesmaximal, complemented
12C2×#4810C212yesmaximal, complemented
13C2×#3284—31nomaximal, Sylow
14C22×#239C2212yes—
15C22×#239C2212yes—
16C22×C3×Q16C2212yes—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×#946C2212yes—
20C22×#215C2212yes—
21C22×#215C2212yes—
22C22×#215C2211yescharacteristic
23C2×#919—216no—
24C2×#919C22116yescomplemented
25C22×D24C2212yes—
26C22×D3×C8C2212yes—
27#5726—24no—
28#5726C2218yescomplemented
29#5577—24no—
30#5577C2218yescomplemented
31C2×#895—21no—
32C2×#895—22no—
33C2×#895C2212yes—
34C2×#894—22no—
35C2×D3×C4⋊C4—21no—
36C2×#874—21no—
37C2×#874C2211yescharacteristic
38C2×C4×D12—21no—
39C2×C4×D12C2211yescharacteristic
40C2×C4×D12C2211yescharacteristic
41C3×#462C2218yescomplemented
42C2×C3×C4⋊C8C2211yescharacteristic
43C2×C3×#103C2212yes—
44C2×#835C2211yescharacteristic
45#5187C22116yescomplemented
46C2×#821C2212yes—
47C2×(C2×D3)⋊C8C2212yes—
48#5085C2218yescomplemented
49C4×#215C2218yescomplemented
50C2×#809C2211yescharacteristic
51C2×#808C2211yescharacteristic
52C2×C4×C3⋊C8C2211yescharacteristic
53#4843C2218yescomplemented
54#4834C2218yescomplemented
55#4810C22116yescomplemented
56#4807C22116yescomplemented
57C22×#135—32no—
58C2×#550—31no—
59C2×#547—31no—
60#3284—316no—
61C2×#492—31no—
62C2×#465—32no—
63C2×#462D311yescharacteristic, complemented
64C2×#459—31no—
65C2×#452—32no—
66C2×#449—32no—
67C2×C4×SD16—31no—
68C2×C8×D4—31no—
69C22×C3×Q8C2312yes—
70C22×C3×Q8C2311yescharacteristic
71C2×#239—24no—
72C2×#239—24no—
73C2×#239—28no—
74C2×#239—28no—
75C2×#239—44no—
76C2×#239—44no—
77C2×#239C2318yes—
78C2×#239C2318yes—
79C22×D12—22no—
80C22×D12—22no—
81C22×D12—21no—
82C22×D12—22no—
83C22×D12—21no—
84C22×D12D411yescharacteristic
85C22×D12C2312yes—
86C22×D12C2311yescharacteristic
87C22×C4×D3—21no—
88C22×C4×D3—22no—
89C22×C4×D3—22no—
90C22×C4×D3C2312yes—
91C2×C3×Q16—28no—
92C2×C3×Q16C2318yes—
93C22×C3×C8C2312yes—
94C3×C4⋊Q8—22no—
95C3×C4⋊Q8C2314yes—
96C3×C4×Q8—22no—
97C3×C4×Q8C2314yes—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C42—21no—
103C2×C3×C42C2311yescharacteristic
104#946—24no—
105#946C2318yes—
106C2×#215—28no—
107C2×#215—44no—
108C2×#215D418yescomplemented
109C2×#215C2318yes—
110C2×#215C2318yes—
111C2×#215C2314yes—
112C2×(C2×D3)⋊C4—22no—
113C2×(C2×D3)⋊C4—21no—
114C2×(C2×D3)⋊C4—22no—
115C2×(C2×D3)⋊C4D412yes—
116C2×C4⋊Dic3D411yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×Dic3⋊C4—22no—
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#919—264no—
122C2×D24—28no—
123C2×D24C2318yes—
124C2×D3×C8—28no—
125C2×D3×C8C2318yes—
126#895—28no—
127#895—24no—
128#895—216no—
129#895C2318yes—
130#894—216no—
131D3×C4⋊C4—28no—
132#874—28no—
133#874—22no—
134#874C2314yes—
135C4×D12—22no—
136C4×D12—28no—
137C4×D12D414yes—
138C4×D12C2314yes—
139C4×D12C2314yes—
140C3×C4⋊C8C2314yes—
141C3×#103C2318yes—
142#835C2314yes—
143#821C2318yes—
144(C2×D3)⋊C8C2318yes—
145#809C2314yes—
146#808C2314yes—
147C4×C3⋊C8C2314yes—
148C22×#41—32no—
149C22×#41—32no—
150C2×#135—316no—
151C2×#135—616no—
152C22×Q16C2×D312yes—
153C22×SD16—31no—
154C22×SD16—32no—
155C22×SD16—32no—
156C22×D8—32no—
157C23×C8—32no—
158#550—38no—
159#550—64no—
160#547—38no—
161#547—64no—
162C2×C4⋊Q8C2×D311yescharacteristic
163C2×#127—31no—
164C2×#127—61no—
165C2×#123—62no—
166C2×C22⋊Q8—32no—
167C2×#121—32no—
168C2×#121—62no—
169C2×#121—61no—
170C2×C4×Q8C2×D311yescharacteristic
171C2×C4×D4—31no—
172C2×C4×D4—31no—
173C2×C4×D4—61no—
174C22×C4⋊C4—61no—
175#492—38no—
176#465—316no—
177#462C2×D318yescomplemented
178#459—38no—
179#452—316no—
180#449—316no—
181C4×SD16—38no—
182C8×D4—38no—
183C2×#106—31no—
184C2×C4⋊C8C2×D311yescharacteristic
185C2×#103—31no—
186C2×#103C2×D312yes—
187C2×#102—32no—
188C2×#102—31no—
189C2×C22⋊C8—32no—
190C2×C4×C8—31no—
191C23×D3—21no—
192C23×D3—22no—
193C23×D3C2×D412yes—
194C2×C3×Q8—21no—
195C2×C3×Q8—22no—
196C2×C3×Q8—22no—
197C2×C3×Q8—24no—
198C2×C3×Q8—24no—
199C2×C3×Q8C2×D414yes—
200C2×C3×Q8C2414yes—
201C2×C3×Q8C2412yes—
202C22×C3×C4—21no—
203C22×C3×C4—22no—
204C22×C3×C4—21no—
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C22×Dic3—21no—
209C22×Dic3C2×D412yes—
210#239—216no—
211#239—216no—
212#239—48no—
213#239—48no—
214#239—416no—
215#239—416no—
216C2×D12—24no—
217C2×D12—21no—
218C2×D12—22no—
219C2×D12—24no—
220C2×D12—24no—
221C2×D12—28no—
222C2×D12—28no—
223C2×D12—24no—
224C2×D12—22no—
225C2×D12—28no—
226C2×D12—24no—
227C2×D12—48no—
228C2×D12—48no—
229C2×D12—44no—
230C2×D12—82no—
231C2×D12—84no—
232C2×D12C2×D414yes—
233C2×D12C2×D414yes—
234C2×D12C2414yes—
235C2×D12C2412yes—
236C2×C4×D3—24no—
237C2×C4×D3—24no—
238C2×C4×D3—24no—
239C2×C4×D3—28no—
240C2×C4×D3—28no—
241C2×C4×D3—48no—
242C2×C4×D3—48no—
243C2×C4×D3—44no—
244C2×C4×D3C2×D414yes—
245C2×C4×D3C2414yes—
246C3×Q16—216no—
247C2×C3×C8—24no—
248C2×C3×C8C2414yes—
249C3×C4⋊C4—24no—
250C3×C4⋊C4—24no—
251C3×C4⋊C4—24no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4C2412yes—
254C3×C42—24no—
255C3×C42C2×D412yes—
256C3×C42C2412yes—
257#215—216no—
258#215—48no—
259#215C2×D4116yescomplemented
260(C2×D3)⋊C4—28no—
261(C2×D3)⋊C4—24no—
262(C2×D3)⋊C4—28no—
263(C2×D3)⋊C4C2×D418yes—
264C4⋊Dic3—21no—
265C4⋊Dic3C2×D414yes—
266C4⋊Dic3C2412yes—
267Dic3⋊C4—28no—
268C2×C3⋊C8—22no—
269C2×C3⋊C8C2×D414yes—
270C2×C3⋊C8C2412yes—
271C2×C3⋊C8C2412yes—
272D24—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×Q8C22×D312yes—
283C22×Q8C22×D311yescharacteristic
284C22×D4—31no—
285C22×D4—31no—
286C22×D4—32no—
287C22×D4—62no—
288C22×D4—62no—
289C22×D4—62no—
290C22×D4—61no—
291C22×D4—61no—
292C23×C4—32no—
293C23×C4—62no—
294C23×C4—62no—
295C23×C4—61no—
296#135—664no—
297C2×Q16—28no—
298C2×Q16C22×D318yes—
299C2×SD16—38no—
300C2×SD16—34no—
301C2×SD16—38no—
302C2×SD16—38no—
303C2×SD16—68no—
304C2×SD16—124no—
305C2×D8—38no—
306C2×D8—68no—
307C22×C8—31no—
308C22×C8—31no—
309C22×C8—38no—
310C22×C8—68no—
311C22×C8C22×D312yes—
312C4⋊Q8—22no—
313C4⋊Q8C22×D314yes—
314#127—34no—
315#127—68no—
316#127—62no—
317#123—616no—
318C22⋊Q8—38no—
319C22⋊Q8—64no—
320#121—38no—
321#121—616no—
322#121—64no—
323#121—68no—
324C4×Q8—22no—
325C4×Q8C22×D314yes—
326C4×D4—34no—
327C4×D4—34no—
328C4×D4—34no—
329C4×D4—68no—
330C4×D4—62no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—31no—
336C2×C4⋊C4—68no—
337C2×C4⋊C4—62no—
338C2×C4⋊C4C22×D311yescharacteristic
339C2×C22⋊C4—32no—
340C2×C22⋊C4—62no—
341C2×C22⋊C4—62no—
342C2×C22⋊C4—61no—
343C2×C42—21no—
344C2×C42C22×D311yescharacteristic
345#106—34no—
346C4⋊C8C22×D314yes—
347#103—34no—
348#103C22×D318yes—
349#102—38no—
350#102—34no—
351C22⋊C8—38no—
352C4×C8—34no—
353C23×C3C22×D411yescharacteristic
354C22×D3—24no—
355C22×D3—24no—
356C22×D3—24no—
357C22×D3—24no—
358C22×D3—22no—
359C22×D3—24no—
360C22×D3—42no—
361C22×D3—42no—
362C22×D3—44no—
363C22×D3—44no—
364C22×D3—44no—
365C22×D3C22×D414yes—
366C3×Q8—24no—
367C3×Q8—22no—
368C3×Q8—24no—
369C3×Q8—24no—
370C3×Q8—44no—
371C3×Q8C22×D414yes—
372C2×C3×C4—24no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—24no—
376C2×C3×C4—22no—
377C2×C3×C4—44no—
378C2×C3×C4—42no—
379C2×C3×C4#13514yescomplemented
380C2×C3×C4C22×D412yes—
381C2×C3×C4C22×D411yescharacteristic
382C2×C3×C4C22×D412yes—
383C2×C3×C4C22×D412yes—
384C2×C3×C4H(5,𝔽2)12yes—
385C2×C3×C4C2511yescharacteristic
386C2×Dic3—24no—
387C2×Dic3—22no—
388C2×Dic3—42no—
389C2×Dic3C22×D414yes—
390D12—24no—
391D12—28no—
392D12—24no—
393D12—22no—
394D12—24no—
395D12—416no—
396D12—416no—
397D12—44no—
398D12—48no—
399D12—84no—
400D12—88no—
401D12C22×D414yes—
402C4×D3—28no—
403C4×D3—24no—
404C4×D3—24no—
405C4×D3—416no—
406C4×D3—416no—
407C4×D3—48no—
408C3×C8—24no—
409C3⋊C8—22no—
410C3⋊C8C22×D414yes—
411C24—32no—
412C24—61no—
413C24—62no—
414#41—616no—
415#41—616no—
416#41—128no—
417#41—128no—
418#41—1216no—
419#41—1216no—
420C2×Q8—21no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8D3×D414yes—
426C2×Q8C23×D314yes—
427C2×Q8C23×D312yes—
428C2×D4—34no—
429C2×D4—34no—
430C2×D4—32no—
431C2×D4—34no—
432C2×D4—68no—
433C2×D4—68no—
434C2×D4—68no—
435C2×D4—64no—
436C2×D4—61no—
437C2×D4—64no—
438C2×D4—64no—
439C2×D4—62no—
440C2×D4—64no—
441C2×D4—62no—
442C2×D4—64no—
443C2×D4—128no—
444C2×D4—128no—
445C2×D4—124no—
446C2×D4—242no—
447C2×D4—244no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—32no—
452C22×C4—34no—
453C22×C4—34no—
454C22×C4—68no—
455C22×C4—68no—
456C22×C4—64no—
457C22×C4—64no—
458C22×C4—61no—
459C22×C4—64no—
460C22×C4—128no—
461C22×C4—128no—
462C22×C4—124no—
463C22×C4D3×D411yescharacteristic
464C22×C4D3×D411yescharacteristic
465C22×C4C23×D311yescharacteristic
466Q16—216no—
467SD16—316no—
468SD16—616no—
469SD16—128no—
470D8—616no—
471C2×C8—24no—
472C2×C8—34no—
473C2×C8—32no—
474C2×C8—32no—
475C2×C8—616no—
476C2×C8—62no—
477C2×C8C23×D314yes—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—24no—
481C4⋊C4—21no—
482C4⋊C4—34no—
483C4⋊C4—32no—
484C4⋊C4—68no—
485C4⋊C4—61no—
486C4⋊C4C23×D312yes—
487C22⋊C4—38no—
488C22⋊C4—68no—
489C22⋊C4—68no—
490C22⋊C4—64no—
491C42—24no—
492C42D3×D412yes—
493C42C23×D312yes—
494C22×C3D4212yes—
495C22×C3C2×#13512yes—
496C22×C3#57712yes—
497C22×C3C23×D411yescharacteristic
498C2×D3—24no—
499C2×D3—24no—
500C2×D3—24no—
501C2×D3—24no—
502C2×D3—48no—
503C2×D3—48no—
504C2×D3—42no—
505C2×D3—42no—
506C2×D3—44no—
507C2×D3—44no—
508C2×D3—44no—
509C2×D3—88no—
510C3×C4—24no—
511C3×C4—22no—
512C3×C4—44no—
513C3×C4—42no—
514C3×C4D4212yes—
515C3×C4C2×#13514yescomplemented
516C3×C4C23×D411yescharacteristic
517C3×C4C2×H(5,𝔽2)11yescharacteristic
518Dic3—24no—
519Dic3—42no—
520C23—34no—
521C23—62no—
522C23—64no—
523C23—64no—
524C23—64no—
525C23—64no—
526C23—64no—
527C23—122no—
528C23—122no—
529C23—124no—
530C23—124no—
531C23—124no—
532C23C2×D3×D411yescharacteristic
533Q8—22no—
534Q8—24no—
535Q8—24no—
536Q8—24no—
537Q8—44no—
538Q8C2×D3×D414yes—
539D4—34no—
540D4—64no—
541D4—62no—
542D4—64no—
543D4—68no—
544D4—64no—
545D4—1216no—
546D4—1216no—
547D4—128no—
548D4—124no—
549D4—244no—
550D4—248no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—34no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—64no—
562C2×C4—68no—
563C2×C4—62no—
564C2×C4—1216no—
565C2×C4—1216no—
566C2×C4—122no—
567C2×C4—128no—
568C2×C4#91914yescomplemented
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#328414yescomplemented
579C2×C3C2×D4211yescharacteristic
580C2×C3C22×#13511yescharacteristic
581C2×C3C2×#57711yescharacteristic
582D3—44no—
583D3—44no—
584D3—84no—
585C22—64no—
586C22—64no—
587C22—64no—
588C22—64no—
589C22—128no—
590C22—128no—
591C22—122no—
592C22—122no—
593C22—124no—
594C22—124no—
595C22—124no—
596C22—248no—
597C22#560912yes—
598C22C2×#91912yes—
599C22#577812yes—
600C22C22×D3×D411yescharacteristic
601C4—22no—
602C4—24no—
603C4—42no—
604C4—44no—
605C4—64no—
606C4—122no—
607C4#560912yes—
608C4C2×#91914yescomplemented
609C4C22×D3×D411yescharacteristic
610C4C2×#100911yescharacteristic
611C3C2×#328411yescharacteristic, complemented, Sylow
612C2—124no—
613C2—124no—
614C2—244no—
615C2#8050214yesdirect product factor
616C2C2×#560911yescharacteristic
617C2C22×#91911yescharacteristic
618C2C2×#577811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#462C2
socleC23×C3C22×D4
hypercenterC2×#462D3
hypocenterC3C2×#3284
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
1. C22×#919C2semidirect
2. C2×#5726C2semidirect
3. C2×#5577C2semidirect
4. C2×C3×#462C2semidirect
5. C2×#5187C2semidirect
6. C2×#5085C2semidirect
7. C2×C4×#215C2semidirect
8. #80502C2direct (N × H)
9. C2×#4843C2semidirect
10. C2×#4834C2semidirect
11. C2×#4807C2semidirect
12. C2×#4810C2semidirect
24. C2×#919C22semidirect
28. #5726C22semidirect
30. #5577C22semidirect
41. C3×#462C22semidirect
45. #5187C22semidirect
48. #5085C22semidirect
49. C4×#215C22semidirect
53. #4843C22semidirect
54. #4834C22semidirect
55. #4810C22semidirect
56. #4807C22semidirect
63. C2×#462D3semidirect
108. C2×#215D4semidirect
177. #462C2×D3semidirect
259. #215C2×D4semidirect
379. C2×C3×C4#135semidirect
515. C3×C4C2×#135semidirect
568. C2×C4#919semidirect
578. C2×C3#3284semidirect
608. C4C2×#919semidirect
611. C3C2×#3284semidirect
615. C2#80502direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9192
C2×#57262
C2×#55772
C2×C3×#4622
C2×#51872
C2×#50852
C2×C4×#2152
#805022
C2×#48432
C2×#48342
C2×#48072
C2×#48102
C2×#32843