Finite Group Databasebeta

← C2×#79437 #11670442

Subgroups — C2×#79437 #11670442

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×#904C212yesmaximal, complemented
2C2×#5551C211yesmaximal, characteristic, complemented
3C2×#5553C211yesmaximal, characteristic, complemented
4C2×C3×C8×D4C211yesmaximal, characteristic, complemented
5C2×#5114C212yesmaximal, complemented
6#79437C2116yesmaximal, direct product factor
7C2×#4849C211yesmaximal, characteristic, complemented
8C2×#4843C211yesmaximal, characteristic, complemented
9C2×#4733C212yesmaximal, complemented
10C2×#4734C212yesmaximal, complemented
11C2×C4×#204C211yesmaximal, characteristic, complemented
12C2×#4693C211yesmaximal, characteristic, complemented
13C2×#3284—31nomaximal, Sylow
14C22×#235C2212yes—
15C22×#235C2212yes—
16C23×C3×C8C2212yes—
17C2×C3×C4×D4C2211yescharacteristic
18C2×#940—21no—
19C2×#938—21no—
20C2×#930C2212yes—
21C2×#924C2212yes—
22C22×Dic12C2212yes—
23C2×#904—216no—
24C2×#904C22116yescomplemented
25C22×D24C2212yes—
26C22×#204C2212yes—
27C22×#204C2212yes—
28C22×#204C2211yescharacteristic
29#5553—24no—
30#5553C2218yescomplemented
31#5551—24no—
32#5551C2218yescomplemented
33C2×#893—21no—
34C2×D3×C4⋊C4—21no—
35C2×#884—22no—
36C2×#883—22no—
37C2×#874C2211yescharacteristic
38C2×C4×D12C2211yescharacteristic
39C2×#869C2211yescharacteristic
40C2×C4×Dic6C2211yescharacteristic
41C3×C8×D4C2218yescomplemented
42C2×C3×C4⋊C8C2211yescharacteristic
43C2×C3×C22⋊C8C2212yes—
44C2×C3×C4×C8C2211yescharacteristic
45#5114C22116yescomplemented
46C2×#821C2211yescharacteristic
47C2×#821C2212yes—
48C2×#817C2211yescharacteristic
49C2×#816C2212yes—
50C2×#816C2211yescharacteristic
51#4849C2218yescomplemented
52#4843C2218yescomplemented
53#4734C22116yescomplemented
54#4733C22116yescomplemented
55#4693C2218yescomplemented
56C4×#204C2218yescomplemented
57C22×#135—32no—
58C2×#550—31no—
59C2×#547—31no—
60#3284—316no—
61C2×#492—31no—
62C2×#465—32no—
63C2×#462—31no—
64C2×#459—31no—
65C2×#452—32no—
66C2×#449—32no—
67C2×C4×SD16—31no—
68C2×C8×D4D311yescharacteristic, complemented
69C22×C3×D4D411yescharacteristic
70C23×C3×C4C2312yes—
71C22×#67—22no—
72C22×#67—22no—
73C22×#67—22no—
74C2×#235—24no—
75C2×#235—24no—
76C2×#235—28no—
77C2×#235—28no—
78C2×#235—44no—
79C2×#235—44no—
80C2×#235C2318yes—
81C2×#235C2318yes—
82C22×D12—21no—
83C22×D12—21no—
84C22×D12C2312yes—
85C22×D12C2311yescharacteristic
86C22×C4×D3—22no—
87C22×C4×D3—22no—
88C22×C4×D3—21no—
89C22×Dic6C2312yes—
90C22×Dic6C2311yescharacteristic
91C22×C3×C8—28no—
92C22×C3×C8C2312yes—
93C22×C3×C8C2318yes—
94C22×C3×C8C2311yescharacteristic
95C22×C3×C8C2311yescharacteristic
96C3×C4×D4D414yes—
97C3×C4×D4C2314yes—
98C2×C3×C4⋊C4D411yescharacteristic
99C2×C3×C22⋊C4D412yes—
100C2×C3×C42C2311yescharacteristic
101C2×#217—22no—
102#940—28no—
103#938—28no—
104#930—24no—
105#930C2318yes—
106C2×(C2×D3)⋊C4—21no—
107C2×(C2×D3)⋊C4—22no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112#924—24no—
113#924C2318yes—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—22no—
116C2×C4×Dic3—21no—
117C2×Dic12—28no—
118C2×Dic12C2318yes—
119#904—264no—
120C2×D24—28no—
121C2×D24C2318yes—
122C2×#204—28no—
123C2×#204—44no—
124C2×#204D418yescomplemented
125C2×#204C2318yes—
126C2×#204C2318yes—
127C2×#204C2314yes—
128#893—28no—
129D3×C4⋊C4—28no—
130#884—216no—
131#883—216no—
132#874—22no—
133#874C2314yes—
134C4×D12—22no—
135C4×D12C2314yes—
136#869—22no—
137#869C2314yes—
138C4×Dic6—22no—
139C4×Dic6C2314yes—
140C3×C4⋊C8C2314yes—
141C3×C22⋊C8C2318yes—
142C3×C4×C8C2314yes—
143#821C2314yes—
144#821C2318yes—
145#817C2314yes—
146#816C2314yes—
147#816C2318yes—
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×D8—32no—
157C23×C8C2×D312yes—
158#550—38no—
159#550—64no—
160#547—38no—
161#547—64no—
162C2×C4⋊Q8—31no—
163C2×#127—31no—
164C2×#127—61no—
165C2×#123—62no—
166C2×C22⋊Q8—32no—
167C2×#121—32no—
168C2×#121—61no—
169C2×#121—62no—
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#459—38no—
179#452—316no—
180#449—316no—
181C4×SD16—38no—
182C8×D4C2×D318yescomplemented
183C2×#106—31no—
184C2×C4⋊C8C2×D311yescharacteristic
185C2×#103—31no—
186C2×#103—32no—
187C2×#102—31no—
188C2×#102—32no—
189C2×C22⋊C8C2×D312yes—
190C2×C4×C8C2×D311yescharacteristic
191C24×C3C2×D412yes—
192C23×D3—21no—
193C23×D3—22no—
194C2×C3×D4—24no—
195C2×C3×D4C2×D414yes—
196C22×C3×C4—24no—
197C22×C3×C4—24no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D412yes—
200C22×C3×C4C2×D414yes—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2414yes—
203C22×C3×C4C2411yescharacteristic
204C2×#67—28no—
205C2×#67—28no—
206C2×#67—28no—
207C2×#67—48no—
208C2×#67—48no—
209C2×#67—84no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—22no—
214#235—216no—
215#235—216no—
216#235—48no—
217#235—48no—
218#235—416no—
219#235—416no—
220C2×D12—21no—
221C2×D12—22no—
222C2×D12—24no—
223C2×D12—24no—
224C2×D12—22no—
225C2×D12—24no—
226C2×D12—24no—
227C2×D12—44no—
228C2×D12—82no—
229C2×D12C2×D414yes—
230C2×D12C2414yes—
231C2×D12C2412yes—
232C2×C4×D3—24no—
233C2×C4×D3—28no—
234C2×C4×D3—28no—
235C2×C4×D3—48no—
236C2×C4×D3—48no—
237C2×C4×D3—44no—
238C2×Dic6—21no—
239C2×Dic6—22no—
240C2×Dic6—24no—
241C2×Dic6—24no—
242C2×Dic6—22no—
243C2×Dic6C2×D414yes—
244C2×Dic6C2414yes—
245C2×Dic6C2412yes—
246C2×C3×C8—24no—
247C2×C3×C8—22no—
248C2×C3×C8—216no—
249C2×C3×C8C2×D414yes—
250C2×C3×C8C2412yes—
251C2×C3×C8C2414yes—
252C2×C3×C8C2412yes—
253C3×C4⋊C4C2×D414yes—
254C3×C22⋊C4C2×D418yes—
255C3×C42C2×D412yes—
256C3×C42C2412yes—
257#217—28no—
258(C2×D3)⋊C4—24no—
259(C2×D3)⋊C4—28no—
260C4⋊Dic3—21no—
261C4⋊Dic3—24no—
262C4⋊Dic3—24no—
263C4⋊Dic3—21no—
264C4⋊Dic3C2412yes—
265C4⋊Dic3C2412yes—
266Dic3⋊C4—24no—
267Dic3⋊C4—28no—
268C4×Dic3—24no—
269Dic12—216no—
270D24—216no—
271#204—216no—
272#204—48no—
273#204C2×D4116yescomplemented
274C2×#41—38no—
275C2×#41—38no—
276C2×#41—68no—
277C2×#41—68no—
278C2×#41—64no—
279C2×#41—64no—
280C2×#41—124no—
281C2×#41—124no—
282C22×Q8—31no—
283C22×Q8—32no—
284C22×D4—31no—
285C22×D4—32no—
286C22×D4—62no—
287C22×D4—62no—
288C22×D4—62no—
289C22×D4—61no—
290C22×D4—61no—
291C22×D4D1211yescharacteristic
292C23×C4—61no—
293C23×C4—62no—
294C23×C4—62no—
295C23×C4C22×D312yes—
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—38no—
306C2×D8—68no—
307C22×C8—28no—
308C22×C8C22×D312yes—
309C22×C8C22×D318yes—
310C22×C8C22×D311yescharacteristic
311C22×C8C22×D311yescharacteristic
312C4⋊Q8—34no—
313C4⋊Q8—62no—
314#127—34no—
315#127—68no—
316#127—62no—
317#123—616no—
318C22⋊Q8—38no—
319C22⋊Q8—64no—
320#121—38no—
321#121—68no—
322#121—64no—
323#121—616no—
324C4×Q8—34no—
325C4×Q8—62no—
326C4×D4—34no—
327C4×D4—68no—
328C4×D4—62no—
329C4×D4D1214yes—
330C4×D4C22×D314yes—
331C2×C4⋊C4—31no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—62no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—68no—
338C2×C4⋊C4D1211yescharacteristic
339C2×C22⋊C4—62no—
340C2×C22⋊C4—61no—
341C2×C22⋊C4—62no—
342C2×C22⋊C4D1212yes—
343C2×C42—61no—
344C2×C42C22×D311yescharacteristic
345#106—34no—
346C4⋊C8C22×D314yes—
347#103—34no—
348#103—38no—
349#102—34no—
350#102—38no—
351C22⋊C8C22×D318yes—
352C4×C8C22×D314yes—
353C23×C3—24no—
354C23×C3—24no—
355C23×C3—24no—
356C23×C3C22×D414yes—
357C23×C3C22×D411yescharacteristic
358C22×D3—24no—
359C22×D3—22no—
360C22×D3—24no—
361C22×D3—42no—
362C22×D3—42no—
363C22×D3—44no—
364C22×D3—44no—
365C22×D3—44no—
366C3×D4—28no—
367C2×C3×C4—24no—
368C2×C3×C4—22no—
369C2×C3×C4—28no—
370C2×C3×C4—24no—
371C2×C3×C4—24no—
372C2×C3×C4#13514yescomplemented
373C2×C3×C4C22×D412yes—
374C2×C3×C4C22×D414yes—
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C22×D412yes—
377C2×C3×C4C22×D412yes—
378C2×C3×C4H(5,𝔽2)12yes—
379C2×C3×C4C2511yescharacteristic
380#67—416no—
381#67—416no—
382#67—88no—
383C2×Dic3—24no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—24no—
388C2×Dic3—44no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391D12—24no—
392D12—24no—
393D12—22no—
394D12—24no—
395D12—44no—
396D12—48no—
397D12—84no—
398D12C22×D414yes—
399C4×D3—416no—
400C4×D3—416no—
401C4×D3—48no—
402Dic6—24no—
403Dic6—22no—
404Dic6—24no—
405Dic6—24no—
406Dic6—44no—
407Dic6C22×D414yes—
408C3×C8—24no—
409C3×C8—22no—
410C3×C8C22×D414yes—
411C24—61no—
412C24—62no—
413C24C2×D1212yes—
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—24no—
429C2×D4—34no—
430C2×D4—32no—
431C2×D4—34no—
432C2×D4—68no—
433C2×D4—68no—
434C2×D4—68no—
435C2×D4—61no—
436C2×D4—64no—
437C2×D4—62no—
438C2×D4—64no—
439C2×D4—64no—
440C2×D4—62no—
441C2×D4—64no—
442C2×D4—128no—
443C2×D4—128no—
444C2×D4—124no—
445C2×D4—242no—
446C2×D4—244no—
447C2×D4C2×D1214yes—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—62no—
454C22×C4—64no—
455C22×C4—68no—
456C22×C4—68no—
457C22×C4—128no—
458C22×C4—128no—
459C22×C4—124no—
460C22×C4C2×D1214yes—
461C22×C4C2×D1211yescharacteristic
462C22×C4C2×D1212yes—
463C22×C4C2×D1211yescharacteristic
464C22×C4C23×D314yes—
465C22×C4C23×D311yescharacteristic
466Q16—616no—
467SD16—316no—
468SD16—616no—
469SD16—128no—
470D8—616no—
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—216no—
474C2×C8D3×D414yes—
475C2×C8C23×D312yes—
476C2×C8C23×D314yes—
477C2×C8C23×D312yes—
478C4⋊C4—32no—
479C4⋊C4—32no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—64no—
483C4⋊C4—68no—
484C4⋊C4—64no—
485C4⋊C4—64no—
486C4⋊C4C2×D1214yes—
487C22⋊C4—68no—
488C22⋊C4—64no—
489C22⋊C4—68no—
490C22⋊C4C2×D1218yes—
491C42—64no—
492C42C2×D1212yes—
493C42C23×D312yes—
494C22×C3—24no—
495C22×C3—24no—
496C22×C3—24no—
497C22×C3—48no—
498C22×C3D4212yes—
499C22×C3C2×#13512yes—
500C22×C3#57712yes—
501C22×C3C23×D411yescharacteristic
502C2×D3—24no—
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×C4D4212yes—
513C3×C4C2×#13514yescomplemented
514C3×C4C23×D411yescharacteristic
515C3×C4C2×H(5,𝔽2)11yescharacteristic
516Dic3—24no—
517Dic3—44no—
518Dic3—42no—
519Dic3—42no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—62no—
524C23—64no—
525C23—64no—
526C23—122no—
527C23—122no—
528C23—124no—
529C23—124no—
530C23—124no—
531C23C22×D1214yes—
532C23C22×D1211yescharacteristic
533Q8—34no—
534Q8—62no—
535Q8—64no—
536Q8—64no—
537Q8—64no—
538Q8—124no—
539D4—28no—
540D4—34no—
541D4—64no—
542D4—62no—
543D4—64no—
544D4—64no—
545D4—1216no—
546D4—1216no—
547D4—128no—
548D4—124no—
549D4—244no—
550D4—248no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—62no—
557C2×C4—64no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—64no—
561C2×C4—1216no—
562C2×C4—1216no—
563C2×C4—122no—
564C2×C4—122no—
565C2×C4—124no—
566C2×C4—128no—
567C2×C4#90414yescomplemented
568C2×C4C22×D1212yes—
569C2×C4C22×D1214yes—
570C2×C4C22×D1212yes—
571C2×C4C22×D1211yescharacteristic
572C2×C4C2×D3×D412yes—
573C2×C4#100912yes—
574C2×C4C24×D311yescharacteristic
575C8—24no—
576C8—22no—
577C8C2×D3×D414yes—
578C2×C3—44no—
579C2×C3#328414yescomplemented
580C2×C3C2×D4211yescharacteristic
581C2×C3C22×#13511yescharacteristic
582C2×C3C2×#57711yescharacteristic
583D3—44no—
584D3—84no—
585C22—24no—
586C22—24no—
587C22—24no—
588C22—48no—
589C22—64no—
590C22—128no—
591C22—122no—
592C22—122no—
593C22—124no—
594C22—124no—
595C22—124no—
596C22—248no—
597C22D4×D1212yes—
598C22C2×#90412yes—
599C22#575112yes—
600C22C23×D1211yescharacteristic
601C4—24no—
602C4—22no—
603C4—64no—
604C4—124no—
605C4—122no—
606C4—122no—
607C4D4×D1212yes—
608C4C2×#90414yescomplemented
609C4C22×D3×D411yescharacteristic
610C4C2×#100911yescharacteristic
611C3C2×#328411yescharacteristic, complemented, Sylow
612C2—44no—
613C2—124no—
614C2—244no—
615C2#7943714yesdirect product factor
616C2C2×D4×D1211yescharacteristic
617C2C22×#90411yescharacteristic
618C2C2×#575111yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×C8×D4C2
socleC23×C3C22×D4
hypercenterC2×C8×D4D3
hypocenterC3C2×#3284
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4

Complements

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

NH ≅ G/Ntype
1. C22×#904C2semidirect
2. C2×#5551C2semidirect
3. C2×#5553C2semidirect
4. C2×C3×C8×D4C2semidirect
5. C2×#5114C2semidirect
6. #79437C2direct (N × H)
7. C2×#4849C2semidirect
8. C2×#4843C2semidirect
9. C2×#4733C2semidirect
10. C2×#4734C2semidirect
11. C2×C4×#204C2semidirect
12. C2×#4693C2semidirect
24. C2×#904C22semidirect
30. #5553C22semidirect
32. #5551C22semidirect
41. C3×C8×D4C22semidirect
45. #5114C22semidirect
51. #4849C22semidirect
52. #4843C22semidirect
53. #4734C22semidirect
54. #4733C22semidirect
55. #4693C22semidirect
56. C4×#204C22semidirect
68. C2×C8×D4D3semidirect
124. C2×#204D4semidirect
182. C8×D4C2×D3semidirect
273. #204C2×D4semidirect
372. C2×C3×C4#135semidirect
513. C3×C4C2×#135semidirect
567. C2×C4#904semidirect
579. C2×C3#3284semidirect
608. C4C2×#904semidirect
611. C3C2×#3284semidirect
615. C2#79437direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9042
C2×#55512
C2×#55532
C2×C3×C8×D42
C2×#51142
#794372
C2×#48492
C2×#48432
C2×#47332
C2×#47342
C2×C4×#2042
C2×#46932
C2×#32843