Finite Group Databasebeta

← C2×#81772 #11670439

Subgroups — C2×#81772 #11670439

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×#916C212yesmaximal, complemented
2C2×#5726C211yesmaximal, characteristic, complemented
3C2×#5718C211yesmaximal, characteristic, complemented
4C2×C3×#492C211yesmaximal, characteristic, complemented
5C2×#5171C212yesmaximal, complemented
6C2×#5173C212yesmaximal, complemented
7C2×#5172C212yesmaximal, complemented
8C2×#5090C211yesmaximal, characteristic, complemented
9C2×#5072C211yesmaximal, characteristic, complemented
10#81772C2116yesmaximal, direct product factor
11C2×C4×#204C211yesmaximal, characteristic, complemented
12C2×C8×D12C211yesmaximal, characteristic, complemented
13C2×#3284—31nomaximal, Sylow
14C22×#239C2212yes—
15C22×#237C2212yes—
16C22×C3×SD16C2212yes—
17C22×C3×SD16C2212yes—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#127C2211yescharacteristic
20C2×#946C2212yes—
21C22×#216C2212yes—
22C2×#940—21no—
23C2×#939—22no—
24C2×#938C2212yes—
25C2×D4×Dic3—21no—
26C22×#213C2212yes—
27C2×#916—216no—
28C2×#916C22116yescomplemented
29C22×#204C2211yescharacteristic
30C22×D3×C8C2212yes—
31#5726—24no—
32#5726C2218yescomplemented
33#5718—24no—
34#5718C2218yescomplemented
35C2×#895—21no—
36C2×#894—22no—
37C2×D3×C4⋊C4—21no—
38C2×C4×D12C2211yescharacteristic
39C2×C4×D12C2211yescharacteristic
40C2×C4×Dic6C2211yescharacteristic
41C3×#492C2218yescomplemented
42C2×C3×C4×C8C2211yescharacteristic
43C2×#835C2212yes—
44C2×#832C2212yes—
45#5173C22116yescomplemented
46#5172C22116yescomplemented
47#5171C22116yescomplemented
48C2×#821C2211yescharacteristic
49C2×(C2×D3)⋊C8C2212yes—
50C2×#817C2211yescharacteristic
51C2×#816C2211yescharacteristic
52#5090C2218yescomplemented
53#5072C2218yescomplemented
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C4×#204C2218yescomplemented
56C8×D12C2218yescomplemented
57C22×#135—32no—
58C2×#550—31no—
59C2×#547—31no—
60#3284—316no—
61C2×#492D311yescharacteristic, complemented
62C2×#465—32no—
63C2×#462—31no—
64C2×#459—31no—
65C2×#452—32no—
66C2×#449—32no—
67C2×C4×SD16—31no—
68C2×C8×D4—31no—
69C22×C3×Q8C2312yes—
70C22×C3×D4—21no—
71C22×C3×D4—21no—
72C22×C3×D4C2312yes—
73C22×#67—22no—
74C22×#67—22no—
75C23×Dic3—22no—
76C2×#239—24no—
77C2×#239—28no—
78C2×#239—44no—
79C2×#239C2318yes—
80C2×#237—24no—
81C2×#237—28no—
82C2×#237—44no—
83C2×#237C2318yes—
84C22×D12—22no—
85C22×D12D411yescharacteristic
86C22×D12C2311yescharacteristic
87C22×C4×D3—21no—
88C22×C4×D3—22no—
89C22×C4×D3C2312yes—
90C22×Dic6C2311yescharacteristic
91C2×C3×SD16—28no—
92C2×C3×SD16—44no—
93C2×C3×SD16C2318yes—
94C2×C3×SD16C2318yes—
95C22×C3×C8C2311yescharacteristic
96C22×C3×C8C2311yescharacteristic
97C3×C4⋊Q8—22no—
98C3×C4⋊Q8C2314yes—
99C3×#127—22no—
100C3×#127C2314yes—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C42C2311yescharacteristic
104C2×#217—22no—
105#946—24no—
106#946C2318yes—
107C2×#216—28no—
108C2×#216C2318yes—
109#940—28no—
110#939—216no—
111#938—24no—
112#938C2318yes—
113D4×Dic3—28no—
114C2×#213—28no—
115C2×#213C2318yes—
116C2×(C2×D3)⋊C4—22no—
117C2×(C2×D3)⋊C4—21no—
118C2×(C2×D3)⋊C4D412yes—
119C2×C4⋊Dic3D411yescharacteristic
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4⋊Dic3C2311yescharacteristic
122C2×Dic3⋊C4—21no—
123C2×Dic3⋊C4—22no—
124C2×C4×Dic3—21no—
125C22×C3⋊C8C2312yes—
126#916—264no—
127C2×#204D418yescomplemented
128C2×#204C2314yes—
129C2×D3×C8—28no—
130C2×D3×C8C2318yes—
131#895—28no—
132#894—216no—
133D3×C4⋊C4—28no—
134C4×D12—22no—
135C4×D12D414yes—
136C4×D12C2314yes—
137C4×D12C2314yes—
138C4×Dic6—22no—
139C4×Dic6C2314yes—
140C3×C4×C8C2314yes—
141#835C2318yes—
142#832C2318yes—
143#821C2314yes—
144(C2×D3)⋊C8C2318yes—
145#817C2314yes—
146#816C2314yes—
147C4⋊(C3⋊C8)C2314yes—
148C22×#41—32no—
149C22×#41—32no—
150C2×#135—316no—
151C2×#135—616no—
152C22×Q16—32no—
153C22×SD16—31no—
154C22×SD16C2×D312yes—
155C22×SD16C2×D312yes—
156C22×D8—32no—
157C23×C8—32no—
158#550—38no—
159#550—64no—
160#547—38no—
161#547—64no—
162C2×C4⋊Q8C2×D311yescharacteristic
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—31no—
173C2×C4×D4—61no—
174C22×C4⋊C4—61no—
175#492C2×D318yescomplemented
176#465—316no—
177#462—38no—
178#459—38no—
179#452—316no—
180#449—316no—
181C4×SD16—38no—
182C8×D4—38no—
183C2×#106—31no—
184C2×C4⋊C8—31no—
185C2×#103—32no—
186C2×#103—31no—
187C2×#102—32no—
188C2×#102—31no—
189C2×C22⋊C8—32no—
190C2×C4×C8C2×D311yescharacteristic
191C24×C3—22no—
192C23×D3—21no—
193C23×D3C2×D412yes—
194C2×C3×Q8—22no—
195C2×C3×Q8—24no—
196C2×C3×Q8—24no—
197C2×C3×Q8C2414yes—
198C2×C3×D4—24no—
199C2×C3×D4—24no—
200C2×C3×D4—22no—
201C2×C3×D4—24no—
202C2×C3×D4—24no—
203C2×C3×D4—44no—
204C2×C3×D4—82no—
205C2×C3×D4C2414yes—
206C22×C3×C4—22no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2411yescharacteristic
210C2×#67—28no—
211C2×#67—28no—
212C2×#67—48no—
213C2×#67—84no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—28no—
218C22×Dic3—48no—
219C22×Dic3C2×D412yes—
220#239—216no—
221#239—48no—
222#239—416no—
223#237—216no—
224#237—48no—
225#237—416no—
226C2×D12—24no—
227C2×D12—21no—
228C2×D12—22no—
229C2×D12—28no—
230C2×D12—48no—
231C2×D12C2×D414yes—
232C2×D12C2×D414yes—
233C2×D12C2412yes—
234C2×C4×D3—24no—
235C2×C4×D3—24no—
236C2×C4×D3—24no—
237C2×C4×D3—28no—
238C2×C4×D3—48no—
239C2×C4×D3—44no—
240C2×C4×D3C2×D414yes—
241C2×C4×D3C2414yes—
242C2×Dic6—21no—
243C2×Dic6—22no—
244C2×Dic6C2×D414yes—
245C2×Dic6C2412yes—
246C3×SD16—216no—
247C3×SD16—48no—
248C2×C3×C8—22no—
249C2×C3×C8C2×D414yes—
250C2×C3×C8C2412yes—
251C2×C3×C8C2412yes—
252C3×C4⋊C4—24no—
253C3×C4⋊C4—24no—
254C3×C42C2×D412yes—
255C3×C42C2412yes—
256#217—28no—
257#216—216no—
258#213—216no—
259(C2×D3)⋊C4—28no—
260(C2×D3)⋊C4—24no—
261(C2×D3)⋊C4C2×D418yes—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3C2×D414yes—
265C4⋊Dic3C2412yes—
266C4⋊Dic3C2412yes—
267Dic3⋊C4—24no—
268Dic3⋊C4—28no—
269C4×Dic3—24no—
270C2×C3⋊C8—24no—
271C2×C3⋊C8C2414yes—
272#204C2×D4116yescomplemented
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×Q8C22×D312yes—
284C22×D4—21no—
285C22×D4—21no—
286C22×D4—31no—
287C22×D4—31no—
288C22×D4—62no—
289C22×D4—62no—
290C22×D4—62no—
291C22×D4C22×D312yes—
292C23×C4—32no—
293C23×C4—62no—
294C23×C4—62no—
295C23×C4—61no—
296#135—664no—
297C2×Q16—38no—
298C2×Q16—68no—
299C2×SD16—28no—
300C2×SD16—38no—
301C2×SD16—34no—
302C2×SD16—44no—
303C2×SD16C22×D318yes—
304C2×SD16C22×D318yes—
305C2×D8—38no—
306C2×D8—68no—
307C22×C8—32no—
308C22×C8—38no—
309C22×C8—68no—
310C22×C8C22×D311yescharacteristic
311C22×C8C22×D311yescharacteristic
312C4⋊Q8—22no—
313C4⋊Q8C22×D314yes—
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—34no—
327C4×D4—34no—
328C4×D4—34no—
329C4×D4—68no—
330C4×D4—62no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—31no—
336C2×C4⋊C4—68no—
337C2×C4⋊C4—62no—
338C2×C4⋊C4—61no—
339C2×C22⋊C4—32no—
340C2×C22⋊C4—62no—
341C2×C22⋊C4—62no—
342C2×C22⋊C4—61no—
343C2×C42—61no—
344C2×C42C22×D311yescharacteristic
345#106—34no—
346C4⋊C8—34no—
347#103—38no—
348#103—34no—
349#102—38no—
350#102—34no—
351C22⋊C8—38no—
352C4×C8C22×D314yes—
353C23×C3—24no—
354C23×C3—44no—
355C23×C3—44no—
356C23×C3—44no—
357C23×C3C22×D411yescharacteristic
358C22×D3—24no—
359C22×D3—24no—
360C22×D3—24no—
361C22×D3—24no—
362C22×D3—22no—
363C22×D3—42no—
364C22×D3—42no—
365C22×D3C22×D414yes—
366C3×Q8—24no—
367C3×Q8—24no—
368C3×Q8—44no—
369C3×D4—24no—
370C3×D4—24no—
371C3×D4—44no—
372C3×D4—48no—
373C3×D4—84no—
374C2×C3×C4—22no—
375C2×C3×C4—24no—
376C2×C3×C4—44no—
377C2×C3×C4#13514yescomplemented
378C2×C3×C4C22×D412yes—
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C22×D412yes—
381C2×C3×C4C22×D412yes—
382C2×C3×C4H(5,𝔽2)12yes—
383C2×C3×C4C2511yescharacteristic
384#67—416no—
385#67—88no—
386C2×Dic3—24no—
387C2×Dic3—24no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—416no—
392C2×Dic3—42no—
393C2×Dic3—42no—
394C2×Dic3C22×D414yes—
395D12—24no—
396D12—28no—
397D12—22no—
398D12—416no—
399D12C22×D414yes—
400C4×D3—28no—
401C4×D3—24no—
402C4×D3—24no—
403C4×D3—416no—
404C4×D3—48no—
405Dic6—24no—
406Dic6—22no—
407Dic6C22×D414yes—
408C3×C8—22no—
409C3×C8C22×D414yes—
410C3⋊C8—24no—
411C24—22no—
412C24—32no—
413C24—61no—
414#41—616no—
415#41—616no—
416#41—128no—
417#41—1216no—
418#41—128no—
419#41—1216no—
420C2×Q8—22no—
421C2×Q8—24no—
422C2×Q8—24no—
423C2×Q8—34no—
424C2×Q8—32no—
425C2×Q8—61no—
426C2×Q8—62no—
427C2×Q8C23×D314yes—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—22no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—34no—
434C2×D4—34no—
435C2×D4—32no—
436C2×D4—44no—
437C2×D4—68no—
438C2×D4—68no—
439C2×D4—68no—
440C2×D4—64no—
441C2×D4—61no—
442C2×D4—62no—
443C2×D4—82no—
444C2×D4—128no—
445C2×D4—128no—
446C2×D4—244no—
447C2×D4C23×D314yes—
448C22×C4—22no—
449C22×C4—32no—
450C22×C4—34no—
451C22×C4—34no—
452C22×C4—68no—
453C22×C4—68no—
454C22×C4—61no—
455C22×C4—64no—
456C22×C4—64no—
457C22×C4—61no—
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—616no—
467SD16—216no—
468SD16—316no—
469SD16—48no—
470D8—616no—
471C2×C8—22no—
472C2×C8—34no—
473C2×C8—64no—
474C2×C8—616no—
475C2×C8D3×D414yes—
476C2×C8C23×D312yes—
477C2×C8C23×D312yes—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—34no—
481C4⋊C4—32no—
482C4⋊C4—32no—
483C4⋊C4—68no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—64no—
487C22⋊C4—38no—
488C22⋊C4—68no—
489C22⋊C4—68no—
490C22⋊C4—64no—
491C42—64no—
492C42D3×D412yes—
493C42C23×D312yes—
494C22×C3—44no—
495C22×C3—44no—
496C22×C3—44no—
497C22×C3—88no—
498C22×C3D4212yes—
499C22×C3C2×#13512yes—
500C22×C3#57712yes—
501C22×C3C23×D411yescharacteristic
502C2×D3—24no—
503C2×D3—24no—
504C2×D3—24no—
505C2×D3—24no—
506C2×D3—48no—
507C2×D3—48no—
508C2×D3—42no—
509C2×D3—42no—
510C3×C4—22no—
511C3×C4—44no—
512C3×C4D4212yes—
513C3×C4C2×#13514yescomplemented
514C3×C4C23×D411yescharacteristic
515C3×C4C2×H(5,𝔽2)11yescharacteristic
516Dic3—24no—
517Dic3—24no—
518Dic3—42no—
519Dic3—42no—
520C23—24no—
521C23—34no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—62no—
526C23—64no—
527C23—64no—
528C23—64no—
529C23—64no—
530C23—122no—
531C23—122no—
532C23C2×D3×D411yescharacteristic
533Q8—24no—
534Q8—24no—
535Q8—34no—
536Q8—44no—
537Q8—62no—
538Q8—64no—
539D4—24no—
540D4—24no—
541D4—34no—
542D4—44no—
543D4—48no—
544D4—62no—
545D4—64no—
546D4—68no—
547D4—84no—
548D4—1216no—
549D4—1216no—
550D4—248no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—34no—
554C2×C4—44no—
555C2×C4—62no—
556C2×C4—64no—
557C2×C4—64no—
558C2×C4—64no—
559C2×C4—64no—
560C2×C4—68no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—1216no—
564C2×C4—1216no—
565C2×C4—122no—
566C2×C4—122no—
567C2×C4—128no—
568C2×C4#91614yescomplemented
569C2×C4C2×D3×D412yes—
570C2×C4C2×D3×D412yes—
571C2×C4C2×D3×D411yescharacteristic
572C2×C4C2×D3×D412yes—
573C2×C4#100412yes—
574C2×C4C24×D311yescharacteristic
575C8—22no—
576C8—64no—
577C8C2×D3×D414yes—
578C2×C3—84no—
579C2×C3#328414yescomplemented
580C2×C3C2×D4211yescharacteristic
581C2×C3C22×#13511yescharacteristic
582C2×C3C2×#57711yescharacteristic
583D3—44no—
584D3—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—64no—
589C22—64no—
590C22—64no—
591C22—64no—
592C22—88no—
593C22—128no—
594C22—128no—
595C22—122no—
596C22—122no—
597C22#571712yes—
598C22C2×#91612yes—
599C22#576212yes—
600C22C22×D3×D411yescharacteristic
601C4—22no—
602C4—44no—
603C4—64no—
604C4—64no—
605C4—122no—
606C4—122no—
607C4#571712yes—
608C4C2×#91614yescomplemented
609C4C22×D3×D411yescharacteristic
610C4C2×#100411yescharacteristic
611C3C2×#328411yescharacteristic, complemented, Sylow
612C2—84no—
613C2—124no—
614C2—124no—
615C2#8177214yesdirect product factor
616C2C2×#571711yescharacteristic
617C2C22×#91611yescharacteristic
618C2C2×#576211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#492C2
socleC23×C3C22×D4
hypercenterC2×#492D3
hypocenterC3C2×#3284
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#492

Complements

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

NH ≅ G/Ntype
1. C22×#916C2semidirect
2. C2×#5726C2semidirect
3. C2×#5718C2semidirect
4. C2×C3×#492C2semidirect
5. C2×#5171C2semidirect
6. C2×#5173C2semidirect
7. C2×#5172C2semidirect
8. C2×#5090C2semidirect
9. C2×#5072C2semidirect
10. #81772C2direct (N × H)
11. C2×C4×#204C2semidirect
12. C2×C8×D12C2semidirect
28. C2×#916C22semidirect
32. #5726C22semidirect
34. #5718C22semidirect
41. C3×#492C22semidirect
45. #5173C22semidirect
46. #5172C22semidirect
47. #5171C22semidirect
52. #5090C22semidirect
53. #5072C22semidirect
55. C4×#204C22semidirect
56. C8×D12C22semidirect
61. C2×#492D3semidirect
127. C2×#204D4semidirect
175. #492C2×D3semidirect
272. #204C2×D4semidirect
377. C2×C3×C4#135semidirect
513. C3×C4C2×#135semidirect
568. C2×C4#916semidirect
579. C2×C3#3284semidirect
608. C4C2×#916semidirect
611. C3C2×#3284semidirect
615. C2#81772direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#9162
C2×#57262
C2×#57182
C2×C3×#4922
C2×#51712
C2×#51732
C2×#51722
C2×#50902
C2×#50722
#817722
C2×C4×#2042
C2×C8×D122
C2×#32843