Finite Group Databasebeta

← C2×#18780 #6348696

Subgroups — C2×#18780 #6348696

Order 512.

3154 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#556C211yesmaximal, characteristic, complemented
2C2×#3226C211yesmaximal, characteristic, complemented
3C2×#3103C211yesmaximal, characteristic, complemented
4C2×#3036C211yesmaximal, characteristic
5#18780C2116yesmaximal, direct product factor
6C2×#1650C214yesmaximal, complemented
7C2×#1628C214yesmaximal, complemented
8C2×#1546C211yesmaximal, characteristic, complemented
9C2×#1487C211yesmaximal, characteristic, complemented
10C2×#1478C211yesmaximal, characteristic, complemented
11C22×AGL(1,ℤ/8)—21no—
12C2×#556—22no—
13C2×#556—24no—
14C2×#556—22no—
15C2×#556—24no—
16C2×#556—22no—
17C2×#556—24no—
18C2×#556C2214yes—
19C2×#556C2218yescomplemented
20C2×#532C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×#126—22no—
23C22×#124—21no—
24C22×C22⋊Q8—21no—
25C22×C4×Q8—21no—
26C22×C4×Q8C2211yescharacteristic
27C22×C4×D4C2211yescharacteristic
28C22×#117—21no—
29#3226—24no—
30#3226C2218yescomplemented
31C2×#500C2212yescomplemented
32C2×#486C2212yescomplemented
33#3103—24no—
34#3103C2218yescomplemented
35C2×#481—21no—
36C2×#480—22no—
37C2×#475C2212yescomplemented
38C2×#474—22no—
39#3036—24no—
40#3036C2218yes—
41C2×#460C2212yescomplemented
42C2×#463—22no—
43C2×#461C2212yes—
44C2×#459C2212yes—
45C2×#448—21no—
46C2×C4×SD16—22no—
47C2×#431C2211yescharacteristic
48C2×#431C2211yescharacteristic
49C2×#423C2211yescharacteristic
50C2×#417—21no—
51#1650C22132yescomplemented
52#1628C22132yescomplemented
53#1546C2218yescomplemented
54C2×#332C2214yescomplemented
55#1487C2218yescomplemented
56#1478C2218yescomplemented
57C2×Q8⋊C8C2214yescomplemented
58C2×D4⋊C8C2214yes—
59C22×#41—21no—
60C23×Q8—21no—
61C23×Q8D411yescharacteristic
62C23×D4D411yescharacteristic
63C24×C4—21no—
64C2×AGL(1,ℤ/8)—28no—
65C2×AGL(1,ℤ/8)—48no—
66C22×SD16—22no—
67C22×SD16—22no—
68C22×D8—22no—
69C22×M16—21no—
70#556—216no—
71#556—216no—
72#556—216no—
73#556—28no—
74#556—216no—
75#556—28no—
76#556—216no—
77#556—28no—
78#556—216no—
79#556C23116yes—
80#532—22no—
81#532—22no—
82#532—22no—
83#532C2314yes—
84C2×C4⋊Q8—21no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—22no—
88C2×C4⋊Q8D414yes—
89C2×C4⋊Q8D412yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2312yes—
92C2×C4⋊Q8C2314yes—
93C2×#127C2311yescharacteristic
94C2×#126—216no—
95C2×#126—28no—
96C2×#126—28no—
97C2×#126—216no—
98C2×#124—28no—
99C2×#124—24no—
100C2×#124—24no—
101C2×#124—28no—
102C2×#124—21no—
103C2×C22⋊Q8—28no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8—28no—
107C2×#121—21no—
108C2×#121—21no—
109C2×C4×Q8—28no—
110C2×C4×Q8—24no—
111C2×C4×Q8—24no—
112C2×C4×Q8—28no—
113C2×C4×Q8—21no—
114C2×C4×Q8—22no—
115C2×C4×Q8—21no—
116C2×C4×Q8—22no—
117C2×C4×Q8D414yes—
118C2×C4×Q8D412yes—
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2314yes—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—22no—
124C2×C4×D4—22no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4D412yes—
128C2×C4×D4D414yes—
129C2×C4×D4C2312yes—
130C2×C4×D4C2314yes—
131C2×#117—28no—
132C2×#117—24no—
133C2×#117—24no—
134C2×#117—28no—
135C22×C4⋊C4—22no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4D412yes—
139C22×C4⋊C4D411yescharacteristic
140C22×C4⋊C4D411yescharacteristic
141C22×C22⋊C4—22no—
142C22×C22⋊C4—22no—
143C22×C22⋊C4—21no—
144C22×C42—22no—
145C22×C42—21no—
146C22×C42C2311yescharacteristic
147#500—24no—
148#500C2318yescomplemented
149#486—24no—
150#486C2318yescomplemented
151#481—28no—
152#480—216no—
153#475—24no—
154#475C2318yescomplemented
155#474—216no—
156#463—216no—
157#461—24no—
158#461C2318yes—
159#460—24no—
160#460C2318yescomplemented
161#459—24no—
162#459C2318yes—
163#448—28no—
164C4×SD16—216no—
165#431—22no—
166#431—22no—
167#431C2314yes—
168#431C2314yes—
169C2×#107—24no—
170C2×#106—22no—
171#423—22no—
172#423C2314yes—
173C2×C4⋊C8—21no—
174C2×C4⋊C8—21no—
175C2×C4⋊C8C2312yes—
176C2×C4⋊C8C2312yes—
177#417—28no—
178C2×#103—22no—
179C2×#103—22no—
180C2×#102—24no—
181C2×#102—22no—
182C2×#102—22no—
183C2×C22⋊C8—21no—
184C2×C22⋊C8—21no—
185C2×C4×C8—21no—
186C2×C4×C8C2312yes—
187#332C23116yescomplemented
188Q8⋊C8C23116yescomplemented
189D4⋊C8C23116yes—
190C25—21no—
191C2×#41—24no—
192C2×#41—42no—
193C2×#41—44no—
194C2×#41—44no—
195C22×Q8—22no—
196C22×Q8—21no—
197C22×Q8—24no—
198C22×Q8—22no—
199C22×Q8—24no—
200C22×Q8—22no—
201C22×Q8—21no—
202C22×Q8—44no—
203C22×Q8—44no—
204C22×Q8—44no—
205C22×Q8C2×D412yes—
206C22×Q8C2×D414yes—
207C22×D4—22no—
208C22×D4—21no—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4—21no—
214C22×D4—41no—
215C22×D4C2×D412yes—
216C22×D4C2×D414yes—
217C23×C4—21no—
218C23×C4—22no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—24no—
222C23×C4—22no—
223C23×C4—24no—
224C23×C4—22no—
225C23×C4—48no—
226C23×C4—41no—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230AGL(1,ℤ/8)—432no—
231C2×SD16—28no—
232C2×SD16—28no—
233C2×SD16—48no—
234C2×SD16—48no—
235C2×D8—28no—
236C2×D8—48no—
237C2×M16—24no—
238C2×M16—44no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C4⋊Q8—24no—
243C4⋊Q8—24no—
244C4⋊Q8—24no—
245C4⋊Q8—24no—
246C4⋊Q8—22no—
247C4⋊Q8—24no—
248C4⋊Q8—22no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8C2×D418yes—
253C4⋊Q8C2×D414yes—
254C4⋊Q8C2412yes—
255C4⋊Q8C2414yes—
256#127—21no—
257#127—21no—
258#127—21no—
259#127C2412yes—
260#126—232no—
261#126—264no—
262#126—232no—
263#124—216no—
264#124—232no—
265#124—216no—
266#124—24no—
267#124—42no—
268C22⋊Q8—216no—
269C22⋊Q8—232no—
270C22⋊Q8—216no—
271#121—24no—
272#121—24no—
273#121—42no—
274#121—42no—
275C4×Q8—216no—
276C4×Q8—232no—
277C4×Q8—216no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—24no—
281C4×Q8—22no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—22no—
285C4×Q8C2×D418yes—
286C4×Q8C2×D414yes—
287C4×Q8C2414yes—
288C4×D4—24no—
289C4×D4—24no—
290C4×D4—24no—
291C4×D4—24no—
292C4×D4—22no—
293C4×D4—24no—
294C4×D4—22no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—42no—
298C4×D4—42no—
299C4×D4C2×D414yes—
300C4×D4C2×D418yes—
301C4×D4C2414yes—
302#117—216no—
303#117—232no—
304#117—216no—
305C2×C4⋊C4—216no—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D414yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D414yes—
327C2×C4⋊C4C2×D414yes—
328C2×C4⋊C4C2×D418yes—
329C2×C22⋊C4—216no—
330C2×C22⋊C4—28no—
331C2×C22⋊C4—216no—
332C2×C22⋊C4—28no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—41no—
338C2×C22⋊C4—41no—
339C2×C42—216no—
340C2×C42—28no—
341C2×C42—24no—
342C2×C42—22no—
343C2×C42—24no—
344C2×C42—22no—
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D412yes—
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2412yes—
353#107—216no—
354#106—28no—
355C4⋊C8—24no—
356C4⋊C8—24no—
357C4⋊C8—22no—
358C4⋊C8—22no—
359C4⋊C8C2414yes—
360C4⋊C8C2414yes—
361#103—28no—
362#103—28no—
363#102—28no—
364#102—28no—
365#102—216no—
366C22⋊C8—24no—
367C22⋊C8—24no—
368C4×C8—24no—
369C4×C8—22no—
370C4×C8C2414yes—
371C24—24no—
372C24—22no—
373C24—44no—
374C24—44no—
375C24—44no—
376C24—41no—
377C24#12011yescharacteristic
378#41—48no—
379#41—48no—
380#41—88no—
381C2×Q8—24no—
382C2×Q8—22no—
383C2×Q8—28no—
384C2×Q8—24no—
385C2×Q8—24no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—22no—
389C2×Q8—48no—
390C2×Q8—44no—
391C2×Q8—48no—
392C2×Q8—41no—
393C2×Q8—44no—
394C2×Q8—48no—
395C2×Q8—44no—
396C2×Q8—48no—
397C2×Q8—48no—
398C2×Q8—416no—
399C2×Q8—42no—
400C2×Q8—42no—
401C2×Q8C22×D414yes—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—28no—
406C2×D4—24no—
407C2×D4—28no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—41no—
414C2×D4—41no—
415C2×D4—41no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—48no—
421C2×D4—84no—
422C2×D4C22×D414yes—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—28no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—42no—
436C22×C4—48no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—41no—
445C22×C4—44no—
446C22×C4—84no—
447C22×C4#12012yes—
448C22×C4#12011yescharacteristic
449C22×C4#12011yescharacteristic
450C22×C4#12012yes—
451C22×C4#12012yes—
452C22×C4#12011yescharacteristic
453C22×C4#13514yes—
454C22×C4AGL(1,ℤ/8)14yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461SD16—416no—
462SD16—416no—
463D8—416no—
464M16—48no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—44no—
469C2×C8—44no—
470C2×C8—44no—
471C4⋊C4—232no—
472C4⋊C4—28no—
473C4⋊C4—24no—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—28no—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4C22×D414yes—
489C4⋊C4C22×D414yes—
490C4⋊C4C22×D418yes—
491C22⋊C4—232no—
492C22⋊C4—232no—
493C22⋊C4—28no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C42—232no—
499C42—28no—
500C42—24no—
501C42—24no—
502C42#12012yes—
503C42#12012yes—
504C42#12012yes—
505C42C22×D412yes—
506C42C22×D411yescharacteristic
507C42C22×D412yes—
508C42C22×D411yescharacteristic
509C42C22×D412yes—
510C42C22×D411yescharacteristic
511C42C2511yescharacteristic
512C23—22no—
513C23—22no—
514C23—24no—
515C23—48no—
516C23—48no—
517C23—48no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—42no—
522C23—816no—
523C23—82no—
524C23—82no—
525C23—82no—
526C23#44914yes—
527C23C2×#12012yes—
528C23C2×#12011yescharacteristic
529Q8—24no—
530Q8—24no—
531Q8—48no—
532Q8—416no—
533Q8—48no—
534Q8—44no—
535Q8—44no—
536Q8—44no—
537Q8—42no—
538Q8—42no—
539Q8—82no—
540D4—24no—
541D4—24no—
542D4—44no—
543D4—48no—
544D4—44no—
545D4—42no—
546D4—42no—
547D4—44no—
548D4—82no—
549D4—88no—
550D4—82no—
551D4—82no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—416no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—416no—
568C2×C4—48no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—42no—
575C2×C4—82no—
576C2×C4—88no—
577C2×C4#44914yes—
578C2×C4#44918yes—
579C2×C4C2×#12012yes—
580C2×C4C2×#12011yescharacteristic
581C2×C4C2×#12012yes—
582C2×C4C2×#12012yes—
583C2×C4C2×#12011yescharacteristic
584C2×C4C2×#12011yescharacteristic
585C2×C4C2×#13512yes—
586C2×C4C2×#13514yes—
587C2×C4C2×AGL(1,ℤ/8)12yes—
588C2×C4C2×AGL(1,ℤ/8)14yes—
589C2×C4#57514yes—
590C2×C4C23×D411yescharacteristic
591C2×C4C23×D411yescharacteristic
592C2×C4C23×D411yescharacteristic
593C8—44no—
594C8—44no—
595C8—44no—
596C22—28no—
597C22—22no—
598C22—22no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—816no—
603C22—88no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—164no—
608C22C2×#44914yes—
609C22C2×#44912yes—
610C22#300512yes—
611C22#302012yes—
612C22C22×#12011yescharacteristic
613C4—22no—
614C4—21no—
615C4—22no—
616C4—21no—
617C4—48no—
618C4—44no—
619C4—44no—
620C4—44no—
621C4—82no—
622C4C2×#44914yes—
623C4C2×#44912yes—
624C4#300512yes—
625C4C22×#13511yescharacteristic
626C4C22×AGL(1,ℤ/8)11yescharacteristic
627C4C2×#57512yes—
628C2—24no—
629C2—84no—
630C2—162no—
631C2#1878014yesdirect product factor
632C2C22×#44911yescharacteristic
633C2C2×#300511yescharacteristic
634C2C2×#302011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
prime radicalC2×#3036C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#556C2semidirect
2. C2×#3226C2semidirect
3. C2×#3103C2semidirect
5. #18780C2direct (N × H)
6. C2×#1650C2semidirect
7. C2×#1628C2semidirect
8. C2×#1546C2semidirect
9. C2×#1487C2semidirect
10. C2×#1478C2semidirect
19. C2×#556C22semidirect
30. #3226C22semidirect
31. C2×#500C22semidirect
32. C2×#486C22semidirect
34. #3103C22semidirect
37. C2×#475C22semidirect
41. C2×#460C22semidirect
51. #1650C22semidirect
52. #1628C22semidirect
53. #1546C22semidirect
54. C2×#332C22semidirect
55. #1487C22semidirect
56. #1478C22semidirect
57. C2×Q8⋊C8C22semidirect
148. #500C23semidirect
150. #486C23semidirect
154. #475C23semidirect
160. #460C23semidirect
187. #332C23semidirect
188. Q8⋊C8C23semidirect
631. C2#18780direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5562
C2×#32262
C2×#31032
C2×#30362
#187802
C2×#16502
C2×#16282
C2×#15462
C2×#14872
C2×#14782