Finite Group Databasebeta

← C2×#13016 #494511

Subgroups — C2×#13016 #494511

Order 512.

1288 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#508C211yesmaximal, characteristic
2C2×#2957C211yesmaximal, characteristic, complemented
3C2×#2145C211yesmaximal, characteristic, complemented
4C2×#2143C211yesmaximal, characteristic, complemented
5C2×#1976C211yesmaximal, characteristic, complemented
6#13016C218yesmaximal, direct product factor
7C2×#1382C211yesmaximal, characteristic, complemented
8C2×#1379C211yesmaximal, characteristic, complemented
9C22×#135C2211yescharacteristic
10C22×#135C2211yescharacteristic
11C2×#568—21no—
12C2×#508—22no—
13C2×#508—22no—
14C2×#508—22no—
15C2×#508—22no—
16C2×#508C414yescomplemented
17C2×#508C2214yes—
18C22×Q32C411yescharacteristic, complemented
19C22×SD32C411yescharacteristic, complemented
20C22×SD32C411yescharacteristic, complemented
21C22×D16C411yescharacteristic, complemented
22C22×M32C2211yescharacteristic
23C23×C16C2211yescharacteristic
24#2957—22no—
25#2957—22no—
26#2957—22no—
27#2957C2214yescomplemented
28C2×#444—22no—
29C2×#443—22no—
30C22×#108C2211yescharacteristic
31C2×#427C2211yescharacteristic
32C2×#421—21no—
33C2×#421—21no—
34C2×#410—21no—
35C2×#405C2211yescharacteristic
36#2145—22no—
37#2145C2214yescomplemented
38C2×#360—22no—
39#2143—22no—
40#2143C2214yescomplemented
41C2×#359—22no—
42#1976—22no—
43#1976C2214yescomplemented
44#1382C2214yescomplemented
45#1379C2214yescomplemented
46C22×#41—21no—
47C22×#41—21no—
48C2×#135—21no—
49C2×#135—21no—
50C2×#135—21no—
51C2×#135—21no—
52C2×#135—21no—
53C2×#135—21no—
54C2×#135—21no—
55C2×#135—21no—
56C2×#135—21no—
57C2×#135—21no—
58C2×#135—21no—
59C2×#135—21no—
60C2×#135—21no—
61C2×#135—21no—
62C2×#135—21no—
63C2×#135—21no—
64C2×#135C2×C412yes—
65C2×#135C2×C412yes—
66C2×#135D412yes—
67C2×#135D412yes—
68C2×#135D412yes—
69C2×#135D412yes—
70C2×#135C2312yes—
71C2×#135C2312yes—
72C22×Q16C2×C411yescharacteristic
73C22×Q16C2×C411yescharacteristic
74C22×SD16—21no—
75C22×SD16—21no—
76C22×D8C2×C411yescharacteristic
77C22×D8C2×C411yescharacteristic
78#568—24no—
79#568—24no—
80#568—42no—
81#568—42no—
82C2×#131—24no—
83C2×#131—42no—
84C22×M16—21no—
85C22×M16—42no—
86C22×M16D411yescharacteristic
87C23×C8C2311yescharacteristic
88C2×#117—21no—
89C2×#117D411yescharacteristic
90#508—28no—
91#508—28no—
92#508—28no—
93#508—28no—
94C2×Q32—22no—
95C2×Q32—22no—
96C2×Q32C2×C414yescomplemented
97C2×SD32—22no—
98C2×SD32—22no—
99C2×SD32—22no—
100C2×SD32—22no—
101C2×SD32C2×C414yescomplemented
102C2×SD32C2×C414yescomplemented
103C2×D16—22no—
104C2×D16—22no—
105C2×D16C2×C414yescomplemented
106C2×M32—21no—
107C2×M32—21no—
108C2×M32—21no—
109C2×M32—21no—
110C2×M32—21no—
111C2×M32C2312yes—
112C22×C16—21no—
113C22×C16—21no—
114C22×C16—21no—
115C22×C16—21no—
116C22×C16—21no—
117C22×C16C2×C411yescharacteristic
118C22×C16C2×C411yescharacteristic
119C22×C16C2×C412yes—
120C22×C16C2312yes—
121#444—28no—
122#444—28no—
123#443—28no—
124#443—28no—
125C2×#108—21no—
126C2×#108—21no—
127C2×#108—21no—
128C2×#108D412yes—
129C2×#108D412yes—
130C2×#108C2312yes—
131#427—21no—
132#427—21no—
133#427—21no—
134#427C2312yes—
135C2×#107—21no—
136C2×#106—21no—
137#421—24no—
138#421—24no—
139#421—24no—
140#421—42no—
141C2×#104—24no—
142C2×#104—42no—
143#410—24no—
144#410—42no—
145C2×#99—42no—
146#405—21no—
147#405—21no—
148#405D412yes—
149#405C2312yes—
150C2×#97—21no—
151C2×C4×C8—21no—
152#360—28no—
153#359—28no—
154C2×#41—22no—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41—22no—
160C2×#41—22no—
161C2×#41—22no—
162C2×#41—41no—
163C2×#41—41no—
164C2×#41—41no—
165C2×#41—41no—
166C2×#41—41no—
167C2×#41—41no—
168C2×#41—41no—
169C2×#41—41no—
170C2×#41—41no—
171C2×#41—41no—
172C2×#41—41no—
173C2×#41—41no—
174C2×#41—41no—
175C2×#41—41no—
176C2×#41—41no—
177C2×#41—41no—
178C22×Q8—21no—
179C22×Q8—21no—
180C22×D4—21no—
181C22×D4—21no—
182C22×D4—41no—
183C22×D4—41no—
184C23×C4—41no—
185C23×C4—41no—
186C23×C4C2×D411yescharacteristic
187#135—22no—
188#135—22no—
189#135—22no—
190#135—22no—
191#135—22no—
192#135—22no—
193#135—22no—
194#135—22no—
195#135—22no—
196#135—22no—
197#135—22no—
198#135—22no—
199#135—22no—
200#135—22no—
201#135—22no—
202#135—22no—
203#135—22no—
204#135—22no—
205#135—22no—
206#135—22no—
207#135—41no—
208#135—41no—
209#135—41no—
210#135—41no—
211#135—41no—
212#135—41no—
213#135—41no—
214#135—41no—
215#135C22⋊C414yes—
216#135C22⋊C414yes—
217#135C2×D414yes—
218#135C2×D414yes—
219C2×Q16—21no—
220C2×Q16—21no—
221C2×Q16—21no—
222C2×Q16—21no—
223C2×Q16—21no—
224C2×Q16—21no—
225C2×Q16—21no—
226C2×Q16—21no—
227C2×Q16C22⋊C412yes—
228C2×Q16C22⋊C412yes—
229C2×Q16C22×C412yes—
230C2×Q16C22×C412yes—
231C2×SD16—22no—
232C2×SD16—22no—
233C2×SD16—22no—
234C2×SD16—22no—
235C2×SD16—22no—
236C2×SD16—22no—
237C2×SD16—22no—
238C2×SD16—22no—
239C2×SD16—41no—
240C2×SD16—41no—
241C2×SD16—41no—
242C2×SD16—41no—
243C2×D8—21no—
244C2×D8—21no—
245C2×D8—21no—
246C2×D8—21no—
247C2×D8—21no—
248C2×D8—21no—
249C2×D8—21no—
250C2×D8—21no—
251C2×D8C22⋊C412yes—
252C2×D8C22⋊C412yes—
253C2×D8C22×C412yes—
254C2×D8C22×C412yes—
255#131—216no—
256#131—48no—
257#131—48no—
258#131—48no—
259C2×M16—21no—
260C2×M16—21no—
261C2×M16—21no—
262C2×M16—22no—
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—44no—
266C2×M16—48no—
267C2×M16—48no—
268C2×M16—41no—
269C2×M16—41no—
270C2×M16—41no—
271C2×M16D812yes—
272C2×M16D812yes—
273C2×M16C2×D412yes—
274C22×C8—21no—
275C22×C8—44no—
276C22×C8—41no—
277C22×C8C22×C411yescharacteristic
278C22×C8C22×C411yescharacteristic
279C22×C8C2×D411yescharacteristic
280C22×C8C2×D411yescharacteristic
281C22×C8C2×D411yescharacteristic
282C22×C8C2×D411yescharacteristic
283C22×C8C2×D411yescharacteristic
284C22×C8C2×D411yescharacteristic
285C22×C8C2×D411yescharacteristic
286C22×C8#4111yescharacteristic
287C22×C8#4111yescharacteristic
288C22×C8#4111yescharacteristic
289C22×C8#4111yescharacteristic
290C22×C8C2411yescharacteristic
291#117—21no—
292#117—21no—
293#117—21no—
294#117—22no—
295#117—41no—
296#117—41no—
297#117—41no—
298#117C2×D412yes—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—41no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—41no—
303C2×C42—21no—
304C2×C42—41no—
305Q32—28no—
306SD32—28no—
307SD32—28no—
308D16—28no—
309M32—22no—
310M32—22no—
311M32—41no—
312M32—41no—
313C2×C16—21no—
314C2×C16—21no—
315C2×C16—21no—
316C2×C16—21no—
317C2×C16—22no—
318C2×C16—22no—
319C2×C16—22no—
320C2×C16—22no—
321C2×C16—22no—
322C2×C16—41no—
323C2×C16—41no—
324C2×C16C22×C412yes—
325C2×C16C22×C412yes—
326#108—22no—
327#108—22no—
328#108—22no—
329#108C2×D414yes—
330#107—22no—
331#107—41no—
332#106—22no—
333#106—41no—
334#104—216no—
335#104—48no—
336#99—48no—
337#97—22no—
338#97—22no—
339C4×C8—22no—
340C4×C8—22no—
341C24—41no—
342C24—41no—
343C24AGL(1,ℤ/8)11yescharacteristic
344#41—24no—
345#41—24no—
346#41—24no—
347#41—24no—
348#41—42no—
349#41—42no—
350#41—42no—
351#41—42no—
352#41—42no—
353#41—42no—
354#41—42no—
355#41—42no—
356#41—42no—
357#41—42no—
358#41—42no—
359#41—42no—
360#41—42no—
361#41—42no—
362#41—42no—
363#41—42no—
364#41—42no—
365#41—42no—
366#41—42no—
367#41—42no—
368#41—81no—
369#41—81no—
370#41—81no—
371#41—81no—
372#41—81no—
373#41—81no—
374#41—81no—
375#41—81no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—41no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—41no—
386C2×Q8—41no—
387C2×Q8—41no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—41no—
401C2×D4—41no—
402C2×D4—41no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—81no—
409C2×D4—81no—
410C2×D4—81no—
411C2×D4—81no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—81no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4C4×D411yescharacteristic
429C22×C4C4×D411yescharacteristic
430C22×C4#12011yescharacteristic
431C22×C4#12011yescharacteristic
432C22×C4#12111yescharacteristic
433C22×C4#12111yescharacteristic
434C22×C4C2×D811yescharacteristic
435C22×C4C2×D811yescharacteristic
436C22×C4#13511yescharacteristic
437C22×C4#13511yescharacteristic
438C22×C4AGL(1,ℤ/8)11yescharacteristic
439C22×C4AGL(1,ℤ/8)11yescharacteristic
440C22×C4AGL(1,ℤ/8)11yescharacteristic
441C22×C4C22×D411yescharacteristic
442Q16—22no—
443Q16—22no—
444Q16—22no—
445Q16—22no—
446Q16—22no—
447Q16—41no—
448Q16—41no—
449Q16C2×C22⋊C414yes—
450SD16—24no—
451SD16—24no—
452SD16—24no—
453SD16—24no—
454SD16—42no—
455SD16—42no—
456SD16—42no—
457SD16—42no—
458D8—22no—
459D8—22no—
460D8—22no—
461D8—22no—
462D8—22no—
463D8—41no—
464D8—41no—
465D8C2×C22⋊C414yes—
466M16—22no—
467M16—22no—
468M16—22no—
469M16—24no—
470M16—416no—
471M16—42no—
472M16—42no—
473M16—42no—
474M16C2×D814yes—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—416no—
479C2×C8—42no—
480C2×C8—42no—
481C2×C8—42no—
482C2×C8C2×C22⋊C411yescharacteristic
483C2×C8C2×C22⋊C411yescharacteristic
484C2×C8C2×C22⋊C411yescharacteristic
485C2×C8C2×C22⋊C411yescharacteristic
486C2×C8C4×D411yescharacteristic
487C2×C8C4×D411yescharacteristic
488C2×C8C4×D411yescharacteristic
489C2×C8C4×D411yescharacteristic
490C2×C8C4×D411yescharacteristic
491C2×C8C4×D411yescharacteristic
492C2×C8#12011yescharacteristic
493C2×C8#12011yescharacteristic
494C2×C8#12111yescharacteristic
495C2×C8#12111yescharacteristic
496C2×C8#12111yescharacteristic
497C2×C8#12111yescharacteristic
498C2×C8#12111yescharacteristic
499C2×C8#12111yescharacteristic
500C2×C8#12311yescharacteristic
501C2×C8#12311yescharacteristic
502C2×C8#12311yescharacteristic
503C2×C8#12311yescharacteristic
504C2×C8C23×C411yescharacteristic
505C2×C8C22×D411yescharacteristic
506C2×C8C22×D411yescharacteristic
507C2×C8C22×D411yescharacteristic
508C2×C8C2×#4111yescharacteristic
509C2×C8C2×#4111yescharacteristic
510C4⋊C4—22no—
511C4⋊C4—41no—
512C4⋊C4—42no—
513C4⋊C4—81no—
514C22⋊C4—22no—
515C22⋊C4—41no—
516C22⋊C4—42no—
517C22⋊C4—81no—
518C42—22no—
519C42—41no—
520C42—42no—
521C42—81no—
522C16—22no—
523C16—22no—
524C16—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—81no—
531C23—81no—
532C23—162no—
533C23—162no—
534C23#44211yescharacteristic
535C23#44211yescharacteristic
536C23#44911yescharacteristic
537C23#44911yescharacteristic
538C23#45911yescharacteristic
539C23#45911yescharacteristic
540C23C2×AGL(1,ℤ/8)11yescharacteristic
541Q8—24no—
542Q8—42no—
543Q8—42no—
544Q8—42no—
545Q8—42no—
546Q8—42no—
547Q8—81no—
548Q8—81no—
549D4—24no—
550D4—44no—
551D4—44no—
552D4—44no—
553D4—44no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—82no—
560D4—82no—
561D4—82no—
562D4—82no—
563D4—81no—
564D4—81no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—82no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—162no—
583C2×C4#38611yescharacteristic
584C2×C4#38611yescharacteristic
585C2×C4#38611yescharacteristic
586C2×C4#38611yescharacteristic
587C2×C4C4×D811yescharacteristic
588C2×C4C4×D811yescharacteristic
589C2×C4C4×D811yescharacteristic
590C2×C4C4×D811yescharacteristic
591C2×C4#44211yescharacteristic
592C2×C4#44211yescharacteristic
593C2×C4#44711yescharacteristic
594C2×C4#44711yescharacteristic
595C2×C4#44711yescharacteristic
596C2×C4#44711yescharacteristic
597C2×C4#44911yescharacteristic
598C2×C4#44911yescharacteristic
599C2×C4#45911yescharacteristic
600C2×C4#45911yescharacteristic
601C2×C4#46311yescharacteristic
602C2×C4#46311yescharacteristic
603C2×C4#46311yescharacteristic
604C2×C4#46311yescharacteristic
605C2×C4C2×C4×D411yescharacteristic
606C2×C4C2×#12011yescharacteristic
607C2×C4C2×#12111yescharacteristic
608C2×C4C22×D811yescharacteristic
609C2×C4C2×#13511yescharacteristic
610C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
611C8—24no—
612C8—44no—
613C8#38611yescharacteristic
614C8#38611yescharacteristic
615C8#38611yescharacteristic
616C8#38611yescharacteristic
617C8C22×C22⋊C411yescharacteristic
618C8C2×C4×D411yescharacteristic
619C8C2×#12111yescharacteristic
620C8C2×#12311yescharacteristic
621C22—22no—
622C22—22no—
623C22—22no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—44no—
629C22—84no—
630C22—82no—
631C22—162no—
632C22—162no—
633C22—162no—
634C22—162no—
635C22—162no—
636C22#187911yescharacteristic
637C22#187911yescharacteristic
638C22#187911yescharacteristic
639C22#187911yescharacteristic
640C22C2×#44211yescharacteristic
641C22C2×#44911yescharacteristic
642C22C2×#45911yescharacteristic
643C4—44no—
644C4—82no—
645C4—82no—
646C4—162no—
647C4#187911yescharacteristic
648C4#187911yescharacteristic
649C4#187911yescharacteristic
650C4#187911yescharacteristic
651C4C2×#38611yescharacteristic
652C4C2×C4×D811yescharacteristic
653C4C2×#44711yescharacteristic
654C4C2×#46311yescharacteristic
655C2—21no—
656C2—21no—
657C2—21no—
658C2—21no—
659C2—21no—
660C2—21no—
661C2—84no—
662C2—162no—
663C2#1301612yesdirect product factor
664C2C2×#187911yescharacteristic
subgroupG/H
centerC2×C4#447
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
socleC22#1879
prime radicalC22×#508C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2957C2semidirect
3. C2×#2145C2semidirect
4. C2×#2143C2semidirect
5. C2×#1976C2semidirect
6. #13016C2direct (N × H)
7. C2×#1382C2semidirect
8. C2×#1379C2semidirect
16. C2×#508C4semidirect
18. C22×Q32C4semidirect
19. C22×SD32C4semidirect
20. C22×SD32C4semidirect
21. C22×D16C4semidirect
27. #2957C22semidirect
37. #2145C22semidirect
40. #2143C22semidirect
43. #1976C22semidirect
44. #1382C22semidirect
45. #1379C22semidirect
96. C2×Q32C2×C4semidirect
101. C2×SD32C2×C4semidirect
102. C2×SD32C2×C4semidirect
105. C2×D16C2×C4semidirect
663. C2#13016direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5082
C2×#29572
C2×#21452
C2×#21432
C2×#19762
#130162
C2×#13822
C2×#13792