Finite Group Databasebeta

← C2×#12806 #493732

Subgroups — C2×#12806 #493732

Order 512.

1132 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2930C211yesmaximal, characteristic
2C2×#2120C212yesmaximal
3C2×#1813C211yesmaximal, characteristic, complemented
4C2×#1812C211yesmaximal, characteristic, complemented
5#12806C218yesmaximal, direct product factor
6C2×#1395C212yesmaximal, complemented
7C22×#131C2211yescharacteristic
8C23×M16C411yescharacteristic, complemented
9C2×#525C2211yescharacteristic
10#2930—22no—
11#2930—22no—
12#2930C414yescomplemented
13#2930C2214yes—
14C2×#436—21no—
15C2×#436—21no—
16C2×#435—21no—
17C2×#435—21no—
18C22×#108—21no—
19C2×#427—21no—
20C2×#424C411yescharacteristic, complemented
21C2×#413C2212yes—
22C2×#410C2212yes—
23C2×#408C2211yescharacteristic
24C2×#407C411yescharacteristic, complemented
25C2×#405C411yescharacteristic, complemented
26#2120—24no—
27#2120C2218yes—
28C2×#352—22no—
29C2×AΣL(1,𝔽16)—22no—
30#1813—22no—
31#1813C2214yescomplemented
32#1812—22no—
33#1812C2214yescomplemented
34C2×#338—21no—
35C2×H(3,ℤ/4)—21no—
36#1395C2218yescomplemented
37C22×#41C2311yescharacteristic
38C24×C4C2×C411yescharacteristic
39C2×#131—41no—
40C2×#131—41no—
41C2×#131—42no—
42C2×#131C2×C412yes—
43C2×#131D412yes—
44C2×#131D412yes—
45C2×#131C2312yes—
46C22×M16—21no—
47C22×M16—22no—
48C22×M16—21no—
49C22×M16—22no—
50C22×M16—41no—
51C22×M16—41no—
52C22×M16—42no—
53C22×M16C2×C411yescharacteristic
54C22×M16C2×C411yescharacteristic
55C22×M16C2×C414yescomplemented
56C22×M16D411yescharacteristic
57C22×M16D411yescharacteristic
58C23×C8C2×C411yescharacteristic
59C23×C8C2×C411yescharacteristic
60C23×C8D411yescharacteristic
61C23×C8D411yescharacteristic
62#525—21no—
63#525—21no—
64#525—21no—
65#525—21no—
66#525C2×C412yes—
67#525D412yes—
68#525Q812yes—
69#525C2312yes—
70C2×#123D411yescharacteristic
71C2×#123Q811yescharacteristic
72C2×C22⋊Q8—21no—
73C2×#121—21no—
74C2×#120D411yescharacteristic
75C2×#120Q811yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×#117—22no—
79C2×#117C2×C411yescharacteristic
80#436—24no—
81#436—24no—
82#436—24no—
83#436—24no—
84#435—24no—
85#435—24no—
86#435—24no—
87#435—24no—
88C2×#108—24no—
89C2×#108—41no—
90C2×#108—41no—
91C2×#108—42no—
92#427—24no—
93#427—42no—
94C2×#107—41no—
95C2×#106—41no—
96#424—22no—
97#424C2×C414yescomplemented
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100#413—22no—
101#413—22no—
102#413—22no—
103#413C2314yes—
104C2×#101—22no—
105C2×#100—22no—
106#410—22no—
107#410—22no—
108#410—22no—
109#410C2314yes—
110C2×#99—22no—
111C2×#99—22no—
112#408—21no—
113#408—21no—
114#408C2×C412yes—
115#408C2312yes—
116#407—22no—
117#407C2×C414yescomplemented
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122#405—22no—
123#405C2×C414yescomplemented
124C2×#97—21no—
125C2×C4×C8—21no—
126#352—28no—
127AΣL(1,𝔽16)—28no—
128#338—24no—
129H(3,ℤ/4)—24no—
130C25C4⋊C411yescharacteristic
131C2×#41—41no—
132C2×#41—41no—
133C2×#41—41no—
134C2×#41—41no—
135C2×#41C22×C411yescharacteristic
136C2×#41C2×D411yescharacteristic
137C2×#41C2×D411yescharacteristic
138C2×#41C2×D411yescharacteristic
139C2×#41C2×Q811yescharacteristic
140C2×#41#4111yescharacteristic
141C2×#41#4111yescharacteristic
142C2×#41C2411yescharacteristic
143C22×Q8#4111yescharacteristic
144C22×D4—21no—
145C22×D4—21no—
146C22×D4C2×D411yescharacteristic
147C22×D4C2×Q811yescharacteristic
148C22×D4#4111yescharacteristic
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—41no—
154C23×C4—41no—
155C23×C4—41no—
156C23×C4—41no—
157C23×C4C4⋊C411yescharacteristic
158C23×C4C4⋊C412yes—
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163#131—42no—
164#131—42no—
165#131—42no—
166#131—42no—
167#131—42no—
168#131—42no—
169#131—42no—
170#131—42no—
171C2×M16—21no—
172C2×M16—24no—
173C2×M16—24no—
174C2×M16—24no—
175C2×M16—41no—
176C2×M16—41no—
177C2×M16—41no—
178C2×M16—41no—
179C2×M16—41no—
180C2×M16—41no—
181C2×M16—42no—
182C2×M16—42no—
183C2×M16—44no—
184C2×M16—44no—
185C2×M16—44no—
186C2×M16—44no—
187C2×M16—42no—
188C2×M16—41no—
189C2×M16—41no—
190C2×M16—42no—
191C2×M16—41no—
192C2×M16—41no—
193C2×M16—42no—
194C2×M16—42no—
195C2×M16—42no—
196C2×M16—42no—
197C2×M16C22⋊C412yes—
198C2×M16C22⋊C412yes—
199C2×M16C22⋊C412yes—
200C2×M16C22×C412yes—
201C2×M16C22×C412yes—
202C2×M16C2×D412yes—
203C2×M16C2×D412yes—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—22no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—22no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—22no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—41no—
217C22×C8—41no—
218C22×C8—42no—
219C22×C8C22⋊C412yes—
220C22×C8C22⋊C411yescharacteristic
221C22×C8C22⋊C411yescharacteristic
222C22×C8C22⋊C412yes—
223C22×C8C22⋊C412yes—
224C22×C8C22⋊C411yescharacteristic
225C22×C8C22⋊C411yescharacteristic
226C22×C8C22×C412yes—
227C22×C8C22×C412yes—
228C22×C8C2×D412yes—
229C22×C8C2×D412yes—
230#123—21no—
231#123—21no—
232#123—21no—
233#123—21no—
234#123C4⋊C412yes—
235#123C4⋊C412yes—
236#123C2×D412yes—
237#123C2×Q812yes—
238C22⋊Q8—22no—
239C22⋊Q8—22no—
240C22⋊Q8—22no—
241C22⋊Q8—22no—
242#121—22no—
243#121—22no—
244#121—22no—
245#121—22no—
246#120—21no—
247#120—21no—
248#120—21no—
249#120—21no—
250#120C4⋊C412yes—
251#120C4⋊C412yes—
252#120C2×D412yes—
253#120C2×Q812yes—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4—22no—
262#117—21no—
263#117—21no—
264#117—24no—
265#117—42no—
266#117—42no—
267#117—42no—
268#117C4⋊C412yes—
269#117C22×C412yes—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—42no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4C4⋊C411yescharacteristic
280C2×C22⋊C4C4⋊C411yescharacteristic
281C2×C42—21no—
282C2×C42—42no—
283#108—44no—
284#108—44no—
285#107—44no—
286#106—44no—
287C4⋊C8—24no—
288C4⋊C8—24no—
289#101—24no—
290#101—42no—
291#100—24no—
292#100—42no—
293#99—24no—
294#99—24no—
295#99—42no—
296#99—42no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8—22no—
300C22⋊C8—22no—
301C22⋊C8—24no—
302C22⋊C8—24no—
303#97—24no—
304C4×C8—24no—
305C24—21no—
306C24—21no—
307C24—41no—
308C24—41no—
309C24—82no—
310C24C2×C4⋊C411yescharacteristic
311C24C2×C4⋊C412yes—
312C24#12111yescharacteristic
313C24C22⋊Q811yescharacteristic
314#41—41no—
315#41—41no—
316#41—41no—
317#41—41no—
318#41—41no—
319#41—41no—
320#41—41no—
321#41—41no—
322#41—41no—
323#41—41no—
324#41—41no—
325#41—41no—
326#41—41no—
327#41—41no—
328#41—41no—
329#41—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8#11711yescharacteristic
333C2×Q8#12111yescharacteristic
334C2×Q8C22⋊Q811yescharacteristic
335C2×Q8C2×#4111yescharacteristic
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—82no—
347C2×D4—82no—
348C2×D4C2×C4⋊C411yescharacteristic
349C2×D4C2×C4⋊C411yescharacteristic
350C2×D4#11711yescharacteristic
351C2×D4#12111yescharacteristic
352C2×D4#12111yescharacteristic
353C2×D4#12111yescharacteristic
354C2×D4C22⋊Q811yescharacteristic
355C2×D4C22⋊Q811yescharacteristic
356C2×D4C22⋊Q811yescharacteristic
357C2×D4C22×D411yescharacteristic
358C2×D4C22×Q811yescharacteristic
359C2×D4C2×#4111yescharacteristic
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—82no—
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C4⋊C411yescharacteristic
402C22×C4C2×C4⋊C412yes—
403C22×C4#11711yescharacteristic
404C22×C4#11711yescharacteristic
405C22×C4#12111yescharacteristic
406C22×C4#12111yescharacteristic
407C22×C4#12111yescharacteristic
408C22×C4C22⋊Q811yescharacteristic
409C22×C4C22⋊Q811yescharacteristic
410C22×C4C22⋊Q811yescharacteristic
411C22×C4C23×C411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414M16—42no—
415M16—42no—
416M16—42no—
417M16—42no—
418M16—42no—
419M16—42no—
420M16—48no—
421M16—42no—
422M16—44no—
423M16—44no—
424M16—42no—
425M16—44no—
426M16—44no—
427M16—82no—
428M16—82no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—42no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—42no—
455C2×C8—44no—
456C2×C8—82no—
457C2×C8—82no—
458C2×C8#9911yescharacteristic
459C2×C8#9911yescharacteristic
460C2×C8#9912yes—
461C2×C8C2×C22⋊C412yes—
462C2×C8C2×C22⋊C412yes—
463C2×C8C2×C22⋊C412yes—
464C2×C8C2×C22⋊C412yes—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—44no—
472C4⋊C4—82no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—44no—
484C22⋊C4—82no—
485C22⋊C4C2×C4⋊C412yes—
486C22⋊C4C2×C4⋊C412yes—
487C42—22no—
488C42—22no—
489C42—44no—
490C42—82no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—22no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—82no—
508C23—82no—
509C23—84no—
510C23—84no—
511C23#38011yescharacteristic
512C23#38011yescharacteristic
513C23#38011yescharacteristic
514C23#38011yescharacteristic
515C23C22×C4⋊C411yescharacteristic
516C23C2×#12111yescharacteristic
517C23C2×C22⋊Q811yescharacteristic
518Q8—41no—
519Q8—41no—
520Q8—41no—
521Q8—41no—
522D4—41no—
523D4—41no—
524D4—41no—
525D4—41no—
526D4—41no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—41no—
532D4—41no—
533D4—41no—
534D4—82no—
535D4—82no—
536D4—82no—
537D4—82no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—84no—
584C2×C4—164no—
585C2×C4#38011yescharacteristic
586C2×C4#38011yescharacteristic
587C2×C4#38011yescharacteristic
588C2×C4#38011yescharacteristic
589C2×C4C2×#9911yescharacteristic
590C2×C4C2×#9911yescharacteristic
591C2×C4#41011yescharacteristic
592C2×C4#41011yescharacteristic
593C2×C4C22×C22⋊C411yescharacteristic
594C2×C4C2×#11711yescharacteristic
595C2×C4C2×#12111yescharacteristic
596C2×C4C2×C22⋊Q811yescharacteristic
597C8—22no—
598C8—22no—
599C8—22no—
600C8—42no—
601C8—42no—
602C8—84no—
603C8C2×#9912yes—
604C8C2×#9912yes—
605C22—22no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—82no—
628C22—84no—
629C22—84no—
630C22—82no—
631C22#179911yescharacteristic
632C22#179911yescharacteristic
633C22C2×#38011yescharacteristic
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—21no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—164no—
644C4#179911yescharacteristic
645C4#179911yescharacteristic
646C4C22×#9911yescharacteristic
647C4C2×#41011yescharacteristic
648C2—21no—
649C2—21no—
650C2—41no—
651C2—41no—
652C2—41no—
653C2—41no—
654C2—41no—
655C2—41no—
656C2—82no—
657C2#1280612yesdirect product factor
658C2C2×#179911yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC22×C4C23×C4
Frattini, cosocleC2×#41C24
socleC22#1799
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1813C2semidirect
4. C2×#1812C2semidirect
5. #12806C2direct (N × H)
6. C2×#1395C2semidirect
8. C23×M16C4semidirect
12. #2930C4semidirect
20. C2×#424C4semidirect
24. C2×#407C4semidirect
25. C2×#405C4semidirect
31. #1813C22semidirect
33. #1812C22semidirect
36. #1395C22semidirect
55. C22×M16C2×C4semidirect
97. #424C2×C4semidirect
117. #407C2×C4semidirect
123. #405C2×C4semidirect
657. C2#12806direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#29302
C2×#21202
C2×#18132
C2×#18122
#128062
C2×#13952