Finite Group Databasebeta

← C2×#24999 #6447946

Subgroups — C2×#24999 #6447946

Order 512.

1880 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3#24999C2116yesmaximal, direct product factor
4C2×#2100C211yesmaximal, characteristic, complemented
5C2×#2101C211yesmaximal, characteristic, complemented
6C2×#2073C211yesmaximal, characteristic
7C2×#2071C211yesmaximal, characteristic
8C2×#1984C211yesmaximal, characteristic, complemented
9C2×#1949C211yesmaximal, characteristic, complemented
10C2×#1926C211yesmaximal, characteristic
11C2×#1925C211yesmaximal, characteristic
12C2×#1920C211yesmaximal, characteristic, complemented
13C2×#1918C211yesmaximal, characteristic, complemented
14C2×#1835C211yesmaximal, characteristic, complemented
15C2×#1804C211yesmaximal, characteristic
16C2×C4×#102C211yesmaximal, characteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27#2580—24no—
28#2580C2218yes—
29C2×#399—21no—
30C2×#393—21no—
31#2355—24no—
32#2355C2218yescomplemented
33C2×#391—21no—
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#387—21no—
39C2×#387—21no—
40C2×#385—21no—
41C2×#385—21no—
42C2×#384C2211yescharacteristic
43C2×#382C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×C4×C4⋊C4—21no—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2101C2218yescomplemented
51#2100C2218yescomplemented
52#2073C2218yes—
53#2071C2218yes—
54#1984C2218yescomplemented
55#1949C2218yescomplemented
56#1926C2218yes—
57#1925C2218yes—
58#1920C2218yescomplemented
59#1918C2218yescomplemented
60#1835C2218yescomplemented
61#1804C2218yes—
62C4×#102C2218yes—
63C2×#340C2211yescharacteristic
64C2×#340C2211yescharacteristic
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8D411yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C23×C8C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C2×C4×Q8—24no—
85C2×C4×Q8—24no—
86C2×C4×Q8D414yes—
87C2×C4×Q8C2314yes—
88C2×C4×D4—22no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4C2314yes—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4D411yescharacteristic
97C22×C4⋊C4D411yescharacteristic
98C22×C4⋊C4D411yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C42—21no—
112C22×C42—21no—
113C22×C42—21no—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C2×#107—24no—
118C2×#107C2314yes—
119C2×#106—24no—
120C2×#106C2314yes—
121C2×C4⋊C8—24no—
122C2×C4⋊C8C2314yes—
123C2×#102—24no—
124C2×#102—24no—
125C2×#102—24no—
126C2×#102—24no—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×C4×C8—24no—
132C2×C4×C8C2314yes—
133#399—28no—
134#393—28no—
135#391—28no—
136#390—22no—
137#390—22no—
138#390C2314yes—
139#390C2314yes—
140#389—22no—
141#389—22no—
142#389C2314yes—
143#389C2314yes—
144#387—28no—
145#387—28no—
146#385—28no—
147#385—28no—
148#384—22no—
149#384C2314yes—
150#382—22no—
151#382—22no—
152#382—22no—
153#382C2314yes—
154#382C2314yes—
155#382C2314yes—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—22no—
158C4×C4⋊C4—22no—
159C4×C4⋊C4—22no—
160C4×C4⋊C4C2314yes—
161C4×C4⋊C4C2314yes—
162C4×C4⋊C4C2314yes—
163C2×#95—21no—
164C2×#95—21no—
165C2×#95—21no—
166C2×#95—21no—
167C2×#95—21no—
168C2×#95—21no—
169C2×#95—21no—
170C2×#95—21no—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#336C2314yes—
185#336C2314yes—
186#336C2314yes—
187#336C2314yes—
188C25—21no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—42no—
192C22×Q8C2×D414yes—
193C22×D4—21no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4#4114yes—
200C22×D4C2412yes—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—21no—
204C23×C4—21no—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—24no—
209C23×C4—44no—
210C23×C4—82no—
211C23×C4—82no—
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2411yescharacteristic
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—22no—
224C22×C8—22no—
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—22no—
228C22×C8—22no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8C2412yes—
232C22×C8C2412yes—
233C22×C8C2412yes—
234C22×C8C2412yes—
235C4×Q8—216no—
236C4×Q8—28no—
237C4×Q8—28no—
238C4×D4—28no—
239C4×D4—28no—
240C4×D4—48no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—82no—
270C2×C4⋊C4—82no—
271C2×C4⋊C4—82no—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4#4114yes—
276C2×C4⋊C4C2412yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—82no—
292C2×C22⋊C4—82no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—44no—
301C2×C42—82no—
302C2×C42—82no—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42#4114yes—
307C2×C42#4114yes—
308C2×C42C2412yes—
309C2×C42C2412yes—
310C2×C42C2412yes—
311#107—28no—
312#106—28no—
313C4⋊C8—28no—
314#102—28no—
315#102—28no—
316#102—28no—
317#102—28no—
318C4×C8—28no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330C24—24no—
331C24—22no—
332C24—42no—
333C24—42no—
334C24—42no—
335C24—42no—
336C24—42no—
337C24—42no—
338C24C22×D411yescharacteristic
339C2×Q8—28no—
340C2×Q8—28no—
341C2×Q8—44no—
342C2×Q8—416no—
343C2×D4—28no—
344C2×D4—24no—
345C2×D4—28no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—42no—
352C2×D4—44no—
353C2×D4—42no—
354C2×D4—44no—
355C2×D4C2×#4114yes—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—42no—
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—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—48no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4—816no—
403C22×C4#12314yes—
404C22×C4#12314yes—
405C22×C4#13514yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C2×#4112yes—
414C22×C4C2×#4112yes—
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C42−1+412yes—
418C22×C42−1+412yes—
419C22×C4C2511yescharacteristic
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—44no—
433C2×C8—44no—
434C2×C8—44no—
435C2×C8—44no—
436C4⋊C4—28no—
437C4⋊C4—28no—
438C4⋊C4—28no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—48no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—84no—
461C4⋊C4—84no—
462C4⋊C4—84no—
463C4⋊C4C2×#4114yes—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—84no—
468C22⋊C4—84no—
469C42—24no—
470C42—24no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—48no—
476C42—84no—
477C42—84no—
478C42#12318yes—
479C42C2×#4114yes—
480C42C2×#4114yes—
481C23—24no—
482C23—48no—
483C23—48no—
484C23—42no—
485C23—48no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—84no—
492C23—84no—
493C23—84no—
494C23—84no—
495C23C2×#12312yes—
496C23#54812yes—
497C23#54812yes—
498C23C2×#13512yes—
499C23#57512yes—
500C23#57612yes—
501C23#57712yes—
502C23C23×D411yescharacteristic
503Q8—416no—
504D4—24no—
505D4—24no—
506D4—48no—
507D4—42no—
508D4—42no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—48no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—816no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4C2×#12314yes—
560C2×C4C2×#12314yes—
561C2×C4C2×#12312yes—
562C2×C4#52914yes—
563C2×C4#54114yes—
564C2×C4#54812yes—
565C2×C4#54812yes—
566C2×C4#55214yes—
567C2×C4#55614yes—
568C2×C4C2×#13514yes—
569C2×C4C22×#4111yescharacteristic
570C2×C4C22×#4111yescharacteristic
571C2×C4C22×#4111yescharacteristic
572C2×C4C22×#4111yescharacteristic
573C2×C4C2×2−1+411yescharacteristic
574C2×C4C2×2−1+411yescharacteristic
575C8—42no—
576C8—42no—
577C8—42no—
578C8—42no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—816no—
587C22#242114yes—
588C22#309114yes—
589C22#309614yes—
590C22#336014yes—
591C22#336114yes—
592C22#337014yes—
593C22#337114yes—
594C22C22×#12311yescharacteristic
595C22C2×#54811yescharacteristic
596C22C2×#54811yescharacteristic
597C22C22×#13511yescharacteristic
598C22C2×#57511yescharacteristic
599C22C2×#57611yescharacteristic
600C22C2×#57711yescharacteristic
601C4—24no—
602C4—24no—
603C4—24no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—44no—
608C4—82no—
609C4—82no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4#242114yes—
615C4C2×#52911yescharacteristic
616C4C2×#54111yescharacteristic
617C4C2×#55211yescharacteristic
618C4C2×#55611yescharacteristic
619C2—84no—
620C2#2499918yesdirect product factor
621C2C2×#242111yescharacteristic
622C2C2×#309111yescharacteristic
623C2C2×#309611yescharacteristic
624C2C2×#336011yescharacteristic
625C2C2×#336111yescharacteristic
626C2C2×#337011yescharacteristic
627C2C2×#337111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2355C2semidirect
3. #24999C2direct (N × H)
4. C2×#2100C2semidirect
5. C2×#2101C2semidirect
8. C2×#1984C2semidirect
9. C2×#1949C2semidirect
12. C2×#1920C2semidirect
13. C2×#1918C2semidirect
14. C2×#1835C2semidirect
32. #2355C22semidirect
50. #2101C22semidirect
51. #2100C22semidirect
54. #1984C22semidirect
55. #1949C22semidirect
58. #1920C22semidirect
59. #1918C22semidirect
60. #1835C22semidirect
620. C2#24999direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25802
C2×#23552
#249992
C2×#21002
C2×#21012
C2×#20732
C2×#20712
C2×#19842
C2×#19492
C2×#19262
C2×#19252
C2×#19202
C2×#19182
C2×#18352
C2×#18042
C2×C4×#1022