Finite Group Databasebeta

← C2×#25065 #6445996

Subgroups — C2×#25065 #6445996

Order 512.

1924 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3#25065C2116yesmaximal, direct product factor
4C2×#2094C211yesmaximal, characteristic
5C2×#2092C211yesmaximal, characteristic
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2083C211yesmaximal, characteristic, complemented
8C2×#1984C211yesmaximal, characteristic, complemented
9C2×#1949C211yesmaximal, characteristic, complemented
10C2×#1925C211yesmaximal, characteristic
11C2×#1925C211yesmaximal, characteristic
12C2×#1919C211yesmaximal, characteristic, complemented
13C2×#1920C211yesmaximal, characteristic, complemented
14C2×#1846C211yesmaximal, characteristic, complemented
15C2×#1804C211yesmaximal, characteristic
16C2×C4×#102C211yesmaximal, characteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31#2580—24no—
32#2580C2218yes—
33C2×#399—21no—
34C2×#393—21no—
35#2355—24no—
36#2355C2218yescomplemented
37C2×#391—21no—
38C2×#390C2211yescharacteristic
39C2×#390C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#387—21no—
43C2×#387—21no—
44C2×#385—21no—
45C2×#385—21no—
46C2×#384C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×#382C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54#2094C2218yes—
55#2092C2218yes—
56#2085C2218yescomplemented
57#2083C2218yescomplemented
58#1984C2218yescomplemented
59#1949C2218yescomplemented
60#1925C2218yes—
61#1925C2218yes—
62#1920C2218yescomplemented
63#1919C2218yescomplemented
64#1846C2218yescomplemented
65#1804C2218yes—
66C4×#102C2218yes—
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—42no—
118C2×#107—42no—
119C2×#107—42no—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#106—24no—
124C2×#106—24no—
125C2×#106—42no—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8—24no—
130C2×C4⋊C8C2314yes—
131C2×#102—24no—
132C2×#102—42no—
133C2×#102—42no—
134C2×#102Q818yescomplemented
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C4×C8—24no—
140C2×C4×C8C2314yes—
141#399—28no—
142#393—28no—
143#391—28no—
144#390—22no—
145#390—22no—
146#390C2314yes—
147#390C2314yes—
148#389—22no—
149#389—22no—
150#389C2314yes—
151#389C2314yes—
152#387—28no—
153#387—28no—
154#385—28no—
155#385—28no—
156#384—22no—
157#384C2314yes—
158#382—22no—
159#382—22no—
160#382—22no—
161#382C2314yes—
162#382C2314yes—
163#382C2314yes—
164C4×C4⋊C4—28no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4C2314yes—
169C4×C4⋊C4C2314yes—
170C4×C4⋊C4C2314yes—
171C2×#95—21no—
172C2×#95—21no—
173C2×#95—21no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#336C2314yes—
189#336C2314yes—
190#336C2314yes—
191#336C2314yes—
192C25—21no—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—42no—
196C22×Q8C2×D414yes—
197C22×D4—21no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4C2×Q814yes—
204C22×D4C2412yes—
205C23×C4—21no—
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—24no—
213C23×C4—44no—
214C23×C4—82no—
215C23×C4—82no—
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2411yescharacteristic
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×C8—22no—
232C22×C8—22no—
233C22×C8C2×Q814yes—
234C22×C8#4114yes—
235C22×C8C2412yes—
236C22×C8C2412yes—
237C22×C8C2412yes—
238C22×C8C2412yes—
239C4×Q8—216no—
240C4×Q8—28no—
241C4×Q8—28no—
242C4×D4—28no—
243C4×D4—28no—
244C4×D4—48no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—21no—
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—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—82no—
274C2×C4⋊C4—82no—
275C2×C4⋊C4—82no—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4C2×Q814yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—82no—
296C2×C22⋊C4—82no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—44no—
305C2×C42—82no—
306C2×C42—82no—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42#4114yes—
311C2×C42#4114yes—
312C2×C42C2412yes—
313C2×C42C2412yes—
314C2×C42C2412yes—
315#107—44no—
316#107—44no—
317#107—44no—
318#106—28no—
319#106—28no—
320#106—44no—
321C4⋊C8—28no—
322#102—28no—
323#102—44no—
324#102—44no—
325#102C2×Q8116yescomplemented
326C4×C8—28no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338C24—24no—
339C24—22no—
340C24—42no—
341C24—42no—
342C24—42no—
343C24—42no—
344C24—42no—
345C24—42no—
346C24C22×D411yescharacteristic
347C2×Q8—28no—
348C2×Q8—28no—
349C2×Q8—44no—
350C2×Q8—416no—
351C2×D4—28no—
352C2×D4—24no—
353C2×D4—28no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—42no—
360C2×D4—44no—
361C2×D4—42no—
362C2×D4—44no—
363C2×D4C22×Q814yes—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
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—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—48no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4—816no—
411C22×C4#12314yes—
412C22×C4#12314yes—
413C22×C4#13514yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×Q812yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C42−1+412yes—
427C22×C4C2511yescharacteristic
428C2×C8—28no—
429C2×C8—28no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—44no—
441C2×C8—44no—
442C2×C8C22×Q814yes—
443C2×C8C2×#4114yes—
444C4⋊C4—28no—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—48no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C4⋊C4—84no—
471C4⋊C4C22×Q814yes—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—84no—
476C22⋊C4—84no—
477C42—24no—
478C42—24no—
479C42—22no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—48no—
484C42—84no—
485C42—84no—
486C42#12318yes—
487C42C2×#4114yes—
488C42C2×#4114yes—
489C23—24no—
490C23—48no—
491C23—48no—
492C23—42no—
493C23—48no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—84no—
500C23—84no—
501C23—84no—
502C23—84no—
503C23C2×#12312yes—
504C23D4×Q812yes—
505C23#55012yes—
506C23C2×#13512yes—
507C23#57512yes—
508C23#57612yes—
509C23#57712yes—
510C23C23×D411yescharacteristic
511Q8—416no—
512D4—24no—
513D4—24no—
514D4—48no—
515D4—42no—
516D4—42no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—48no—
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—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—816no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4C2×#12314yes—
568C2×C4C2×#12314yes—
569C2×C4C2×#12312yes—
570C2×C4#53314yes—
571C2×C4#54214yes—
572C2×C4D4×Q812yes—
573C2×C4#55012yes—
574C2×C4#55214yes—
575C2×C4#55414yes—
576C2×C4C2×#13514yes—
577C2×C4C23×Q811yescharacteristic
578C2×C4C22×#4111yescharacteristic
579C2×C4C22×#4111yescharacteristic
580C2×C4C22×#4111yescharacteristic
581C2×C4C2×H(5,𝔽2)11yescharacteristic
582C2×C4C2×2−1+411yescharacteristic
583C8—24no—
584C8—24no—
585C8—42no—
586C8—42no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—816no—
595C22#242314yes—
596C22#309114yes—
597C22#309614yes—
598C22#338014yes—
599C22#338814yes—
600C22#339414yes—
601C22#340314yes—
602C22C22×#12311yescharacteristic
603C22C2×D4×Q811yescharacteristic
604C22C2×#55011yescharacteristic
605C22C22×#13511yescharacteristic
606C22C2×#57511yescharacteristic
607C22C2×#57611yescharacteristic
608C22C2×#57711yescharacteristic
609C4—24no—
610C4—24no—
611C4—24no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—44no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4#242314yes—
623C4C2×#53311yescharacteristic
624C4C2×#54211yescharacteristic
625C4C2×#55211yescharacteristic
626C4C2×#55411yescharacteristic
627C2—84no—
628C2#2506518yesdirect product factor
629C2C2×#242311yescharacteristic
630C2C2×#309111yescharacteristic
631C2C2×#309611yescharacteristic
632C2C2×#338011yescharacteristic
633C2C2×#338811yescharacteristic
634C2C2×#339411yescharacteristic
635C2C2×#340311yescharacteristic
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. #25065C2direct (N × H)
6. C2×#2085C2semidirect
7. C2×#2083C2semidirect
8. C2×#1984C2semidirect
9. C2×#1949C2semidirect
12. C2×#1919C2semidirect
13. C2×#1920C2semidirect
14. C2×#1846C2semidirect
36. #2355C22semidirect
56. #2085C22semidirect
57. #2083C22semidirect
58. #1984C22semidirect
59. #1949C22semidirect
62. #1920C22semidirect
63. #1919C22semidirect
64. #1846C22semidirect
134. C2×#102Q8semidirect
325. #102C2×Q8semidirect
628. C2#25065direct (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
#250652
C2×#20942
C2×#20922
C2×#20852
C2×#20832
C2×#19842
C2×#19492
C2×#19252
C2×#19192
C2×#19202
C2×#18462
C2×#18042
C2×C4×#1022