Finite Group Databasebeta

← #625762

Subgroups — #625762

Order 512.

6426 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#534C211yesmaximal, characteristic
2C2×#2854C212yesmaximal, complemented
3C2×#2405C212yesmaximal, complemented
4C22×#388C211yesmaximal, characteristic, complemented
5#15397C218yesmaximal, complemented
6#15281C218yesmaximal, complemented
7#14926C212yesmaximal, complemented
8#14719C212yesmaximal, complemented
9#14338C211yesmaximal, characteristic, complemented
10#14271C212yesmaximal, complemented
11#13867C212yesmaximal, complemented
12C24×D4C2212yes—
13C2×#534—216no—
14C2×#534—28no—
15C22×#123C2214yes—
16C22×#123C2214yescomplemented
17C22×#121C2212yes—
18C22×#121C2212yes—
19C22×#120C2214yes—
20C23×C4⋊C4C2211yescharacteristic, complemented
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C42C2211yescharacteristic, complemented
25#2854—216no—
26#2405—216no—
27C2×#394C2218yes—
28C2×#394C2214yescomplemented
29C2×#390C2214yescomplemented
30C2×#389C2214yescomplemented
31C2×#388—28no—
32C2×#388C2218yescomplemented
33C2×#388C2218yescomplemented
34C2×#386C2214yescomplemented
35C2×#386C2214yescomplemented
36C2×#386C2214yescomplemented
37C2×#385C2214yescomplemented
38C2×#380C2212yescomplemented
39C2×#379C2212yes—
40C2×#379C2214yescomplemented
41C2×#379C2214yescomplemented
42C22×#95C2211yescharacteristic, complemented
43C22×#95C2211yescharacteristic, complemented
44#1445C2214yescomplemented
45#1444C2214yescomplemented
46#1441C2214yescomplemented
47#1439C2218yescomplemented
48#1439C2218yescomplemented
49#1439C2218yescomplemented
50#1439C2218yescomplemented
51#1438C2218yes—
52#1438C2214yescomplemented
53#1438C2214yescomplemented
54C22⋊#95C2212yescomplemented
55C22⋊#95C2214yescomplemented
56C26C2312yes—
57C26C2311yescharacteristic
58C23×D4—28no—
59C23×D4—216no—
60C23×D4—28no—
61C23×D4—28no—
62C23×D4—48no—
63C23×D4C2314yes—
64C23×D4C2314yes—
65C23×D4C2312yes—
66C23×D4C2312yes—
67C24×C4C2311yescharacteristic
68C24×C4C2312yes—
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74#534—2128no—
75C2×#123—216no—
76C2×#123—216no—
77C2×#123—216no—
78C2×#123—216no—
79C2×#123—216no—
80C2×#123—48no—
81C2×#121—216no—
82C2×#121—28no—
83C2×#121—28no—
84C2×#121—28no—
85C2×#121D4116yes—
86C2×#120—216no—
87C2×#120—216no—
88C2×#120—216no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2318yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C22⋊C4—24no—
97C22×C22⋊C4—24no—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—44no—
101C22×C22⋊C4D418yes—
102C22×C22⋊C4D418yescomplemented
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2318yes—
106C22×C22⋊C4C2318yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2318yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2318yes—
115C22×C42—28no—
116C22×C42—24no—
117C22×C42C2314yes—
118#394—232no—
119#394—48no—
120#390—216no—
121#389—216no—
122#388—232no—
123#388—232no—
124#386—216no—
125#386—216no—
126#386—48no—
127#385—216no—
128#380—28no—
129#379—28no—
130#379—216no—
131#379—48no—
132C2×#95—24no—
133C2×#95—24no—
134C2×#95C2318yes—
135C2×#95C2318yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2318yes—
141C2×#95C2314yes—
142C25—24no—
143C25—24no—
144C25—28no—
145C25—24no—
146C25—24no—
147C25—24no—
148C25—42no—
149C25—44no—
150C25—42no—
151C25—42no—
152C25—42no—
153C25—44no—
154C25—44no—
155C25—44no—
156C25—44no—
157C25C2×D414yes—
158C25C2×D414yes—
159C25#4114yes—
160C25C2414yes—
161C25C2412yes—
162C25C2411yescharacteristic
163C22×D4—264no—
164C22×D4—28no—
165C22×D4—28no—
166C22×D4—28no—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—28no—
174C22×D4—28no—
175C22×D4—28no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4—464no—
186C22×D4—464no—
187C22×D4—432no—
188C22×D4—432no—
189C22×D4C2×D4116yes—
190C22×D4C2×D418yes—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×D418yes—
221C23×C4C2×D414yes—
222C23×C4C2418yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230#123—2128no—
231#123—464no—
232#121—264no—
233#121—264no—
234#120—2128no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—216no—
241C2×C4⋊C4—216no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—48no—
248C2×C4⋊C4—416no—
249C2×C4⋊C4C2×D418yes—
250C2×C22⋊C4—232no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—232no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—216no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—216no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—216no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—216no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—216no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—48no—
278C2×C22⋊C4—416no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4C2×D4116yes—
287C2×C22⋊C4C2×D4116yes—
288C2×C22⋊C4C2×D4116yes—
289C2×C42—28no—
290C2×C42—28no—
291C2×C42—28no—
292C2×C42—416no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—216no—
297#95—216no—
298#95—44no—
299#95—48no—
300#95—44no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—28no—
312C24—28no—
313C24—216no—
314C24—216no—
315C24—216no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—28no—
321C24—24no—
322C24—24no—
323C24—28no—
324C24—28no—
325C24—28no—
326C24—44no—
327C24—48no—
328C24—44no—
329C24—44no—
330C24—44no—
331C24—48no—
332C24—48no—
333C24—48no—
334C24—416no—
335C24—416no—
336C24—48no—
337C24—416no—
338C24—416no—
339C24—48no—
340C24—88no—
341C24—88no—
342C24—88no—
343C24—88no—
344C24—84no—
345C24—84no—
346C24—84no—
347C24—84no—
348C24—88no—
349C24—88no—
350C24—816no—
351C24—816no—
352C24—816no—
353C24—816no—
354C24#121116yes—
355C24C22×D418yes—
356C24C22×D414yes—
357C24C22×D414yes—
358C24C22×D412yes—
359C24C22×D414yes—
360C24C22×D412yes—
361C24C2×#4114yes—
362C24C2×#4112yes—
363C24H(5,𝔽2)14yes—
364C24C2511yescharacteristic
365C2×D4—232no—
366C2×D4—232no—
367C2×D4—232no—
368C2×D4—232no—
369C2×D4—216no—
370C2×D4—216no—
371C2×D4—416no—
372C2×D4—416no—
373C2×D4—416no—
374C2×D4—416no—
375C2×D4—416no—
376C2×D4—4256no—
377C2×D4—416no—
378C2×D4—416no—
379C2×D4—416no—
380C2×D4—416no—
381C2×D4—416no—
382C2×D4—416no—
383C2×D4—84no—
384C2×D4—88no—
385C2×D4—84no—
386C2×D4—84no—
387C2×D4—84no—
388C2×D4—816no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—216no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—216no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—24no—
429C22×C4—416no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—416no—
436C22×C4—416no—
437C22×C4—416no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—88no—
452C22×C4—88no—
453C22×C4#120116yes—
454C22×C4C22×D414yes—
455C22×C4C22×D414yes—
456C22×C4C22×D418yes—
457C22×C4C22×D414yes—
458C22×C4C22×D414yes—
459C22×C4C22×D414yes—
460C22×C4C22×D414yes—
461C22×C4H(5,𝔽2)14yes—
462C4⋊C4—216no—
463C4⋊C4—216no—
464C4⋊C4—416no—
465C4⋊C4—432no—
466C4⋊C4—416no—
467C22⋊C4—232no—
468C22⋊C4—232no—
469C22⋊C4—232no—
470C22⋊C4—232no—
471C22⋊C4—432no—
472C22⋊C4—416no—
473C22⋊C4—432no—
474C22⋊C4—416no—
475C22⋊C4—416no—
476C22⋊C4—816no—
477C22⋊C4—88no—
478C22⋊C4—816no—
479C42—416no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—28no—
487C23—28no—
488C23—24no—
489C23—24no—
490C23—28no—
491C23—28no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—28no—
504C23—232no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—216no—
509C23—216no—
510C23—416no—
511C23—416no—
512C23—432no—
513C23—48no—
514C23—48no—
515C23—48no—
516C23—48no—
517C23—44no—
518C23—44no—
519C23—48no—
520C23—48no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—48no—
526C23—44no—
527C23—44no—
528C23—48no—
529C23—48no—
530C23—48no—
531C23—48no—
532C23—48no—
533C23—48no—
534C23—48no—
535C23—832no—
536C23—88no—
537C23—88no—
538C23—832no—
539C23—88no—
540C23—88no—
541C23—816no—
542C23—816no—
543C23—816no—
544C23—816no—
545C23—864no—
546C23—864no—
547C23—816no—
548C23—816no—
549C23C2×#12014yes—
550C23C2×#12118yes—
551C23C2×#12114yes—
552C23#53514yes—
553C23D4218yes—
554C23#54618yes—
555C23C23×D411yescharacteristic
556C23C23×D412yes—
557C23C23×D411yescharacteristic
558C23C22×#4111yescharacteristic
559C23C2×H(5,𝔽2)12yes—
560C23C2×H(5,𝔽2)12yes—
561C23#58512yes—
562C23#58512yes—
563C23#58512yes—
564D4—464no—
565D4—464no—
566D4—816no—
567D4—816no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—416no—
585C2×C4—416no—
586C2×C4—416no—
587C2×C4—432no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—416no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—42no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—48no—
608C2×C4—42no—
609C2×C4—48no—
610C2×C4—48no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—88no—
615C2×C4—88no—
616C2×C4C2×#12018yes—
617C22—24no—
618C22—216no—
619C22—24no—
620C22—24no—
621C22—216no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—28no—
626C22—28no—
627C22—28no—
628C22—416no—
629C22—44no—
630C22—44no—
631C22—416no—
632C22—44no—
633C22—44no—
634C22—416no—
635C22—48no—
636C22—416no—
637C22—48no—
638C22—48no—
639C22—832no—
640C22—864no—
641C22—84no—
642C22—88no—
643C22—84no—
644C22—84no—
645C22—88no—
646C22—84no—
647C22#240818yes—
648C22C22×#12011yescharacteristic
649C22C22×#12111yescharacteristic
650C22C2×#53511yescharacteristic
651C22#344912yes—
652C22#345912yes—
653C22C2×D4212yes—
654C22C2×#54612yes—
655C22#348411yescharacteristic
656C22#348511yescharacteristic
657C22#348511yescharacteristic
658C22#348514yes—
659C22#348511yescharacteristic
660C22#348814yes—
661C22#349112yes—
662C22#349612yes—
663C22#355814yes—
664C22#356014yes—
665C4—28no—
666C4—44no—
667C4—48no—
668C4—44no—
669C4—44no—
670C4—44no—
671C4—88no—
672C4—82no—
673C2—28no—
674C2—48no—
675C2—88no—
676C2C2×#240811yescharacteristic
677C2#3498411yescharacteristic
678C2#3508911yescharacteristic
679C2#3544611yescharacteristic
680C2#3551211yescharacteristic
681C2#3696711yescharacteristic
682C2#3696911yescharacteristic
683C2#3767814yes—
684C2#3774614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#534C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2854C2semidirect
3. C2×#2405C2semidirect
4. C22×#388C2semidirect
5. #15397C2semidirect
6. #15281C2semidirect
7. #14926C2semidirect
8. #14719C2semidirect
9. #14338C2semidirect
10. #14271C2semidirect
11. #13867C2semidirect
16. C22×#123C22semidirect
20. C23×C4⋊C4C22semidirect
24. C23×C42C22semidirect
28. C2×#394C22semidirect
29. C2×#390C22semidirect
30. C2×#389C22semidirect
32. C2×#388C22semidirect
33. C2×#388C22semidirect
34. C2×#386C22semidirect
35. C2×#386C22semidirect
36. C2×#386C22semidirect
37. C2×#385C22semidirect
38. C2×#380C22semidirect
40. C2×#379C22semidirect
41. C2×#379C22semidirect
42. C22×#95C22semidirect
43. C22×#95C22semidirect
44. #1445C22semidirect
45. #1444C22semidirect
46. #1441C22semidirect
47. #1439C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
52. #1438C22semidirect
53. #1438C22semidirect
54. C22⋊#95C22semidirect
55. C22⋊#95C22semidirect
102. C22×C22⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5342
C2×#28542
C2×#24052
C22×#3882
#153972
#152812
#149262
#147192
#143382
#142712
#138672