Finite Group Databasebeta

← #636802

Subgroups — #636802

Order 512.

5717 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
1C23×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2849C211yesmaximal, characteristic
3C2×#2412C212yesmaximal
4#16364C214yesmaximal, complemented
5#16251C214yesmaximal, complemented
6#15073C211yesmaximal, characteristic, complemented
7#15069C211yesmaximal, characteristic, complemented
8#14874C211yesmaximal, characteristic, complemented
9#14873C211yesmaximal, characteristic, complemented
10#14759C214yesmaximal, complemented
11#14758C214yesmaximal, complemented
12#14263C212yesmaximal, complemented
13#14256C212yesmaximal, complemented
14#13869C212yesmaximal, complemented
15C2×C22⋊#95C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C25×C4C2211yescharacteristic
18C22×#123C2212yes—
19C22×#123C2212yes—
20C22×C22⋊Q8—28no—
21C22×C22⋊Q8—216no—
22C22×C22⋊Q8C2214yes—
23C22×C22⋊Q8C2214yes—
24C22×#120C2211yescharacteristic
25C23×C4⋊C4C2212yes—
26C23×C4⋊C4C2211yescharacteristic
27C23×C22⋊C4C2212yes—
28C23×C22⋊C4C2212yes—
29#2849—28no—
30#2412—216no—
31C2×#397C2214yes—
32C2×#395C2214yescomplemented
33C2×#394C2214yes—
34C2×#393C2214yescomplemented
35C2×#386C2211yescharacteristic
36C2×#386C2212yes—
37C2×#386C2211yescharacteristic
38C2×#386C2212yes—
39C2×#386C2214yes—
40C2×#385C2211yescharacteristic
41C2×#385C2212yes—
42C2×#385C2211yescharacteristic
43C2×#385C2212yes—
44C2×#381C2212yes—
45C2×#381C2212yes—
46C2×#381C2212yes—
47C2×#381C2212yes—
48C2×#380C2214yes—
49C2×#379C2211yescharacteristic
50C22×#95C2214yes—
51#1446C2214yescomplemented
52#1446C2214yescomplemented
53#1444C2218yescomplemented
54#1444C2214yescomplemented
55#1442C2218yescomplemented
56#1442C2214yescomplemented
57#1441C2214yescomplemented
58#1441C2218yescomplemented
59#1440C2218yes—
60#1439C2218yescomplemented
61#1439C2214yescomplemented
62C22⋊#95C2212yes—
63C22⋊#95C2211yescharacteristic
64C22⋊#95C2212yes—
65C22⋊#95C2211yescharacteristic
66C22⋊#95C2218yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C26C2311yescharacteristic
70C23×Q8—28no—
71C23×Q8—24no—
72C23×Q8—48no—
73C23×Q8C2314yes—
74C23×D4C2311yescharacteristic
75C23×D4C2311yescharacteristic
76C23×D4C2311yescharacteristic
77C24×C4—28no—
78C24×C4—48no—
79C24×C4D418yes—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2314yes—
83C24×C4C2312yes—
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C2×#123—28no—
88C2×#123—28no—
89C2×#123—28no—
90C2×#123—48no—
91C2×#123—44no—
92C2×C22⋊Q8—2128no—
93C2×C22⋊Q8—216no—
94C2×C22⋊Q8—216no—
95C2×C22⋊Q8—232no—
96C2×C22⋊Q8—232no—
97C2×#120—24no—
98C2×#120—28no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—28no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2318yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4—28no—
111C22×C22⋊C4—28no—
112C22×C22⋊C4—28no—
113C22×C22⋊C4—28no—
114C22×C22⋊C4—44no—
115C22×C22⋊C4D4116yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132#397—216no—
133#395—48no—
134#394—216no—
135#393—48no—
136#386—24no—
137#386—28no—
138#386—24no—
139#386—28no—
140#386—216no—
141#385—24no—
142#385—28no—
143#385—24no—
144#385—28no—
145#381—28no—
146#381—28no—
147#381—28no—
148#381—44no—
149#380—216no—
150#379—24no—
151C2×#95—48no—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2318yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2318yes—
164C2×#95C2318yes—
165C2×#95C2318yes—
166C2×#95C2318yes—
167C25—24no—
168C25—22no—
169C25—28no—
170C25—22no—
171C25—44no—
172C25C2×D418yes—
173C25C2×D414yes—
174C25C2411yescharacteristic
175C25C2411yescharacteristic
176C25C2411yescharacteristic
177C25C2411yescharacteristic
178C22×Q8—232no—
179C22×Q8—216no—
180C22×Q8—28no—
181C22×Q8—28no—
182C22×Q8—28no—
183C22×Q8—48no—
184C22×Q8—464no—
185C22×Q8—432no—
186C22×D4—22no—
187C22×D4—24no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—22no—
193C22×D4—28no—
194C22×D4—22no—
195C22×D4—24no—
196C22×D4—44no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—232no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—28no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—216no—
215C23×C4—28no—
216C23×C4—432no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—48no—
221C23×C4—44no—
222C23×C4—48no—
223C23×C4—464no—
224C23×C4C2×D418yes—
225C23×C4C2×D414yes—
226C23×C4C2×D418yes—
227C23×C4C2×D418yes—
228C23×C4C2×D418yes—
229C23×C4C2×D414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2414yes—
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241#123—432no—
242#123—432no—
243C22⋊Q8—2128no—
244C22⋊Q8—2128no—
245#120—416no—
246C2×C4⋊C4—232no—
247C2×C4⋊C4—264no—
248C2×C4⋊C4—216no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—216no—
252C2×C4⋊C4—216no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—48no—
262C2×C22⋊C4—264no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—432no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—48no—
300C2×C22⋊C4C2×D4116yes—
301C2×C22⋊C4C2×D4116yes—
302C2×C22⋊C4#4118yes—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—28no—
310#95—28no—
311#95—216no—
312#95—28no—
313#95—28no—
314#95—48no—
315#95—48no—
316#95—48no—
317#95—48no—
318C24—22no—
319C24—24no—
320C24—24no—
321C24—21no—
322C24—24no—
323C24—21no—
324C24—28no—
325C24—216no—
326C24—232no—
327C24—28no—
328C24—216no—
329C24—216no—
330C24—22no—
331C24—21no—
332C24—24no—
333C24—24no—
334C24—21no—
335C24—21no—
336C24—22no—
337C24—22no—
338C24—24no—
339C24—21no—
340C24—24no—
341C24—21no—
342C24—48no—
343C24—416no—
344C24—432no—
345C24—416no—
346C24—416no—
347C24#120116yes—
348C24C22×D412yes—
349C24C22×D414yes—
350C24C22×D412yes—
351C24C22×D412yes—
352C24C22×D414yes—
353C24C22×D414yes—
354C24C2×#4114yes—
355C24H(5,𝔽2)12yes—
356C242−1+418yes—
357C242−1+412yes—
358C24C2511yescharacteristic
359C2×Q8—232no—
360C2×Q8—432no—
361C2×Q8—432no—
362C2×Q8—4128no—
363C2×Q8—416no—
364C2×Q8—416no—
365C2×D4—48no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—48no—
371C2×D4—416no—
372C2×D4—44no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—88no—
376C22×C4—232no—
377C22×C4—28no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—216no—
381C22×C4—216no—
382C22×C4—216no—
383C22×C4—28no—
384C22×C4—216no—
385C22×C4—216no—
386C22×C4—216no—
387C22×C4—216no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—28no—
405C22×C4—216no—
406C22×C4—216no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—28no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—416no—
436C22×C4—432no—
437C22×C4—432no—
438C22×C4—4128no—
439C22×C4—416no—
440C22×C4—432no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—416no—
449C22×C4—88no—
450C22×C4—816no—
451C22×C4C22×D418yes—
452C22×C4C22×D418yes—
453C22×C4C22×D414yes—
454C22×C4C22×D414yes—
455C22×C4C2×#4118yes—
456C22×C4H(5,𝔽2)18yes—
457C4⋊C4—232no—
458C4⋊C4—264no—
459C4⋊C4—416no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—88no—
463C22⋊C4—232no—
464C22⋊C4—232no—
465C22⋊C4—216no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—48no—
475C22⋊C4—44no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—88no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—28no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—28no—
488C23—28no—
489C23—216no—
490C23—216no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—232no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—22no—
501C23—232no—
502C23—24no—
503C23—24no—
504C23—28no—
505C23—416no—
506C23—432no—
507C23—432no—
508C23—464no—
509C23—464no—
510C23—48no—
511C23—44no—
512C23—44no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—44no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—48no—
523C23—44no—
524C23—44no—
525C23C2×#12014yes—
526C23C2×#12014yes—
527C23C2×#12014yes—
528C23#53214yes—
529C23#53618yes—
530C23#536116yes—
531C23#53714yes—
532C23#54618yes—
533C23#54718yes—
534C23C23×D411yescharacteristic
535C23C23×D412yes—
536C23C22×#4112yes—
537C23C2×H(5,𝔽2)14yes—
538C23C2×H(5,𝔽2)11yescharacteristic
539C23C2×2−1+414yes—
540C23C2×2−1+411yescharacteristic
541Q8—464no—
542D4—88no—
543D4—88no—
544D4—88no—
545C2×C4—232no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—216no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—216no—
560C2×C4—432no—
561C2×C4—432no—
562C2×C4—416no—
563C2×C4—432no—
564C2×C4—44no—
565C2×C4—48no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—432no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—48no—
596C2×C4—48no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—832no—
601C2×C4#53418yes—
602C22—28no—
603C22—28no—
604C22—24no—
605C22—28no—
606C22—28no—
607C22—216no—
608C22—216no—
609C22—216no—
610C22—216no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—24no—
617C22—464no—
618C22—44no—
619C22—44no—
620C22—48no—
621C22—48no—
622C22—82no—
623C22—81no—
624C22—82no—
625C22—84no—
626C22—84no—
627C22—81no—
628C22—81no—
629C22#243018yes—
630C22#261418yes—
631C22C22×#12011yescharacteristic
632C22C2×#53211yescharacteristic
633C22C2×#53412yes—
634C22C2×#53612yes—
635C22C2×#53614yes—
636C22C2×#53711yescharacteristic
637C22C2×#54612yes—
638C22C2×#54712yes—
639C22#348414yes—
640C22#352018yes—
641C22#353614yes—
642C22#353814yes—
643C4—28no—
644C4—48no—
645C4—44no—
646C4—48no—
647C4—48no—
648C4—82no—
649C4—81no—
650C4—82no—
651C4—84no—
652C4—81no—
653C4—81no—
654C2—28no—
655C2—28no—
656C2—44no—
657C2—161no—
658C2C2×#261411yescharacteristic
659C2C2×#243011yescharacteristic
660C2#3498011yescharacteristic
661C2#3508512yes—
662C2#3744212yes—
663C2#3802114yes—
664C2#3803714yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊Q8C2semidirect
4. #16364C2semidirect
5. #16251C2semidirect
6. #15073C2semidirect
7. #15069C2semidirect
8. #14874C2semidirect
9. #14873C2semidirect
10. #14759C2semidirect
11. #14758C2semidirect
12. #14263C2semidirect
13. #14256C2semidirect
14. #13869C2semidirect
15. C2×C22⋊#95C2semidirect
32. C2×#395C22semidirect
34. C2×#393C22semidirect
51. #1446C22semidirect
52. #1446C22semidirect
53. #1444C22semidirect
54. #1444C22semidirect
55. #1442C22semidirect
56. #1442C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
60. #1439C22semidirect
61. #1439C22semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#28492
C2×#24122
#163642
#162512
#150732
#150692
#148742
#148732
#147592
#147582
#142632
#142562
#138692
C2×C22⋊#952