Finite Group Databasebeta

← #635997

Subgroups — #635997

Order 512.

5861 conjugacy classes of subgroups.

The subgroup lattice has 701 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
4C2×#2614C211yesmaximal, characteristic, complemented
5C22×#381C211yesmaximal, characteristic, complemented
6#16263C214yesmaximal, complemented
7#16262C214yesmaximal, complemented
8#15073C212yesmaximal, complemented
9#15069C212yesmaximal, complemented
10#14759C214yesmaximal, complemented
11#14758C214yesmaximal, complemented
12#14336C211yesmaximal, characteristic, complemented
13#14263C211yesmaximal, characteristic, complemented
14#14256C211yesmaximal, characteristic, complemented
15#13869C212yesmaximal, complemented
16C24×Q8C2211yescharacteristic
17C25×C4C2211yescharacteristic
18C22×#123C2211yescharacteristic
19C22×#123C2212yes—
20C22×#123C2211yescharacteristic
21C22×C22⋊Q8—28no—
22C22×C22⋊Q8—216no—
23C22×C22⋊Q8C2214yes—
24C22×C22⋊Q8C2214yes—
25C22×#120C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C23×C4⋊C4C2212yes—
28C23×C22⋊C4C2212yes—
29C23×C22⋊C4C2212yes—
30#2849—28no—
31#2614—28no—
32#2412—216no—
33C2×#397C2214yes—
34C2×#395C2214yescomplemented
35C2×#394C2214yes—
36C2×#393C2214yescomplemented
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2214yes—
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2212yes—
44C2×#381—28no—
45C2×#381C2211yescharacteristic
46C2×#381C2212yes—
47C2×#381C2211yescharacteristic
48C2×#381C2211yescharacteristic
49C2×#381C2212yes—
50C2×#381C2214yes—
51C2×#381C2212yes—
52C2×#381C2212yes—
53C2×#381C2211yescharacteristic
54C2×#380C2214yes—
55C2×#379C2211yescharacteristic
56C22×#95C2214yes—
57#1446C2214yescomplemented
58#1446C2214yescomplemented
59#1444C2218yescomplemented
60#1444C2214yescomplemented
61#1442C2214yescomplemented
62#1442C2218yescomplemented
63#1441C2214yescomplemented
64#1441C2218yescomplemented
65#1440C2218yes—
66#1439C2214yescomplemented
67#1439C2218yescomplemented
68C22⋊#95C2218yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C26C2311yescharacteristic
72C23×Q8—28no—
73C23×Q8—24no—
74C23×Q8—24no—
75C23×Q8—24no—
76C23×Q8—44no—
77C23×Q8C2314yes—
78C23×D4C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4—28no—
82C24×C4—24no—
83C24×C4—24no—
84C24×C4—44no—
85C24×C4D418yes—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2314yes—
89C24×C4C2312yes—
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C24×C4C2312yes—
93C2×#123—24no—
94C2×#123—24no—
95C2×#123—24no—
96C2×#123—28no—
97C2×#123—24no—
98C2×#123—28no—
99C2×#123—44no—
100C2×#123—44no—
101C2×C22⋊Q8—2128no—
102C2×C22⋊Q8—216no—
103C2×C22⋊Q8—216no—
104C2×C22⋊Q8—232no—
105C2×C22⋊Q8—232no—
106C2×#120—24no—
107C2×#120—28no—
108C22×C4⋊C4—28no—
109C22×C4⋊C4—24no—
110C22×C4⋊C4—28no—
111C22×C4⋊C4—28no—
112C22×C4⋊C4—28no—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2318yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4—28no—
120C22×C22⋊C4—28no—
121C22×C22⋊C4—28no—
122C22×C22⋊C4—28no—
123C22×C22⋊C4—44no—
124C22×C22⋊C4D4116yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2314yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2314yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141#397—216no—
142#395—216no—
143#394—216no—
144#393—216no—
145#386—28no—
146#386—28no—
147#386—28no—
148#386—216no—
149#385—28no—
150#385—28no—
151#385—28no—
152#381—24no—
153#381—24no—
154#381—28no—
155#381—24no—
156#381—28no—
157#381—28no—
158#381—216no—
159#381—44no—
160#381D418yescomplemented
161#380—216no—
162#379—24no—
163C2×#95—216no—
164C2×#95C2318yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2318yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2318yes—
176C2×#95C2318yes—
177C2×#95C2318yes—
178C2×#95C2314yes—
179C25—22no—
180C25—22no—
181C25—24no—
182C25—28no—
183C25—44no—
184C25C2×D418yes—
185C25C2×D414yes—
186C25C2411yescharacteristic
187C25C2411yescharacteristic
188C25C2411yescharacteristic
189C25C2411yescharacteristic
190C22×Q8—232no—
191C22×Q8—216no—
192C22×Q8—28no—
193C22×Q8—28no—
194C22×Q8—28no—
195C22×Q8—48no—
196C22×Q8—432no—
197C22×Q8—432no—
198C22×Q8—416no—
199C22×Q8—416no—
200C22×D4—22no—
201C22×D4—24no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—24no—
208C22×D4—28no—
209C22×D4—24no—
210C22×D4—44no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—232no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—28no—
220C23×C4—28no—
221C23×C4—24no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—232no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—28no—
234C23×C4—216no—
235C23×C4—28no—
236C23×C4—416no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—48no—
240C23×C4—44no—
241C23×C4—48no—
242C23×C4—432no—
243C23×C4—432no—
244C23×C4C2×D418yes—
245C23×C4C2×D414yes—
246C23×C4C2×D418yes—
247C23×C4C2×D418yes—
248C23×C4C2×D414yes—
249C23×C4C2×D414yes—
250C23×C4C2×D418yes—
251C23×C4#4114yes—
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2414yes—
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C23×C4C2411yescharacteristic
258C23×C4C2414yes—
259C23×C4C2414yes—
260C23×C4C2414yes—
261C23×C4C2412yes—
262C23×C4C2412yes—
263#123—432no—
264#123—416no—
265#123—416no—
266C22⋊Q8—2128no—
267C22⋊Q8—2128no—
268#120—416no—
269C2×C4⋊C4—232no—
270C2×C4⋊C4—264no—
271C2×C4⋊C4—216no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—216no—
274C2×C4⋊C4—216no—
275C2×C4⋊C4—216no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—48no—
286C2×C22⋊C4—264no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—432no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—48no—
325C2×C22⋊C4C2×D418yes—
326C2×C22⋊C4C2×D4116yes—
327C2×C22⋊C4C2×D4116yes—
328#95—216no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—216no—
338#95—28no—
339#95—48no—
340#95—48no—
341#95—48no—
342#95C2×D4116yes—
343C24—21no—
344C24—21no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—28no—
350C24—28no—
351C24—216no—
352C24—232no—
353C24—216no—
354C24—216no—
355C24—21no—
356C24—21no—
357C24—22no—
358C24—24no—
359C24—24no—
360C24—21no—
361C24—21no—
362C24—22no—
363C24—21no—
364C24—22no—
365C24—24no—
366C24—24no—
367C24—48no—
368C24—416no—
369C24—416no—
370C24—416no—
371C24—432no—
372C24#120116yes—
373C24C22×D412yes—
374C24C22×D414yes—
375C24C22×D412yes—
376C24C22×D412yes—
377C24C22×D414yes—
378C24C22×D414yes—
379C24C22×D412yes—
380C24C2×#4112yes—
381C24H(5,𝔽2)14yes—
382C242−1+418yes—
383C24C2511yescharacteristic
384C2×Q8—232no—
385C2×Q8—432no—
386C2×Q8—432no—
387C2×Q8—4128no—
388C2×Q8—416no—
389C2×Q8—416no—
390C2×D4—48no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—48no—
397C2×D4—416no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—88no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—216no—
404C22×C4—216no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—28no—
410C22×C4—216no—
411C22×C4—216no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—216no—
415C22×C4—216no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—216no—
435C22×C4—216no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—28no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—28no—
460C22×C4—24no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—416no—
464C22×C4—4128no—
465C22×C4—416no—
466C22×C4—416no—
467C22×C4—416no—
468C22×C4—416no—
469C22×C4—416no—
470C22×C4—432no—
471C22×C4—416no—
472C22×C4—416no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—416no—
480C22×C4—88no—
481C22×C4—816no—
482C22×C4#121116yes—
483C22×C4C22×D418yes—
484C22×C4C22×D418yes—
485C22×C4C22×D418yes—
486C22×C4C22×D414yes—
487C22×C4C22×D414yes—
488C22×C4C22×D414yes—
489C22×C4C2×#4114yes—
490C4⋊C4—232no—
491C4⋊C4—264no—
492C4⋊C4—416no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—88no—
496C22⋊C4—232no—
497C22⋊C4—232no—
498C22⋊C4—216no—
499C22⋊C4—416no—
500C22⋊C4—416no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—88no—
512C23—22no—
513C23—22no—
514C23—28no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—28no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—216no—
527C23—216no—
528C23—232no—
529C23—22no—
530C23—232no—
531C23—22no—
532C23—28no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—28no—
538C23—416no—
539C23—464no—
540C23—432no—
541C23—464no—
542C23—432no—
543C23—44no—
544C23—44no—
545C23—48no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—44no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—48no—
558C23C2×#12014yes—
559C23C2×#12014yes—
560C23C2×#12014yes—
561C23C2×#12114yes—
562C23#53414yes—
563C23#53618yes—
564C23#536116yes—
565C23D4218yes—
566C23#54618yes—
567C23C23×D411yescharacteristic
568C23C23×D412yes—
569C23C23×D411yescharacteristic
570C23C22×#4111yescharacteristic
571C23C2×H(5,𝔽2)12yes—
572C23C2×2−1+414yes—
573C23#58514yes—
574Q8—464no—
575D4—88no—
576D4—88no—
577D4—88no—
578C2×C4—24no—
579C2×C4—232no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—216no—
594C2×C4—432no—
595C2×C4—432no—
596C2×C4—432no—
597C2×C4—44no—
598C2×C4—416no—
599C2×C4—432no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—48no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—48no—
632C2×C4—44no—
633C2×C4—48no—
634C2×C4—48no—
635C2×C4—832no—
636C2×C4C2×#12118yes—
637C22—216no—
638C22—216no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—28no—
643C22—28no—
644C22—24no—
645C22—216no—
646C22—216no—
647C22—28no—
648C22—28no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—464no—
653C22—48no—
654C22—44no—
655C22—48no—
656C22—44no—
657C22—84no—
658C22—81no—
659C22—81no—
660C22—82no—
661C22—81no—
662C22—84no—
663C22—82no—
664C22#241018yes—
665C22#261418yescomplemented
666C22C22×#12011yescharacteristic
667C22C22×#12111yescharacteristic
668C22C2×#53411yescharacteristic
669C22C2×#53614yes—
670C22C2×#53612yes—
671C22#344912yes—
672C22C2×D4212yes—
673C22C2×#54612yes—
674C22#348514yes—
675C22#348914yes—
676C22#349514yes—
677C22#353518yes—
678C4—28no—
679C4—48no—
680C4—48no—
681C4—44no—
682C4—48no—
683C4—84no—
684C4—81no—
685C4—81no—
686C4—82no—
687C4—81no—
688C4—82no—
689C2—28no—
690C2—28no—
691C2—44no—
692C2—161no—
693C2C2×#261411yescharacteristic
694C2C2×#241011yescharacteristic
695C2#3508811yescharacteristic
696C2#3530612yes—
697C2#3744712yes—
698C2#3764714yes—
699C2#3783514yes—
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. C2×#2614C2semidirect
5. C22×#381C2semidirect
6. #16263C2semidirect
7. #16262C2semidirect
8. #15073C2semidirect
9. #15069C2semidirect
10. #14759C2semidirect
11. #14758C2semidirect
12. #14336C2semidirect
13. #14263C2semidirect
14. #14256C2semidirect
15. #13869C2semidirect
34. C2×#395C22semidirect
36. C2×#393C22semidirect
57. #1446C22semidirect
58. #1446C22semidirect
59. #1444C22semidirect
60. #1444C22semidirect
61. #1442C22semidirect
62. #1442C22semidirect
63. #1441C22semidirect
64. #1441C22semidirect
66. #1439C22semidirect
67. #1439C22semidirect
160. #381D4semidirect
665. C22#2614semidirect

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
C2×#26142
C22×#3812
#162632
#162622
#150732
#150692
#147592
#147582
#143362
#142632
#142562
#138692