Finite Group Databasebeta

← #640582

Subgroups — #640582

Order 512.

3794 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic
3#16892C214yesmaximal, complemented
4#16815C214yesmaximal, complemented
5#15236C212yesmaximal, complemented
6#15233C212yesmaximal, complemented
7#15182C212yesmaximal, complemented
8#15150C212yesmaximal, complemented
9#14727C214yesmaximal
10#14265C214yesmaximal, complemented
11#14139C211yesmaximal, characteristic, complemented
12#14139C211yesmaximal, characteristic, complemented
13#13881C211yesmaximal, characteristic, complemented
14#13880C211yesmaximal, characteristic, complemented
15C4×#379C211yesmaximal, characteristic
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C22⋊C4C2212yes—
23#2848—28no—
24C2×#397C2218yes—
25C2×#391C2212yescomplemented
26C2×#391C2212yescomplemented
27C2×#385C2212yes—
28C2×#385C2212yes—
29C2×#385C2212yes—
30C2×#385C2214yes—
31C2×#384C2212yescomplemented
32C2×#382C2212yescomplemented
33C2×#380—28no—
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#379C2212yes—
42C2×#379C2212yes—
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47C22×#95C2212yes—
48#1445C2218yescomplemented
49#1444C2214yescomplemented
50#1443C2218yescomplemented
51#1442C2214yescomplemented
52#1442C2218yescomplemented
53#1441C2214yescomplemented
54#1441C2214yescomplemented
55#1441C2218yescomplemented
56#1440C2218yes—
57#1440C2218yes—
58#1439C2218yes—
59C4⋊#95C2212yescomplemented
60C4⋊#95C2212yescomplemented
61C4⋊#95C2212yescomplemented
62C4⋊#95C2212yescomplemented
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C26C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C24×C4—24no—
72C24×C4—24no—
73C24×C4—44no—
74C24×C4—44no—
75C24×C4D418yes—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—44no—
90C2×C22⋊Q8—44no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2318yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4—28no—
107C22×C22⋊C4—48no—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122#397—416no—
123#391—44no—
124#391—44no—
125#385—28no—
126#385—28no—
127#385—28no—
128#385—216no—
129#384—44no—
130#382—44no—
131#380—24no—
132#380—24no—
133#380—24no—
134#380—28no—
135#380—28no—
136#380—28no—
137#380—28no—
138#379—28no—
139#379—28no—
140C4×C22⋊C4—24no—
141C4×C22⋊C4—24no—
142C4×C22⋊C4—44no—
143C4×C22⋊C4—44no—
144C2×#95—28no—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2318yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2318yes—
154C2×#95C2318yes—
155C2×#95C2318yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2318yes—
159C2×#95C2318yes—
160C2×#95C2318yes—
161C25—22no—
162C25—22no—
163C25—24no—
164C25—24no—
165C25—24no—
166C25—24no—
167C25—42no—
168C25—42no—
169C25C2×D414yes—
170C25C2411yescharacteristic
171C25C2411yescharacteristic
172C25C2411yescharacteristic
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—24no—
177C22×Q8—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—232no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—416no—
200C23×C4—416no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—432no—
212C23×C4—432no—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4#4114yes—
220C23×C4#4114yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C22⋊Q8—416no—
234C22⋊Q8—416no—
235C22⋊Q8—416no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—216no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—416no—
262C2×C4⋊C4—416no—
263C2×C4⋊C4—48no—
264C2×C4⋊C4—48no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—216no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—432no—
294C2×C22⋊C4#4118yes—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301#95—28no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—216no—
309#95—44no—
310#95—44no—
311#95—48no—
312#95—48no—
313#95—48no—
314#95—48no—
315#95—48no—
316#95—48no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—216no—
322C24—216no—
323C24—28no—
324C24—28no—
325C24—28no—
326C24—22no—
327C24—22no—
328C24—21no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—28no—
341C24—28no—
342C24—44no—
343C24—44no—
344C24—416no—
345C24—416no—
346C24—416no—
347C24—416no—
348C24—48no—
349C24—48no—
350C24—48no—
351C24—48no—
352C24C22×D412yes—
353C24C22×D412yes—
354C24C22×D412yes—
355C24C2×#4112yes—
356C24C2×#4112yes—
357C24C2×#4114yes—
358C24H(5,𝔽2)14yes—
359C242−1+414yes—
360C242−1+414yes—
361C24C2511yescharacteristic
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—48no—
366C2×Q8—48no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—216no—
372C22×C4—216no—
373C22×C4—28no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—28no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—28no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—28no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—4128no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—816no—
455C22×C4—816no—
456C22×C4#123116yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C2×#4118yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4H(5,𝔽2)14yes—
467C22×C4H(5,𝔽2)14yes—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—816no—
479C22⋊C4—216no—
480C22⋊C4—416no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—416no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C42—48no—
492C42—48no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—232no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—416no—
527C23—416no—
528C23—416no—
529C23—416no—
530C23—416no—
531C23—464no—
532C23—464no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23—416no—
544C23—44no—
545C23—44no—
546C23C2×#12314yes—
547C23#53214yes—
548C23#53514yes—
549C23#53614yes—
550C23#53614yes—
551C23#55218yes—
552C23#55618yes—
553C23C23×D411yescharacteristic
554C23C22×#4111yescharacteristic
555C23C22×#4111yescharacteristic
556C23C22×#4112yes—
557C23C2×H(5,𝔽2)12yes—
558C23C2×H(5,𝔽2)12yes—
559C23C2×H(5,𝔽2)12yes—
560C23C2×2−1+412yes—
561C23C2×2−1+412yes—
562Q8—88no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—232no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—416no—
578C2×C4—416no—
579C2×C4—416no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—416no—
589C2×C4—416no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—48no—
617C2×C4—48no—
618C2×C4—48no—
619C2×C4—44no—
620C2×C4—48no—
621C2×C4—44no—
622C2×C4—48no—
623C2×C4—48no—
624C2×C4—44no—
625C2×C4—816no—
626C2×C4—816no—
627C2×C4C2×#12314yes—
628C2×C4C2×#12314yes—
629C2×C4#53414yes—
630C2×C4#53414yes—
631C2×C4#53818yes—
632C2×C4#53818yes—
633C22—24no—
634C22—24no—
635C22—216no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—216no—
640C22—24no—
641C22—24no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—464no—
648C22—42no—
649C22—42no—
650C22—48no—
651C22—48no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—48no—
657C22—48no—
658C22—42no—
659C22—42no—
660C22#249918yes—
661C22C22×#12311yescharacteristic
662C22C2×#53211yescharacteristic
663C22C2×#53411yescharacteristic
664C22C2×#53411yescharacteristic
665C22C2×#53511yescharacteristic
666C22C2×#53611yescharacteristic
667C22C2×#53611yescharacteristic
668C22C2×#53812yes—
669C22C2×#53812yes—
670C22C2×#55212yes—
671C22C2×#55612yes—
672C22#349212yes—
673C22#349212yes—
674C22#351914yes—
675C22#352014yes—
676C22#353614yes—
677C22#353814yes—
678C4—24no—
679C4—48no—
680C4—42no—
681C4—42no—
682C4—82no—
683C4—84no—
684C4—82no—
685C4—84no—
686C4—82no—
687C4—84no—
688C4—82no—
689C4#262218yes—
690C2—28no—
691C2—44no—
692C2—44no—
693C2C2×#262211yescharacteristic
694C2C2×#249911yescharacteristic
695C2#3507811yescharacteristic
696C2#3508511yescharacteristic
697C2#3522711yescharacteristic
698C2#3745111yescharacteristic
699C2#3745311yescharacteristic
700C2#3892814yes—
701C2#3915414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16892C2semidirect
4. #16815C2semidirect
5. #15236C2semidirect
6. #15233C2semidirect
7. #15182C2semidirect
8. #15150C2semidirect
10. #14265C2semidirect
11. #14139C2semidirect
12. #14139C2semidirect
13. #13881C2semidirect
14. #13880C2semidirect
25. C2×#391C22semidirect
26. C2×#391C22semidirect
31. C2×#384C22semidirect
32. C2×#382C22semidirect
48. #1445C22semidirect
49. #1444C22semidirect
50. #1443C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
59. C4⋊#95C22semidirect
60. C4⋊#95C22semidirect
61. C4⋊#95C22semidirect
62. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C22×#3802
#168922
#168152
#152362
#152332
#151822
#151502
#147272
#142652
#141392
#138812
#138802
C4×#3792