Finite Group Databasebeta

← #636217

Subgroups — #636217

Order 512.

4074 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C211yesmaximal, characteristic
2C2×#2502C211yesmaximal, characteristic, complemented
3C2×#2280C211yesmaximal, characteristic
4C22×#380C211yesmaximal, characteristic
5#16399C214yesmaximal, complemented
6#16393C214yesmaximal, complemented
7#15236C212yesmaximal, complemented
8#15150C212yesmaximal, complemented
9#14920C214yesmaximal, complemented
10#14727C214yesmaximal
11#14340C212yesmaximal, complemented
12#14258C212yesmaximal, complemented
13#14139C211yesmaximal, characteristic, complemented
14#13924C211yesmaximal, characteristic, complemented
15#13881C211yesmaximal, characteristic, complemented
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×#117C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C22⋊C4C2212yes—
24#2848—28no—
25#2502—28no—
26C2×#398C2214yescomplemented
27C2×#397C2218yes—
28C2×#391C2212yescomplemented
29C2×#391C2212yescomplemented
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#385C2214yes—
34C2×#384C2212yescomplemented
35C2×#382C2212yescomplemented
36#2280—28no—
37C2×#381C2212yes—
38C2×#380—28no—
39C2×#380C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×#379C2212yes—
49C2×#379C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C22×#95C2212yes—
54#1445C2214yescomplemented
55#1445C2218yescomplemented
56#1442C2214yescomplemented
57#1442C2218yescomplemented
58#1441C2218yescomplemented
59#1441C2214yescomplemented
60#1441C2218yescomplemented
61#1440C2218yes—
62#1440C2218yes—
63#1439C2218yes—
64C4⋊#95C2212yescomplemented
65C4⋊#95C2212yescomplemented
66C4⋊#95C2212yescomplemented
67C4⋊#95C2212yescomplemented
68C22⋊#95C2214yes—
69C22⋊#95C2214yes—
70C22⋊#95C2214yes—
71C26C2311yescharacteristic
72C23×Q8C2311yescharacteristic
73C24×C4—24no—
74C24×C4—24no—
75C24×C4—28no—
76C24×C4—44no—
77C24×C4D418yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2312yes—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—44no—
93C2×#117—28no—
94C2×#117—28no—
95C2×#117—44no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—44no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2318yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C22⋊C4—28no—
114C22×C22⋊C4—48no—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129#398—216no—
130#397—416no—
131#391—28no—
132#391—44no—
133#385—28no—
134#385—28no—
135#385—28no—
136#385—216no—
137#384—28no—
138#382—44no—
139#381—28no—
140#380—24no—
141#380—24no—
142#380—24no—
143#380—28no—
144#380—28no—
145#380—28no—
146#380—28no—
147#380—28no—
148#380—28no—
149#379—28no—
150#379—28no—
151C4×C22⋊C4—24no—
152C4×C22⋊C4—28no—
153C4×C22⋊C4D418yes—
154C2×#95—28no—
155C2×#95C2318yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2318yes—
164C2×#95C2318yes—
165C2×#95C2318yes—
166C2×#95C2318yes—
167C2×#95C2318yes—
168C2×#95C2318yes—
169C2×#95C2312yes—
170C25—24no—
171C25—24no—
172C25—24no—
173C25—22no—
174C25—22no—
175C25—24no—
176C25—42no—
177C25—42no—
178C25C2×D414yes—
179C25C2411yescharacteristic
180C25C2411yescharacteristic
181C25C2411yescharacteristic
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—232no—
196C23×C4—232no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—28no—
212C23×C4—416no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—432no—
217C23×C4—432no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4#4114yes—
230C23×C4#4118yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2414yes—
237C23×C4C2414yes—
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C22⋊Q8—416no—
244C22⋊Q8—416no—
245C22⋊Q8—416no—
246#117—432no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—216no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—416no—
277C2×C4⋊C4—416no—
278C2×C4⋊C4—48no—
279C2×C4⋊C4—48no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—216no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—48no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—432no—
309C2×C22⋊C4C2×D418yes—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42C2×D418yes—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—216no—
324#95—48no—
325#95—48no—
326#95—48no—
327#95—48no—
328#95—48no—
329#95—48no—
330#95C2×D418yes—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—232no—
336C24—28no—
337C24—28no—
338C24—28no—
339C24—21no—
340C24—21no—
341C24—21no—
342C24—21no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—28no—
354C24—28no—
355C24—44no—
356C24—44no—
357C24—48no—
358C24—48no—
359C24—48no—
360C24—48no—
361C24—416no—
362C24—416no—
363C24—416no—
364C24—48no—
365C24—48no—
366C24C22×D412yes—
367C24C22×D412yes—
368C24C22×D412yes—
369C24C22×D412yes—
370C24C2×#4114yes—
371C24C2×#4112yes—
372C24H(5,𝔽2)14yes—
373C242−1+414yes—
374C242−1+414yes—
375C24C2511yescharacteristic
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—48no—
380C2×Q8—48no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—216no—
391C22×C4—216no—
392C22×C4—216no—
393C22×C4—216no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—28no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—28no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—28no—
447C22×C4—24no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—4128no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—88no—
467C22×C4—88no—
468C22×C4—816no—
469C22×C4#121116yes—
470C22×C4#123116yes—
471C22×C4C22×D412yes—
472C22×C4C22×D418yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4114yes—
482C22×C4H(5,𝔽2)14yes—
483C4⋊C4—416no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—44no—
494C4⋊C4—816no—
495C22⋊C4—216no—
496C22⋊C4—416no—
497C22⋊C4—416no—
498C22⋊C4—48no—
499C22⋊C4—416no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—416no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C42—216no—
508C42—48no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—28no—
517C23—24no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—232no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—28no—
538C23—28no—
539C23—24no—
540C23—28no—
541C23—28no—
542C23—416no—
543C23—416no—
544C23—416no—
545C23—416no—
546C23—416no—
547C23—464no—
548C23—464no—
549C23—416no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23C2×#12114yes—
563C23C2×#12314yes—
564C23#53514yes—
565C23#53618yes—
566C23#53614yes—
567C23#53614yes—
568C23#54118yes—
569C23C23×D411yescharacteristic
570C23C23×D411yescharacteristic
571C23C22×#4111yescharacteristic
572C23C22×#4112yes—
573C23C2×H(5,𝔽2)12yes—
574C23C2×H(5,𝔽2)12yes—
575C23C2×2−1+412yes—
576C23C2×2−1+412yes—
577C23#58512yes—
578Q8—88no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—232no—
588C2×C4—232no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—416no—
596C2×C4—416no—
597C2×C4—416no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—416no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—48no—
630C2×C4—48no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—48no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—48no—
637C2×C4—48no—
638C2×C4—48no—
639C2×C4—44no—
640C2×C4—816no—
641C2×C4—816no—
642C2×C4C2×#12114yes—
643C2×C4C2×#12114yes—
644C2×C4C2×#12314yes—
645C2×C4C2×#12314yes—
646C2×C4#53414yes—
647C2×C4#54618yes—
648C2×C4#55118yes—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—216no—
655C22—216no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—464no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—48no—
672C22—48no—
673C22—48no—
674C22—48no—
675C22—42no—
676C22#262118yes—
677C22C22×#12111yescharacteristic
678C22C22×#12311yescharacteristic
679C22C2×#53411yescharacteristic
680C22C2×#53511yescharacteristic
681C22C2×#53611yescharacteristic
682C22C2×#53611yescharacteristic
683C22C2×#53612yes—
684C22#344911yescharacteristic
685C22C2×#54112yes—
686C22C2×#54612yes—
687C22C2×#55112yes—
688C22#348812yes—
689C22#349214yes—
690C22#349412yes—
691C22#349514yes—
692C22#353614yes—
693C22#354114yes—
694C4—24no—
695C4—24no—
696C4—42no—
697C4—48no—
698C4—82no—
699C4—84no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—84no—
704C4—84no—
705C4#241218yes—
706C2—28no—
707C2—44no—
708C2—44no—
709C2C2×#262111yescharacteristic
710C2C2×#241211yescharacteristic
711C2#3522711yescharacteristic
712C2#3525211yescharacteristic
713C2#3531111yescharacteristic
714C2#3744911yescharacteristic
715C2#3745111yescharacteristic
716C2#3783814yes—
717C2#3785014yes—
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
2. C2×#2502C2semidirect
5. #16399C2semidirect
6. #16393C2semidirect
7. #15236C2semidirect
8. #15150C2semidirect
9. #14920C2semidirect
11. #14340C2semidirect
12. #14258C2semidirect
13. #14139C2semidirect
14. #13924C2semidirect
15. #13881C2semidirect
26. C2×#398C22semidirect
28. C2×#391C22semidirect
29. C2×#391C22semidirect
34. C2×#384C22semidirect
35. C2×#382C22semidirect
54. #1445C22semidirect
55. #1445C22semidirect
56. #1442C22semidirect
57. #1442C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
60. #1441C22semidirect
64. C4⋊#95C22semidirect
65. C4⋊#95C22semidirect
66. C4⋊#95C22semidirect
67. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C2×#25022
C2×#22802
C22×#3802
#163992
#163932
#152362
#151502
#149202
#147272
#143402
#142582
#141392
#139242
#138812