Finite Group Databasebeta

← #4327719

Subgroups — #4327719

Order 512.

3092 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2432C211yesmaximal, characteristic
2C2×#2311C211yesmaximal, characteristic
3#16393C212yesmaximal, complemented
4#16365C212yesmaximal, complemented
5#16308C214yesmaximal, complemented
6#16110C212yesmaximal, complemented
7#16093C212yesmaximal, complemented
8#15764C214yesmaximal, complemented
9#15114C212yesmaximal, complemented
10#15061C212yesmaximal, complemented
11#14786C212yesmaximal, complemented
12#14766C212yesmaximal, complemented
13#14473C212yesmaximal, complemented
14#14401C212yesmaximal, complemented
15#13956C211yesmaximal, characteristic
16C23×#41C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2432—28no—
23C2×#398C2212yes—
24C2×#398C2212yes—
25C2×#397C2214yes—
26C2×#394C2212yes—
27C2×#394C2212yes—
28#2311—28no—
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#390C2211yescharacteristic
32C2×#390C2212yes—
33C2×#387C2214yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#385C2214yes—
41C2×#381C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×#380C2214yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46C22×#95C2211yescharacteristic
47#1446C2212yes—
48#1446C2214yes—
49#1445C2214yes—
50#1445C2212yes—
51#1445C2212yes—
52#1444C2212yes—
53#1444C2214yes—
54#1444C2214yes—
55#1442C2212yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2212yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2212yes—
70#1436C2214yes—
71#1434C2214yes—
72#1434C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C22×#41—28no—
78C22×#41—42no—
79C22×#41—42no—
80C22×#41—44no—
81C22×#41D418yes—
82C23×Q8C2311yescharacteristic
83C23×Q8C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C23×D4C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C24×C4C2312yes—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—44no—
94C2×#124—24no—
95C2×#124—28no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—42no—
98C2×C22⋊Q8—42no—
99C2×#117—24no—
100C2×#117—44no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—42no—
103C22×C4⋊C4—42no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#398—44no—
130#398—44no—
131#397—48no—
132#394—28no—
133#394—28no—
134#391—28no—
135#391—24no—
136#390—28no—
137#390—24no—
138#387—48no—
139#386—28no—
140#386—28no—
141#386—28no—
142#386—44no—
143#385—28no—
144#385—28no—
145#385—48no—
146#381—42no—
147#380—28no—
148#380—28no—
149#380—48no—
150C4×C22⋊C4—24no—
151C2×#95—42no—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C25C2411yescharacteristic
177C25C2411yescharacteristic
178C25C2411yescharacteristic
179C2×#41—232no—
180C2×#41—416no—
181C2×#41—416no—
182C2×#41—432no—
183C2×#41—432no—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—41no—
189C22×Q8—41no—
190C22×Q8—41no—
191C22×Q8—41no—
192C22×Q8—42no—
193C22×Q8C2×D414yes—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—42no—
198C22×D4—41no—
199C22×D4—41no—
200C22×D4—41no—
201C22×D4—41no—
202C22×D4—41no—
203C22×D4—41no—
204C22×D4—42no—
205C22×D4C2×D414yes—
206C22×D4C2×D414yes—
207C22×D4C2×D414yes—
208C22×D4#4118yes—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—28no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—42no—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2414yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C4⋊Q8—416no—
247#124—416no—
248C22⋊Q8—416no—
249#117—416no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—416no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—28no—
317C2×C42—42no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—24no—
325#95—28no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—42no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—21no—
346C24—22no—
347C24—21no—
348C24—21no—
349C24—21no—
350C24—24no—
351C24—24no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—21no—
357C24—21no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24C22×D412yes—
362C24C22×D412yes—
363C24C22×D412yes—
364C24C2×#4114yes—
365C24C2×#4114yes—
366C24C2511yescharacteristic
367#41—4128no—
368C2×Q8—28no—
369C2×Q8—28no—
370C2×Q8—44no—
371C2×Q8—48no—
372C2×Q8—48no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—84no—
377C2×Q8—84no—
378C2×D4—28no—
379C2×D4—28no—
380C2×D4—28no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4#123116yes—
393C22×C4—216no—
394C22×C4—216no—
395C22×C4—28no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—416no—
470C22×C4—48no—
471C22×C4—88no—
472C22×C4—84no—
473C22×C4—88no—
474C22×C4—84no—
475C22×C4—88no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C2×#4114yes—
483C22×C4H(5,𝔽2)14yes—
484C22×C4H(5,𝔽2)14yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C42−1+418yes—
488C4⋊C4—416no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—88no—
497C4⋊C4—88no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—416no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C22⋊C4—84no—
510C42—44no—
511C42—44no—
512C42—48no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—44no—
535C23—44no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23—41no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—42no—
554C23—42no—
555C23—44no—
556C23—41no—
557C23—42no—
558C23—41no—
559C23C2×#12314yes—
560C23C2×#12314yes—
561C23C23×D411yescharacteristic
562C23C22×#4112yes—
563C23C2×H(5,𝔽2)12yes—
564C23C2×H(5,𝔽2)11yescharacteristic
565C23C2×H(5,𝔽2)11yescharacteristic
566C23C2×H(5,𝔽2)12yes—
567C23C2×2−1+414yes—
568C23#58512yes—
569Q8—416no—
570Q8—88no—
571D4—416no—
572D4—416no—
573D4—416no—
574C2×C4—232no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—416no—
596C2×C4—416no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—816no—
645C2×C4—816no—
646C2×C4C2×#12314yes—
647C2×C4#53414yes—
648C2×C4#53414yes—
649C22—28no—
650C22—28no—
651C22—24no—
652C22—24no—
653C22—42no—
654C22—42no—
655C22—44no—
656C22—44no—
657C22—48no—
658C22—48no—
659C22—42no—
660C22—42no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—42no—
666C22—42no—
667C22—44no—
668C22—42no—
669C22—44no—
670C22—82no—
671C22—84no—
672C22—82no—
673C22C22×#12311yescharacteristic
674C22C2×#53411yescharacteristic
675C22C2×#53411yescharacteristic
676C22#348812yes—
677C22#349012yes—
678C22#349012yes—
679C22#349912yes—
680C22#351514yes—
681C22#352014yes—
682C22#353614yes—
683C22#353812yes—
684C22#354612yes—
685C22#354714yes—
686C22#354911yescharacteristic
687C22#355012yes—
688C22#355111yescharacteristic
689C4—24no—
690C4—24no—
691C4—24no—
692C4—42no—
693C4—44no—
694C4—48no—
695C4—84no—
696C4—82no—
697C4—84no—
698C4—82no—
699C4—82no—
700C4—84no—
701C2—44no—
702C2—44no—
703C2—82no—
704C2#3499211yescharacteristic
705C2#3599812yes—
706C2#3727311yescharacteristic
707C2#3738611yescharacteristic
708C2#4298412yes—
709C2#4673914yes—
710C2#4966012yes—
711C2#5020012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16393C2semidirect
4. #16365C2semidirect
5. #16308C2semidirect
6. #16110C2semidirect
7. #16093C2semidirect
8. #15764C2semidirect
9. #15114C2semidirect
10. #15061C2semidirect
11. #14786C2semidirect
12. #14766C2semidirect
13. #14473C2semidirect
14. #14401C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24322
C2×#23112
#163932
#163652
#163082
#161102
#160932
#157642
#151142
#150612
#147862
#147662
#144732
#144012
#139562