Finite Group Databasebeta

← #919766

Subgroups — #919766

Order 512.

2978 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2406C211yesmaximal, characteristic
2C22×#400C211yesmaximal, characteristic, complemented
3#17144C212yesmaximal, complemented
4#17141C212yesmaximal, complemented
5#17032C212yesmaximal, complemented
6#16932C212yesmaximal, complemented
7#16414C214yesmaximal, complemented
8#15680C214yesmaximal, complemented
9#14921C212yesmaximal, complemented
10#14919C212yesmaximal, complemented
11#14912C212yesmaximal, complemented
12#14256C212yesmaximal, complemented
13#14020C212yesmaximal, complemented
14#13956C212yesmaximal
15#13904C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C22×#123C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2406—28no—
21C2×#400—28no—
22C2×#400C2212yes—
23C2×#400C2212yes—
24C2×#400C2212yes—
25C2×#400C2212yes—
26C2×#395C2211yescharacteristic
27C2×#395C2211yescharacteristic
28C2×#394C2211yescharacteristic
29C2×#394C2211yescharacteristic
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#381C2211yescharacteristic
39C2×#381C2212yes—
40C2×#381C2212yes—
41C2×#380C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2212yes—
45C22×#95C2212yes—
46#1446C2214yes—
47#1445C2212yes—
48#1445C2214yes—
49#1445C2212yes—
50#1443C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1440C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1438C2214yes—
68#1437C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22×#41—28no—
78C22×#41—24no—
79C22×#41—42no—
80C22×#41—42no—
81C22×#41—44no—
82C23×Q8C2311yescharacteristic
83C23×D4C2312yes—
84C23×D4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2312yes—
90C24×C4C2312yes—
91C2×#123—28no—
92C2×#123—28no—
93C2×#123—44no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—42no—
96C22×C4⋊C4—42no—
97C22×C4⋊C4—42no—
98C22×C4⋊C4—42no—
99C22×C4⋊C4D418yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C42C2312yes—
122#400—28no—
123#400—28no—
124#400—28no—
125#400—28no—
126#395—24no—
127#395—24no—
128#394—24no—
129#394—24no—
130#386—28no—
131#386—28no—
132#386—44no—
133#386—44no—
134#386—44no—
135#385—28no—
136#385—28no—
137#385—44no—
138#381—24no—
139#381—44no—
140#381—44no—
141#380—42no—
142C4×C22⋊C4—42no—
143C2×#95—28no—
144C2×#95—28no—
145C2×#95—24no—
146C2×#95C2311yescharacteristic
147C2×#95C2311yescharacteristic
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C25C2412yes—
175C25C2411yescharacteristic
176C2×#41—432no—
177C2×#41—416no—
178C2×#41—416no—
179C2×#41—432no—
180C2×#41—416no—
181C22×Q8—22no—
182C22×Q8—24no—
183C22×Q8—42no—
184C22×Q8—41no—
185C22×Q8—41no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—42no—
195C22×D4—41no—
196C22×D4—41no—
197C22×D4#4118yes—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2414yes—
235C23×C4C2414yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243#123—432no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—416no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—416no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D414yes—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
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—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C42—24no—
308C2×C42—42no—
309C2×C42—42no—
310#95—24no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—24no—
317#95—22no—
318#95—22no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—42no—
329#95—42no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24C22×D412yes—
352C24C2×#4114yes—
353C24C2×#4114yes—
354C24H(5,𝔽2)12yes—
355C24H(5,𝔽2)12yes—
356C24H(5,𝔽2)14yes—
357C24C2511yescharacteristic
358#41—4128no—
359C2×Q8—48no—
360C2×Q8—44no—
361C2×Q8—44no—
362C2×Q8—48no—
363C2×Q8—44no—
364C2×D4—216no—
365C2×D4—216no—
366C2×D4—216no—
367C2×D4—48no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—48no—
371C2×D4—44no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—48no—
375C2×D4—48no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
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—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C2×#4114yes—
482C22×C42−1+414yes—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—84no—
492C4⋊C4—88no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C42—84no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—28no—
509C23—28no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—28no—
515C23—28no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—28no—
520C23—28no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23#53414yes—
540C23#53818yes—
541C23#54618yes—
542C23#55114yes—
543C23#56014yes—
544C23C23×D411yescharacteristic
545C23C22×#4112yes—
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×H(5,𝔽2)12yes—
549C23C2×2−1+412yes—
550C23#58512yes—
551C23#58514yes—
552Q8—416no—
553D4—416no—
554D4—432no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—416no—
574C2×C4—432no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—416no—
583C2×C4—416no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
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—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
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—44no—
632C2×C4—44no—
633C2×C4—816no—
634C2×C4—816no—
635C2×C4#53614yes—
636C22—216no—
637C22—28no—
638C22—24no—
639C22—24no—
640C22—28no—
641C22—28no—
642C22—28no—
643C22—48no—
644C22—44no—
645C22—44no—
646C22—48no—
647C22—44no—
648C22—84no—
649C22—84no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22#269718yes—
655C22C2×#53411yescharacteristic
656C22C2×#53611yescharacteristic
657C22#344911yescharacteristic
658C22C2×#53812yes—
659C22C2×#54612yes—
660C22C2×#55111yescharacteristic
661C22#348712yes—
662C22#351514yes—
663C22#352114yes—
664C22C2×#56011yescharacteristic
665C22#353512yes—
666C22#353512yes—
667C22#354512yes—
668C22#354512yes—
669C22#355914yes—
670C22#356014yes—
671C4—24no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—44no—
678C4—84no—
679C4—84no—
680C4—84no—
681C4—82no—
682C4—84no—
683C4—82no—
684C2—28no—
685C2—84no—
686C2#3718711yescharacteristic
687C2#3744711yescharacteristic
688C2#3803112yes—
689C2#3890712yes—
690C2#4153414yes—
691C2#4372712yes—
692C2C2×#269711yescharacteristic
693C2#5279912yes—
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
2. C22×#400C2semidirect
3. #17144C2semidirect
4. #17141C2semidirect
5. #17032C2semidirect
6. #16932C2semidirect
7. #16414C2semidirect
8. #15680C2semidirect
9. #14921C2semidirect
10. #14919C2semidirect
11. #14912C2semidirect
12. #14256C2semidirect
13. #14020C2semidirect
15. #13904C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24062
C22×#4002
#171442
#171412
#170322
#169322
#164142
#156802
#149212
#149192
#149122
#142562
#140202
#139562
#139042