Finite Group Databasebeta

← #767346

Subgroups — #767346

Order 512.

2846 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#400C211yesmaximal, characteristic, complemented
2C2×#2325C211yesmaximal, characteristic
3#15870C214yesmaximal, complemented
4#15843C214yesmaximal, complemented
5#15461C214yesmaximal, complemented
6#15357C214yesmaximal, complemented
7#15019C211yesmaximal, characteristic, complemented
8#15014C211yesmaximal, characteristic, complemented
9#14926C211yesmaximal, characteristic
10#14913C211yesmaximal, characteristic
11#14019C212yesmaximal, complemented
12#14015C212yesmaximal, complemented
13#13996C212yesmaximal, complemented
14#13968C212yesmaximal, complemented
15#13796C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#121C2211yescharacteristic
18C22×C4×D4C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C2×#400—28no—
22C2×#400C2214yes—
23C2×#400C2214yes—
24#2325—28no—
25C2×#388C2212yes—
26C2×#388C2214yes—
27C2×#387C2212yes—
28C2×#386C2212yes—
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#385C2212yes—
34C2×#385C2214yes—
35C2×#385C2212yes—
36C2×#381C2212yes—
37C2×#380C2211yescharacteristic
38C2×C4×C22⋊C4C2212yes—
39C22×#95C2211yescharacteristic
40C22×#95C2211yescharacteristic
41C22×#95C2211yescharacteristic
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44#1445C2212yes—
45#1445C2212yes—
46#1445C2214yes—
47#1444C2212yes—
48#1444C2212yes—
49#1444C2214yes—
50#1443C2214yes—
51#1442C2212yes—
52#1442C2214yes—
53#1442C2212yes—
54#1442C2214yes—
55#1441C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2212yes—
63#1438C2212yes—
64#1437C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2214yes—
69#1434C2214yes—
70#1434C2214yes—
71C22⋊#95C2211yescharacteristic
72C22⋊#95C2211yescharacteristic
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C2×#123—28no—
88C2×#123—28no—
89C2×#123—44no—
90C2×#121—28no—
91C2×#121—24no—
92C2×C4×D4—28no—
93C2×C4×D4—44no—
94C2×C4×D4—44no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C42C2312yes—
119C22×C42C2314yes—
120#400—216no—
121#400—216no—
122#388—28no—
123#388—48no—
124#387—28no—
125#386—28no—
126#386—28no—
127#386—28no—
128#386—28no—
129#386—44no—
130#385—28no—
131#385—48no—
132#385—44no—
133#381—44no—
134#380—24no—
135C4×C22⋊C4—28no—
136C2×#95—24no—
137C2×#95—24no—
138C2×#95—24no—
139C2×#95—24no—
140C2×#95D418yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C25C2412yes—
170C25C2411yescharacteristic
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—22no—
179C22×D4—42no—
180C22×D4—42no—
181C22×D4#4118yes—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231#123—432no—
232#121—232no—
233C4×D4—432no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—416no—
251C2×C4⋊C4—416no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4#4118yes—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—44no—
293C2×C42—44no—
294#95—28no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—42no—
309#95—44no—
310#95—42no—
311#95—44no—
312#95—42no—
313#95—42no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95C2×D418yes—
321#95C2×D418yes—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24C22×D412yes—
337C24C2×#4114yes—
338C24C2×#4114yes—
339C24C2×#4114yes—
340C24H(5,𝔽2)12yes—
341C24H(5,𝔽2)12yes—
342C24C2511yescharacteristic
343C2×D4—216no—
344C2×D4—216no—
345C2×D4—216no—
346C2×D4—28no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—82no—
352C2×D4—82no—
353C2×D4—88no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—416no—
427C22×C4—416no—
428C22×C4—48no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—88no—
444C22×C4—88no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4—84no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D414yes—
454C22×C4C22×D414yes—
455C22×C4C2×#4118yes—
456C22×C4C2×#4114yes—
457C22×C4H(5,𝔽2)14yes—
458C22×C42−1+414yes—
459C4⋊C4—28no—
460C4⋊C4—416no—
461C4⋊C4—416no—
462C4⋊C4—416no—
463C4⋊C4—48no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—88no—
467C22⋊C4—216no—
468C22⋊C4—216no—
469C22⋊C4—416no—
470C22⋊C4—416no—
471C22⋊C4—416no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—84no—
476C22⋊C4—84no—
477C22⋊C4—88no—
478C42—48no—
479C42—88no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—28no—
496C23—28no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23#52514yes—
516C23#53214yes—
517C23#53514yes—
518C23#55118yes—
519C23#55318yes—
520C23C23×D411yescharacteristic
521C23C22×#4112yes—
522C23C22×#4112yes—
523C23C2×H(5,𝔽2)11yescharacteristic
524C23C2×H(5,𝔽2)11yescharacteristic
525C23C2×H(5,𝔽2)12yes—
526C23C2×2−1+412yes—
527C23#58512yes—
528C23#58512yes—
529D4—432no—
530D4—88no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—416no—
554C2×C4—416no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—88no—
603C2×C4—88no—
604C2×C4—88no—
605C2×C4—816no—
606C2×C4—88no—
607C2×C4—88no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—216no—
615C22—216no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—48no—
620C22—48no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—84no—
626C22—84no—
627C22#260318yes—
628C22C2×#52511yescharacteristic
629C22C2×#53211yescharacteristic
630C22C2×#53511yescharacteristic
631C22C2×#55112yes—
632C22C2×#55312yes—
633C22#348411yescharacteristic
634C22#348511yescharacteristic
635C22#348511yescharacteristic
636C22#348611yescharacteristic
637C22#351314yes—
638C22#351514yes—
639C22#352014yes—
640C22#352114yes—
641C22#355812yes—
642C22#355812yes—
643C22#355912yes—
644C22#355912yes—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—42no—
651C4—42no—
652C4—48no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—84no—
658C2—28no—
659C2—84no—
660C2C2×#260311yescharacteristic
661C2#3535711yescharacteristic
662C2#3539411yescharacteristic
663C2#3545211yescharacteristic
664C2#3546811yescharacteristic
665C2#3695811yescharacteristic
666C2#3696711yescharacteristic
667C2#4322614yes—
668C2#4324514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#400C2semidirect
3. #15870C2semidirect
4. #15843C2semidirect
5. #15461C2semidirect
6. #15357C2semidirect
7. #15019C2semidirect
8. #15014C2semidirect
11. #14019C2semidirect
12. #14015C2semidirect
13. #13996C2semidirect
14. #13968C2semidirect
15. #13796C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4002
C2×#23252
#158702
#158432
#154612
#153572
#150192
#150142
#149262
#149132
#140192
#140152
#139962
#139682
#137962