Finite Group Databasebeta

← #764905

Subgroups — #764905

Order 512.

2700 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C22×#400C211yesmaximal, characteristic, complemented
3C2×#2322C211yesmaximal, characteristic
4C2×#2313C211yesmaximal, characteristic, complemented
5#15895C214yesmaximal, complemented
6#15584C214yesmaximal, complemented
7#15530C214yesmaximal, complemented
8#15525C214yesmaximal, complemented
9#14919C212yesmaximal, complemented
10#14912C212yesmaximal
11#14149C211yesmaximal, characteristic, complemented
12#14044C212yesmaximal, complemented
13#14020C212yesmaximal
14#13960C211yesmaximal, characteristic
15#13924C211yesmaximal, characteristic, complemented
16C22×#123C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2487—28no—
21C2×#400—28no—
22C2×#400C2214yes—
23C2×#400C2212yes—
24C2×#400C2214yes—
25C2×#399C2212yes—
26#2322—28no—
27#2313—28no—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#388C2212yes—
31C2×#387C2212yes—
32C2×#387C2214yes—
33C2×#387C2214yes—
34C2×#386C2211yescharacteristic
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#385C2212yes—
38C2×#385C2211yescharacteristic
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#381C2211yescharacteristic
42C2×#381C2212yes—
43C2×#380C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C22×#95C2212yes—
49C22×#95C2211yescharacteristic
50C22×#95C2212yes—
51#1445C2214yes—
52#1445C2214yes—
53#1443C2214yes—
54#1443C2214yes—
55#1443C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2212yes—
62#1441C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2212yes—
67#1437C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70C4⋊#95C2214yes—
71C4⋊#95C2214yes—
72#1434C2212yes—
73#1434C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C2×#123—24no—
87C2×#123—24no—
88C2×#123—24no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4D418yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2314yes—
121C22×C42C2312yes—
122#400—216no—
123#400—216no—
124#400—28no—
125#399—28no—
126#389—28no—
127#389—28no—
128#388—28no—
129#387—216no—
130#387—28no—
131#387—216no—
132#386—28no—
133#386—24no—
134#386—28no—
135#385—24no—
136#385—28no—
137#385—28no—
138#385—28no—
139#381—24no—
140#381—28no—
141#380—24no—
142#380—28no—
143#380D418yes—
144C4×C22⋊C4—24no—
145C4×C22⋊C4—28no—
146C2×#95—28no—
147C2×#95—28no—
148C2×#95—24no—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C25C2411yescharacteristic
178C25C2411yescharacteristic
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4#4114yes—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4#4114yes—
215C23×C4C2411yescharacteristic
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228#123—232no—
229C4×D4—416no—
230C2×C4⋊C4—232no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—416no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D418yes—
260C2×C4⋊C4C2×D418yes—
261C2×C4⋊C4C2×D418yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4C2×D418yes—
290C2×C22⋊C4#4114yes—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—28no—
295C2×C42—24no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42—44no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95C2×D418yes—
327C24—24no—
328C24—24no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—21no—
333C24—21no—
334C24—21no—
335C24—24no—
336C24—24no—
337C24C22×D412yes—
338C24C22×D412yes—
339C24C2×#4112yes—
340C24C2×#4112yes—
341C24C2×#4112yes—
342C24C2×#4112yes—
343C242−1+412yes—
344C242−1+412yes—
345C24C2511yescharacteristic
346C2×D4—28no—
347C2×D4—28no—
348C2×D4—28no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—44no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—28no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4—88no—
443C22×C4#121116yes—
444C22×C4C22×D414yes—
445C22×C4C22×D414yes—
446C22×C4C22×D418yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C4⋊C4—216no—
456C4⋊C4—216no—
457C4⋊C4—216no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—84no—
463C4⋊C4—84no—
464C4⋊C4—88no—
465C4⋊C4—88no—
466C22⋊C4—216no—
467C22⋊C4—28no—
468C22⋊C4—216no—
469C22⋊C4—44no—
470C22⋊C4—44no—
471C22⋊C4—48no—
472C22⋊C4—44no—
473C22⋊C4—84no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C42—44no—
477C42—84no—
478C42—88no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—28no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—28no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23C2×#12114yes—
507C23#52514yes—
508C23#53614yes—
509C23#54714yes—
510C23#54814yes—
511C23#55214yes—
512C23#55614yes—
513C23C23×D411yescharacteristic
514C23C23×D411yescharacteristic
515C23C22×#4111yescharacteristic
516C23C22×#4111yescharacteristic
517C23C22×#4111yescharacteristic
518C23C2×2−1+411yescharacteristic
519C23C2×2−1+411yescharacteristic
520C23#58514yes—
521C23#58514yes—
522D4—416no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—416no—
544C2×C4—41no—
545C2×C4—416no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—88no—
597C2×C4—88no—
598C2×C4—816no—
599C2×C4—816no—
600C2×C4C2×#12114yes—
601C2×C4C2×#12114yes—
602C22—28no—
603C22—24no—
604C22—24no—
605C22—28no—
606C22—28no—
607C22—28no—
608C22—28no—
609C22—28no—
610C22—44no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—82no—
615C22—81no—
616C22—81no—
617C22—82no—
618C22—82no—
619C22#241518yes—
620C22C22×#12111yescharacteristic
621C22C2×#52511yescharacteristic
622C22C2×#53611yescharacteristic
623C22C2×#54711yescharacteristic
624C22C2×#54811yescharacteristic
625C22C2×#55211yescharacteristic
626C22C2×#55611yescharacteristic
627C22#348512yes—
628C22#348512yes—
629C22#348712yes—
630C22#349112yes—
631C22#350012yes—
632C22#351012yes—
633C22#352114yes—
634C22#352114yes—
635C22#355314yes—
636C22#355314yes—
637C4—24no—
638C4—24no—
639C4—44no—
640C4—44no—
641C4—44no—
642C4—42no—
643C4—48no—
644C4—82no—
645C4—81no—
646C4—84no—
647C4—81no—
648C4—84no—
649C4—84no—
650C2—28no—
651C2—82no—
652C2C2×#241511yescharacteristic
653C2#3544711yescharacteristic
654C2#3545412yes—
655C2#3567711yescharacteristic
656C2#3712012yes—
657C2#4349512yes—
658C2#4349812yes—
659C2#4351312yes—
660C2#4351612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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
4. C2×#2313C2semidirect
5. #15895C2semidirect
6. #15584C2semidirect
7. #15530C2semidirect
8. #15525C2semidirect
9. #14919C2semidirect
11. #14149C2semidirect
12. #14044C2semidirect
15. #13924C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C22×#4002
C2×#23222
C2×#23132
#158952
#155842
#155302
#155252
#149192
#149122
#141492
#140442
#140202
#139602
#139242