Finite Group Databasebeta

← #849318

Subgroups — #849318

Order 512.

2840 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2591C211yesmaximal, characteristic
2C22×#382C211yesmaximal, characteristic, complemented
3#16689C212yesmaximal, complemented
4#16684C212yesmaximal, complemented
5#16084C214yesmaximal, complemented
6#15946C212yesmaximal, complemented
7#15941C212yesmaximal, complemented
8#15069C212yesmaximal, complemented
9#14927C212yesmaximal, complemented
10#14926C212yesmaximal
11#14696C212yesmaximal, complemented
12#14399C214yesmaximal, complemented
13#13996C212yesmaximal, complemented
14#13992C212yesmaximal, complemented
15#13923C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#123C2212yes—
18C22×#121C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2591—28no—
23C2×#400C2211yescharacteristic
24C2×#400C2211yescharacteristic
25C2×#394C2211yescharacteristic
26C2×#394C2211yescharacteristic
27C2×#388C2212yes—
28C2×#388C2212yes—
29C2×#388C2214yes—
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#385C2212yes—
35C2×#382—28no—
36C2×#382C2212yes—
37C2×#382C2212yes—
38C2×#382C2212yes—
39C2×#382C2212yes—
40C2×#381C2212yes—
41C2×#381C2212yes—
42C2×#380C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C22×#95C2212yes—
46C22×#95C2212yes—
47#1445C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1444C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1441C2212yes—
55#1441C2214yes—
56#1441C2212yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1438C2214yes—
65#1437C2214yes—
66#1437C2214yes—
67#1436C2212yes—
68#1436C2214yes—
69#1436C2212yes—
70#1436C2214yes—
71C4⋊#95C2214yes—
72#1434C2214yes—
73#1434C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×#123—28no—
85C2×#123—44no—
86C2×#123—44no—
87C2×#123—44no—
88C2×#123—44no—
89C2×#123—44no—
90C2×#121—24no—
91C2×#121—42no—
92C2×#121—42no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—42no—
95C22×C4⋊C4—42no—
96C22×C4⋊C4—42no—
97C22×C4⋊C4—42no—
98C22×C4⋊C4D418yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
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⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C42—24no—
123C22×C42—28no—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128#400—24no—
129#400—24no—
130#394—24no—
131#394—24no—
132#388—28no—
133#388—28no—
134#388—216no—
135#386—28no—
136#386—28no—
137#386—28no—
138#386—44no—
139#385—44no—
140#382—28no—
141#382—28no—
142#382—28no—
143#382—28no—
144#381—28no—
145#381—44no—
146#380—42no—
147C4×C22⋊C4—24no—
148C4×C22⋊C4—42no—
149C2×#95—28no—
150C2×#95—28no—
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
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×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C25C2412yes—
177C25C2411yescharacteristic
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—22no—
186C22×D4—41no—
187C22×D4—41no—
188C22×D4—44no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227#123—432no—
228#123—432no—
229#121—416no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—416no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—416no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×D414yes—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4#4118yes—
296C2×C42—232no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—28no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—42no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42#4118yes—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—22no—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—42no—
326#95—42no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—22no—
335C24—24no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24C22×D412yes—
346C24C2×#4114yes—
347C24H(5,𝔽2)12yes—
348C24H(5,𝔽2)12yes—
349C24H(5,𝔽2)14yes—
350C24C2511yescharacteristic
351C2×D4—44no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—84no—
360C2×D4—84no—
361C2×D4—84no—
362C22×C4—216no—
363C22×C4—216no—
364C22×C4—28no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—216no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C2×#4118yes—
455C22×C4C2×#4118yes—
456C22×C4C2×#4114yes—
457C22×C42−1+414yes—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—44no—
465C4⋊C4—416no—
466C4⋊C4—416no—
467C4⋊C4—88no—
468C4⋊C4—84no—
469C22⋊C4—216no—
470C22⋊C4—416no—
471C22⋊C4—48no—
472C22⋊C4—416no—
473C22⋊C4—416no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—84no—
479C22⋊C4—84no—
480C22⋊C4—88no—
481C42—216no—
482C42—216no—
483C42—84no—
484C42—84no—
485C23—22no—
486C23—22no—
487C23—28no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—24no—
494C23—24no—
495C23—28no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—44no—
505C23—44no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23#53414yes—
520C23#53818yes—
521C23#55018yes—
522C23#55514yes—
523C23H(3,𝔽4)14yes—
524C23C23×D411yescharacteristic
525C23C22×#4112yes—
526C23C22×#4112yes—
527C23C2×H(5,𝔽2)11yescharacteristic
528C23C2×H(5,𝔽2)11yescharacteristic
529C23C2×H(5,𝔽2)12yes—
530C23C2×2−1+412yes—
531C23#58514yes—
532D4—88no—
533D4—816no—
534C2×C4—232no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—432no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—88no—
606C2×C4—88no—
607C2×C4—816no—
608C2×C4#52514yes—
609C2×C4#53614yes—
610C22—216no—
611C22—28no—
612C22—28no—
613C22—28no—
614C22—28no—
615C22—24no—
616C22—24no—
617C22—84no—
618C22—84no—
619C22—82no—
620C22—82no—
621C22—84no—
622C22—82no—
623C22—84no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22#273318yes—
629C22C2×#52511yescharacteristic
630C22C2×#53411yescharacteristic
631C22C2×#53611yescharacteristic
632C22C2×#53812yes—
633C22C2×#55012yes—
634C22C2×#55511yescharacteristic
635C22#347514yes—
636C22#348512yes—
637C22#349212yes—
638C22#349212yes—
639C22#349414yes—
640C22#351514yes—
641C22C2×H(3,𝔽4)11yescharacteristic
642C22#354512yes—
643C22#354512yes—
644C22#355914yes—
645C4—24no—
646C4—24no—
647C4—48no—
648C4—48no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—82no—
654C4—84no—
655C4—84no—
656C4—82no—
657C4—82no—
658C2—28no—
659C2—162no—
660C2#3505711yescharacteristic
661C2#3651712yes—
662C2#3711911yescharacteristic
663C2#3790514yes—
664C2#3913412yes—
665C2#4599112yes—
666C2C2×#273311yescharacteristic
667C2#5933912yes—
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
2. C22×#382C2semidirect
3. #16689C2semidirect
4. #16684C2semidirect
5. #16084C2semidirect
6. #15946C2semidirect
7. #15941C2semidirect
8. #15069C2semidirect
9. #14927C2semidirect
11. #14696C2semidirect
12. #14399C2semidirect
13. #13996C2semidirect
14. #13992C2semidirect
15. #13923C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25912
C22×#3822
#166892
#166842
#160842
#159462
#159412
#150692
#149272
#149262
#146962
#143992
#139962
#139922
#139232