Finite Group Databasebeta

← #4862606

Subgroups — #4862606

Order 512.

2552 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic, complemented
2C2×#2635C211yesmaximal, characteristic
3#17168C212yesmaximal, complemented
4#17088C212yesmaximal, complemented
5#16602C212yesmaximal, complemented
6#16539C214yesmaximal, complemented
7#16530C214yesmaximal, complemented
8#16364C212yesmaximal, complemented
9#16263C212yesmaximal, complemented
10#16253C212yesmaximal, complemented
11#15798C212yesmaximal, complemented
12#15643C212yesmaximal, complemented
13#15100C212yesmaximal, complemented
14#14765C212yesmaximal, complemented
15#13940C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×#121C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2635—28no—
22#2542—28no—
23C2×#400C2211yescharacteristic
24C2×#399C2211yescharacteristic
25C2×#399C2212yes—
26C2×#399C2212yes—
27C2×#397C2214yes—
28C2×#394C2211yescharacteristic
29C2×#394C2212yes—
30C2×#394C2214yes—
31C2×#394C2214yes—
32C2×#393C2212yes—
33C2×#393C2212yes—
34C2×#392C2212yes—
35C2×#392C2211yescharacteristic
36C2×#390C2212yes—
37C2×#388C2212yes—
38C2×#388C2214yes—
39C2×#386C2212yes—
40C2×#384C2212yes—
41C2×#381C2211yescharacteristic
42C2×#381C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45#1446C2212yes—
46#1446C2212yes—
47#1444C2212yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2212yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1440C2212yes—
57#1440C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2212yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1438C2212yes—
67#1438C2214yes—
68#1438C2214yes—
69#1438C2214yes—
70#1437C2214yes—
71#1437C2212yes—
72#1436C2212yes—
73#1436C2214yes—
74#1436C2214yes—
75#1434C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2312yes—
78C23×D4C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C23×D4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—42no—
86C2×C4⋊Q8—42no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—48no—
90C2×C22⋊Q8—48no—
91C2×#121—24no—
92C2×#121—42no—
93C2×#121—42no—
94C2×C4×D4—42no—
95C2×C4×D4—42no—
96C2×C4×D4D418yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126#400—24no—
127#399—24no—
128#399—28no—
129#399—28no—
130#397—48no—
131#394—24no—
132#394—28no—
133#394—48no—
134#394—48no—
135#393—44no—
136#393—44no—
137#392—28no—
138#392—24no—
139#390—28no—
140#388—28no—
141#388—48no—
142#386—28no—
143#384—28no—
144#381—24no—
145#381—42no—
146C4×C4⋊C4—24no—
147C4×C22⋊C4—42no—
148C2×#95C2312yes—
149C2×#95C2311yescharacteristic
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2311yescharacteristic
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×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C25C2412yes—
172C25C2411yescharacteristic
173C25C2411yescharacteristic
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—44no—
178C22×Q8—44no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—22no—
182C22×D4—24no—
183C22×D4—41no—
184C22×D4—41no—
185C22×D4—42no—
186C22×D4—41no—
187C22×D4—41no—
188C22×D4—42no—
189C22×D4—42no—
190C22×D4—42no—
191C22×D4—42no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4C2×D414yes—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4C2×D414yes—
205C23×C4C2×D414yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C4⋊Q8—416no—
219C22⋊Q8—432no—
220C22⋊Q8—432no—
221#121—416no—
222C4×D4—416no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4C2×D414yes—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
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—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
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—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4C2×D414yes—
292C2×C22⋊C4C2×D414yes—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—44no—
297C2×C42—42no—
298C2×C42—44no—
299C2×C42C2×D414yes—
300#95—24no—
301#95—24no—
302#95—22no—
303#95—22no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—42no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323C24—24no—
324C24—24no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24C22×D412yes—
347C24C22×D412yes—
348C24C2511yescharacteristic
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—88no—
353C2×Q8—88no—
354C2×D4—28no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—44no—
360C2×D4—48no—
361C2×D4—48no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—84no—
365C2×D4—84no—
366C2×D4—84no—
367C2×D4—84no—
368C2×D4—84no—
369C2×D4—88no—
370C2×D4—88no—
371C2×D4—88no—
372C2×D4—88no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—28no—
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—24no—
404C22×C4—22no—
405C22×C4—22no—
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—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4—84no—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4H(5,𝔽2)18yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C4H(5,𝔽2)12yes—
444C22×C42−1+414yes—
445C4⋊C4—216no—
446C4⋊C4—44no—
447C4⋊C4—416no—
448C4⋊C4—416no—
449C4⋊C4—44no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—84no—
453C4⋊C4—88no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—416no—
457C22⋊C4—416no—
458C22⋊C4—416no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—84no—
463C22⋊C4—84no—
464C22⋊C4—88no—
465C22⋊C4—84no—
466C22⋊C4—84no—
467C22⋊C4—88no—
468C22⋊C4—84no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C42—28no—
472C42—48no—
473C42—84no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—44no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23C23×D411yescharacteristic
528C23C2×H(5,𝔽2)11yescharacteristic
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×H(5,𝔽2)14yes—
531C23C2×2−1+412yes—
532C23#58512yes—
533C23#58514yes—
534Q8—816no—
535D4—416no—
536D4—88no—
537D4—816no—
538D4—816no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
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—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—88no—
602C2×C4—88no—
603C2×C4#53414yes—
604C2×C4#53614yes—
605C22—42no—
606C22—42no—
607C22—44no—
608C22—44no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—44no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—84no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—84no—
625C22—84no—
626C22—82no—
627C22—82no—
628C22—84no—
629C22—84no—
630C22C2×#53411yescharacteristic
631C22C2×#53611yescharacteristic
632C22#344911yescharacteristic
633C22#348412yes—
634C22#350012yes—
635C22#353314yes—
636C22#354012yes—
637C22#354112yes—
638C22#354112yes—
639C22#354312yes—
640C22#354312yes—
641C22#354714yes—
642C22#355112yes—
643C22#355414yes—
644C22#356014yes—
645C4—24no—
646C4—24no—
647C4—42no—
648C4—48no—
649C4—42no—
650C4—42no—
651C4—82no—
652C4—82no—
653C4—84no—
654C4—84no—
655C4—82no—
656C4—84no—
657C2—82no—
658C2—82no—
659C2—162no—
660C2#3723211yescharacteristic
661C2#3741911yescharacteristic
662C2#3743811yescharacteristic
663C2#4050012yes—
664C2#4243112yes—
665C2#5344914yes—
666C2#5571714yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2635C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2542C2semidirect
3. #17168C2semidirect
4. #17088C2semidirect
5. #16602C2semidirect
6. #16539C2semidirect
7. #16530C2semidirect
8. #16364C2semidirect
9. #16263C2semidirect
10. #16253C2semidirect
11. #15798C2semidirect
12. #15643C2semidirect
13. #15100C2semidirect
14. #14765C2semidirect
15. #13940C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#26352
#171682
#170882
#166022
#165392
#165302
#163642
#162632
#162532
#157982
#156432
#151002
#147652
#139402