Finite Group Databasebeta

← #4861935

Subgroups — #4861935

Order 512.

2838 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2C2×#2633C211yesmaximal, characteristic
3#17132C214yesmaximal, complemented
4#16831C212yesmaximal, complemented
5#16815C212yesmaximal, complemented
6#16696C212yesmaximal, complemented
7#16608C212yesmaximal, complemented
8#16470C212yesmaximal, complemented
9#16321C212yesmaximal, complemented
10#16303C212yesmaximal, complemented
11#16264C212yesmaximal, complemented
12#15390C214yesmaximal
13#14874C212yesmaximal, complemented
14#14758C212yesmaximal, complemented
15#13960C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20#2633—28no—
21#2409—28no—
22C2×#399C2211yescharacteristic
23C2×#399C2211yescharacteristic
24C2×#398C2212yes—
25C2×#398C2214yes—
26C2×#397C2211yescharacteristic
27C2×#397C2211yescharacteristic
28C2×#397C2214yes—
29C2×#395C2212yes—
30C2×#395C2212yes—
31C2×#395C2212yes—
32C2×#395C2214yes—
33C2×#394C2212yes—
34C2×#393C2212yes—
35C2×#391C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#385C2214yes—
42C2×#383C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2214yes—
45C2×#380C2211yescharacteristic
46C22×#95C2211yescharacteristic
47#1446C2212yes—
48#1446C2212yes—
49#1446C2214yes—
50#1445C2214yes—
51#1444C2212yes—
52#1444C2214yes—
53#1444C2214yes—
54#1443C2214yes—
55#1443C2214yes—
56#1443C2214yes—
57#1442C2214yes—
58#1442C2212yes—
59#1442C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2212yes—
65#1441C2212yes—
66#1441C2212yes—
67#1441C2214yes—
68#1440C2214yes—
69#1440C2214yes—
70#1439C2214yes—
71#1439C2212yes—
72#1434C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C23×Q8—24no—
78C23×Q8—42no—
79C23×Q8—42no—
80C23×Q8—48no—
81C23×Q8C2312yes—
82C23×Q8C2312yes—
83C23×D4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×#125—24no—
89C2×#125—42no—
90C2×#125—42no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—48no—
93C2×C4×D4—42no—
94C2×C4×D4—42no—
95C2×C4×D4D418yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#399—24no—
121#399—24no—
122#398—28no—
123#398—48no—
124#397—24no—
125#397—24no—
126#397—48no—
127#395—28no—
128#395—44no—
129#395—44no—
130#395—48no—
131#394—28no—
132#393—28no—
133#391—28no—
134#386—28no—
135#386—44no—
136#385—28no—
137#385—28no—
138#385—28no—
139#385—48no—
140#383—24no—
141#380—24no—
142#380—42no—
143#380—48no—
144C2×#95—42no—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2311yescharacteristic
155C2×#95C2311yescharacteristic
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
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—
173C25C2411yescharacteristic
174C25C2411yescharacteristic
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—28no—
178C22×Q8—28no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8—42no—
184C22×Q8—42no—
185C22×Q8—416no—
186C22×Q8—432no—
187C22×Q8—432no—
188C22×Q8—432no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—41no—
194C22×D4—41no—
195C22×D4C2×D414yes—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—44no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—42no—
210C23×C4—44no—
211C23×C4—42no—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2411yescharacteristic
227#125—416no—
228C22⋊Q8—432no—
229C4×D4—416no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2×D414yes—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4C2×D414yes—
292C2×C22⋊C4C2×D414yes—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42C2×D414yes—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—22no—
300#95—22no—
301#95—28no—
302#95—24no—
303#95—24no—
304#95—28no—
305#95—24no—
306#95—28no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—42no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324C24—24no—
325C24—21no—
326C24—24no—
327C24—21no—
328C24—21no—
329C24—21no—
330C24—21no—
331C24—24no—
332C24—24no—
333C24—21no—
334C24—21no—
335C24—21no—
336C24C22×D412yes—
337C24C22×D412yes—
338C24H(5,𝔽2)12yes—
339C242−1+412yes—
340C242−1+412yes—
341C242−1+412yes—
342C24C2511yescharacteristic
343C2×Q8—216no—
344C2×Q8—416no—
345C2×Q8—416no—
346C2×Q8—416no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8—416no—
351C2×Q8—4128no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×D4—28no—
355C2×D4—28no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—416no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—84no—
439C22×C4—84no—
440C22×C4—88no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4H(5,𝔽2)18yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C42−1+418yes—
452C4⋊C4—44no—
453C4⋊C4—416no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C4⋊C4—88no—
462C4⋊C4—88no—
463C4⋊C4—88no—
464C4⋊C4—84no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—84no—
473C22⋊C4—84no—
474C22⋊C4—84no—
475C42—28no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—28no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—28no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—28no—
495C23—28no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23#53614yes—
509C23#53714yes—
510C23C23×D411yescharacteristic
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×H(5,𝔽2)11yescharacteristic
513C23C2×H(5,𝔽2)11yescharacteristic
514C23C2×H(5,𝔽2)14yes—
515C23C2×2−1+411yescharacteristic
516C23C2×2−1+411yescharacteristic
517C23C2×2−1+411yescharacteristic
518C23C2×2−1+414yes—
519Q8—432no—
520Q8—432no—
521D4—416no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—416no—
541C2×C4—432no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
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—816no—
598C2×C4#53414yes—
599C22—28no—
600C22—28no—
601C22—24no—
602C22—24no—
603C22—42no—
604C22—42no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22C2×#53411yescharacteristic
618C22C2×#53611yescharacteristic
619C22C2×#53711yescharacteristic
620C22#348412yes—
621C22#349912yes—
622C22#353212yes—
623C22#353212yes—
624C22#353614yes—
625C22#353611yescharacteristic
626C22#353611yescharacteristic
627C22#353812yes—
628C22#353814yes—
629C22#353811yescharacteristic
630C22#353811yescharacteristic
631C22#355012yes—
632C22#355214yes—
633C22#355214yes—
634C4—24no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—48no—
639C4—42no—
640C4—48no—
641C4—84no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4—82no—
646C4—84no—
647C2—44no—
648C2—44no—
649C2#3670211yescharacteristic
650C2#3738611yescharacteristic
651C2#3744211yescharacteristic
652C2#4049312yes—
653C2#4049512yes—
654C2#5366912yes—
655C2#5796612yes—
656C2#5895612yes—
657C2#5898712yes—
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
3. #17132C2semidirect
4. #16831C2semidirect
5. #16815C2semidirect
6. #16696C2semidirect
7. #16608C2semidirect
8. #16470C2semidirect
9. #16321C2semidirect
10. #16303C2semidirect
11. #16264C2semidirect
13. #14874C2semidirect
14. #14758C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C2×#26332
#171322
#168312
#168152
#166962
#166082
#164702
#163212
#163032
#162642
#153902
#148742
#147582
#139602