Finite Group Databasebeta

← #4375187

Subgroups — #4375187

Order 512.

3200 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic, complemented
2C2×#2328C211yesmaximal, characteristic
3#16816C212yesmaximal, complemented
4#16368C214yesmaximal, complemented
5#16263C212yesmaximal, complemented
6#16074C212yesmaximal, complemented
7#15944C212yesmaximal, complemented
8#15662C212yesmaximal, complemented
9#15313C212yesmaximal
10#15119C212yesmaximal, complemented
11#14873C212yesmaximal, complemented
12#14865C212yesmaximal, complemented
13#14765C212yesmaximal, complemented
14#14517C214yesmaximal, complemented
15#13940C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2409—28no—
22C2×#395C2212yes—
23C2×#394C2214yes—
24C2×#394C2214yes—
25C2×#393C2212yes—
26C2×#393C2212yes—
27C2×#393C2212yes—
28C2×#393C2214yes—
29#2328—28no—
30C2×#391C2212yes—
31C2×#390C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#388C2214yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2214yes—
40C2×#385C2212yes—
41C2×#382C2211yescharacteristic
42C2×#381C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×#376C2211yescharacteristic
46#1446C2212yes—
47#1446C2212yes—
48#1446C2214yes—
49#1445C2214yes—
50#1444C2212yes—
51#1443C2214yes—
52#1442C2212yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2214yes—
56#1440C2214yes—
57#1440C2212yes—
58#1440C2214yes—
59#1440C2214yes—
60#1440C2214yes—
61#1439C2212yes—
62#1439C2212yes—
63#1439C2214yes—
64#1438C2214yes—
65#1438C2212yes—
66#1438C2214yes—
67#1437C2212yes—
68#1436C2214yes—
69#1436C2212yes—
70#1436C2214yes—
71#1434C2212yes—
72#1434C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C23×Q8—24no—
78C23×Q8—42no—
79C23×Q8—42no—
80C23×Q8—48no—
81C23×Q8C2312yes—
82C23×Q8C2312yes—
83C23×D4C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C23×D4C2312yes—
86C23×D4C2312yes—
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×#127—24no—
92C2×#127—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—48no—
95C2×C4×D4—24no—
96C2×C4×D4—42no—
97C2×C4×D4—42no—
98C2×C4×D4—42no—
99C2×C4×D4—42no—
100C2×C4×D4D418yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#395—28no—
127#394—216no—
128#394—48no—
129#393—28no—
130#393—44no—
131#393—44no—
132#393—48no—
133#391—28no—
134#390—24no—
135#390—24no—
136#388—48no—
137#386—28no—
138#386—28no—
139#386—28no—
140#386—28no—
141#386—28no—
142#386—48no—
143#385—28no—
144#382—24no—
145#381—42no—
146#380—24no—
147C4×C22⋊C4—42no—
148#376—24no—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
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×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C25C2411yescharacteristic
174C25C2412yes—
175C25C2411yescharacteristic
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—28no—
179C22×Q8—28no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—44no—
183C22×Q8—44no—
184C22×Q8—42no—
185C22×Q8—42no—
186C22×Q8—416no—
187C22×Q8—432no—
188C22×Q8—432no—
189C22×Q8—432no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—41no—
201C22×D4—41no—
202C22×D4—41no—
203C22×D4—41no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4C2×D414yes—
209C22×D4#4118yes—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—44no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—44no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240#127—232no—
241C22⋊Q8—432no—
242C4×D4—416no—
243C4×D4—416no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C2×D414yes—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—42no—
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—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4C2×D414yes—
298C2×C22⋊C4C2×D414yes—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—42no—
306C2×C42—44no—
307C2×C42C2×D414yes—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—42no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332C24—21no—
333C24—21no—
334C24—24no—
335C24—22no—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—22no—
344C24—22no—
345C24—21no—
346C24—21no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24C22×D412yes—
354C24C22×D412yes—
355C24C2×#4114yes—
356C24C2×#4114yes—
357C24C2511yescharacteristic
358C2×Q8—216no—
359C2×Q8—416no—
360C2×Q8—416no—
361C2×Q8—416no—
362C2×Q8—48no—
363C2×Q8—48no—
364C2×Q8—48no—
365C2×Q8—416no—
366C2×Q8—4128no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×D4—28no—
370C2×D4—28no—
371C2×D4—216no—
372C2×D4—216no—
373C2×D4—28no—
374C2×D4—44no—
375C2×D4—48no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—44no—
380C2×D4—416no—
381C2×D4—44no—
382C2×D4—88no—
383C2×D4—84no—
384C2×D4—84no—
385C2×D4—84no—
386C2×D4—84no—
387C2×D4—88no—
388C2×D4—88no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
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—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—416no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—416no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4—84no—
458C22×C4—84no—
459C22×C4—88no—
460C22×C4—88no—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C2×#4114yes—
467C22×C4H(5,𝔽2)18yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C4H(5,𝔽2)12yes—
470C22×C42−1+414yes—
471C4⋊C4—44no—
472C4⋊C4—416no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—88no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—416no—
479C22⋊C4—416no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—84no—
484C22⋊C4—84no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—84no—
488C22⋊C4—84no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C22⋊C4—84no—
492C42—28no—
493C42—28no—
494C42—44no—
495C42—84no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—28no—
502C23—24no—
503C23—24no—
504C23—22no—
505C23—24no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—28no—
511C23—28no—
512C23—24no—
513C23—24no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—44no—
518C23—42no—
519C23—44no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—44no—
536C23—44no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—42no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23#52514yes—
549C23C23×D411yescharacteristic
550C23C22×#4112yes—
551C23C2×H(5,𝔽2)11yescharacteristic
552C23C2×H(5,𝔽2)11yescharacteristic
553C23C2×H(5,𝔽2)14yes—
554C23C2×2−1+412yes—
555C23#58514yes—
556Q8—432no—
557Q8—432no—
558D4—416no—
559D4—416no—
560D4—816no—
561D4—816no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—416no—
581C2×C4—432no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—88no—
627C2×C4—816no—
628C2×C4#53414yes—
629C2×C4#53614yes—
630C22—24no—
631C22—28no—
632C22—48no—
633C22—42no—
634C22—42no—
635C22—48no—
636C22—44no—
637C22—48no—
638C22—42no—
639C22—44no—
640C22—42no—
641C22—42no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—48no—
646C22—44no—
647C22—84no—
648C22—84no—
649C22—82no—
650C22—82no—
651C22—84no—
652C22—84no—
653C22—82no—
654C22—82no—
655C22C2×#52511yescharacteristic
656C22C2×#53411yescharacteristic
657C22C2×#53611yescharacteristic
658C22#348412yes—
659C22#350012yes—
660C22#350514yes—
661C22#351414yes—
662C22#352012yes—
663C22#352012yes—
664C22#353212yes—
665C22#353212yes—
666C22#354012yes—
667C22#354714yes—
668C22#355112yes—
669C22#355414yes—
670C4—24no—
671C4—24no—
672C4—42no—
673C4—42no—
674C4—48no—
675C4—42no—
676C4—44no—
677C4—84no—
678C4—82no—
679C4—84no—
680C4—84no—
681C4—82no—
682C2—44no—
683C2—84no—
684C2—82no—
685C2#3508411yescharacteristic
686C2#3570312yes—
687C2#3670211yescharacteristic
688C2#3741911yescharacteristic
689C2#4243112yes—
690C2#4711514yes—
691C2#5035614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2409C2semidirect
3. #16816C2semidirect
4. #16368C2semidirect
5. #16263C2semidirect
6. #16074C2semidirect
7. #15944C2semidirect
8. #15662C2semidirect
10. #15119C2semidirect
11. #14873C2semidirect
12. #14865C2semidirect
13. #14765C2semidirect
14. #14517C2semidirect
15. #13940C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C2×#23282
#168162
#163682
#162632
#160742
#159442
#156622
#153132
#151192
#148732
#148652
#147652
#145172
#139402