Finite Group Databasebeta

← #640922

Subgroups — #640922

Order 512.

2484 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2850C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3#16695C211yesmaximal, characteristic, complemented
4#16694C211yesmaximal, characteristic, complemented
5#16692C211yesmaximal, characteristic, complemented
6#16691C211yesmaximal, characteristic, complemented
7#16364C212yesmaximal, complemented
8#16361C212yesmaximal
9#15162C214yesmaximal, complemented
10#15023C214yesmaximal, complemented
11#14842C212yesmaximal, complemented
12#14839C212yesmaximal, complemented
13#14306C212yesmaximal, complemented
14#14305C212yesmaximal, complemented
15#14169C212yesmaximal, complemented
16#13963C212yesmaximal, complemented
17#13753C211yesmaximal, characteristic, complemented
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×#121C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2850—28no—
23C2×#399C2212yes—
24C2×#397C2212yes—
25C2×#397C2214yes—
26C2×#394C2214yes—
27C2×#391C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#388C2212yes—
30C2×#388C2212yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33#2283—28no—
34C2×#383C2212yes—
35C2×#382C2211yescharacteristic
36C2×#382C2211yescharacteristic
37C2×#382C2211yescharacteristic
38C2×#381C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×#376C2212yes—
46C2×#376C2212yes—
47#1446C2212yes—
48#1446C2212yes—
49#1445C2212yes—
50#1445C2212yes—
51#1444C2212yes—
52#1444C2212yes—
53#1444C2212yes—
54#1443C2214yes—
55#1442C2212yes—
56#1442C2214yes—
57#1442C2212yes—
58#1442C2212yes—
59#1442C2214yes—
60#1441C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2212yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2212yes—
69#1438C2212yes—
70#1438C2214yes—
71#1437C2214yes—
72#1437C2214yes—
73#1437C2214yes—
74#1437C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2214yes—
78#1434C2211yescharacteristic
79#1434C2211yescharacteristic
80#1434C2214yes—
81#1434C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×#124—24no—
92C2×#124—28no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—44no—
95C2×#121—24no—
96C2×#121—44no—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C42—28no—
120C22×C42D418yes—
121C22×C42C2314yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128#399—28no—
129#397—28no—
130#397—48no—
131#394—48no—
132#391—24no—
133#390—24no—
134#388—28no—
135#388—28no—
136#386—28no—
137#386—28no—
138#383—28no—
139#382—24no—
140#382—24no—
141#382—24no—
142#381—28no—
143#380—24no—
144#380—24no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—24no—
147C4×C22⋊C4—28no—
148C4×C22⋊C4—28no—
149#376—28no—
150#376—28no—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C25C2412yes—
182C25C2411yescharacteristic
183C22×Q8—22no—
184C22×Q8—24no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×D4—24no—
188C22×D4—28no—
189C22×D4—22no—
190C22×D4—24no—
191C22×D4—42no—
192C22×D4—42no—
193C22×D4—42no—
194C22×D4—44no—
195C22×D4—42no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—42no—
202C23×C4—41no—
203C23×C4—41no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224#124—416no—
225C22⋊Q8—416no—
226#121—416no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—232no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—28no—
286C2×C42—28no—
287C2×C42—44no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42C2×D414yes—
293C2×C42C2×D414yes—
294C2×C42C2×D414yes—
295C2×C42C2×D414yes—
296C2×C42#4118yes—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—44no—
315#95—44no—
316#95—42no—
317#95—42no—
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#95#4118yes—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—21no—
332C24—21no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—22no—
344C24—22no—
345C24C22×D412yes—
346C24H(5,𝔽2)14yes—
347C24C2511yescharacteristic
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—88no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—44no—
358C2×D4—88no—
359C2×D4—88no—
360C2×D4—88no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—28no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C2×#4118yes—
445C22×C4C2×#4114yes—
446C22×C4C2×#4114yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4114yes—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—48no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—88no—
457C4⋊C4—88no—
458C22⋊C4—416no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C42—28no—
472C42—28no—
473C42—28no—
474C42—416no—
475C42—48no—
476C42—48no—
477C42#123116yes—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—28no—
485C23—28no—
486C23—24no—
487C23—24no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23#53514yes—
518C23C23×D411yescharacteristic
519C23C22×#4112yes—
520C23C22×#4112yes—
521C23C2×H(5,𝔽2)12yes—
522C23#58512yes—
523C23#58511yescharacteristic
524C23#58511yescharacteristic
525C23#58512yes—
526C23#58512yes—
527Q8—88no—
528D4—88no—
529D4—816no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—416no—
553C2×C4—416no—
554C2×C4—416no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
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—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
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—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4C2×#12314yes—
605C2×C4C2×#12314yes—
606C2×C4C2×#12314yes—
607C2×C4#53214yes—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—48no—
612C22—48no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—84no—
618C22—84no—
619C22—84no—
620C22—84no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C22×#12311yescharacteristic
629C22C2×#53211yescharacteristic
630C22C2×#53511yescharacteristic
631C22#344911yescharacteristic
632C22#344911yescharacteristic
633C22#344911yescharacteristic
634C22#344911yescharacteristic
635C22#345512yes—
636C22#345512yes—
637C22#347514yes—
638C22#349414yes—
639C22#349414yes—
640C22#352112yes—
641C22#352412yes—
642C22#354512yes—
643C22#354512yes—
644C22#356614yes—
645C4—24no—
646C4—24no—
647C4—24no—
648C4—24no—
649C4—48no—
650C4—42no—
651C4—42no—
652C4—84no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—82no—
660C2—44no—
661C2—162no—
662C2#3463511yescharacteristic
663C2#3481011yescharacteristic
664C2#3521211yescharacteristic
665C2#3526211yescharacteristic
666C2#3530311yescharacteristic
667C2#3755811yescharacteristic
668C2#3760611yescharacteristic
669C2#3841314yes—
670C2#3841414yes—
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. C2×#2283C2semidirect
3. #16695C2semidirect
4. #16694C2semidirect
5. #16692C2semidirect
6. #16691C2semidirect
7. #16364C2semidirect
9. #15162C2semidirect
10. #15023C2semidirect
11. #14842C2semidirect
12. #14839C2semidirect
13. #14306C2semidirect
14. #14305C2semidirect
15. #14169C2semidirect
16. #13963C2semidirect
17. #13753C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28502
C2×#22832
#166952
#166942
#166922
#166912
#163642
#163612
#151622
#150232
#148422
#148392
#143062
#143052
#141692
#139632
#137532