Finite Group Databasebeta

← #763383

Subgroups — #763383

Order 512.

3352 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2442C212yesmaximal, complemented
2C2×#2311C212yesmaximal
3#15958C214yesmaximal, complemented
4#15509C214yesmaximal, complemented
5#15329C214yesmaximal, complemented
6#15326C214yesmaximal, complemented
7#15223C211yesmaximal, characteristic, complemented
8#15220C212yesmaximal, complemented
9#15216C211yesmaximal, characteristic, complemented
10#14129C212yesmaximal, complemented
11#13951C212yesmaximal, complemented
12#13881C212yesmaximal, complemented
13#13851C211yesmaximal, characteristic
14C23×#41C2211yescharacteristic
15C22×#125C2211yescharacteristic
16C22×#117C2212yes—
17C23×C4⋊C4C2212yes—
18#2442—216no—
19C2×#400C2214yes—
20C2×#398C2214yes—
21#2311—216no—
22C2×#391C2214yes—
23C2×#390C2214yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#386C2214yes—
28C2×#386C2214yes—
29C2×#385C2214yes—
30C2×#382C2212yes—
31C2×#382C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2214yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41#1446C2212yes—
42#1445C2214yes—
43#1443C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1441C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1438C2212yes—
57#1438C2212yes—
58#1438C2212yes—
59#1436C2214yes—
60#1436C2214yes—
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2214yes—
66#1434C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C22×#41—28no—
70C22×#41—28no—
71C22×#41—42no—
72C22×#41—42no—
73C22×#41Q818yes—
74C23×Q8C2311yescharacteristic
75C23×D4C2311yescharacteristic
76C23×D4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2314yes—
80C2×#125—28no—
81C2×#125—42no—
82C2×#117—28no—
83C2×#117—28no—
84C2×#117—44no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2314yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2314yes—
113#400—216no—
114#398—216no—
115#391—216no—
116#390—216no—
117#389—28no—
118#389—28no—
119#389—44no—
120#386—216no—
121#386—216no—
122#385—216no—
123#382—28no—
124#382—42no—
125#380—28no—
126#380—28no—
127#380—28no—
128#380—216no—
129#380D4116yes—
130C4×C22⋊C4—28no—
131C4×C22⋊C4—28no—
132C4×C22⋊C4—42no—
133C4×C22⋊C4—42no—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C25C2411yescharacteristic
157C25C2412yes—
158C2×#41—264no—
159C2×#41—416no—
160C2×#41—416no—
161C2×#41—432no—
162C2×#41—416no—
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—41no—
166C22×Q8—41no—
167C22×Q8C2×Q814yes—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—42no—
175C22×D4—42no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4C2×Q814yes—
179C22×D4C2×Q818yes—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—28no—
189C23×C4—28no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4C2×D418yes—
196C23×C4C2×Q814yes—
197C23×C4C2×Q818yes—
198C23×C4#4118yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210#125—416no—
211#117—432no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—432no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4C2×D418yes—
239C2×C4⋊C4C2×D418yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—41no—
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⋊C4C2×D4116yes—
274C2×C42—28no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—28no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—44no—
284#95—24no—
285#95—24no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—42no—
302#95—42no—
303#95—44no—
304#95—44no—
305#95C2×D4116yes—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—21no—
321C24—21no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24C22×D414yes—
327C24C22×Q812yes—
328C24C22×Q814yes—
329C24C2×#4114yes—
330C24C2511yescharacteristic
331#41—4128no—
332C2×Q8—216no—
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—48no—
336C2×Q8—44no—
337C2×D4—216no—
338C2×D4—216no—
339C2×D4—216no—
340C2×D4—48no—
341C2×D4—44no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—44no—
347C2×D4—48no—
348C2×D4—44no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—216no—
354C22×C4—216no—
355C22×C4—216no—
356C22×C4—216no—
357C22×C4—216no—
358C22×C4—216no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—216no—
376C22×C4—28no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—416no—
420C22×C4—44no—
421C22×C4—48no—
422C22×C4—44no—
423C22×C4—416no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4—88no—
429C22×C4C22⋊Q8132yes—
430C22×C4C22×D418yes—
431C22×C4C22×D418yes—
432C22×C4C22×D414yes—
433C22×C4C22×D414yes—
434C22×C4C22×Q812yes—
435C22×C4C22×Q814yes—
436C22×C4C22×Q812yes—
437C22×C4C2×#4114yes—
438C22×C4C2×#4114yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C4H(5,𝔽2)12yes—
441C4⋊C4—216no—
442C4⋊C4—216no—
443C4⋊C4—416no—
444C4⋊C4—416no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—88no—
448C4⋊C4—88no—
449C4⋊C4—88no—
450C22⋊C4—416no—
451C22⋊C4—416no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—48no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C42—416no—
463C42—44no—
464C42—48no—
465C42—44no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—22no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—42no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23C2×C22⋊Q818yes—
522C23C23×D412yes—
523C23C23×Q811yescharacteristic
524C23C22×#4112yes—
525C23C2×H(5,𝔽2)11yescharacteristic
526C23C2×H(5,𝔽2)11yescharacteristic
527C23#58512yes—
528C23#58512yes—
529C23#58512yes—
530C23#58512yes—
531Q8—416no—
532D4—432no—
533D4—416no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—264no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—416no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—832no—
604C2×C4C2×C22⋊Q818yes—
605C2×C4C2×C22⋊Q818yes—
606C2×C4#54314yes—
607C2×C4D4214yes—
608C2×C4#54618yes—
609C2×C4#55114yes—
610C22—28no—
611C22—28no—
612C22—28no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—42no—
627C22—42no—
628C22—44no—
629C22—42no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22C22×C22⋊Q812yes—
635C22C2×#54311yescharacteristic
636C22C2×D4211yescharacteristic
637C22C2×#54612yes—
638C22C2×#55111yescharacteristic
639C22#348512yes—
640C22#348712yes—
641C22#349312yes—
642C22#349312yes—
643C22#350012yes—
644C22#350012yes—
645C22#350411yescharacteristic
646C22#350412yes—
647C22#350411yescharacteristic
648C22#351212yes—
649C22#351212yes—
650C22#353911yescharacteristic
651C22#353911yescharacteristic
652C22#354012yes—
653C22#355112yes—
654C22#355112yes—
655C4—28no—
656C4—48no—
657C4—48no—
658C4—42no—
659C4—82no—
660C4—82no—
661C4—84no—
662C4—82no—
663C4—84no—
664C4—82no—
665C4#243718yes—
666C2—48no—
667C2—82no—
668C2C2×#243711yescharacteristic
669C2#3562012yes—
670C2#3563912yes—
671C2#3742511yescharacteristic
672C2#3742611yescharacteristic
673C2#4509212yes—
674C2#4509312yes—
675C2#4512512yes—
676C2#4523612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2442C2semidirect
3. #15958C2semidirect
4. #15509C2semidirect
5. #15329C2semidirect
6. #15326C2semidirect
7. #15223C2semidirect
8. #15220C2semidirect
9. #15216C2semidirect
10. #14129C2semidirect
11. #13951C2semidirect
12. #13881C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24422
C2×#23112
#159582
#155092
#153292
#153262
#152232
#152202
#152162
#141292
#139512
#138812
#138512