Finite Group Databasebeta

← #641994

Subgroups — #641994

Order 512.

2536 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2296C211yesmaximal, characteristic, complemented
2C2×#2290C211yesmaximal, characteristic
3#16894C214yesmaximal, complemented
4#16893C214yesmaximal
5#15221C212yesmaximal, complemented
6#15220C212yesmaximal, complemented
7#15096C214yesmaximal, complemented
8#15095C214yesmaximal, complemented
9#14682C211yesmaximal, characteristic, complemented
10#14266C212yesmaximal, complemented
11#14264C212yesmaximal, complemented
12#14129C211yesmaximal, characteristic, complemented
13#14129C211yesmaximal, characteristic, complemented
14#13871C211yesmaximal, characteristic
15#13871C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2296—28no—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26#2290—28no—
27C2×#388C2212yes—
28C2×#387C2212yes—
29C2×#386C2212yes—
30C2×#385C2212yes—
31C2×#385C2214yes—
32C2×#382C2212yes—
33C2×#380C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2214yes—
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×#376C2211yescharacteristic
42#1445C2214yes—
43#1445C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1440C2212yes—
55#1440C2212yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2212yes—
63#1438C2212yes—
64#1437C2212yes—
65#1437C2212yes—
66#1436C2212yes—
67#1436C2212yes—
68#1436C2214yes—
69#1436C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75#1434C2214yes—
76C22⋊#95C2214yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C2×C4×D4—24no—
82C2×C4×D4—24no—
83C2×C4×D4Q818yes—
84C2×#117—24no—
85C2×#117—24no—
86C2×#117—28no—
87C2×#117—24no—
88C2×#117—44no—
89C2×#117—44no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C42—44no—
113C22×C42D418yes—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2312yes—
122#389—28no—
123#389—28no—
124#389—28no—
125#389—44no—
126#388—28no—
127#387—28no—
128#386—28no—
129#385—28no—
130#385—216no—
131#382—44no—
132#380—24no—
133#380—24no—
134#380—28no—
135#380—44no—
136C4×C4⋊C4—24no—
137C4×C4⋊C4—216no—
138C4×C22⋊C4—24no—
139C4×C22⋊C4—24no—
140#376—24no—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C25C2411yescharacteristic
167C25C2411yescharacteristic
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—42no—
171C22×D4—42no—
172C22×D4C2×Q814yes—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4C2×D414yes—
186C23×C4C2×D418yes—
187C23×C4C2×Q814yes—
188C23×C4C2×Q814yes—
189C23×C4#4114yes—
190C23×C4C2412yes—
191C23×C4C2414yes—
192C23×C4C2414yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2414yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C4×D4—232no—
204#117—232no—
205#117—416no—
206#117—416no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4C2×Q814yes—
239C2×C4⋊C4#4118yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
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—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4C2×Q814yes—
268C2×C22⋊C4C2×Q814yes—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—28no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—416no—
281C2×C42—44no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42C2×D414yes—
285C2×C42C2×D414yes—
286C2×C42C2×D418yes—
287C2×C42C2×Q814yes—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—28no—
293#95—28no—
294#95—24no—
295#95—28no—
296#95—24no—
297#95—24no—
298#95—28no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—21no—
320C24—22no—
321C24—21no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24C22×D412yes—
328C24C22×Q812yes—
329C24C22×Q812yes—
330C24C2×#4112yes—
331C24C2511yescharacteristic
332C2×D4—28no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—48no—
336C2×D4—48no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—216no—
342C22×C4—216no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—24no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—28no—
374C22×C4—216no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—48no—
420C22×C4C22⋊Q8116yes—
421C22×C4C22×D414yes—
422C22×C4C22×D414yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×Q812yes—
426C22×C4C22×Q812yes—
427C22×C4C22×Q812yes—
428C22×C4C22×Q812yes—
429C22×C4C22×Q812yes—
430C22×C4C2×#4118yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4114yes—
434C22×C4H(5,𝔽2)14yes—
435C4⋊C4—28no—
436C4⋊C4—216no—
437C4⋊C4—48no—
438C4⋊C4—416no—
439C4⋊C4—416no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—88no—
445C4⋊C4—88no—
446C22⋊C4—216no—
447C22⋊C4—28no—
448C22⋊C4—28no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—88no—
454C22⋊C4—84no—
455C22⋊C4—84no—
456C22⋊C4—88no—
457C42—216no—
458C42—28no—
459C42—28no—
460C42—28no—
461C42—48no—
462C42—48no—
463C42—44no—
464C42—44no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23C2×C22⋊Q814yes—
507C23C23×D411yescharacteristic
508C23C23×Q811yescharacteristic
509C23C22×#4111yescharacteristic
510C23C22×#4112yes—
511C23C2×H(5,𝔽2)12yes—
512C23#58512yes—
513C23#58512yes—
514C23#58512yes—
515C23#58512yes—
516D4—416no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—232no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—432no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—416no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4C2×C22⋊Q814yes—
596C2×C4C2×C22⋊Q814yes—
597C2×C4#52514yes—
598C2×C4#53414yes—
599C2×C4#55318yes—
600C2×C4#55418yes—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—44no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22C22×C22⋊Q811yescharacteristic
620C22C2×#52511yescharacteristic
621C22C2×#53411yescharacteristic
622C22#344912yes—
623C22#344912yes—
624C22#345512yes—
625C22#345811yescharacteristic
626C22#345811yescharacteristic
627C22C2×#55312yes—
628C22C2×#55412yes—
629C22#350412yes—
630C22#351011yescharacteristic
631C22#351011yescharacteristic
632C22#352112yes—
633C22#352112yes—
634C22#352412yes—
635C22#352412yes—
636C22#353912yes—
637C22#353912yes—
638C22#354012yes—
639C22#354012yes—
640C4—24no—
641C4—42no—
642C4—42no—
643C4—44no—
644C4—44no—
645C4—42no—
646C4—48no—
647C4—82no—
648C4—84no—
649C4—82no—
650C4—84no—
651C4—84no—
652C4#257918yes—
653C2—44no—
654C2—82no—
655C2#3463411yescharacteristic
656C2C2×#257911yescharacteristic
657C2#3521012yes—
658C2#3533411yescharacteristic
659C2#3744512yes—
660C2#3810612yes—
661C2#3810712yes—
662C2#3813012yes—
663C2#3813112yes—
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
1. C2×#2296C2semidirect
3. #16894C2semidirect
5. #15221C2semidirect
6. #15220C2semidirect
7. #15096C2semidirect
8. #15095C2semidirect
9. #14682C2semidirect
10. #14266C2semidirect
11. #14264C2semidirect
12. #14129C2semidirect
13. #14129C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22962
C2×#22902
#168942
#168932
#152212
#152202
#150962
#150952
#146822
#142662
#142642
#141292
#138712