Finite Group Databasebeta

← #613770

Subgroups — #613770

Order 512.

2844 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2500C211yesmaximal, characteristic
2C2×C4×#117C211yesmaximal, characteristic, complemented
3#15115C214yesmaximal, complemented
4#14798C214yesmaximal, complemented
5#14750C214yesmaximal, complemented
6#14701C214yesmaximal, complemented
7#14698C211yesmaximal, characteristic
8#14141C212yesmaximal, complemented
9#14136C211yesmaximal, characteristic
10#14129C211yesmaximal, characteristic, complemented
11#14022C212yesmaximal, complemented
12#13981C212yesmaximal, complemented
13#13939C211yesmaximal, characteristic, complemented
14#13885C212yesmaximal, complemented
15C4×#385C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2500—28no—
21C2×#394C2214yes—
22C2×#390C2212yes—
23C2×#390C2211yescharacteristic
24C2×#390C2212yes—
25C2×#390C2211yescharacteristic
26C2×#389C2212yes—
27C2×#388C2214yes—
28C2×#388C2214yes—
29C2×#388C2214yes—
30C2×#387C2214yes—
31C2×#386C2214yes—
32C2×#385C2212yes—
33C2×#384C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C4×#117—28no—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2214yes—
43C2×#376C2212yes—
44C2×C43C2212yes—
45#1445C2214yes—
46#1443C2214yes—
47#1442C2214yes—
48#1441C2214yes—
49#1440C2214yes—
50#1440C2214yes—
51#1440C2212yes—
52#1440C2212yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1438C2212yes—
57#1438C2212yes—
58#1437C2212yes—
59#1437C2214yes—
60#1437C2214yes—
61#1437C2212yes—
62#1437C2214yes—
63#1436C2212yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2212yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73#1434C2214yes—
74#1434C2214yes—
75C22⋊#95C2212yes—
76C4×#95C2212yes—
77C23×D4C2311yescharacteristic
78C23×D4C2312yes—
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C2×#121—24no—
82C2×#121—24no—
83C2×#121—24no—
84C2×#117—24no—
85C2×#117—24no—
86C2×#117—24no—
87C2×#117—28no—
88C2×#117D418yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C42—24no—
112C22×C42—28no—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2314yes—
120C22×C42C2314yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123#394—216no—
124#390—24no—
125#390—24no—
126#390—28no—
127#390—28no—
128#389—28no—
129#388—216no—
130#388—216no—
131#388—48no—
132#387—48no—
133#386—48no—
134#385—28no—
135#384—24no—
136#382—24no—
137#380—24no—
138C4×C4⋊C4—28no—
139C4×C4⋊C4—28no—
140C4×C4⋊C4—28no—
141C4×C22⋊C4—24no—
142C4×C22⋊C4—24no—
143C4×C22⋊C4—216no—
144#376—28no—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C43—28no—
169C25C2412yes—
170C25C2411yescharacteristic
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—44no—
180C22×D4—44no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4C2×D414yes—
191C23×C4#4114yes—
192C23×C4#4114yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206#121—416no—
207#117—232no—
208#117—232no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4#4118yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4C2×D418yes—
271C2×C42—232no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—28no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—28no—
283C2×C42—28no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—28no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42C2×D418yes—
296C2×C42#4118yes—
297#95—28no—
298#95—28no—
299#95—24no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—42no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95#4118yes—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24C22×D412yes—
336C24C2×#4112yes—
337C24C2×#4112yes—
338C24C2511yescharacteristic
339C2×D4—416no—
340C2×D4—48no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—88no—
347C2×D4—84no—
348C2×D4—84no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—216no—
358C22×C4—216no—
359C22×C4—216no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4#123116yes—
425C22×C4C22×D414yes—
426C22×C4C22×D414yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C2×#4114yes—
430C22×C4C2×#4114yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4118yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4114yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4H(5,𝔽2)14yes—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C4⋊C4—216no—
445C4⋊C4—48no—
446C4⋊C4—416no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—88no—
451C22⋊C4—28no—
452C22⋊C4—216no—
453C22⋊C4—216no—
454C22⋊C4—416no—
455C22⋊C4—416no—
456C22⋊C4—48no—
457C22⋊C4—44no—
458C22⋊C4—48no—
459C22⋊C4—416no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C22⋊C4—88no—
463C22⋊C4—84no—
464C42—28no—
465C42—28no—
466C42—216no—
467C42—216no—
468C42—216no—
469C42—416no—
470C42—416no—
471C42—44no—
472C42—44no—
473C42—48no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—24no—
478C23—22no—
479C23—22no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23C2×#12314yes—
516C23C23×D411yescharacteristic
517C23C22×#4111yescharacteristic
518C23C22×#4111yescharacteristic
519C23C22×#4112yes—
520C23C22×#4112yes—
521C23C22×#4112yes—
522C23C2×H(5,𝔽2)12yes—
523C23#58512yes—
524C23#58512yes—
525D4—816no—
526D4—88no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—232no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—432no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4C2×#12314yes—
604C2×C4C2×#12314yes—
605C2×C4#52514yes—
606C2×C4#52514yes—
607C2×C4#52918yes—
608C2×C4#52918yes—
609C2×C4#55518yes—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—82no—
620C22—84no—
621C22—84no—
622C22—84no—
623C22—84no—
624C22—82no—
625C22—82no—
626C22—84no—
627C22—84no—
628C22C22×#12311yescharacteristic
629C22C2×#52511yescharacteristic
630C22C2×#52511yescharacteristic
631C22C2×#52912yes—
632C22C2×#52912yes—
633C22#344411yescharacteristic
634C22#345511yescharacteristic
635C22#346012yes—
636C22D4×#4112yes—
637C22C2×#55512yes—
638C22#347512yes—
639C22#348012yes—
640C22#348012yes—
641C22#348112yes—
642C22#348812yes—
643C22#349312yes—
644C22#349412yes—
645C22#350811yescharacteristic
646C22#351011yescharacteristic
647C22#351712yes—
648C22#352112yes—
649C4—24no—
650C4—28no—
651C4—48no—
652C4—44no—
653C4—44no—
654C4—44no—
655C4—44no—
656C4—44no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4—84no—
661C4#285318yes—
662C2—44no—
663C2—162no—
664C2#3474411yescharacteristic
665C2#3480012yes—
666C2#3516112yes—
667C2#3525411yescharacteristic
668C2#3623512yes—
669C2#3630212yes—
670C2#3644212yes—
671C2#3651512yes—
672C2C2×#285311yescharacteristic
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×C4×#117C2semidirect
3. #15115C2semidirect
4. #14798C2semidirect
5. #14750C2semidirect
6. #14701C2semidirect
8. #14141C2semidirect
10. #14129C2semidirect
11. #14022C2semidirect
12. #13981C2semidirect
13. #13939C2semidirect
14. #13885C2semidirect
15. C4×#385C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25002
C2×C4×#1172
#151152
#147982
#147502
#147012
#146982
#141412
#141362
#141292
#140222
#139812
#139392
#138852
C4×#3852