Finite Group Databasebeta

← #644665

Subgroups — #644665

Order 512.

2556 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic, complemented
3#17058C211yesmaximal, characteristic, complemented
4#16885C214yesmaximal, complemented
5#16366C211yesmaximal, characteristic, complemented
6#16362C211yesmaximal, characteristic
7#16253C211yesmaximal, characteristic
8#15110C212yesmaximal, complemented
9#15108C212yesmaximal, complemented
10#15107C212yesmaximal, complemented
11#14902C212yesmaximal, complemented
12#14894C212yesmaximal, complemented
13#14882C212yesmaximal, complemented
14#14326C212yesmaximal, complemented
15#14325C212yesmaximal, complemented
16#14167C214yesmaximal, complemented
17#13753C211yesmaximal, characteristic, complemented
18C22×#123C2212yes—
19C22×C22⋊Q8C2211yescharacteristic
20C22×#121C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2854—28no—
24C2×#400C2212yes—
25C2×#397C2212yes—
26C2×#394C2212yes—
27C2×#393C2212yes—
28C4×C22⋊Q8—28no—
29C2×#391C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#388C2212yes—
36C2×#388C2212yes—
37C2×#387C2212yes—
38C2×#386C2212yes—
39C2×#385C2212yes—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×#376C2212yes—
47#1446C2212yes—
48#1445C2212yes—
49#1445C2212yes—
50#1443C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2212yes—
54#1442C2212yes—
55#1442C2212yes—
56#1442C2212yes—
57#1442C2214yes—
58#1441C2212yes—
59#1441C2212yes—
60#1440C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2214yes—
65#1439C2212yes—
66#1439C2212yes—
67#1439C2214yes—
68#1438C2212yes—
69#1438C2214yes—
70#1438C2214yes—
71#1437C2214yes—
72#1437C2214yes—
73#1437C2214yes—
74#1437C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79#1434C2212yes—
80#1434C2212yes—
81C22⋊#95C2211yescharacteristic
82C22⋊#95C2211yescharacteristic
83C22⋊#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C23×D4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×#123—28no—
92C2×#123—216no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—28no—
95C2×#121—24no—
96C2×#121—44no—
97C2×C4×Q8—24no—
98C2×C4×Q8—44no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C42—44no—
125C22×C42D418yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2314yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#400—28no—
134#397—44no—
135#394—28no—
136#393—44no—
137#391—24no—
138#390—24no—
139#390—24no—
140#389—24no—
141#389—24no—
142#389—24no—
143#388—28no—
144#388—28no—
145#387—28no—
146#386—28no—
147#385—28no—
148#380—24no—
149#380—24no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—24no—
152C4×C22⋊C4—28no—
153C4×C22⋊C4—28no—
154#376—44no—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C25C2412yes—
183C25C2411yescharacteristic
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—42no—
188C22×D4—24no—
189C22×D4—28no—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—42no—
193C22×D4—42no—
194C22×D4—42no—
195C22×D4—42no—
196C22×D4—44no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—42no—
200C23×C4—42no—
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×C4#4118yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225#123—432no—
226C22⋊Q8—232no—
227#121—416no—
228C4×Q8—416no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4#4118yes—
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—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—416no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—44no—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307C2×C42C2×D414yes—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—24no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—21no—
339C24—21no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—24no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—24no—
353C24C22×D412yes—
354C24C2×#4114yes—
355C24C2511yescharacteristic
356C2×Q8—28no—
357C2×Q8—416no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—44no—
366C2×D4—88no—
367C2×D4—88no—
368C2×D4—88no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—28no—
396C22×C4—28no—
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—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4#124116yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4114yes—
458C22×C4H(5,𝔽2)14yes—
459C4⋊C4—28no—
460C4⋊C4—216no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C22⋊C4—216no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—88no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C22⋊C4—88no—
484C42—28no—
485C42—28no—
486C42—44no—
487C42—44no—
488C42—48no—
489C42—416no—
490C42—48no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—28no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—44no—
530C23C2×#12414yes—
531C23C23×D411yescharacteristic
532C23C22×#4112yes—
533C23C22×#4112yes—
534C23C2×H(5,𝔽2)12yes—
535C23#58511yescharacteristic
536C23#58511yescharacteristic
537C23#58512yes—
538C23#58512yes—
539C23#58512yes—
540Q8—416no—
541D4—88no—
542D4—816no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—416no—
567C2×C4—416no—
568C2×C4—416no—
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—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
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×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#52514yes—
620C2×C4C2×#12414yes—
621C2×C4C2×#12414yes—
622C2×C4#53414yes—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—42no—
627C22—42no—
628C22—48no—
629C22—48no—
630C22—42no—
631C22—42no—
632C22—82no—
633C22—82no—
634C22—84no—
635C22—84no—
636C22—82no—
637C22—84no—
638C22—84no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22C2×#52511yescharacteristic
644C22C22×#12411yescharacteristic
645C22C2×#53411yescharacteristic
646C22#344911yescharacteristic
647C22#344911yescharacteristic
648C22#344911yescharacteristic
649C22#344911yescharacteristic
650C22#345512yes—
651C22#345512yes—
652C22#348014yes—
653C22#350514yes—
654C22#351012yes—
655C22#351212yes—
656C22#351314yes—
657C22#353912yes—
658C22#354012yes—
659C22#356614yes—
660C4—24no—
661C4—24no—
662C4—24no—
663C4—24no—
664C4—44no—
665C4—42no—
666C4—42no—
667C4—44no—
668C4—82no—
669C4—84no—
670C4—82no—
671C4—84no—
672C4—82no—
673C4—82no—
674C4—82no—
675C2—44no—
676C2—162no—
677C2#3463711yescharacteristic
678C2#3495511yescharacteristic
679C2#3501911yescharacteristic
680C2#3506211yescharacteristic
681C2#3534211yescharacteristic
682C2#3744511yescharacteristic
683C2#3755211yescharacteristic
684C2#4001214yes—
685C2#4001314yes—
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×C22⋊Q8C2semidirect
3. #17058C2semidirect
4. #16885C2semidirect
5. #16366C2semidirect
8. #15110C2semidirect
9. #15108C2semidirect
10. #15107C2semidirect
11. #14902C2semidirect
12. #14894C2semidirect
13. #14882C2semidirect
14. #14326C2semidirect
15. #14325C2semidirect
16. #14167C2semidirect
17. #13753C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C2×C4×C22⋊Q82
#170582
#168852
#163662
#163622
#162532
#151102
#151082
#151072
#149022
#148942
#148822
#143262
#143252
#141672
#137532