Finite Group Databasebeta

← #4422962

Subgroups — #4422962

Order 512.

2626 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic, complemented
2C2×#2486C211yesmaximal, characteristic
3C2×#2638C211yesmaximal, characteristic
4C2×#2353C211yesmaximal, characteristic
5#17061C212yesmaximal, complemented
6#16806C212yesmaximal, complemented
7#16677C212yesmaximal, complemented
8#16399C212yesmaximal, complemented
9#15929C214yesmaximal, complemented
10#15269C214yesmaximal
11#15150C212yesmaximal, complemented
12#15050C212yesmaximal, complemented
13#14903C212yesmaximal, complemented
14#14766C212yesmaximal, complemented
15#14660C211yesmaximal, characteristic, complemented
16#14355C211yesmaximal, characteristic, complemented
17#14342C211yesmaximal, characteristic, complemented
18C22×#123C2212yes—
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2638—28no—
23#2502—28no—
24#2486—28no—
25C2×#398C2211yescharacteristic
26C2×#398C2212yes—
27C2×#398C2212yes—
28C2×#398C2212yes—
29C2×#397C2211yescharacteristic
30C2×#397C2212yes—
31C2×#397C2212yes—
32C2×#397C2214yes—
33C2×#395C2211yescharacteristic
34C2×#394C2212yes—
35C2×#393C2211yescharacteristic
36#2353—28no—
37C2×#391C2212yes—
38C2×#389C2212yes—
39C2×#388C2212yes—
40C2×#387C2212yes—
41C2×#387C2212yes—
42C2×#386C2212yes—
43C2×#385C2212yes—
44C2×#385C2214yes—
45C2×#384C2212yes—
46C2×#383C2211yescharacteristic
47C2×#381C2211yescharacteristic
48C2×#381C2211yescharacteristic
49C2×#381C2212yes—
50C2×#380C2211yescharacteristic
51C2×#380C2212yes—
52C2×#380C2211yescharacteristic
53C2×#380C2214yes—
54C2×C4×C22⋊C4C2212yes—
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×#376C2211yescharacteristic
57C22×#95C2211yescharacteristic
58#1446C2212yes—
59#1445C2212yes—
60#1445C2212yes—
61#1445C2212yes—
62#1445C2212yes—
63#1445C2214yes—
64#1445C2212yes—
65#1444C2212yes—
66#1444C2214yes—
67#1444C2214yes—
68#1443C2214yes—
69#1442C2212yes—
70#1442C2214yes—
71#1442C2214yes—
72#1441C2212yes—
73#1441C2214yes—
74#1441C2214yes—
75#1441C2212yes—
76#1441C2214yes—
77#1441C2214yes—
78#1440C2214yes—
79#1440C2214yes—
80#1439C2214yes—
81C4⋊#95C2212yes—
82C4⋊#95C2214yes—
83#1434C2212yes—
84#1434C2214yes—
85C22⋊#95C2214yes—
86C22⋊#95C2214yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C2×#123—28no—
94C2×#123—44no—
95C2×#123—44no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4D418yes—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132#398—24no—
133#398—28no—
134#398—28no—
135#398—28no—
136#397—24no—
137#397—28no—
138#397—28no—
139#397—48no—
140#395—24no—
141#394—28no—
142#393—24no—
143#391—28no—
144#389—28no—
145#388—28no—
146#387—28no—
147#387—28no—
148#386—28no—
149#385—28no—
150#385—216no—
151#384—28no—
152#383—24no—
153#381—24no—
154#381—24no—
155#381—28no—
156#380—28no—
157#380—24no—
158#380—24no—
159#380—48no—
160C4×C22⋊C4—28no—
161C4×C22⋊C4D418yes—
162#376—24no—
163C2×#95—24no—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2311yescharacteristic
195C2×#95C2311yescharacteristic
196C25C2411yescharacteristic
197C25C2411yescharacteristic
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—44no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—41no—
207C22×D4—41no—
208C22×D4C2×D414yes—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—28no—
217C23×C4—44no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—44no—
221C23×C4—42no—
222C23×C4—44no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4#4114yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244#123—432no—
245C22⋊Q8—416no—
246C4×D4—232no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—416no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4C2×D414yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4C2×D414yes—
311C2×C22⋊C4C2×D414yes—
312C2×C22⋊C4C2×D418yes—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—28no—
317C2×C42—44no—
318C2×C42C2×D414yes—
319C2×C42C2×D418yes—
320#95—22no—
321#95—24no—
322#95—22no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—22no—
331#95—22no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95C2×D414yes—
351#95C2×D414yes—
352C24—21no—
353C24—21no—
354C24—21no—
355C24—21no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—21no—
360C24—24no—
361C24—21no—
362C24—21no—
363C24—21no—
364C24C22×D412yes—
365C24C22×D412yes—
366C24C22×D412yes—
367C24C2×#4112yes—
368C24H(5,𝔽2)12yes—
369C24H(5,𝔽2)12yes—
370C24C2511yescharacteristic
371C2×Q8—48no—
372C2×Q8—44no—
373C2×Q8—84no—
374C2×Q8—84no—
375C2×D4—28no—
376C2×D4—28no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—28no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
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—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—416no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—84no—
461C22×C4—84no—
462C22×C4—84no—
463C22×C4—88no—
464C22×C4—84no—
465C22×C4—84no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4#121116yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D414yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D414yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C42−1+418yes—
481C4⋊C4—28no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—416no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—88no—
490C4⋊C4—88no—
491C4⋊C4—84no—
492C4⋊C4—84no—
493C4⋊C4—88no—
494C22⋊C4—28no—
495C22⋊C4—28no—
496C22⋊C4—216no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—416no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C42—28no—
506C42—48no—
507C42—48no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—28no—
529C23—28no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23C2×#12114yes—
541C23#53514yes—
542C23C23×D411yescharacteristic
543C23C23×D411yescharacteristic
544C23C22×#4111yescharacteristic
545C23C2×H(5,𝔽2)11yescharacteristic
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×2−1+414yes—
548C23#58511yescharacteristic
549C23#58511yescharacteristic
550C23#58514yes—
551Q8—88no—
552D4—416no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—416no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—88no—
631C2×C4—816no—
632C2×C4—816no—
633C2×C4C2×#12114yes—
634C2×C4C2×#12114yes—
635C2×C4#53618yes—
636C22—24no—
637C22—28no—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—42no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—42no—
654C22C22×#12111yescharacteristic
655C22C2×#53511yescharacteristic
656C22C2×#53612yes—
657C22#344911yescharacteristic
658C22#348512yes—
659C22#348512yes—
660C22#348811yescharacteristic
661C22#348811yescharacteristic
662C22#349112yes—
663C22#349411yescharacteristic
664C22#349411yescharacteristic
665C22#349912yes—
666C22#350912yes—
667C22#353514yes—
668C22#355914yes—
669C22#356414yes—
670C22#356514yes—
671C4—24no—
672C4—24no—
673C4—28no—
674C4—42no—
675C4—42no—
676C4—44no—
677C4—42no—
678C4—82no—
679C4—82no—
680C4—84no—
681C4—82no—
682C4—82no—
683C4—84no—
684C4—82no—
685C4—82no—
686C2—44no—
687C2—44no—
688C2#3525211yescharacteristic
689C2#3531611yescharacteristic
690C2#3544711yescharacteristic
691C2#4137512yes—
692C2#4141212yes—
693C2#4231012yes—
694C2#4268212yes—
695C2#5066212yes—
696C2#5104412yes—
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×#2502C2semidirect
5. #17061C2semidirect
6. #16806C2semidirect
7. #16677C2semidirect
8. #16399C2semidirect
9. #15929C2semidirect
11. #15150C2semidirect
12. #15050C2semidirect
13. #14903C2semidirect
14. #14766C2semidirect
15. #14660C2semidirect
16. #14355C2semidirect
17. #14342C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
C2×#24862
C2×#26382
C2×#23532
#170612
#168062
#166772
#163992
#159292
#152692
#151502
#150502
#149032
#147662
#146602
#143552
#143422