Finite Group Databasebeta

← #764381

Subgroups — #764381

Order 512.

3990 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#539C211yesmaximal, characteristic
2C2×#2413C212yesmaximal, complemented
3C22×#400C211yesmaximal, characteristic, complemented
4#15940C214yesmaximal, complemented
5#15847C214yesmaximal, complemented
6#15672C214yesmaximal, complemented
7#15667C214yesmaximal, complemented
8#14934C211yesmaximal, characteristic, complemented
9#14926C211yesmaximal, characteristic
10#14914C211yesmaximal, characteristic, complemented
11#14913C211yesmaximal, characteristic
12#14135C212yesmaximal, complemented
13#14016C212yesmaximal, complemented
14#13968C212yesmaximal, complemented
15#13795C211yesmaximal, characteristic, complemented
16C2×#539—216no—
17C2×#539—28no—
18C22×#126C2212yes—
19C22×#125C2212yes—
20C22×#123C2212yes—
21C22×#123C2212yes—
22C22×#123C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×#121C2211yescharacteristic
26C22×C4×D4C2212yes—
27C22×#117C2212yes—
28C23×C4⋊C4C2211yescharacteristic
29C23×C4⋊C4C2211yescharacteristic
30#2413—216no—
31C2×#400—28no—
32C2×#400C2214yes—
33C2×#400C2214yes—
34C2×#398C2214yes—
35C2×#396C2214yes—
36C2×#389C2214yes—
37C2×#388C2214yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#381C2212yes—
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45C22×#95C2211yescharacteristic
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48#1446C2212yes—
49#1446C2212yes—
50#1444C2212yes—
51#1444C2212yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2212yes—
55#1442C2212yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1440C2214yes—
60#1440C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1438C2212yes—
65#1438C2212yes—
66#1437C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70C4⋊#95C2214yes—
71C4⋊#95C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2211yescharacteristic
77C22⋊#95C2211yescharacteristic
78C22⋊#95C2212yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90#539—2128no—
91C2×#126—28no—
92C2×#126—28no—
93C2×#126—28no—
94C2×#125—28no—
95C2×#125—28no—
96C2×#125—28no—
97C2×#123—28no—
98C2×#123—28no—
99C2×#123—28no—
100C2×#123—28no—
101C2×#123—28no—
102C2×#123—28no—
103C2×#123—28no—
104C2×#123—44no—
105C2×#123D4116yes—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—28no—
108C2×C22⋊Q8—28no—
109C2×C22⋊Q8—28no—
110C2×C22⋊Q8—28no—
111C2×#121—28no—
112C2×#121—24no—
113C2×C4×D4—28no—
114C2×C4×D4—28no—
115C2×C4×D4—28no—
116C2×#117—28no—
117C2×#117—28no—
118C2×#117—44no—
119C22×C4⋊C4—28no—
120C22×C4⋊C4—24no—
121C22×C4⋊C4—28no—
122C22×C4⋊C4D418yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2314yes—
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2314yes—
139C22×C22⋊C4C2314yes—
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2312yes—
142C22×C22⋊C4C2312yes—
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2314yes—
145C22×C22⋊C4C2314yes—
146C22×C42C2312yes—
147C22×C42C2314yes—
148#400—216no—
149#400—216no—
150#398—216no—
151#396—216no—
152#389—216no—
153#388—216no—
154#386—28no—
155#386—28no—
156#386—44no—
157#385—28no—
158#381—28no—
159C2×#95—24no—
160C2×#95—24no—
161C2×#95—24no—
162C2×#95—24no—
163C2×#95—24no—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C25C2412yes—
188C25C2411yescharacteristic
189C22×Q8—22no—
190C22×Q8—24no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—22no—
201C22×D4—42no—
202C22×D4—42no—
203C22×D4C2×D418yes—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4C2×D418yes—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2×D414yes—
240C23×C4C2414yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2411yescharacteristic
250C23×C4C2414yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253#126—264no—
254#125—264no—
255#123—264no—
256#123—264no—
257#123—432no—
258C22⋊Q8—264no—
259C22⋊Q8—232no—
260#121—232no—
261C4×D4—264no—
262#117—432no—
263C2×C4⋊C4—232no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—416no—
295C2×C4⋊C4C2×D418yes—
296C2×C4⋊C4C2×D418yes—
297C2×C4⋊C4C2×D418yes—
298C2×C4⋊C4C2×D418yes—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4C2×D418yes—
329C2×C22⋊C4C2×D418yes—
330C2×C22⋊C4C2×D418yes—
331C2×C22⋊C4#4118yes—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—28no—
336C2×C42—28no—
337C2×C42—44no—
338#95—28no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—42no—
351#95—44no—
352#95—44no—
353#95—42no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—42no—
358#95—44no—
359#95—42no—
360#95—44no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—24no—
371C24—22no—
372C24—24no—
373C24—24no—
374C24—24no—
375C24C22×D414yes—
376C24C22×D412yes—
377C24C22×D414yes—
378C24C2×#4114yes—
379C24H(5,𝔽2)12yes—
380C242−1+412yes—
381C24C2511yescharacteristic
382C2×Q8—28no—
383C2×Q8—82no—
384C2×Q8—82no—
385C2×Q8—88no—
386C2×D4—216no—
387C2×D4—216no—
388C2×D4—216no—
389C2×D4—28no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—82no—
395C2×D4—82no—
396C2×D4—88no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—216no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—416no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—84no—
484C22×C4—84no—
485C22×C4—88no—
486C22×C4—88no—
487C22×C4—88no—
488C22×C4—88no—
489C22×C4—88no—
490C22×C4—84no—
491C22×C4—84no—
492C22×C4#120116yes—
493C22×C4C22×D414yes—
494C22×C4C22×D412yes—
495C22×C4C22×D414yes—
496C22×C4C22×D412yes—
497C22×C4C22×D414yes—
498C22×C4C22×D414yes—
499C22×C4C22×D414yes—
500C22×C4C22×D414yes—
501C22×C4C22×D414yes—
502C22×C4C2×#4118yes—
503C22×C42−1+414yes—
504C4⋊C4—216no—
505C4⋊C4—28no—
506C4⋊C4—216no—
507C4⋊C4—28no—
508C4⋊C4—216no—
509C4⋊C4—216no—
510C4⋊C4—216no—
511C4⋊C4—216no—
512C4⋊C4—416no—
513C4⋊C4—416no—
514C4⋊C4—416no—
515C4⋊C4—416no—
516C22⋊C4—216no—
517C22⋊C4—216no—
518C22⋊C4—216no—
519C22⋊C4—216no—
520C22⋊C4—216no—
521C22⋊C4—216no—
522C22⋊C4—416no—
523C22⋊C4—416no—
524C22⋊C4—48no—
525C22⋊C4—88no—
526C22⋊C4—88no—
527C42—216no—
528C42—416no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—28no—
534C23—28no—
535C23—28no—
536C23—28no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—28no—
545C23—28no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23C2×#12014yes—
565C23#53214yes—
566C23#53714yes—
567C23#54618yes—
568C23#54718yes—
569C23C23×D411yescharacteristic
570C23C23×D412yes—
571C23C22×#4112yes—
572C23C2×H(5,𝔽2)11yescharacteristic
573C23C2×2−1+411yescharacteristic
574C23C2×2−1+412yes—
575C23#58512yes—
576C23#58512yes—
577C23#58512yes—
578Q8—88no—
579D4—432no—
580D4—88no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—232no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—28no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—416no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
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—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—88no—
654C2×C4—88no—
655C2×C4—816no—
656C2×C4—88no—
657C2×C4—88no—
658C2×C4—88no—
659C2×C4C2×#12018yes—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—24no—
666C22—216no—
667C22—216no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—48no—
672C22—48no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—84no—
678C22—84no—
679C22#243018yes—
680C22C22×#12011yescharacteristic
681C22C2×#53211yescharacteristic
682C22C2×#53711yescharacteristic
683C22C2×#54612yes—
684C22C2×#54712yes—
685C22#348511yescharacteristic
686C22#348511yescharacteristic
687C22#348611yescharacteristic
688C22#348711yescharacteristic
689C22#348914yes—
690C22#350014yes—
691C22#351514yes—
692C22#352114yes—
693C22#355912yes—
694C22#356012yes—
695C22#356312yes—
696C22#356412yes—
697C4—28no—
698C4—44no—
699C4—42no—
700C4—42no—
701C4—44no—
702C4—44no—
703C4—44no—
704C4—44no—
705C4—48no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—84no—
710C2—28no—
711C2—84no—
712C2C2×#243011yescharacteristic
713C2#3546811yescharacteristic
714C2#3546911yescharacteristic
715C2#3561611yescharacteristic
716C2#3565911yescharacteristic
717C2#3696811yescharacteristic
718C2#3697111yescharacteristic
719C2#4447514yes—
720C2#4451314yes—
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×#2413C2semidirect
3. C22×#400C2semidirect
4. #15940C2semidirect
5. #15847C2semidirect
6. #15672C2semidirect
7. #15667C2semidirect
8. #14934C2semidirect
10. #14914C2semidirect
12. #14135C2semidirect
13. #14016C2semidirect
14. #13968C2semidirect
15. #13795C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5392
C2×#24132
C22×#4002
#159402
#158472
#156722
#156672
#149342
#149262
#149142
#149132
#141352
#140162
#139682
#137952