Finite Group Databasebeta

← #4452723

Subgroups — #4452723

Order 512.

3256 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2639C211yesmaximal, characteristic, complemented
2C22×#397C211yesmaximal, characteristic, complemented
3#16837C214yesmaximal, complemented
4#16748C214yesmaximal, complemented
5#16552C214yesmaximal, complemented
6#16386C214yesmaximal, complemented
7#14914C211yesmaximal, characteristic, complemented
8#14913C211yesmaximal, characteristic, complemented
9#14729C212yesmaximal, complemented
10#14725C212yesmaximal, complemented
11#14717C211yesmaximal, characteristic, complemented
12#14716C211yesmaximal, characteristic
13#14020C212yesmaximal, complemented
14#13995C212yesmaximal, complemented
15#13778C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#120C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23#2639—28no—
24C2×#400C2212yescomplemented
25C2×#397—28no—
26C2×#397C2214yes—
27C2×#397C2214yes—
28C2×#397C2212yescomplemented
29C2×#396C2214yescomplemented
30C2×#394C2214yes—
31C2×#394C2212yescomplemented
32C2×#393C2212yes—
33C2×#393C2212yescomplemented
34C2×#392C2212yes—
35C2×#390C2214yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#380C2212yes—
43C2×#379C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C22×#95C2211yescharacteristic
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48#1446C2212yescomplemented
49#1444C2212yescomplemented
50#1443C2214yescomplemented
51#1442C2212yescomplemented
52#1441C2214yescomplemented
53#1441C2214yescomplemented
54#1441C2214yescomplemented
55#1440C2214yes—
56#1440C2214yescomplemented
57#1440C2214yescomplemented
58#1439C2212yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2212yes—
62#1439C2214yes—
63#1439C2214yescomplemented
64#1438C2214yes—
65#1438C2214yes—
66#1438C2214yes—
67#1438C2212yescomplemented
68#1437C2214yes—
69#1437C2214yescomplemented
70#1436C2214yescomplemented
71#1436C2214yescomplemented
72#1434C2214yescomplemented
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2314yes—
81C23×D4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—44no—
90C2×#121—24no—
91C2×#121—44no—
92C2×#120—28no—
93C2×#120—44no—
94C2×#120—44no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—24no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—44no—
107C22×C22⋊C4—44no—
108C22×C22⋊C4—44no—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C42C2314yes—
128#400—28no—
129#397—28no—
130#397—216no—
131#397—216no—
132#396—48no—
133#394—28no—
134#394—48no—
135#393—28no—
136#393—44no—
137#392—44no—
138#390—48no—
139#386—28no—
140#386—28no—
141#386—28no—
142#386—44no—
143#386—44no—
144#385—44no—
145#380—28no—
146#379—44no—
147C4×C22⋊C4—44no—
148C2×#95—24no—
149C2×#95—24no—
150C2×#95D418yescomplemented
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C25—22no—
175C25—24no—
176C25—24no—
177C25—22no—
178C25—24no—
179C25—22no—
180C25—42no—
181C25—42no—
182C25—42no—
183C25C2412yes—
184C25C2412yes—
185C25C2411yescharacteristic
186C22×Q8—22no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8—42no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—28no—
198C22×D4—22no—
199C22×D4—28no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—42no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—42no—
207C22×D4—42no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2414yes—
245C23×C4C2412yes—
246C22⋊Q8—416no—
247#121—416no—
248#120—432no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—416no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—416no—
300C2×C22⋊C4—416no—
301C2×C22⋊C4—416no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4#4118yes—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42—44no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—42no—
332#95—44no—
333#95—42no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—42no—
339#95—42no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95C2×D418yes—
344#95C2×D418yes—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—28no—
350C24—22no—
351C24—24no—
352C24—24no—
353C24—22no—
354C24—24no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—24no—
364C24—22no—
365C24—24no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—24no—
371C24—416no—
372C24—416no—
373C24—48no—
374C24—48no—
375C24—44no—
376C24—44no—
377C24—44no—
378C24—44no—
379C24—44no—
380C24—44no—
381C24—48no—
382C24—48no—
383C24—88no—
384C24—88no—
385C24—84no—
386C24—84no—
387C24C22×D412yes—
388C24C2×#4114yes—
389C24H(5,𝔽2)14yes—
390C24H(5,𝔽2)14yes—
391C24H(5,𝔽2)12yes—
392C242−1+412yes—
393C24C2511yescharacteristic
394C2×Q8—44no—
395C2×Q8—82no—
396C2×Q8—82no—
397C2×Q8—88no—
398C2×D4—416no—
399C2×D4—416no—
400C2×D4—44no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—82no—
409C2×D4—82no—
410C2×D4—88no—
411C2×D4—88no—
412C2×D4—88no—
413C2×D4—88no—
414C2×D4—88no—
415C2×D4—88no—
416C22×C4—28no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
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—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—28no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—416no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—88no—
494C22×C4—88no—
495C22×C4—84no—
496C22×C4—84no—
497C22×C4—88no—
498C22×C4—84no—
499C22×C4—84no—
500C22×C4C22×D414yes—
501C22×C4C22×D414yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C2×#4118yes—
505C22×C4H(5,𝔽2)14yes—
506C4⋊C4—416no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—88no—
511C4⋊C4—88no—
512C22⋊C4—216no—
513C22⋊C4—416no—
514C22⋊C4—416no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—416no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—84no—
522C22⋊C4—84no—
523C22⋊C4—84no—
524C22⋊C4—84no—
525C22⋊C4—88no—
526C22⋊C4—88no—
527C22⋊C4—88no—
528C22⋊C4—88no—
529C22⋊C4—88no—
530C42—88no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—28no—
545C23—28no—
546C23—24no—
547C23—24no—
548C23—416no—
549C23—416no—
550C23—416no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23—42no—
567C23—44no—
568C23—44no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—832no—
593C23—832no—
594C23—88no—
595C23—88no—
596C23—88no—
597C23—88no—
598C23#53214yes—
599C23#53514yes—
600C23#53714yes—
601C23#53818yes—
602C23#54218yes—
603C23C23×D411yescharacteristic
604C23C22×#4112yes—
605C23C2×H(5,𝔽2)12yes—
606C23C2×H(5,𝔽2)11yescharacteristic
607C23C2×H(5,𝔽2)12yes—
608C23C2×H(5,𝔽2)12yes—
609C23C2×2−1+411yescharacteristic
610C23#58512yes—
611C23#58512yes—
612Q8—88no—
613D4—832no—
614D4—88no—
615D4—816no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—28no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—28no—
631C2×C4—28no—
632C2×C4—28no—
633C2×C4—416no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—88no—
682C2×C4—88no—
683C2×C4—88no—
684C2×C4—88no—
685C2×C4—88no—
686C22—216no—
687C22—24no—
688C22—216no—
689C22—24no—
690C22—24no—
691C22—24no—
692C22—24no—
693C22—24no—
694C22—48no—
695C22—44no—
696C22—48no—
697C22—44no—
698C22—44no—
699C22—832no—
700C22—82no—
701C22—84no—
702C22—84no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—84no—
708C22—84no—
709C22—82no—
710C22—84no—
711C22—84no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22#266218yes—
718C22C2×#53211yescharacteristic
719C22C2×#53511yescharacteristic
720C22C2×#53711yescharacteristic
721C22C2×#53812yes—
722C22C2×#54212yes—
723C22#348411yescharacteristic
724C22#348411yescharacteristic
725C22#348511yescharacteristic
726C22#348711yescharacteristic
727C22#351914yes—
728C22#352114yes—
729C22#353414yes—
730C22#353414yes—
731C22#353412yes—
732C22#353512yes—
733C22#355812yes—
734C22#356112yes—
735C4—44no—
736C4—48no—
737C4—44no—
738C4—44no—
739C4—42no—
740C4—42no—
741C4—82no—
742C4—82no—
743C4—84no—
744C4—82no—
745C4—82no—
746C4—82no—
747C2—28no—
748C2—84no—
749C2—162no—
750C2C2×#266211yescharacteristic
751C2#3539511yescharacteristic
752C2#3546911yescharacteristic
753C2#3671011yescharacteristic
754C2#3672111yescharacteristic
755C2#3672211yescharacteristic
756C2#3695711yescharacteristic
757C2#4073414yes—
758C2#4079014yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#14716C2
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×#2639C2semidirect
2. C22×#397C2semidirect
3. #16837C2semidirect
4. #16748C2semidirect
5. #16552C2semidirect
6. #16386C2semidirect
7. #14914C2semidirect
8. #14913C2semidirect
9. #14729C2semidirect
10. #14725C2semidirect
11. #14717C2semidirect
13. #14020C2semidirect
14. #13995C2semidirect
15. #13778C2semidirect
24. C2×#400C22semidirect
28. C2×#397C22semidirect
29. C2×#396C22semidirect
31. C2×#394C22semidirect
33. C2×#393C22semidirect
48. #1446C22semidirect
49. #1444C22semidirect
50. #1443C22semidirect
51. #1442C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
56. #1440C22semidirect
57. #1440C22semidirect
63. #1439C22semidirect
67. #1438C22semidirect
69. #1437C22semidirect
70. #1436C22semidirect
71. #1436C22semidirect
72. #1434C22semidirect
150. C2×#95D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26392
C22×#3972
#168372
#167482
#165522
#163862
#149142
#149132
#147292
#147252
#147172
#147162
#140202
#139952
#137782