Finite Group Databasebeta

← #4462520

Subgroups — #4462520

Order 512.

3180 conjugacy classes of subgroups.

The subgroup lattice has 753 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#17159C214yesmaximal, complemented
4#17158C214yesmaximal, complemented
5#17098C214yesmaximal, complemented
6#17094C214yesmaximal, complemented
7#15091C212yesmaximal, complemented
8#14934C211yesmaximal, characteristic, complemented
9#14927C212yesmaximal, complemented
10#14926C211yesmaximal, characteristic, complemented
11#14911C212yesmaximal, complemented
12#14725C212yesmaximal, complemented
13#14717C211yesmaximal, characteristic, complemented
14#14716C211yesmaximal, characteristic
15#14347C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×#120C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24#2639—28no—
25C2×#400C2212yescomplemented
26C2×#397—28no—
27C2×#397C2214yes—
28C2×#397C2214yes—
29C2×#397C2212yescomplemented
30C2×#394C2214yes—
31C2×#394C2212yescomplemented
32C2×#393C2212yes—
33C2×#393C2212yescomplemented
34C2×#392C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#381C2212yes—
40C2×#380C2212yes—
41C2×#379C2212yes—
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45#1446C2212yescomplemented
46#1444C2212yescomplemented
47#1442C2214yescomplemented
48#1442C2214yescomplemented
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1442C2214yescomplemented
52#1442C2212yescomplemented
53#1442C2214yescomplemented
54#1441C2214yescomplemented
55#1440C2214yes—
56#1440C2214yescomplemented
57#1439C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2212yes—
63#1439C2214yescomplemented
64#1439C2214yescomplemented
65#1439C2214yescomplemented
66#1439C2214yescomplemented
67#1438C2214yes—
68#1438C2214yes—
69#1438C2214yes—
70#1438C2214yes—
71#1438C2212yescomplemented
72C22⋊#95C2212yes—
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C23×D4C2314yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×#123—44no—
89C2×#123—44no—
90C2×#123—44no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—44no—
93C2×#121—24no—
94C2×#121—44no—
95C2×#120—44no—
96C2×#120—44no—
97C2×#120—44no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—44no—
110C22×C22⋊C4—44no—
111C22×C22⋊C4—44no—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130#400—28no—
131#397—28no—
132#397—216no—
133#397—216no—
134#394—28no—
135#394—48no—
136#393—28no—
137#393—44no—
138#392—44no—
139#386—28no—
140#386—28no—
141#386—44no—
142#386—44no—
143#381—44no—
144#380—44no—
145#379—44no—
146C2×#95—24no—
147C2×#95—24no—
148C2×#95D418yescomplemented
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C25—22no—
174C25—24no—
175C25—24no—
176C25—22no—
177C25—24no—
178C25—22no—
179C25—42no—
180C25—42no—
181C25—42no—
182C25C2412yes—
183C25C2412yes—
184C25C2411yescharacteristic
185C22×Q8—22no—
186C22×Q8—42no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—28no—
197C22×D4—22no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—42no—
202C22×D4—42no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—42no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—24no—
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×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2414yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245#123—432no—
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—48no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—416no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
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—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—44no—
325#95—44no—
326#95—42no—
327#95—42no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—42no—
336#95—42no—
337#95—44no—
338#95—44no—
339#95C2×D418yes—
340#95C2×D418yes—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—28no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—24no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—24no—
354C24—24no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—416no—
368C24—416no—
369C24—48no—
370C24—48no—
371C24—44no—
372C24—44no—
373C24—44no—
374C24—44no—
375C24—44no—
376C24—44no—
377C24—48no—
378C24—48no—
379C24—88no—
380C24—88no—
381C24—84no—
382C24—84no—
383C24C22×D412yes—
384C24H(5,𝔽2)14yes—
385C24H(5,𝔽2)14yes—
386C24H(5,𝔽2)14yes—
387C24H(5,𝔽2)12yes—
388C242−1+412yes—
389C24C2511yescharacteristic
390C2×Q8—44no—
391C2×Q8—82no—
392C2×Q8—82no—
393C2×Q8—88no—
394C2×D4—44no—
395C2×D4—416no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—82no—
404C2×D4—82no—
405C2×D4—88no—
406C2×D4—88no—
407C2×D4—88no—
408C2×D4—88no—
409C2×D4—88no—
410C2×D4—88no—
411C2×D4—88no—
412C22×C4—28no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—416no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—84no—
493C22×C4—88no—
494C22×C4—84no—
495C22×C4—88no—
496C22×C4—88no—
497C22×C4C22×D414yes—
498C22×C4C22×D414yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C4⋊C4—416no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—416no—
506C4⋊C4—88no—
507C22⋊C4—416no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—416no—
511C22⋊C4—48no—
512C22⋊C4—416no—
513C22⋊C4—416no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C22⋊C4—88no—
522C22⋊C4—88no—
523C22⋊C4—88no—
524C22⋊C4—88no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—28no—
530C23—28no—
531C23—28no—
532C23—24no—
533C23—24no—
534C23—28no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—28no—
539C23—24no—
540C23—28no—
541C23—24no—
542C23—416no—
543C23—416no—
544C23—416no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—42no—
557C23—44no—
558C23—42no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—832no—
587C23—832no—
588C23—88no—
589C23—88no—
590C23—88no—
591C23—88no—
592C23#53514yes—
593C23#53514yes—
594C23#53714yes—
595C23H(3,𝔽4)18yes—
596C23#56218yes—
597C23C23×D411yescharacteristic
598C23C2×H(5,𝔽2)12yes—
599C23C2×H(5,𝔽2)12yes—
600C23C2×H(5,𝔽2)11yescharacteristic
601C23C2×H(5,𝔽2)12yes—
602C23C2×2−1+411yescharacteristic
603C23#58512yes—
604C23#58512yes—
605C23#58512yes—
606Q8—88no—
607D4—88no—
608D4—832no—
609D4—816no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—416no—
623C2×C4—48no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—88no—
675C2×C4—88no—
676C2×C4—88no—
677C2×C4—88no—
678C2×C4—88no—
679C22—216no—
680C22—216no—
681C22—24no—
682C22—24no—
683C22—24no—
684C22—24no—
685C22—24no—
686C22—24no—
687C22—48no—
688C22—48no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—832no—
693C22—84no—
694C22—84no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—84no—
700C22—84no—
701C22—84no—
702C22—84no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22#281918yes—
711C22C2×#53511yescharacteristic
712C22C2×#53511yescharacteristic
713C22C2×#53711yescharacteristic
714C22#348411yescharacteristic
715C22#348511yescharacteristic
716C22#348511yescharacteristic
717C22#348711yescharacteristic
718C22C2×H(3,𝔽4)12yes—
719C22C2×#56212yes—
720C22#353314yes—
721C22#353414yes—
722C22#353414yes—
723C22#353412yes—
724C22#353512yes—
725C22#356014yes—
726C22#356012yes—
727C22#356412yes—
728C4—44no—
729C4—44no—
730C4—42no—
731C4—42no—
732C4—84no—
733C4—84no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4—82no—
740C2—28no—
741C2—84no—
742C2—162no—
743C2C2×#281911yescharacteristic
744C2#3670911yescharacteristic
745C2#3672111yescharacteristic
746C2#3695611yescharacteristic
747C2#3696111yescharacteristic
748C2#3697111yescharacteristic
749C2#3718511yescharacteristic
750C2#5251014yes—
751C2#5253014yes—
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. #17159C2semidirect
4. #17158C2semidirect
5. #17098C2semidirect
6. #17094C2semidirect
7. #15091C2semidirect
8. #14934C2semidirect
9. #14927C2semidirect
10. #14926C2semidirect
11. #14911C2semidirect
12. #14725C2semidirect
13. #14717C2semidirect
15. #14347C2semidirect
25. C2×#400C22semidirect
29. C2×#397C22semidirect
31. C2×#394C22semidirect
33. C2×#393C22semidirect
45. #1446C22semidirect
46. #1444C22semidirect
47. #1442C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
54. #1441C22semidirect
56. #1440C22semidirect
63. #1439C22semidirect
64. #1439C22semidirect
65. #1439C22semidirect
66. #1439C22semidirect
71. #1438C22semidirect
148. 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
#171592
#171582
#170982
#170942
#150912
#149342
#149272
#149262
#149112
#147252
#147172
#147162
#143472