Finite Group Databasebeta

← #643898

Subgroups — #643898

Order 512.

7198 conjugacy classes of subgroups.

The subgroup lattice has 789 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#121C211yesmaximal, characteristic
2C2×#2848C211yesmaximal, characteristic, complemented
3C2×#2639C212yesmaximal, complemented
4#16557C214yesmaximal, complemented
5#16362C212yesmaximal, complemented
6#16253C212yesmaximal, complemented
7#14843C212yesmaximal, complemented
8#14820C212yesmaximal, complemented
9#14747C214yesmaximal, complemented
10#14724C212yesmaximal, complemented
11#14717C212yesmaximal, complemented
12#14271C212yesmaximal, complemented
13#13867C214yesmaximal, complemented
14C2×C22⋊#95C211yesmaximal, characteristic, complemented
15C24×D4C2212yes—
16C24×D4C2211yescharacteristic
17C25×C4C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic, complemented
20C22×C22⋊Q8C2211yescharacteristic, complemented
21C22×#121—28no—
22C22×#121—216no—
23C22×#121C2212yes—
24C22×#121C2212yes—
25C22×#121C2212yes—
26C22×#121C2212yes—
27C23×C4⋊C4C2211yescharacteristic
28C23×C22⋊C4C2212yes—
29C23×C22⋊C4C2212yes—
30#2848—28no—
31#2639—216no—
32C2×#400C2214yescomplemented
33C2×#397C2214yescomplemented
34C2×#394C2214yescomplemented
35C2×#394C2214yescomplemented
36C2×#393C2214yescomplemented
37C2×#386C2218yescomplemented
38C2×#386C2218yescomplemented
39C2×#385C2212yes—
40C2×#385C2212yescomplemented
41C2×#385C2212yescomplemented
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic, complemented
44C2×#380C2211yescharacteristic, complemented
45C2×#379C2212yes—
46C2×#379C2214yes—
47C2×#379C2214yes—
48C22×#95C2212yescomplemented
49C22×#95C2212yescomplemented
50#1445C2214yescomplemented
51#1442C2214yescomplemented
52#1441C2214yescomplemented
53#1440C2218yescomplemented
54#1439C2218yescomplemented
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1438C2218yescomplemented
58#1438C2218yescomplemented
59#1438C2214yescomplemented
60#1438C2214yescomplemented
61C22⋊#95C2214yes—
62C22⋊#95C2214yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2212yescomplemented
66C22⋊#95C2212yescomplemented
67C26C2312yes—
68C26C2311yescharacteristic
69C23×Q8C2311yescharacteristic
70C23×D4—28no—
71C23×D4—28no—
72C23×D4—216no—
73C23×D4—216no—
74C23×D4—24no—
75C23×D4—44no—
76C23×D4—44no—
77C23×D4—48no—
78C23×D4C2314yes—
79C23×D4C2312yes—
80C23×D4C2314yes—
81C23×D4C2312yes—
82C24×C4—28no—
83C24×C4—44no—
84C24×C4—44no—
85C24×C4D418yes—
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2314yes—
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—44no—
99C2×C22⋊Q8—44no—
100C2×#121—2128no—
101C2×#121—28no—
102C2×#121—216no—
103C2×#121—28no—
104C2×#121—28no—
105C2×#121—28no—
106C2×#121—28no—
107C2×#121—28no—
108C2×#121—28no—
109C2×#121—216no—
110C2×#121D4116yes—
111C22×C4⋊C4—24no—
112C22×C4⋊C4—28no—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—28no—
122C22×C22⋊C4—216no—
123C22×C22⋊C4—28no—
124C22×C22⋊C4—44no—
125C22×C22⋊C4—44no—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2318yes—
134C22×C22⋊C4C2318yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2318yes—
139C22×C22⋊C4C2318yes—
140#400—216no—
141#397—216no—
142#394—216no—
143#394—48no—
144#393—216no—
145#386—232no—
146#386—416no—
147#385—28no—
148#385—28no—
149#385—28no—
150#380—24no—
151#380—24no—
152#380—24no—
153#379—28no—
154#379—48no—
155#379—48no—
156C2×#95—28no—
157C2×#95—44no—
158C2×#95C2314yes—
159C2×#95C2318yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2318yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2318yes—
168C2×#95C2318yes—
169C25—24no—
170C25—22no—
171C25—24no—
172C25—24no—
173C25—22no—
174C25—28no—
175C25—24no—
176C25—28no—
177C25—28no—
178C25—24no—
179C25—24no—
180C25—42no—
181C25—42no—
182C25—44no—
183C25—44no—
184C25—44no—
185C25C2×D414yes—
186C25#4118yes—
187C25C2414yes—
188C25C2411yescharacteristic
189C25C2411yescharacteristic
190C25C2411yescharacteristic
191C22×Q8—22no—
192C22×Q8—24no—
193C22×Q8—42no—
194C22×Q8—42no—
195C22×D4—232no—
196C22×D4—264no—
197C22×D4—28no—
198C22×D4—28no—
199C22×D4—28no—
200C22×D4—28no—
201C22×D4—28no—
202C22×D4—216no—
203C22×D4—216no—
204C22×D4—216no—
205C22×D4—28no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—28no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4—48no—
214C22×D4—464no—
215C22×D4—416no—
216C22×D4—416no—
217C22×D4—4128no—
218C22×D4—432no—
219C22×D4—432no—
220C22×D4C2×D4116yes—
221C22×D4C2×D418yes—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—232no—
229C23×C4—24no—
230C23×C4—28no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—28no—
236C23×C4—28no—
237C23×C4—28no—
238C23×C4—28no—
239C23×C4—416no—
240C23×C4—416no—
241C23×C4—42no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4—42no—
245C23×C4—42no—
246C23×C4—42no—
247C23×C4—42no—
248C23×C4—44no—
249C23×C4—44no—
250C23×C4—44no—
251C23×C4—44no—
252C23×C4—44no—
253C23×C4—464no—
254C23×C4C2×D414yes—
255C23×C4C2×D414yes—
256C23×C4C2×D414yes—
257C23×C4C2×D414yes—
258C23×C4C2×D414yes—
259C23×C4C2×D414yes—
260C23×C4C2×D418yes—
261C23×C4C2412yes—
262C23×C4C2412yes—
263C23×C4C2412yes—
264C23×C4C2412yes—
265C23×C4C2418yes—
266C23×C4C2411yescharacteristic
267C23×C4C2414yes—
268C23×C4C2411yescharacteristic
269C23×C4C2411yescharacteristic
270C23×C4C2411yescharacteristic
271C22⋊Q8—416no—
272C22⋊Q8—416no—
273C22⋊Q8—416no—
274#121—264no—
275#121—264no—
276#121—264no—
277#121—264no—
278C2×C4⋊C4—232no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4C2×D418yes—
296C2×C22⋊C4—264no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—216no—
318C2×C22⋊C4—216no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—432no—
324C2×C22⋊C4—48no—
325C2×C22⋊C4—48no—
326C2×C22⋊C4—48no—
327C2×C22⋊C4—48no—
328C2×C22⋊C4—48no—
329C2×C22⋊C4—48no—
330C2×C22⋊C4—48no—
331C2×C22⋊C4—48no—
332C2×C22⋊C4—48no—
333C2×C22⋊C4—48no—
334C2×C22⋊C4C2×D4116yes—
335#95—28no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—216no—
341#95—216no—
342#95—28no—
343#95—44no—
344#95—48no—
345#95—48no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—21no—
351C24—22no—
352C24—21no—
353C24—22no—
354C24—22no—
355C24—28no—
356C24—28no—
357C24—28no—
358C24—28no—
359C24—216no—
360C24—216no—
361C24—232no—
362C24—22no—
363C24—21no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—21no—
368C24—22no—
369C24—216no—
370C24—24no—
371C24—24no—
372C24—28no—
373C24—24no—
374C24—28no—
375C24—28no—
376C24—28no—
377C24—24no—
378C24—24no—
379C24—24no—
380C24—44no—
381C24—44no—
382C24—416no—
383C24—416no—
384C24—48no—
385C24—48no—
386C24—416no—
387C24—48no—
388C24—48no—
389C24—432no—
390C24—432no—
391C24—416no—
392C24—416no—
393C24—48no—
394C24—48no—
395C24—48no—
396C24—48no—
397C24—48no—
398C24—48no—
399C24—416no—
400C24—416no—
401C24—816no—
402C24—816no—
403C24—88no—
404C24—88no—
405C24#124116yes—
406C24C22×D418yes—
407C24C22×D412yes—
408C24C22×D412yes—
409C24C22×D414yes—
410C24C22×D412yes—
411C24C2×#4114yes—
412C24C2×#4114yes—
413C24C2×#4114yes—
414C24H(5,𝔽2)14yes—
415C24H(5,𝔽2)14yes—
416C24H(5,𝔽2)12yes—
417C242−1+412yes—
418C24C2511yescharacteristic
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—44no—
422C2×Q8—88no—
423C2×D4—216no—
424C2×D4—232no—
425C2×D4—232no—
426C2×D4—264no—
427C2×D4—216no—
428C2×D4—416no—
429C2×D4—432no—
430C2×D4—432no—
431C2×D4—4128no—
432C2×D4—48no—
433C2×D4—48no—
434C2×D4—48no—
435C2×D4—416no—
436C2×D4—432no—
437C2×D4—4256no—
438C2×D4—416no—
439C2×D4—416no—
440C2×D4—416no—
441C2×D4—416no—
442C2×D4—432no—
443C2×D4—48no—
444C22×C4—232no—
445C22×C4—28no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—28no—
453C22×C4—28no—
454C22×C4—216no—
455C22×C4—28no—
456C22×C4—28no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—28no—
461C22×C4—22no—
462C22×C4—28no—
463C22×C4—22no—
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—28no—
467C22×C4—28no—
468C22×C4—28no—
469C22×C4—28no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—28no—
477C22×C4—28no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—28no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—28no—
484C22×C4—28no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—28no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—416no—
502C22×C4—416no—
503C22×C4—416no—
504C22×C4—416no—
505C22×C4—432no—
506C22×C4—4128no—
507C22×C4—416no—
508C22×C4—48no—
509C22×C4—48no—
510C22×C4—416no—
511C22×C4—416no—
512C22×C4—416no—
513C22×C4—88no—
514C22×C4—88no—
515C22×C4—88no—
516C22×C4—88no—
517C22×C4#120116yes—
518C22×C4C22×D418yes—
519C22×C4C22×D412yes—
520C22×C4C22×D414yes—
521C22×C4C22×D412yes—
522C22×C4C22×D414yes—
523C22×C4C22×D412yes—
524C22×C4C22×D412yes—
525C22×C4H(5,𝔽2)14yes—
526C4⋊C4—216no—
527C4⋊C4—216no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—48no—
531C4⋊C4—44no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C22⋊C4—232no—
535C22⋊C4—232no—
536C22⋊C4—48no—
537C22⋊C4—416no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—416no—
544C22⋊C4—816no—
545C22⋊C4—816no—
546C22⋊C4—816no—
547C22⋊C4—816no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—24no—
557C23—24no—
558C23—28no—
559C23—28no—
560C23—28no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—28no—
566C23—28no—
567C23—28no—
568C23—28no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—232no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—24no—
581C23—24no—
582C23—232no—
583C23—28no—
584C23—28no—
585C23—28no—
586C23—216no—
587C23—216no—
588C23—28no—
589C23—28no—
590C23—416no—
591C23—416no—
592C23—464no—
593C23—464no—
594C23—416no—
595C23—416no—
596C23—416no—
597C23—432no—
598C23—432no—
599C23—432no—
600C23—44no—
601C23—42no—
602C23—42no—
603C23—44no—
604C23—42no—
605C23—42no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—44no—
610C23—44no—
611C23—44no—
612C23—48no—
613C23—48no—
614C23—48no—
615C23—48no—
616C23—44no—
617C23—44no—
618C23—48no—
619C23—48no—
620C23—48no—
621C23—48no—
622C23—48no—
623C23—48no—
624C23—864no—
625C23—864no—
626C23—816no—
627C23—816no—
628C23—816no—
629C23—816no—
630C23C2×#12014yes—
631C23C2×#12414yes—
632C23C2×#12414yes—
633C23C2×#12414yes—
634C23#53414yes—
635C23#53514yes—
636C23#53714yes—
637C23#53818yes—
638C23#53918yes—
639C23#546116yes—
640C23C23×D411yescharacteristic
641C23C23×D412yes—
642C23C22×#4112yes—
643C23C2×H(5,𝔽2)12yes—
644C23C2×H(5,𝔽2)12yes—
645C23C2×H(5,𝔽2)12yes—
646C23C2×H(5,𝔽2)11yescharacteristic
647C23C2×2−1+411yescharacteristic
648C23#58512yes—
649Q8—88no—
650D4—432no—
651D4—432no—
652D4—464no—
653D4—464no—
654C2×C4—232no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—28no—
662C2×C4—28no—
663C2×C4—28no—
664C2×C4—22no—
665C2×C4—28no—
666C2×C4—22no—
667C2×C4—22no—
668C2×C4—216no—
669C2×C4—416no—
670C2×C4—416no—
671C2×C4—416no—
672C2×C4—416no—
673C2×C4—416no—
674C2×C4—432no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—48no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—48no—
692C2×C4—48no—
693C2×C4—48no—
694C2×C4—48no—
695C2×C4—44no—
696C2×C4—48no—
697C2×C4—44no—
698C2×C4—48no—
699C2×C4—48no—
700C2×C4—48no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—48no—
704C2×C4—832no—
705C2×C4C2×#12014yes—
706C2×C4C2×#12014yes—
707C2×C4#53414yes—
708C22—24no—
709C22—216no—
710C22—216no—
711C22—216no—
712C22—216no—
713C22—24no—
714C22—24no—
715C22—24no—
716C22—24no—
717C22—24no—
718C22—24no—
719C22—24no—
720C22—24no—
721C22—24no—
722C22—24no—
723C22—24no—
724C22—24no—
725C22—24no—
726C22—24no—
727C22—464no—
728C22—48no—
729C22—48no—
730C22—42no—
731C22—42no—
732C22—42no—
733C22—42no—
734C22—416no—
735C22—416no—
736C22—48no—
737C22—48no—
738C22—48no—
739C22—864no—
740C22—84no—
741C22—88no—
742C22—88no—
743C22—84no—
744C22—84no—
745C22—84no—
746C22#243518yes—
747C22#263918yescomplemented
748C22C22×#12011yescharacteristic
749C22C22×#12411yescharacteristic
750C22C2×#53411yescharacteristic
751C22C2×#53411yescharacteristic
752C22C2×#53511yescharacteristic
753C22C2×#53711yescharacteristic
754C22#344911yescharacteristic
755C22C2×#53812yes—
756C22C2×#53912yes—
757C22C2×#54614yes—
758C22#348414yes—
759C22#348812yes—
760C22#348912yes—
761C22#350514yes—
762C22#353414yes—
763C22#353612yes—
764C22#353712yes—
765C4—24no—
766C4—24no—
767C4—24no—
768C4—48no—
769C4—42no—
770C4—88no—
771C4—82no—
772C4—82no—
773C4—82no—
774C4—82no—
775C2—28no—
776C2—28no—
777C2—44no—
778C2—88no—
779C2C2×#263911yescharacteristic
780C2C2×#243511yescharacteristic
781C2#3498011yescharacteristic
782C2#3501911yescharacteristic
783C2#3525111yescharacteristic
784C2#3743911yescharacteristic
785C2#3744611yescharacteristic
786C2#3769214yes—
787C2#3769314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#121C2
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×#2848C2semidirect
3. C2×#2639C2semidirect
4. #16557C2semidirect
5. #16362C2semidirect
6. #16253C2semidirect
7. #14843C2semidirect
8. #14820C2semidirect
9. #14747C2semidirect
10. #14724C2semidirect
11. #14717C2semidirect
12. #14271C2semidirect
13. #13867C2semidirect
14. C2×C22⋊#95C2semidirect
19. C22×C22⋊Q8C22semidirect
20. C22×C22⋊Q8C22semidirect
32. C2×#400C22semidirect
33. C2×#397C22semidirect
34. C2×#394C22semidirect
35. C2×#394C22semidirect
36. C2×#393C22semidirect
37. C2×#386C22semidirect
38. C2×#386C22semidirect
40. C2×#385C22semidirect
41. C2×#385C22semidirect
43. C2×#380C22semidirect
44. C2×#380C22semidirect
48. C22×#95C22semidirect
49. C22×#95C22semidirect
50. #1445C22semidirect
51. #1442C22semidirect
52. #1441C22semidirect
53. #1440C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1438C22semidirect
58. #1438C22semidirect
59. #1438C22semidirect
60. #1438C22semidirect
65. C22⋊#95C22semidirect
66. C22⋊#95C22semidirect
747. C22#2639semidirect

Maximal subgroups

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

subgroupindex
C23×#1212
C2×#28482
C2×#26392
#165572
#163622
#162532
#148432
#148202
#147472
#147242
#147172
#142712
#138672
C2×C22⋊#952