Finite Group Databasebeta

← #4308802

Subgroups — #4308802

Order 512.

4649 conjugacy classes of subgroups.

The subgroup lattice has 788 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2613C211yesmaximal, characteristic
2C22×#386C211yesmaximal, characteristic, complemented
3#16426C214yesmaximal, complemented
4#16308C214yesmaximal, complemented
5#16270C214yesmaximal, complemented
6#16266C214yesmaximal, complemented
7#14919C212yesmaximal, complemented
8#14912C212yesmaximal, complemented
9#14727C212yesmaximal, complemented
10#14718C212yesmaximal, complemented
11#14694C211yesmaximal, characteristic, complemented
12#14693C211yesmaximal, characteristic, complemented
13#14335C211yesmaximal, characteristic, complemented
14#14262C211yesmaximal, characteristic, complemented
15#14255C211yesmaximal, characteristic, complemented
16C24×D4C2211yescharacteristic
17C25×C4C2211yescharacteristic
18C22×#123C2211yescharacteristic
19C22×#120C2211yescharacteristic
20C22×#120C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23#2613—28no—
24C2×#397C2214yescomplemented
25C2×#394C2212yes—
26C2×#394C2212yes—
27C2×#394C2214yescomplemented
28C2×#392C2212yes—
29C2×#392C2212yes—
30C2×#386—28no—
31C2×#386C2214yes—
32C2×#386C2214yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2214yescomplemented
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#385C2214yescomplemented
39C2×#381C2211yescharacteristic
40C2×#381C2212yes—
41C2×#379C2211yescharacteristic
42C2×#379C2211yescharacteristic
43C2×#379C2211yescharacteristic
44C2×#379C2211yescharacteristic
45C2×#379C2212yes—
46C22×#95C2212yes—
47#1445C2214yescomplemented
48#1443C2214yescomplemented
49#1443C2214yescomplemented
50#1442C2214yescomplemented
51#1442C2214yescomplemented
52#1441C2214yescomplemented
53#1440C2214yescomplemented
54#1440C2214yescomplemented
55#1440C2214yescomplemented
56#1440C2214yescomplemented
57#1440C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1439C2214yescomplemented
62#1439C2214yescomplemented
63#1438C2214yes—
64#1438C2214yes—
65#1438C2214yescomplemented
66#1438C2214yescomplemented
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2214yescomplemented
72C22⋊#95C2214yescomplemented
73C26C2311yescharacteristic
74C26C2311yescharacteristic
75C23×D4—24no—
76C23×D4—24no—
77C23×D4—44no—
78C23×D4—44no—
79C23×D4D418yes—
80C23×D4C2311yescharacteristic
81C23×D4C2312yes—
82C23×D4C2312yes—
83C23×D4C2312yes—
84C23×D4C2311yescharacteristic
85C23×D4C2311yescharacteristic
86C24×C4—24no—
87C24×C4—48no—
88C24×C4—44no—
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C24×C4C2314yes—
92C24×C4C2312yes—
93C2×#123—24no—
94C2×#123—44no—
95C2×#120—24no—
96C2×#120—24no—
97C2×#120—24no—
98C2×#120—24no—
99C2×#120—44no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—48no—
106C22×C22⋊C4—44no—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129#397—48no—
130#394—28no—
131#394—28no—
132#394—48no—
133#392—28no—
134#392—28no—
135#386—216no—
136#386—216no—
137#386—48no—
138#386—44no—
139#386—44no—
140#385—48no—
141#385—44no—
142#385—44no—
143#381—24no—
144#381—44no—
145#379—24no—
146#379—24no—
147#379—24no—
148#379—24no—
149#379—44no—
150C2×#95—28no—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C25—24no—
170C25—24no—
171C25—22no—
172C25—22no—
173C25—24no—
174C25—24no—
175C25—44no—
176C25—44no—
177C25—44no—
178C25—44no—
179C25C2×D414yes—
180C25C2×D414yes—
181C25#4114yes—
182C25#4114yes—
183C25C2411yescharacteristic
184C25C2411yescharacteristic
185C25C2411yescharacteristic
186C25C2412yes—
187C25C2411yescharacteristic
188C22×D4—232no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—24no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—416no—
204C22×D4—416no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4—42no—
208C22×D4—42no—
209C22×D4—42no—
210C22×D4—42no—
211C22×D4—42no—
212C22×D4—42no—
213C22×D4—42no—
214C22×D4—42no—
215C22×D4—44no—
216C22×D4—44no—
217C22×D4—44no—
218C22×D4—44no—
219C22×D4—432no—
220C22×D4—432no—
221C22×D4C2×D418yes—
222C22×D4C2×D418yes—
223C23×C4—24no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—28no—
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—44no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4—44no—
241C23×C4—44no—
242C23×C4—44no—
243C23×C4—44no—
244C23×C4—432no—
245C23×C4—416no—
246C23×C4—464no—
247C23×C4C2×D414yes—
248C23×C4C2412yes—
249C23×C4C2414yes—
250C23×C4C2414yes—
251C23×C4C2414yes—
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2412yes—
255C23×C4C2411yescharacteristic
256C23×C4C2412yes—
257C23×C4C2414yes—
258#123—416no—
259#120—416no—
260#120—416no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—432no—
312C2×C22⋊C4—416no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
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⋊C4—44no—
329C2×C22⋊C4—44no—
330#95—28no—
331#95—28no—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—28no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348C24—24no—
349C24—24no—
350C24—28no—
351C24—28no—
352C24—216no—
353C24—216no—
354C24—24no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—24no—
362C24—28no—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—24no—
368C24—24no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—24no—
375C24—28no—
376C24—28no—
377C24—21no—
378C24—21no—
379C24—21no—
380C24—24no—
381C24—22no—
382C24—22no—
383C24—24no—
384C24—48no—
385C24—48no—
386C24—416no—
387C24—416no—
388C24—48no—
389C24—48no—
390C24—416no—
391C24—416no—
392C24—416no—
393C24—416no—
394C24—416no—
395C24—416no—
396C24—48no—
397C24—48no—
398C24—88no—
399C24—88no—
400C24—88no—
401C24—88no—
402C24#123116yes—
403C24C22×D412yes—
404C24C22×D412yes—
405C24C22×D414yes—
406C24C22×D412yes—
407C24C2×#4114yes—
408C24C2×#4114yes—
409C24C2×#4112yes—
410C24C2×#4112yes—
411C24H(5,𝔽2)14yes—
412C24H(5,𝔽2)14yes—
413C24C2511yescharacteristic
414C2×D4—216no—
415C2×D4—216no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—4128no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—416no—
432C2×D4—48no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—48no—
437C2×D4—48no—
438C2×D4—84no—
439C2×D4—84no—
440C2×D4—84no—
441C2×D4—84no—
442C2×D4—88no—
443C2×D4—88no—
444C2×D4—88no—
445C22×C4—216no—
446C22×C4—28no—
447C22×C4—28no—
448C22×C4—28no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—24no—
499C22×C4—24no—
500C22×C4—24no—
501C22×C4—24no—
502C22×C4—24no—
503C22×C4—416no—
504C22×C4—416no—
505C22×C4—416no—
506C22×C4—416no—
507C22×C4—416no—
508C22×C4—416no—
509C22×C4—416no—
510C22×C4—48no—
511C22×C4—416no—
512C22×C4—4128no—
513C22×C4—416no—
514C22×C4—48no—
515C22×C4—48no—
516C22×C4—48no—
517C22×C4—48no—
518C22×C4—48no—
519C22×C4—48no—
520C22×C4—88no—
521C22×C4—88no—
522C22×C4C22×D414yes—
523C22×C4H(5,𝔽2)18yes—
524C4⋊C4—48no—
525C4⋊C4—88no—
526C4⋊C4—88no—
527C22⋊C4—416no—
528C22⋊C4—416no—
529C22⋊C4—48no—
530C22⋊C4—416no—
531C22⋊C4—416no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—48no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—88no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—88no—
546C22⋊C4—88no—
547C22⋊C4—88no—
548C22⋊C4—88no—
549C23—28no—
550C23—24no—
551C23—24no—
552C23—28no—
553C23—28no—
554C23—28no—
555C23—28no—
556C23—28no—
557C23—232no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—28no—
565C23—28no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—28no—
571C23—28no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—24no—
579C23—24no—
580C23—24no—
581C23—24no—
582C23—432no—
583C23—416no—
584C23—464no—
585C23—432no—
586C23—416no—
587C23—464no—
588C23—432no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—416no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—44no—
610C23—44no—
611C23—44no—
612C23—44no—
613C23—44no—
614C23—41no—
615C23—41no—
616C23—41no—
617C23—42no—
618C23—42no—
619C23—44no—
620C23—42no—
621C23—42no—
622C23—44no—
623C23—44no—
624C23—44no—
625C23—44no—
626C23—44no—
627C23—816no—
628C23—88no—
629C23—88no—
630C23—832no—
631C23—832no—
632C23C2×#12318yes—
633C23C2×#12314yes—
634C23#53414yes—
635C23#53414yes—
636C23#53818yes—
637C23#53818yes—
638C23C23×D411yescharacteristic
639C23C22×#4111yescharacteristic
640C23C22×#4111yescharacteristic
641C23C2×H(5,𝔽2)12yes—
642C23C2×H(5,𝔽2)12yes—
643C23C2×H(5,𝔽2)14yes—
644C23#58514yes—
645D4—432no—
646D4—432no—
647D4—816no—
648D4—88no—
649D4—88no—
650D4—88no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—28no—
655C2×C4—28no—
656C2×C4—28no—
657C2×C4—432no—
658C2×C4—464no—
659C2×C4—416no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—41no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—41no—
675C2×C4—41no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—44no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—44no—
704C2×C4—44no—
705C2×C4—44no—
706C2×C4—44no—
707C2×C4—44no—
708C2×C4—44no—
709C2×C4—44no—
710C2×C4—816no—
711C22—216no—
712C22—28no—
713C22—24no—
714C22—24no—
715C22—216no—
716C22—28no—
717C22—28no—
718C22—28no—
719C22—28no—
720C22—464no—
721C22—48no—
722C22—48no—
723C22—44no—
724C22—44no—
725C22—44no—
726C22—44no—
727C22—48no—
728C22—48no—
729C22—44no—
730C22—44no—
731C22—42no—
732C22—42no—
733C22—42no—
734C22—42no—
735C22—44no—
736C22—44no—
737C22—42no—
738C22—42no—
739C22—832no—
740C22—84no—
741C22—84no—
742C22—84no—
743C22—84no—
744C22—84no—
745C22—82no—
746C22—81no—
747C22—81no—
748C22—84no—
749C22—82no—
750C22—81no—
751C22#262218yes—
752C22C22×#12311yescharacteristic
753C22C2×#53411yescharacteristic
754C22C2×#53411yescharacteristic
755C22C2×#53812yes—
756C22C2×#53812yes—
757C22#345914yes—
758C22#348412yes—
759C22#348512yes—
760C22#350514yes—
761C22#353214yes—
762C22#353314yes—
763C22#354214yes—
764C22#354214yes—
765C4—48no—
766C4—44no—
767C4—84no—
768C4—82no—
769C4—84no—
770C4—81no—
771C4—81no—
772C4—84no—
773C4—82no—
774C4—81no—
775C2—28no—
776C2—48no—
777C2—82no—
778C2—82no—
779C2—161no—
780C2C2×#262211yescharacteristic
781C2#3498312yes—
782C2#3670412yes—
783C2#3671411yescharacteristic
784C2#3675814yes—
785C2#3696511yescharacteristic
786C2#3745614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2613C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×#386C2semidirect
3. #16426C2semidirect
4. #16308C2semidirect
5. #16270C2semidirect
6. #16266C2semidirect
7. #14919C2semidirect
8. #14912C2semidirect
9. #14727C2semidirect
10. #14718C2semidirect
11. #14694C2semidirect
12. #14693C2semidirect
13. #14335C2semidirect
14. #14262C2semidirect
15. #14255C2semidirect
24. C2×#397C22semidirect
27. C2×#394C22semidirect
35. C2×#386C22semidirect
38. C2×#385C22semidirect
47. #1445C22semidirect
48. #1443C22semidirect
49. #1443C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1441C22semidirect
53. #1440C22semidirect
54. #1440C22semidirect
55. #1440C22semidirect
56. #1440C22semidirect
57. #1440C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
65. #1438C22semidirect
66. #1438C22semidirect
71. C22⋊#95C22semidirect
72. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26132
C22×#3862
#164262
#163082
#162702
#162662
#149192
#149122
#147272
#147182
#146942
#146932
#143352
#142622
#142552