Finite Group Databasebeta

← #4328653

Subgroups — #4328653

Order 512.

3694 conjugacy classes of subgroups.

The subgroup lattice has 797 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2404C211yesmaximal, characteristic
2C2×#2311C211yesmaximal, characteristic, complemented
3#16699C212yesmaximal, complemented
4#16609C214yesmaximal, complemented
5#16392C212yesmaximal, complemented
6#15966C212yesmaximal, complemented
7#15376C214yesmaximal, complemented
8#15266C212yesmaximal, complemented
9#14888C212yesmaximal, complemented
10#14836C212yesmaximal, complemented
11#14777C212yesmaximal, complemented
12#14734C212yesmaximal, complemented
13#14401C212yesmaximal, complemented
14#14357C212yesmaximal, complemented
15#13956C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C22×#120C2211yescharacteristic
18C22×#120C2211yescharacteristic
19C22×#117C2211yescharacteristic, complemented
20C23×C4⋊C4C2211yescharacteristic, complemented
21C23×C22⋊C4C2211yescharacteristic
22#2404—28no—
23C2×#399C2212yes—
24C2×#399C2212yes—
25C2×#394C2214yes—
26C2×#393C2212yes—
27C2×#392C2212yes—
28#2311—28no—
29C2×#391C2212yescomplemented
30C2×#390C2212yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#386C2211yescharacteristic
35C2×#386C2211yescharacteristic
36C2×#386C2212yes—
37C2×#386C2214yes—
38C2×#386C2214yes—
39C2×#386C2212yescomplemented
40C2×#386C2212yescomplemented
41C2×#385C2214yes—
42C2×#385C2214yescomplemented
43C2×#381C2211yescharacteristic
44C2×#381C2211yescharacteristic, complemented
45C2×#380C2212yes—
46C2×#380C2212yescomplemented
47C2×#380C2214yescomplemented
48C2×#379C2214yes—
49C22×#95C2211yescharacteristic, complemented
50#1446C2212yescomplemented
51#1445C2212yescomplemented
52#1445C2212yescomplemented
53#1445C2214yescomplemented
54#1444C2212yescomplemented
55#1444C2214yescomplemented
56#1443C2214yescomplemented
57#1442C2212yescomplemented
58#1442C2214yescomplemented
59#1442C2214yescomplemented
60#1440C2214yes—
61#1440C2214yes—
62#1440C2214yescomplemented
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yescomplemented
71#1438C2214yes—
72#1438C2212yes—
73#1437C2214yes—
74#1436C2214yescomplemented
75C22⋊#95C2212yes—
76C22⋊#95C2212yescomplemented
77C26C2311yescharacteristic
78C22×#41—28no—
79C22×#41—42no—
80C22×#41—42no—
81C22×#41—44no—
82C22×#41D418yes—
83C23×Q8C2311yescharacteristic
84C23×D4C2312yes—
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2312yes—
93C24×C4C2311yescharacteristic
94C24×C4C2312yes—
95C2×#120—24no—
96C2×#120—24no—
97C2×#120—28no—
98C2×#120—44no—
99C2×#117—24no—
100C2×#117—44no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—42no—
103C22×C4⋊C4—42no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—42no—
112C22×C22⋊C4—42no—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C42C2312yes—
132#399—44no—
133#399—44no—
134#394—48no—
135#393—28no—
136#392—28no—
137#391—28no—
138#390—28no—
139#386—28no—
140#386—28no—
141#386—24no—
142#386—24no—
143#386—28no—
144#386—28no—
145#386—28no—
146#386—48no—
147#386—48no—
148#386—44no—
149#385—48no—
150#385—48no—
151#381—24no—
152#381—42no—
153#380—28no—
154#380—28no—
155#380—48no—
156#379—48no—
157C2×#95—42no—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C25—22no—
180C25—24no—
181C25—24no—
182C25—41no—
183C25—41no—
184C25—42no—
185C25—42no—
186C25—44no—
187C25C2412yes—
188C25C2411yescharacteristic
189C25C2411yescharacteristic
190C25C2411yescharacteristic
191C2×#41—232no—
192C2×#41—416no—
193C2×#41—416no—
194C2×#41—432no—
195C2×#41—432no—
196C22×Q8—24no—
197C22×Q8—42no—
198C22×Q8—41no—
199C22×Q8—41no—
200C22×Q8C2×D414yes—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—28no—
210C22×D4—28no—
211C22×D4—42no—
212C22×D4—41no—
213C22×D4—41no—
214C22×D4—41no—
215C22×D4—41no—
216C22×D4—41no—
217C22×D4—41no—
218C22×D4—42no—
219C22×D4—42no—
220C22×D4—44no—
221C22×D4—44no—
222C22×D4C2×D414yes—
223C22×D4C2×D414yes—
224C22×D4C2×D414yes—
225C22×D4#4118yes—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—22no—
233C23×C4—22no—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—24no—
237C23×C4—22no—
238C23×C4—24no—
239C23×C4—28no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4—41no—
243C23×C4—41no—
244C23×C4—42no—
245C23×C4—42no—
246C23×C4—44no—
247C23×C4—42no—
248C23×C4—44no—
249C23×C4—44no—
250C23×C4—42no—
251C23×C4—44no—
252C23×C4C2×D414yes—
253C23×C4C2×D414yes—
254C23×C4C2×D414yes—
255C23×C4C2414yes—
256C23×C4C2414yes—
257C23×C4C2412yes—
258C23×C4C2414yes—
259C23×C4C2412yes—
260C23×C4C2412yes—
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C23×C4C2414yes—
264C23×C4C2411yescharacteristic
265C23×C4C2411yescharacteristic
266#120—416no—
267#120—416no—
268#117—416no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—416no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—416no—
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—42no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—42no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C42—24no—
336C2×C42—42no—
337C2×C42—42no—
338#95—28no—
339#95—28no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—28no—
345#95—24no—
346#95—28no—
347#95—28no—
348#95—24no—
349#95—28no—
350#95—42no—
351#95—44no—
352#95—42no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359C24—22no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—21no—
364C24—22no—
365C24—22no—
366C24—21no—
367C24—21no—
368C24—21no—
369C24—21no—
370C24—21no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—24no—
379C24—22no—
380C24—22no—
381C24—24no—
382C24—24no—
383C24—24no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—48no—
388C24—44no—
389C24—416no—
390C24—44no—
391C24—48no—
392C24—44no—
393C24—44no—
394C24—44no—
395C24—84no—
396C24—84no—
397C24—84no—
398C24—88no—
399C24—88no—
400C24—88no—
401C24—84no—
402C24—84no—
403C24C22×D412yes—
404C24C22×D412yes—
405C24C22×D412yes—
406C24C2×#4114yes—
407C24C2×#4114yes—
408C24H(5,𝔽2)14yes—
409C242−1+414yes—
410C24C2511yescharacteristic
411#41—4128no—
412C2×Q8—28no—
413C2×Q8—44no—
414C2×Q8—44no—
415C2×Q8—48no—
416C2×Q8—48no—
417C2×D4—28no—
418C2×D4—28no—
419C2×D4—28no—
420C2×D4—28no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—44no—
425C2×D4—48no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—48no—
430C2×D4—48no—
431C2×D4—48no—
432C2×D4—48no—
433C2×D4—416no—
434C2×D4—416no—
435C2×D4—48no—
436C2×D4—48no—
437C2×D4—84no—
438C2×D4—84no—
439C2×D4—88no—
440C2×D4—88no—
441C2×D4#123116yes—
442C22×C4—216no—
443C22×C4—216no—
444C22×C4—28no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—28no—
451C22×C4—28no—
452C22×C4—28no—
453C22×C4—28no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—28no—
469C22×C4—28no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
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—24no—
493C22×C4—24no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—24no—
499C22×C4—24no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—24no—
504C22×C4—24no—
505C22×C4—22no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4—48no—
509C22×C4—48no—
510C22×C4—48no—
511C22×C4—48no—
512C22×C4—48no—
513C22×C4—416no—
514C22×C4—44no—
515C22×C4—48no—
516C22×C4—88no—
517C22×C4—84no—
518C22×C4—84no—
519C22×C4—88no—
520C22×C4—84no—
521C22×C4—88no—
522C22×C4—84no—
523C22×C4—84no—
524C22×C4—84no—
525C22×C4—84no—
526C22×C4C22×D412yes—
527C22×C4C22×D412yes—
528C22×C4C22×D412yes—
529C22×C4C22×D412yes—
530C22×C4C2×#4114yes—
531C22×C4H(5,𝔽2)18yes—
532C22×C4H(5,𝔽2)12yes—
533C22×C4H(5,𝔽2)12yes—
534C4⋊C4—48no—
535C4⋊C4—48no—
536C4⋊C4—48no—
537C4⋊C4—88no—
538C4⋊C4—88no—
539C4⋊C4—88no—
540C22⋊C4—416no—
541C22⋊C4—416no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—48no—
547C22⋊C4—48no—
548C22⋊C4—48no—
549C22⋊C4—88no—
550C22⋊C4—88no—
551C22⋊C4—84no—
552C22⋊C4—88no—
553C22⋊C4—88no—
554C22⋊C4—84no—
555C22⋊C4—84no—
556C22⋊C4—88no—
557C22⋊C4—88no—
558C42—48no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—28no—
578C23—28no—
579C23—28no—
580C23—28no—
581C23—28no—
582C23—28no—
583C23—24no—
584C23—28no—
585C23—28no—
586C23—24no—
587C23—24no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—41no—
598C23—41no—
599C23—41no—
600C23—41no—
601C23—41no—
602C23—41no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—416no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—44no—
615C23—44no—
616C23—42no—
617C23—42no—
618C23—44no—
619C23—44no—
620C23—44no—
621C23—44no—
622C23—44no—
623C23—44no—
624C23—44no—
625C23—42no—
626C23—42no—
627C23—44no—
628C23—816no—
629C23—88no—
630C23—816no—
631C23—832no—
632C23—816no—
633C23—816no—
634C23—88no—
635C23C2×#12314yes—
636C23C2×#12314yes—
637C23#53414yes—
638C23C23×D411yescharacteristic
639C23C22×#4112yes—
640C23C2×H(5,𝔽2)12yes—
641C23C2×H(5,𝔽2)11yescharacteristic
642C23C2×H(5,𝔽2)11yescharacteristic
643C23C2×H(5,𝔽2)14yes—
644C23C2×2−1+412yes—
645C23#58512yes—
646Q8—416no—
647D4—416no—
648D4—416no—
649D4—416no—
650D4—816no—
651D4—816no—
652C2×C4—232no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—24no—
658C2×C4—24no—
659C2×C4—24no—
660C2×C4—24no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—28no—
668C2×C4—28no—
669C2×C4—28no—
670C2×C4—416no—
671C2×C4—416no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—42no—
695C2×C4—42no—
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—44no—
711C2×C4—44no—
712C2×C4—44no—
713C2×C4—44no—
714C2×C4—44no—
715C2×C4—44no—
716C2×C4—816no—
717C2×C4—816no—
718C2×C4—88no—
719C2×C4C2×#12314yes—
720C2×C4#53414yes—
721C22—28no—
722C22—28no—
723C22—24no—
724C22—24no—
725C22—24no—
726C22—28no—
727C22—48no—
728C22—48no—
729C22—48no—
730C22—44no—
731C22—44no—
732C22—44no—
733C22—44no—
734C22—44no—
735C22—44no—
736C22—44no—
737C22—42no—
738C22—42no—
739C22—42no—
740C22—44no—
741C22—44no—
742C22—42no—
743C22—42no—
744C22—42no—
745C22—48no—
746C22—44no—
747C22—48no—
748C22—44no—
749C22—44no—
750C22—832no—
751C22—84no—
752C22—84no—
753C22—82no—
754C22—82no—
755C22—82no—
756C22—82no—
757C22C22×#12311yescharacteristic
758C22C2×#53411yescharacteristic
759C22C2×#53411yescharacteristic
760C22#348912yes—
761C22#349012yes—
762C22#349012yes—
763C22#349812yes—
764C22#351314yes—
765C22#352014yes—
766C22#353414yes—
767C22#353614yes—
768C22#353712yes—
769C22#353811yescharacteristic
770C22#354011yescharacteristic
771C22#354112yes—
772C22#354912yes—
773C4—24no—
774C4—24no—
775C4—48no—
776C4—42no—
777C4—42no—
778C4—84no—
779C4—82no—
780C4—82no—
781C4—84no—
782C4—84no—
783C4—82no—
784C2—44no—
785C2—44no—
786C2—44no—
787C2—84no—
788C2#3499211yescharacteristic
789C2#3598712yes—
790C2#3729711yescharacteristic
791C2#3737411yescharacteristic
792C2#4195212yes—
793C2#4661814yes—
794C2#5162912yes—
795C2#5181512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2404C2
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×#2311C2semidirect
3. #16699C2semidirect
4. #16609C2semidirect
5. #16392C2semidirect
6. #15966C2semidirect
7. #15376C2semidirect
8. #15266C2semidirect
9. #14888C2semidirect
10. #14836C2semidirect
11. #14777C2semidirect
12. #14734C2semidirect
13. #14401C2semidirect
14. #14357C2semidirect
15. #13956C2semidirect
19. C22×#117C22semidirect
20. C23×C4⋊C4C22semidirect
29. C2×#391C22semidirect
39. C2×#386C22semidirect
40. C2×#386C22semidirect
42. C2×#385C22semidirect
44. C2×#381C22semidirect
46. C2×#380C22semidirect
47. C2×#380C22semidirect
49. C22×#95C22semidirect
50. #1446C22semidirect
51. #1445C22semidirect
52. #1445C22semidirect
53. #1445C22semidirect
54. #1444C22semidirect
55. #1444C22semidirect
56. #1443C22semidirect
57. #1442C22semidirect
58. #1442C22semidirect
59. #1442C22semidirect
62. #1440C22semidirect
70. #1439C22semidirect
74. #1436C22semidirect
76. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24042
C2×#23112
#166992
#166092
#163922
#159662
#153762
#152662
#148882
#148362
#147772
#147342
#144012
#143572
#139562