Finite Group Databasebeta

← #622783

Subgroups — #622783

Order 512.

4008 conjugacy classes of subgroups.

The subgroup lattice has 787 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2617C211yesmaximal, characteristic
2C22×#385C211yesmaximal, characteristic, complemented
3#15554C214yesmaximal, complemented
4#15544C214yesmaximal, complemented
5#15391C214yesmaximal, complemented
6#15377C214yesmaximal, complemented
7#14945C212yesmaximal, complemented
8#14776C212yesmaximal, complemented
9#14694C212yesmaximal, complemented
10#14255C212yesmaximal, complemented
11#14152C211yesmaximal, characteristic, complemented
12#14137C211yesmaximal, characteristic, complemented
13#13966C211yesmaximal, characteristic, complemented
14#13881C211yesmaximal, characteristic, complemented
15#13775C211yesmaximal, characteristic, complemented
16C25×C4C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#120C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23#2617—28no—
24C2×#397C2212yes—
25C2×#397C2214yescomplemented
26C2×#394C2212yes—
27C2×#394C2212yes—
28C2×#392C2212yes—
29C2×#390C2212yescomplemented
30C2×#386C2212yes—
31C2×#386C2214yes—
32C2×#386C2214yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#385—28no—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#382C2212yescomplemented
41C2×#380C2211yescharacteristic
42C2×#379C2212yes—
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C22×#95C2211yescharacteristic
46C22×#95C2211yescharacteristic
47#1445C2214yescomplemented
48#1445C2212yescomplemented
49#1443C2214yescomplemented
50#1443C2214yescomplemented
51#1442C2214yescomplemented
52#1442C2212yescomplemented
53#1442C2212yescomplemented
54#1442C2214yescomplemented
55#1440C2214yes—
56#1440C2214yes—
57#1440C2214yescomplemented
58#1440C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1438C2214yes—
62#1438C2214yes—
63#1438C2212yescomplemented
64#1437C2214yes—
65#1437C2214yes—
66#1437C2214yescomplemented
67C4⋊#95C2212yescomplemented
68C4⋊#95C2212yescomplemented
69#1434C2214yescomplemented
70C22⋊#95C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C26C2311yescharacteristic
78C23×D4C2312yes—
79C23×D4C2311yescharacteristic
80C23×D4C2312yes—
81C23×D4C2312yes—
82C24×C4—24no—
83C24×C4—44no—
84C24×C4—44no—
85C24×C4—44no—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2314yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C2×#121—24no—
94C2×#121—24no—
95C2×#121—24no—
96C2×#120—28no—
97C2×#120—28no—
98C2×#120—28no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—44no—
102C22×C4⋊C4—44no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4—24no—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4—44no—
113C22×C22⋊C4D418yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
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—
130C22×C22⋊C4C2314yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133#397—28no—
134#397—48no—
135#394—28no—
136#394—28no—
137#392—28no—
138#390—44no—
139#386—28no—
140#386—48no—
141#386—48no—
142#386—44no—
143#386—44no—
144#386—44no—
145#386—44no—
146#385—216no—
147#385—216no—
148#382—44no—
149#380—24no—
150#379—28no—
151C4×C22⋊C4—24no—
152C4×C22⋊C4—24no—
153C2×#95—24no—
154C2×#95—24no—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C25—22no—
178C25—22no—
179C25—24no—
180C25—42no—
181C25—42no—
182C25—42no—
183C25—42no—
184C25C2×D414yes—
185C25#4118yes—
186C25C2412yes—
187C25C2412yes—
188C25C2411yescharacteristic
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—42no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—42no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—28no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—28no—
225C23×C4—28no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—44no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—416no—
241C23×C4—416no—
242C23×C4—416no—
243C23×C4—464no—
244C23×C4C2×D414yes—
245C23×C4C2×D414yes—
246C23×C4#4114yes—
247C23×C4#4114yes—
248C23×C4C2414yes—
249C23×C4C2414yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2412yes—
255C23×C4C2412yes—
256C23×C4C2412yes—
257C23×C4C2412yes—
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260#121—416no—
261#120—432no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—416no—
277C2×C4⋊C4—416no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C22⋊C4—232no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
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—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
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⋊C4C2×D418yes—
330C2×C22⋊C4C2×D418yes—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42—42no—
336C2×C42—42no—
337#95—28no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—28no—
364C24—28no—
365C24—216no—
366C24—216no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—24no—
372C24—24no—
373C24—24no—
374C24—24no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—24no—
381C24—24no—
382C24—22no—
383C24—22no—
384C24—24no—
385C24—24no—
386C24—44no—
387C24—44no—
388C24—44no—
389C24—44no—
390C24—416no—
391C24—416no—
392C24—416no—
393C24—48no—
394C24—48no—
395C24—48no—
396C24—48no—
397C24—416no—
398C24#124116yes—
399C24C22×D414yes—
400C24C22×D412yes—
401C24C2×#4114yes—
402C24C2×#4114yes—
403C24C2×#4112yes—
404C24C2×#4112yes—
405C24C2×#4114yes—
406C24H(5,𝔽2)14yes—
407C24C2511yescharacteristic
408C2×D4—48no—
409C2×D4—416no—
410C2×D4—416no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—44no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—84no—
424C2×D4—84no—
425C2×D4—84no—
426C2×D4—84no—
427C2×D4—84no—
428C2×D4—88no—
429C2×D4—88no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
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—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—416no—
497C22×C4—416no—
498C22×C4—416no—
499C22×C4—416no—
500C22×C4—416no—
501C22×C4—416no—
502C22×C4—416no—
503C22×C4—48no—
504C22×C4—48no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—416no—
508C22×C4—4128no—
509C22×C4—416no—
510C22×C4—44no—
511C22×C4—44no—
512C22×C4—44no—
513C22×C4—44no—
514C22×C4—44no—
515C22×C4—44no—
516C22×C4—44no—
517C22×C4—44no—
518C22×C4—88no—
519C22×C4—88no—
520C22×C4#123116yes—
521C22×C4C22×D414yes—
522C22×C4C22×D412yes—
523C22×C4C22×D412yes—
524C22×C4C2×#4112yes—
525C22×C4C2×#4112yes—
526C22×C4C2×#4112yes—
527C22×C4C2×#4112yes—
528C22×C4H(5,𝔽2)14yes—
529C22×C4H(5,𝔽2)14yes—
530C4⋊C4—48no—
531C4⋊C4—48no—
532C4⋊C4—416no—
533C4⋊C4—44no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C22⋊C4—216no—
537C22⋊C4—216no—
538C22⋊C4—416no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C22⋊C4—88no—
548C22⋊C4—88no—
549C22⋊C4—88no—
550C22⋊C4—88no—
551C22⋊C4—88no—
552C22⋊C4—88no—
553C42—48no—
554C42—84no—
555C23—24no—
556C23—24no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—24no—
562C23—28no—
563C23—28no—
564C23—24no—
565C23—28no—
566C23—28no—
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—232no—
582C23—416no—
583C23—416no—
584C23—416no—
585C23—416no—
586C23—416no—
587C23—464no—
588C23—464no—
589C23—416no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—44no—
602C23—44no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—44no—
612C23—44no—
613C23—44no—
614C23—44no—
615C23—44no—
616C23—44no—
617C23—44no—
618C23—44no—
619C23—44no—
620C23—44no—
621C23C2×#12314yes—
622C23C2×#12418yes—
623C23C2×#12414yes—
624C23#53514yes—
625C23#55318yes—
626C23#55318yes—
627C23C23×D411yescharacteristic
628C23C22×#4111yescharacteristic
629C23C22×#4111yescharacteristic
630C23C22×#4112yes—
631C23C2×H(5,𝔽2)12yes—
632C23C2×H(5,𝔽2)12yes—
633C23C2×H(5,𝔽2)12yes—
634C23#58512yes—
635C23#58512yes—
636D4—88no—
637D4—816no—
638D4—816no—
639D4—816no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—28no—
654C2×C4—28no—
655C2×C4—416no—
656C2×C4—416no—
657C2×C4—416no—
658C2×C4—416no—
659C2×C4—464no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
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—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—88no—
710C2×C4—88no—
711C2×C4C2×#12314yes—
712C2×C4C2×#12314yes—
713C22—24no—
714C22—24no—
715C22—24no—
716C22—24no—
717C22—24no—
718C22—216no—
719C22—216no—
720C22—24no—
721C22—28no—
722C22—28no—
723C22—28no—
724C22—464no—
725C22—44no—
726C22—44no—
727C22—44no—
728C22—48no—
729C22—48no—
730C22—48no—
731C22—48no—
732C22—44no—
733C22—82no—
734C22—82no—
735C22—82no—
736C22—82no—
737C22—82no—
738C22—84no—
739C22—84no—
740C22—84no—
741C22—84no—
742C22—82no—
743C22—82no—
744C22#250418yes—
745C22C22×#12311yescharacteristic
746C22C22×#12411yescharacteristic
747C22C2×#53511yescharacteristic
748C22#345912yes—
749C22#346012yes—
750C22C2×#55312yes—
751C22C2×#55312yes—
752C22#348411yescharacteristic
753C22#348411yescharacteristic
754C22#348511yescharacteristic
755C22#348511yescharacteristic
756C22#350514yes—
757C22#350514yes—
758C22#350512yes—
759C22#350612yes—
760C22#353214yes—
761C22#353314yes—
762C4—24no—
763C4—24no—
764C4—44no—
765C4—44no—
766C4—44no—
767C4—82no—
768C4—82no—
769C4—82no—
770C4—82no—
771C4—84no—
772C4—84no—
773C4—82no—
774C2—28no—
775C2—48no—
776C2—162no—
777C2C2×#250411yescharacteristic
778C2#3498311yescharacteristic
779C2#3515211yescharacteristic
780C2#3537711yescharacteristic
781C2#3537711yescharacteristic
782C2#3670311yescharacteristic
783C2#3695611yescharacteristic
784C2#3981214yes—
785C2#3981314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2617C2
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×#385C2semidirect
3. #15554C2semidirect
4. #15544C2semidirect
5. #15391C2semidirect
6. #15377C2semidirect
7. #14945C2semidirect
8. #14776C2semidirect
9. #14694C2semidirect
10. #14255C2semidirect
11. #14152C2semidirect
12. #14137C2semidirect
13. #13966C2semidirect
14. #13881C2semidirect
15. #13775C2semidirect
25. C2×#397C22semidirect
29. C2×#390C22semidirect
40. C2×#382C22semidirect
47. #1445C22semidirect
48. #1445C22semidirect
49. #1443C22semidirect
50. #1443C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
57. #1440C22semidirect
58. #1440C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
63. #1438C22semidirect
66. #1437C22semidirect
67. C4⋊#95C22semidirect
68. C4⋊#95C22semidirect
69. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26172
C22×#3852
#155542
#155442
#153912
#153772
#149452
#147762
#146942
#142552
#141522
#141372
#139662
#138812
#137752