Finite Group Databasebeta

← #4308786

Subgroups — #4308786

Order 512.

4044 conjugacy classes of subgroups.

The subgroup lattice has 791 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2618C211yesmaximal, characteristic, complemented
2C22×#385C211yesmaximal, characteristic, complemented
3#16402C214yesmaximal, complemented
4#16352C214yesmaximal, complemented
5#16319C214yesmaximal, complemented
6#16266C214yesmaximal, complemented
7#14912C212yesmaximal, complemented
8#14911C212yesmaximal, complemented
9#14822C211yesmaximal, characteristic, complemented
10#14734C211yesmaximal, characteristic, complemented
11#14719C211yesmaximal, characteristic, complemented
12#14716C211yesmaximal, characteristic
13#14694C212yesmaximal, complemented
14#14342C211yesmaximal, characteristic, complemented
15#14259C212yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C22×#123C2212yes—
18C22×#121C2211yescharacteristic
19C22×#120C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24#2618—28no—
25C2×#400C2212yescomplemented
26C2×#399C2212yes—
27C2×#398C2212yescomplemented
28C2×#396C2212yescomplemented
29C2×#396C2212yescomplemented
30C2×#394C2212yescomplemented
31C2×#394C2212yescomplemented
32C2×#392C2212yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2214yes—
38C2×#386C2214yes—
39C2×#385—28no—
40C2×#385C2214yes—
41C2×#385C2214yes—
42C2×#385C2212yes—
43C2×#381C2212yes—
44C2×#380C2211yescharacteristic
45C2×#379C2212yes—
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48#1445C2214yescomplemented
49#1445C2214yescomplemented
50#1443C2214yescomplemented
51#1442C2214yescomplemented
52#1442C2214yescomplemented
53#1442C2214yescomplemented
54#1441C2212yescomplemented
55#1441C2214yescomplemented
56#1441C2212yescomplemented
57#1440C2214yescomplemented
58#1440C2214yescomplemented
59#1439C2212yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2212yes—
64#1439C2214yescomplemented
65#1439C2214yescomplemented
66#1439C2214yescomplemented
67#1438C2214yes—
68#1438C2214yes—
69#1438C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2211yescharacteristic
72C22⋊#95C2211yescharacteristic
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C23×D4C2312yes—
79C23×D4C2314yes—
80C23×D4C2311yescharacteristic
81C24×C4—24no—
82C24×C4—44no—
83C24×C4—44no—
84C24×C4—44no—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2314yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×#123—28no—
93C2×#123—44no—
94C2×#123—44no—
95C2×#121—24no—
96C2×#121—44no—
97C2×#120—28no—
98C2×#120—44no—
99C2×#120—44no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—44no—
102C22×C4⋊C4—44no—
103C22×C4⋊C4D418yescomplemented
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—24no—
115C22×C22⋊C4—24no—
116C22×C22⋊C4—24no—
117C22×C22⋊C4—24no—
118C22×C22⋊C4—44no—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2314yes—
136#400—44no—
137#399—44no—
138#398—28no—
139#396—28no—
140#396—28no—
141#394—28no—
142#394—44no—
143#392—44no—
144#386—28no—
145#386—28no—
146#386—48no—
147#386—44no—
148#386—44no—
149#386—48no—
150#385—216no—
151#385—216no—
152#385—44no—
153#381—44no—
154#380—24no—
155#379—44no—
156C2×#95—24no—
157C2×#95—24no—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C25—22no—
180C25—22no—
181C25—24no—
182C25—24no—
183C25—24no—
184C25—24no—
185C25—42no—
186C25—42no—
187C25#4114yes—
188C25C2412yes—
189C25C2412yes—
190C25C2411yescharacteristic
191C22×D4—24no—
192C22×D4—28no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—28no—
200C22×D4—28no—
201C22×D4—22no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—42no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—42no—
208C22×D4—42no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4—28no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—28no—
224C23×C4—28no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4—416no—
239C23×C4—416no—
240C23×C4—416no—
241C23×C4—464no—
242C23×C4C2×D414yes—
243C23×C4C2×D414yes—
244C23×C4C2×D414yes—
245C23×C4#4114yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2414yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2414yes—
252C23×C4C2411yescharacteristic
253C23×C4C2412yes—
254C23×C4C2411yescharacteristic
255C23×C4C2412yes—
256C23×C4C2412yes—
257C23×C4C2412yes—
258#123—432no—
259#121—416no—
260#120—432no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—416no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—416no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4C2×D418yes—
289C2×C4⋊C4C2×D418yes—
290C2×C22⋊C4—232no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
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—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—42no—
325C2×C22⋊C4—42no—
326C2×C22⋊C4—42no—
327C2×C22⋊C4—416no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341#95—28no—
342#95—28no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—28no—
347#95—28no—
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—42no—
358#95—42no—
359#95—42no—
360#95—44no—
361#95—42no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—28no—
367C24—28no—
368C24—216no—
369C24—28no—
370C24—216no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—22no—
375C24—24no—
376C24—24no—
377C24—22no—
378C24—24no—
379C24—216no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—24no—
384C24—22no—
385C24—22no—
386C24—24no—
387C24—22no—
388C24—22no—
389C24—24no—
390C24—24no—
391C24—48no—
392C24—416no—
393C24—416no—
394C24—416no—
395C24—416no—
396C24—44no—
397C24—44no—
398C24—48no—
399C24—44no—
400C24—44no—
401C24—48no—
402C24C22×D412yes—
403C24C2×#4114yes—
404C24C2×#4112yes—
405C24C2×#4112yes—
406C24H(5,𝔽2)14yes—
407C24H(5,𝔽2)14yes—
408C24H(5,𝔽2)14yes—
409C242−1+414yes—
410C24C2511yescharacteristic
411C2×D4—44no—
412C2×D4—416no—
413C2×D4—416no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—416no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4—88no—
425C2×D4—88no—
426C2×D4—88no—
427C2×D4—88no—
428C2×D4—88no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—28no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—48no—
501C22×C4—416no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—416no—
505C22×C4—416no—
506C22×C4—416no—
507C22×C4—416no—
508C22×C4—416no—
509C22×C4—44no—
510C22×C4—48no—
511C22×C4—44no—
512C22×C4—44no—
513C22×C4—416no—
514C22×C4—416no—
515C22×C4—4128no—
516C22×C4—416no—
517C22×C4—44no—
518C22×C4—48no—
519C22×C4—48no—
520C22×C4—44no—
521C22×C4—44no—
522C22×C4—44no—
523C22×C4—88no—
524C22×C4—88no—
525C22×C4C22×D414yes—
526C22×C4C22×D414yes—
527C22×C4C22×D412yes—
528C22×C4C22×D412yes—
529C22×C4C2×#4114yes—
530C22×C4H(5,𝔽2)14yes—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—48no—
535C4⋊C4—44no—
536C4⋊C4—48no—
537C4⋊C4—48no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4—88no—
541C22⋊C4—232no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—416no—
548C22⋊C4—48no—
549C22⋊C4—48no—
550C22⋊C4—48no—
551C22⋊C4—84no—
552C22⋊C4—84no—
553C22⋊C4—88no—
554C22⋊C4—88no—
555C22⋊C4—88no—
556C22⋊C4—88no—
557C22⋊C4—88no—
558C23—24no—
559C23—24no—
560C23—22no—
561C23—22no—
562C23—22no—
563C23—22no—
564C23—24no—
565C23—24no—
566C23—28no—
567C23—28no—
568C23—28no—
569C23—28no—
570C23—232no—
571C23—24no—
572C23—28no—
573C23—24no—
574C23—28no—
575C23—24no—
576C23—24no—
577C23—24no—
578C23—24no—
579C23—28no—
580C23—28no—
581C23—28no—
582C23—28no—
583C23—464no—
584C23—416no—
585C23—416no—
586C23—416no—
587C23—416no—
588C23—464no—
589C23—416no—
590C23—416no—
591C23—42no—
592C23—42no—
593C23—44no—
594C23—42no—
595C23—42no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—44no—
607C23—42no—
608C23—42no—
609C23—44no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—44no—
615C23—44no—
616C23—44no—
617C23—44no—
618C23—44no—
619C23—44no—
620C23—44no—
621C23—44no—
622C23—44no—
623C23—44no—
624C23#52514yes—
625C23#53414yes—
626C23#53514yes—
627C23#53818yes—
628C23#53918yes—
629C23#54018yes—
630C23C23×D411yescharacteristic
631C23C22×#4111yescharacteristic
632C23C22×#4111yescharacteristic
633C23C2×H(5,𝔽2)12yes—
634C23C2×H(5,𝔽2)12yes—
635C23C2×H(5,𝔽2)12yes—
636C23C2×H(5,𝔽2)12yes—
637C23C2×2−1+412yes—
638C23#58512yes—
639D4—88no—
640D4—832no—
641D4—816no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—22no—
654C2×C4—24no—
655C2×C4—22no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—28no—
659C2×C4—416no—
660C2×C4—416no—
661C2×C4—416no—
662C2×C4—464no—
663C2×C4—416no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
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—44no—
687C2×C4—42no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—44no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—42no—
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—88no—
716C2×C4—88no—
717C22—24no—
718C22—24no—
719C22—24no—
720C22—24no—
721C22—216no—
722C22—216no—
723C22—24no—
724C22—24no—
725C22—216no—
726C22—28no—
727C22—464no—
728C22—44no—
729C22—44no—
730C22—44no—
731C22—44no—
732C22—44no—
733C22—48no—
734C22—48no—
735C22—48no—
736C22—44no—
737C22—82no—
738C22—82no—
739C22—82no—
740C22—82no—
741C22—82no—
742C22—84no—
743C22—84no—
744C22—84no—
745C22—84no—
746C22—82no—
747C22—82no—
748C22#263518yes—
749C22C2×#52511yescharacteristic
750C22C2×#53411yescharacteristic
751C22C2×#53511yescharacteristic
752C22C2×#53812yes—
753C22C2×#53912yes—
754C22C2×#54012yes—
755C22#345912yes—
756C22#348411yescharacteristic
757C22#348411yescharacteristic
758C22#348611yescharacteristic
759C22#348711yescharacteristic
760C22#349212yes—
761C22#350512yes—
762C22#353214yes—
763C22#353414yes—
764C22#353514yes—
765C22#353614yes—
766C4—44no—
767C4—44no—
768C4—44no—
769C4—42no—
770C4—42no—
771C4—44no—
772C4—82no—
773C4—82no—
774C4—82no—
775C4—82no—
776C4—84no—
777C4—84no—
778C2—28no—
779C2—48no—
780C2—162no—
781C2C2×#263511yescharacteristic
782C2#3498711yescharacteristic
783C2#3498911yescharacteristic
784C2#3670711yescharacteristic
785C2#3671011yescharacteristic
786C2#3671411yescharacteristic
787C2#3677714yes—
788C2#3718611yescharacteristic
789C2#3748214yes—
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×#2618C2semidirect
2. C22×#385C2semidirect
3. #16402C2semidirect
4. #16352C2semidirect
5. #16319C2semidirect
6. #16266C2semidirect
7. #14912C2semidirect
8. #14911C2semidirect
9. #14822C2semidirect
10. #14734C2semidirect
11. #14719C2semidirect
13. #14694C2semidirect
14. #14342C2semidirect
15. #14259C2semidirect
25. C2×#400C22semidirect
27. C2×#398C22semidirect
28. C2×#396C22semidirect
29. C2×#396C22semidirect
30. C2×#394C22semidirect
31. C2×#394C22semidirect
48. #1445C22semidirect
49. #1445C22semidirect
50. #1443C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
57. #1440C22semidirect
58. #1440C22semidirect
64. #1439C22semidirect
65. #1439C22semidirect
66. #1439C22semidirect
103. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26182
C22×#3852
#164022
#163522
#163192
#162662
#149122
#149112
#148222
#147342
#147192
#147162
#146942
#143422
#142592