Finite Group Databasebeta

← #647614

Subgroups — #647614

Order 512.

3054 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic, complemented
2C22×#388C211yesmaximal, characteristic
3#17018C214yesmaximal, complemented
4#16701C212yesmaximal, complemented
5#16604C212yesmaximal, complemented
6#15015C211yesmaximal, characteristic, complemented
7#15014C211yesmaximal, characteristic, complemented
8#14920C212yesmaximal, complemented
9#14820C211yesmaximal, characteristic
10#14805C214yesmaximal, complemented
11#14734C211yesmaximal, characteristic
12#14726C212yesmaximal
13#14324C212yesmaximal, complemented
14#14322C212yesmaximal, complemented
15#14043C212yesmaximal, complemented
16#13971C212yesmaximal
17C4×#380C211yesmaximal, characteristic, complemented
18C22×#125C2211yescharacteristic, complemented
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2851—28no—
25C2×#399C2212yes—
26C2×#397C2212yes—
27C2×#396C2214yes—
28C2×#389C2211yescharacteristic, complemented
29C2×#389C2211yescharacteristic, complemented
30C2×#389C2214yescomplemented
31C2×#388—28no—
32C2×#388C2212yes—
33C2×#388C2212yes—
34C2×#388C2212yes—
35C2×#388C2212yes—
36C2×#388C2214yes—
37C2×#387C2212yescomplemented
38C2×#387C2212yescomplemented
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#382C2211yescharacteristic, complemented
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2212yes—
47C2×#379C2212yes—
48C2×#379C2212yes—
49C2×C4×C4⋊C4C2212yescomplemented
50C2×C4×C22⋊C4C2212yes—
51C22×#95C2211yescharacteristic
52C22×#95C2211yescharacteristic
53#1445C2212yescomplemented
54#1445C2212yescomplemented
55#1445C2214yescomplemented
56#1444C2214yescomplemented
57#1444C2212yescomplemented
58#1443C2214yescomplemented
59#1442C2214yescomplemented
60#1442C2214yescomplemented
61#1442C2212yescomplemented
62#1441C2214yescomplemented
63#1441C2214yescomplemented
64#1441C2214yescomplemented
65#1440C2214yes—
66#1440C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2212yes—
73#1439C2212yes—
74#1437C2214yes—
75#1436C2214yescomplemented
76C4⋊#95C2212yescomplemented
77#1434C2212yescomplemented
78C22⋊#95C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2211yescharacteristic
83C22⋊#95C2211yescharacteristic
84C4×#95C2212yescomplemented
85C26C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×#125—24no—
93C2×#125—44no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—28no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—44no—
114C22×C22⋊C4—44no—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C42—44no—
130C22×C42D418yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2314yes—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#399—28no—
137#397—28no—
138#396—48no—
139#389—24no—
140#389—24no—
141#389—48no—
142#388—28no—
143#388—28no—
144#388—28no—
145#388—28no—
146#388—48no—
147#387—28no—
148#387—28no—
149#385—28no—
150#385—28no—
151#385—28no—
152#385—44no—
153#382—24no—
154#380—24no—
155#380—24no—
156#380—28no—
157#379—28no—
158#379—44no—
159C4×C4⋊C4—44no—
160C4×C22⋊C4—44no—
161C2×#95—24no—
162C2×#95—24no—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C25—22no—
190C25—24no—
191C25—24no—
192C25—22no—
193C25—24no—
194C25—22no—
195C25—42no—
196C25—42no—
197C25—42no—
198C25C2412yes—
199C25C2411yescharacteristic
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241#125—416no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—416no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C22⋊C4—232no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
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—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—416no—
303C2×C22⋊C4—416no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4#4118yes—
312C2×C22⋊C4#4118yes—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—28no—
316C2×C42—416no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42C2×D414yes—
322C2×C42C2×D414yes—
323C2×C42C2×D414yes—
324C2×C42C2×D414yes—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—28no—
337#95—24no—
338#95—28no—
339#95—24no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—42no—
344#95—42no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—28no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—22no—
362C24—22no—
363C24—24no—
364C24—22no—
365C24—22no—
366C24—416no—
367C24—48no—
368C24—48no—
369C24—416no—
370C24—44no—
371C24—44no—
372C24—44no—
373C24—44no—
374C24—44no—
375C24—44no—
376C24—48no—
377C24—48no—
378C24—88no—
379C24—84no—
380C24—84no—
381C24—88no—
382C24C22×D412yes—
383C24C2×#4114yes—
384C24C2×#4114yes—
385C24H(5,𝔽2)12yes—
386C242−1+414yes—
387C242−1+412yes—
388C24C2511yescharacteristic
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—416no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4—88no—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C2×#4118yes—
486C22×C4C2×#4114yes—
487C22×C4C2×#4114yes—
488C22×C4H(5,𝔽2)14yes—
489C22×C4H(5,𝔽2)14yes—
490C4⋊C4—416no—
491C4⋊C4—416no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—88no—
503C22⋊C4—216no—
504C22⋊C4—216no—
505C22⋊C4—216no—
506C22⋊C4—416no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—88no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C42—28no—
518C42—28no—
519C42—44no—
520C42—44no—
521C42—88no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—28no—
527C23—24no—
528C23—24no—
529C23—28no—
530C23—28no—
531C23—24no—
532C23—24no—
533C23—28no—
534C23—24no—
535C23—28no—
536C23—24no—
537C23—24no—
538C23—28no—
539C23—416no—
540C23—416no—
541C23—416no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—42no—
552C23—42no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—832no—
561C23—832no—
562C23—88no—
563C23—88no—
564C23—88no—
565C23—88no—
566C23#53214yes—
567C23#53614yes—
568C23#53714yes—
569C23#53918yes—
570C23#54118yes—
571C23C23×D411yescharacteristic
572C23C22×#4112yes—
573C23C22×#4112yes—
574C23C2×H(5,𝔽2)12yes—
575C23C2×H(5,𝔽2)12yes—
576C23C2×H(5,𝔽2)11yescharacteristic
577C23C2×2−1+412yes—
578C23C2×2−1+411yescharacteristic
579C23#58512yes—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—416no—
605C2×C4—416no—
606C2×C4—416no—
607C2×C4—416no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—88no—
657C2×C4—88no—
658C2×C4#52514yes—
659C2×C4#53414yes—
660C2×C4#53514yes—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—216no—
666C22—216no—
667C22—24no—
668C22—24no—
669C22—44no—
670C22—44no—
671C22—48no—
672C22—48no—
673C22—44no—
674C22—832no—
675C22—82no—
676C22—82no—
677C22—82no—
678C22—84no—
679C22—84no—
680C22—82no—
681C22#266418yes—
682C22C2×#52511yescharacteristic
683C22C2×#53211yescharacteristic
684C22C2×#53411yescharacteristic
685C22C2×#53511yescharacteristic
686C22C2×#53611yescharacteristic
687C22C2×#53711yescharacteristic
688C22#344911yescharacteristic
689C22C2×#53912yes—
690C22C2×#54112yes—
691C22#348114yes—
692C22#349412yes—
693C22#349512yes—
694C22#351314yes—
695C22#352014yes—
696C22#353612yes—
697C22#353712yes—
698C22#356114yes—
699C4—24no—
700C4—24no—
701C4—24no—
702C4—42no—
703C4—42no—
704C4—44no—
705C4—48no—
706C4—42no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—84no—
712C4—82no—
713C4—82no—
714C2—28no—
715C2—84no—
716C2C2×#266411yescharacteristic
717C2#3496511yescharacteristic
718C2#3506411yescharacteristic
719C2#3508411yescharacteristic
720C2#3530211yescharacteristic
721C2#3743911yescharacteristic
722C2#3753711yescharacteristic
723C2#3857614yes—
724C2#3857914yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2851C2semidirect
3. #17018C2semidirect
4. #16701C2semidirect
5. #16604C2semidirect
6. #15015C2semidirect
7. #15014C2semidirect
8. #14920C2semidirect
10. #14805C2semidirect
13. #14324C2semidirect
14. #14322C2semidirect
15. #14043C2semidirect
17. C4×#380C2semidirect
18. C22×#125C22semidirect
28. C2×#389C22semidirect
29. C2×#389C22semidirect
30. C2×#389C22semidirect
37. C2×#387C22semidirect
38. C2×#387C22semidirect
43. C2×#382C22semidirect
49. C2×C4×C4⋊C4C22semidirect
53. #1445C22semidirect
54. #1445C22semidirect
55. #1445C22semidirect
56. #1444C22semidirect
57. #1444C22semidirect
58. #1443C22semidirect
59. #1442C22semidirect
60. #1442C22semidirect
61. #1442C22semidirect
62. #1441C22semidirect
63. #1441C22semidirect
64. #1441C22semidirect
75. #1436C22semidirect
76. C4⋊#95C22semidirect
77. #1434C22semidirect
84. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C22×#3882
#170182
#167012
#166042
#150152
#150142
#149202
#148202
#148052
#147342
#147262
#143242
#143222
#140432
#139712
C4×#3802