Finite Group Databasebeta

← #647640

Subgroups — #647640

Order 512.

3054 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2C22×#387C211yesmaximal, characteristic, complemented
3#17027C214yesmaximal, complemented
4#16567C212yesmaximal, complemented
5#16519C212yesmaximal, complemented
6#15101C212yesmaximal, complemented
7#14936C211yesmaximal, characteristic, complemented
8#14927C212yesmaximal, complemented
9#14926C211yesmaximal, characteristic
10#14860C212yesmaximal, complemented
11#14822C211yesmaximal, characteristic, complemented
12#14729C212yesmaximal, complemented
13#14719C211yesmaximal, characteristic
14#14682C211yesmaximal, characteristic, complemented
15#14315C214yesmaximal, complemented
16#14016C212yesmaximal, complemented
17#13968C212yesmaximal, complemented
18C22×#123C2212yes—
19C22×#121C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2212yes—
22C23×C4⋊C4C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C23×C42C2211yescharacteristic
25C2×#400C2212yes—
26C2×#396C2214yes—
27C2×#394C2212yes—
28C4×#121—28no—
29C2×#390C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#388C2212yes—
32C2×#387—28no—
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#387C2212yes—
36C2×#387C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#382C2212yes—
43C2×#381C2212yes—
44C2×#380C2211yescharacteristic
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51C22×#95C2211yescharacteristic
52#1445C2212yes—
53#1445C2214yes—
54#1444C2212yes—
55#1444C2212yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1441C2214yes—
60#1441C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2212yes—
64#1441C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1438C2212yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76#1434C2212yes—
77#1434C2212yes—
78C22⋊#95C2214yes—
79C22⋊#95C2214yes—
80C22⋊#95C2211yescharacteristic
81C22⋊#95C2212yes—
82C22⋊#95C2211yescharacteristic
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C24×C4C2312yes—
95C2×#123—28no—
96C2×#123—44no—
97C2×#123—44no—
98C2×#121—28no—
99C2×#121—24no—
100C2×C4×D4—28no—
101C2×C4×D4—24no—
102C2×C4×D4—44no—
103C2×C4×D4—44no—
104C2×C4×D4—44no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—44no—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C42—44no—
132C22×C42D418yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#400—44no—
139#396—216no—
140#394—44no—
141#390—24no—
142#389—24no—
143#388—28no—
144#387—28no—
145#387—28no—
146#387—28no—
147#387—28no—
148#386—28no—
149#386—28no—
150#386—44no—
151#386—44no—
152#385—44no—
153#382—44no—
154#381—44no—
155#380—24no—
156#380—44no—
157C4×C4⋊C4—24no—
158C4×C22⋊C4—28no—
159C2×#95—24no—
160C2×#95—24no—
161C2×#95—24no—
162C2×#95—24no—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C25C2412yes—
192C25C2411yescharacteristic
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—22no—
201C22×D4—24no—
202C22×D4—42no—
203C22×D4#4118yes—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—28no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—44no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2×D414yes—
240C23×C4C2412yes—
241C23×C4C2414yes—
242C23×C4C2414yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255#123—432no—
256#121—232no—
257C4×D4—416no—
258C4×D4—432no—
259C2×C4⋊C4—232no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—416no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4#4118yes—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
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—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—42no—
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×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—416no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
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—24no—
348#95—24no—
349#95—28no—
350#95—42no—
351#95—42no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—24no—
373C24—24no—
374C24—22no—
375C24—24no—
376C24C22×D412yes—
377C24C2×#4114yes—
378C24C2×#4114yes—
379C24H(5,𝔽2)14yes—
380C24H(5,𝔽2)12yes—
381C242−1+412yes—
382C24C2511yescharacteristic
383C2×D4—28no—
384C2×D4—216no—
385C2×D4—216no—
386C2×D4—216no—
387C2×D4—416no—
388C2×D4—44no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—44no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—416no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—416no—
476C22×C4—416no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—48no—
494C22×C4—88no—
495C22×C4—88no—
496C22×C4—88no—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C2×#4114yes—
504C22×C4C2×#4114yes—
505C22×C4C2×#4114yes—
506C22×C4C2×#4114yes—
507C22×C4H(5,𝔽2)14yes—
508C22×C42−1+414yes—
509C4⋊C4—216no—
510C4⋊C4—28no—
511C4⋊C4—216no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—416no—
516C4⋊C4—48no—
517C4⋊C4—44no—
518C4⋊C4—88no—
519C4⋊C4—88no—
520C22⋊C4—216no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—416no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—88no—
529C22⋊C4—88no—
530C22⋊C4—88no—
531C42—28no—
532C42—28no—
533C42—44no—
534C42—44no—
535C42—48no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—28no—
541C23—28no—
542C23—28no—
543C23—24no—
544C23—24no—
545C23—28no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—28no—
552C23—28no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23#52514yes—
572C23#53514yes—
573C23#53714yes—
574C23#53818yes—
575C23#53918yes—
576C23C23×D411yescharacteristic
577C23C22×#4112yes—
578C23C22×#4112yes—
579C23C2×H(5,𝔽2)12yes—
580C23C2×H(5,𝔽2)12yes—
581C23C2×H(5,𝔽2)11yescharacteristic
582C23C2×2−1+412yes—
583C23C2×2−1+411yescharacteristic
584C23#58512yes—
585D4—416no—
586D4—432no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—28no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—416no—
615C2×C4—416no—
616C2×C4—416no—
617C2×C4—416no—
618C2×C4—416no—
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—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
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—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—816no—
669C2×C4—88no—
670C2×C4—88no—
671C2×C4#52514yes—
672C2×C4#53414yes—
673C2×C4#53614yes—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—216no—
679C22—216no—
680C22—24no—
681C22—24no—
682C22—44no—
683C22—44no—
684C22—48no—
685C22—48no—
686C22—44no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—84no—
691C22—84no—
692C22—82no—
693C22#263618yes—
694C22C2×#52511yescharacteristic
695C22C2×#52511yescharacteristic
696C22C2×#53411yescharacteristic
697C22C2×#53511yescharacteristic
698C22C2×#53611yescharacteristic
699C22C2×#53711yescharacteristic
700C22#344911yescharacteristic
701C22C2×#53812yes—
702C22C2×#53912yes—
703C22#348214yes—
704C22#349414yes—
705C22#351314yes—
706C22#351912yes—
707C22#352012yes—
708C22#353614yes—
709C22#355912yes—
710C22#356312yes—
711C4—24no—
712C4—24no—
713C4—24no—
714C4—42no—
715C4—44no—
716C4—44no—
717C4—42no—
718C4—44no—
719C4—42no—
720C4—82no—
721C4—82no—
722C4—84no—
723C4—84no—
724C4—82no—
725C4—82no—
726C2—28no—
727C2—84no—
728C2C2×#263611yescharacteristic
729C2#3497111yescharacteristic
730C2#3506211yescharacteristic
731C2#3507711yescharacteristic
732C2#3530111yescharacteristic
733C2#3745111yescharacteristic
734C2#3754911yescharacteristic
735C2#3860814yes—
736C2#3860914yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#387C2semidirect
3. #17027C2semidirect
4. #16567C2semidirect
5. #16519C2semidirect
6. #15101C2semidirect
7. #14936C2semidirect
8. #14927C2semidirect
10. #14860C2semidirect
11. #14822C2semidirect
12. #14729C2semidirect
14. #14682C2semidirect
15. #14315C2semidirect
16. #14016C2semidirect
17. #13968C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
C22×#3872
#170272
#165672
#165192
#151012
#149362
#149272
#149262
#148602
#148222
#147292
#147192
#146822
#143152
#140162
#139682