Finite Group Databasebeta

← #4416648

Subgroups — #4416648

Order 512.

2834 conjugacy classes of subgroups.

The subgroup lattice has 740 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2826C212yesmaximal, complemented
2#16866C214yesmaximal, complemented
3#16851C214yesmaximal, complemented
4#16338C212yesmaximal, complemented
5#16288C212yesmaximal, complemented
6#15147C212yesmaximal, complemented
7#15140C212yesmaximal, complemented
8#15139C212yesmaximal, complemented
9#15135C211yesmaximal, characteristic, complemented
10#15134C212yesmaximal, complemented
11#15126C212yesmaximal, complemented
12#15124C211yesmaximal, characteristic
13#14698C212yesmaximal, complemented
14#14621C212yesmaximal, complemented
15C4×#385C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#121C2212yes—
18C23×C42C2211yescharacteristic
19#2826—216no—
20C2×#400C2214yes—
21C2×#399C2212yes—
22C2×#397C2214yes—
23C2×#396C2214yes—
24C2×#396C2214yes—
25C2×#394C2214yes—
26C2×#394C2214yes—
27C2×#394C2214yes—
28C2×#392C2212yes—
29C2×#392C2212yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#386C2214yes—
38C2×#386C2214yes—
39C2×#385C2211yescharacteristic
40C2×#385C2211yescharacteristic
41C2×#384C2211yescharacteristic
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46#1445C2212yes—
47#1444C2212yes—
48#1443C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2212yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2212yes—
60#1438C2212yes—
61#1438C2214yes—
62#1438C2214yes—
63#1438C2214yes—
64#1437C2214yes—
65#1437C2214yes—
66#1436C2212yes—
67#1436C2212yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2211yescharacteristic
75C4×#95C2212yes—
76C23×D4C2312yes—
77C23×D4C2314yes—
78C23×D4C2312yes—
79C23×D4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×#127—42no—
84C2×#127—44no—
85C2×#121—28no—
86C2×#121—44no—
87C2×#121—44no—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4C2311yescharacteristic
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C42—44no—
114C22×C42—42no—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120#400—216no—
121#399—28no—
122#397—216no—
123#396—216no—
124#396—48no—
125#394—216no—
126#394—216no—
127#394—48no—
128#392—28no—
129#392—28no—
130#390—44no—
131#390—44no—
132#390—44no—
133#390—44no—
134#390—44no—
135#389—44no—
136#389—44no—
137#386—216no—
138#386—48no—
139#385—24no—
140#385D418yescomplemented
141#384—42no—
142#380—44no—
143C4×C4⋊C4—44no—
144C4×C22⋊C4—24no—
145C4×C22⋊C4—42no—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2311yescharacteristic
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C25C2414yes—
169C25C2411yescharacteristic
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4—42no—
186C22×D4—42no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—42no—
192C22×D4—42no—
193C22×D4—44no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—42no—
201C23×C4—41no—
202C23×C4—41no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—41no—
206C23×C4—41no—
207C23×C4C2×D414yes—
208C23×C4#4114yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222#127—416no—
223#121—432no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C2×D418yes—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4C2×D414yes—
296C2×C22⋊C4C2×D414yes—
297C2×C22⋊C4#4114yes—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—416no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—28no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—24no—
322#95—42no—
323#95—42no—
324#95—44no—
325#95—41no—
326#95—41no—
327#95—42no—
328#95—42no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95C2×D418yes—
334C24—21no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24C22×D412yes—
358C24C2×#4112yes—
359C24C2×#4112yes—
360C24C2511yescharacteristic
361C2×D4—416no—
362C2×D4—416no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—416no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—84no—
374C2×D4—88no—
375C2×D4—84no—
376C2×D4—88no—
377C2×D4—88no—
378C2×D4—88no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—84no—
382C2×D4—84no—
383C2×D4—88no—
384C2×D4—88no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—216no—
424C22×C4—28no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—48no—
466C22×C4C22×D414yes—
467C22×C4C22×D412yes—
468C22×C4C22×D414yes—
469C22×C4C22×D412yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4H(5,𝔽2)14yes—
475C22×C4H(5,𝔽2)14yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C42−1+412yes—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—84no—
485C4⋊C4—88no—
486C4⋊C4—88no—
487C4⋊C4—84no—
488C22⋊C4—28no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—416no—
492C22⋊C4—44no—
493C22⋊C4—88no—
494C22⋊C4—88no—
495C22⋊C4—88no—
496C22⋊C4—88no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C42—48no—
505C42—48no—
506C42—44no—
507C42—48no—
508C42—44no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—22no—
514C23—24no—
515C23—22no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23—41no—
525C23—42no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23#53214yes—
567C23C23×D411yescharacteristic
568C23C22×#4111yescharacteristic
569C23C22×#4111yescharacteristic
570C23C2×H(5,𝔽2)12yes—
571C23C2×H(5,𝔽2)12yes—
572C23C2×H(5,𝔽2)11yescharacteristic
573C23C2×H(5,𝔽2)12yes—
574C23C2×2−1+411yescharacteristic
575C23#58511yescharacteristic
576C23#58511yescharacteristic
577C23#58512yes—
578D4—816no—
579D4—816no—
580D4—832no—
581D4—816no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—416no—
602C2×C4—416no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—416no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
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—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4#53818yes—
660C2×C4#53914yes—
661C2×C4#56014yes—
662C2×C4#56014yes—
663C2×C4#56218yes—
664C22—24no—
665C22—28no—
666C22—42no—
667C22—42no—
668C22—44no—
669C22—44no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—44no—
674C22—42no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22—84no—
679C22—84no—
680C22—84no—
681C22—84no—
682C22—84no—
683C22—84no—
684C22—84no—
685C22—84no—
686C22—84no—
687C22—84no—
688C22—84no—
689C22—84no—
690C22C2×#53211yescharacteristic
691C22C2×#53812yes—
692C22C2×#53911yescharacteristic
693C22#345511yescharacteristic
694C22#346012yes—
695C22#349612yes—
696C22#349611yescharacteristic
697C22#349711yescharacteristic
698C22#349812yes—
699C22#350511yescharacteristic
700C22#350711yescharacteristic
701C22#351412yes—
702C22#351912yes—
703C22C2×#56011yescharacteristic
704C22C2×#56011yescharacteristic
705C22C2×#56212yes—
706C22#353711yescharacteristic
707C22#353912yes—
708C22#354011yescharacteristic
709C22#354212yes—
710C22#354612yes—
711C22#355412yes—
712C22#356212yes—
713C4—42no—
714C4—44no—
715C4—42no—
716C4—42no—
717C4—42no—
718C4—44no—
719C4—42no—
720C4—44no—
721C4—82no—
722C4—84no—
723C4—84no—
724C4—84no—
725C4#282618yescomplemented
726C2—44no—
727C2—164no—
728C2#3500611yescharacteristic
729C2#3500712yes—
730C2C2×#282611yescharacteristic
731C2#3749812yes—
732C2#3752311yescharacteristic
733C2#4547811yescharacteristic
734C2#4548412yes—
735C2#4548711yescharacteristic
736C2#4559911yescharacteristic
737C2#4563912yes—
738C2#4569711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#15124C2
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×#2826C2semidirect
2. #16866C2semidirect
3. #16851C2semidirect
4. #16338C2semidirect
5. #16288C2semidirect
6. #15147C2semidirect
7. #15140C2semidirect
8. #15139C2semidirect
9. #15135C2semidirect
10. #15134C2semidirect
11. #15126C2semidirect
13. #14698C2semidirect
14. #14621C2semidirect
15. C4×#385C2semidirect
140. #385D4semidirect
725. C4#2826semidirect

Maximal subgroups

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

subgroupindex
C2×#28262
#168662
#168512
#163382
#162882
#151472
#151402
#151392
#151352
#151342
#151262
#151242
#146982
#146212
C4×#3852