Finite Group Databasebeta

← #628067

Subgroups — #628067

Order 512.

3356 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2617C211yesmaximal, characteristic
2C2×#2312C211yesmaximal, characteristic, complemented
3#15716C214yesmaximal, complemented
4#15705C214yesmaximal, complemented
5#15655C214yesmaximal, complemented
6#15643C214yesmaximal, complemented
7#14725C214yesmaximal, complemented
8#14698C211yesmaximal, characteristic, complemented
9#14686C211yesmaximal, characteristic, complemented
10#14347C211yesmaximal, characteristic, complemented
11#14271C211yesmaximal, characteristic, complemented
12#13999C214yesmaximal, complemented
13C2×#1437C211yesmaximal, characteristic, complemented
14C22×#121C2211yescharacteristic
15C22×#120C2212yes—
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2212yes—
18C23×C42C2211yescharacteristic
19#2617—28no—
20C2×#397C2212yes—
21C2×#394C2214yes—
22C2×#394C2214yes—
23C2×#394C2214yes—
24C2×#392C2212yes—
25#2312—28no—
26C2×#390C2212yescomplemented
27C2×#388C2214yes—
28C2×#388C2214yes—
29C2×#388C2214yescomplemented
30C2×#386C2214yes—
31C2×#386C2212yes—
32C2×#386C2214yes—
33C2×#384C2212yescomplemented
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2214yes—
38C2×#379C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C22×#95C2211yescharacteristic
41C22×#95C2212yes—
42#1445C2212yescomplemented
43#1443C2214yescomplemented
44#1442C2214yescomplemented
45#1442C2214yescomplemented
46#1442C2214yescomplemented
47#1440C2214yes—
48#1440C2214yes—
49#1440C2214yescomplemented
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yescomplemented
53#1439C2214yescomplemented
54#1438C2214yes—
55#1438C2214yes—
56#1438C2212yescomplemented
57#1437C2212yes—
58#1437C2214yes—
59#1437C2212yes—
60#1437C2212yescomplemented
61#1437C2214yescomplemented
62#1437C2212yescomplemented
63#1436C2214yescomplemented
64C4⋊#95C2214yescomplemented
65C4⋊#95C2212yescomplemented
66C4⋊#95C2212yescomplemented
67#1434C2214yescomplemented
68C22⋊#95C2214yes—
69C26C2311yescharacteristic
70C23×D4C2312yes—
71C23×D4C2311yescharacteristic
72C23×D4C2314yes—
73C24×C4C2312yes—
74C24×C4C2312yes—
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C2×#121—24no—
78C2×#121—24no—
79C2×#121—24no—
80C2×#120—28no—
81C2×#120—48no—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4—24no—
89C22×C22⋊C4—24no—
90C22×C22⋊C4—28no—
91C22×C22⋊C4—44no—
92C22×C22⋊C4—44no—
93C22×C22⋊C4D418yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C42—24no—
111C22×C42—44no—
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2314yes—
116#397—28no—
117#394—216no—
118#394—48no—
119#394—48no—
120#392—28no—
121#390—44no—
122#388—216no—
123#388—216no—
124#388—48no—
125#386—28no—
126#386—216no—
127#386—48no—
128#384—44no—
129#380—24no—
130#380—24no—
131#380—24no—
132#380—48no—
133#379—28no—
134C4×C22⋊C4—28no—
135C2×#95—24no—
136C2×#95—44no—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C25—24no—
157C25—22no—
158C25—22no—
159C25—24no—
160C25—42no—
161C25—42no—
162C25—42no—
163C25—42no—
164C25C2×D414yes—
165C25C2412yes—
166C25C2412yes—
167C25C2411yescharacteristic
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—28no—
172C22×D4—28no—
173C22×D4—28no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—44no—
178C22×D4—44no—
179C22×D4—42no—
180C22×D4—42no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4—44no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4C2×D414yes—
205C23×C4C2×D414yes—
206C23×C4#4118yes—
207C23×C4#4114yes—
208C23×C4#4114yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219#121—416no—
220#120—432no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C22⋊C4—232no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
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—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—432no—
270C2×C22⋊C4—44no—
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⋊C4C2×D418yes—
285C2×C22⋊C4C2×D418yes—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—28no—
292C2×C42—416no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—44no—
296C2×C42—44no—
297#95—28no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—24no—
303#95—24no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—28no—
321C24—22no—
322C24—22no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—28no—
330C24—28no—
331C24—22no—
332C24—24no—
333C24—22no—
334C24—22no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—48no—
341C24—48no—
342C24—48no—
343C24—48no—
344C24—48no—
345C24—48no—
346C24—48no—
347C24—48no—
348C24—82no—
349C24—82no—
350C24—84no—
351C24—84no—
352C24—82no—
353C24—82no—
354C24—84no—
355C24—84no—
356C24—84no—
357C24—84no—
358C24C22×D414yes—
359C24C22×D412yes—
360C24C2×#4112yes—
361C24C2×#4112yes—
362C24C2×#4114yes—
363C24H(5,𝔽2)14yes—
364C24C2511yescharacteristic
365C2×D4—48no—
366C2×D4—416no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—44no—
373C2×D4—82no—
374C2×D4—82no—
375C2×D4—84no—
376C2×D4—88no—
377C2×D4—88no—
378C2×D4—88no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—88no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—216no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—216no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—84no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—84no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4#123116yes—
457C22×C4C22×D414yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4114yes—
466C22×C4H(5,𝔽2)14yes—
467C22×C4H(5,𝔽2)14yes—
468C4⋊C4—44no—
469C4⋊C4—84no—
470C4⋊C4—84no—
471C4⋊C4—84no—
472C4⋊C4—88no—
473C4⋊C4—88no—
474C22⋊C4—216no—
475C22⋊C4—216no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—416no—
483C22⋊C4—416no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C42—28no—
491C42—48no—
492C42—44no—
493C42—88no—
494C23—24no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—432no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—42no—
527C23—42no—
528C23—416no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—88no—
555C23—816no—
556C23—88no—
557C23—816no—
558C23—816no—
559C23—816no—
560C23—816no—
561C23C2×#12314yes—
562C23#53214yes—
563C23#53514yes—
564C23#55318yes—
565C23#55518yes—
566C23C23×D411yescharacteristic
567C23C22×#4111yescharacteristic
568C23C22×#4111yescharacteristic
569C23C22×#4112yes—
570C23C2×H(5,𝔽2)12yes—
571C23C2×H(5,𝔽2)12yes—
572C23C2×H(5,𝔽2)12yes—
573C23#58512yes—
574C23#58512yes—
575D4—88no—
576D4—832no—
577D4—816no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—416no—
598C2×C4—432no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—88no—
640C2×C4—816no—
641C2×C4C2×#12314yes—
642C2×C4C2×#12314yes—
643C2×C4#53818yes—
644C22—28no—
645C22—28no—
646C22—24no—
647C22—28no—
648C22—28no—
649C22—28no—
650C22—28no—
651C22—24no—
652C22—44no—
653C22—48no—
654C22—48no—
655C22—44no—
656C22—44no—
657C22—832no—
658C22—84no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22—84no—
664C22—82no—
665C22—84no—
666C22—84no—
667C22—84no—
668C22—84no—
669C22—82no—
670C22—82no—
671C22#250018yes—
672C22C22×#12311yescharacteristic
673C22C2×#53211yescharacteristic
674C22C2×#53511yescharacteristic
675C22C2×#53812yes—
676C22#345512yes—
677C22C2×#55312yes—
678C22C2×#55512yes—
679C22#349011yescharacteristic
680C22#349111yescharacteristic
681C22#349412yes—
682C22#349612yes—
683C22#349612yes—
684C22#350512yes—
685C22#350512yes—
686C22#350612yes—
687C22#350811yescharacteristic
688C22#351011yescharacteristic
689C22#353412yes—
690C22#353712yes—
691C22#355812yes—
692C22#356012yes—
693C4—28no—
694C4—24no—
695C4—24no—
696C4—44no—
697C4—44no—
698C4—84no—
699C4—84no—
700C4—84no—
701C4—84no—
702C4—82no—
703C2—28no—
704C2—84no—
705C2—162no—
706C2C2×#250011yescharacteristic
707C2#3500512yes—
708C2#3536011yescharacteristic
709C2#3554111yescharacteristic
710C2#3673912yes—
711C2#3789112yes—
712C2#3789512yes—
713C2#3949912yes—
714C2#3950312yes—
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. C2×#2312C2semidirect
3. #15716C2semidirect
4. #15705C2semidirect
5. #15655C2semidirect
6. #15643C2semidirect
7. #14725C2semidirect
8. #14698C2semidirect
9. #14686C2semidirect
10. #14347C2semidirect
11. #14271C2semidirect
12. #13999C2semidirect
13. C2×#1437C2semidirect
26. C2×#390C22semidirect
29. C2×#388C22semidirect
33. C2×#384C22semidirect
42. #1445C22semidirect
43. #1443C22semidirect
44. #1442C22semidirect
45. #1442C22semidirect
46. #1442C22semidirect
49. #1440C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
56. #1438C22semidirect
60. #1437C22semidirect
61. #1437C22semidirect
62. #1437C22semidirect
63. #1436C22semidirect
64. C4⋊#95C22semidirect
65. C4⋊#95C22semidirect
66. C4⋊#95C22semidirect
67. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26172
C2×#23122
#157162
#157052
#156552
#156432
#147252
#146982
#146862
#143472
#142712
#139992
C2×#14372