Finite Group Databasebeta

← #630044

Subgroups — #630044

Order 512.

3552 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2617C211yesmaximal, characteristic
2C2×#2315C211yesmaximal, characteristic, complemented
3#15554C214yesmaximal, complemented
4#15544C214yesmaximal, complemented
5#15391C214yesmaximal, complemented
6#15377C214yesmaximal, complemented
7#14735C214yesmaximal, complemented
8#14698C211yesmaximal, characteristic, complemented
9#14682C211yesmaximal, characteristic, complemented
10#14347C211yesmaximal, characteristic, complemented
11#14271C211yesmaximal, characteristic, complemented
12#14134C214yesmaximal, complemented
13C2×#1437C211yesmaximal, characteristic, complemented
14C22×#121C2211yescharacteristic
15C22×#120C2212yes—
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2212yes—
18C23×C22⋊C4C2212yes—
19C23×C42C2211yescharacteristic
20#2617—28no—
21C2×#397C2212yes—
22C2×#397C2214yescomplemented
23C2×#394C2214yes—
24C2×#392C2212yes—
25#2315—28no—
26C2×#390C2212yescomplemented
27C2×#386C2212yes—
28C2×#386C2214yes—
29C2×#386C2214yes—
30C2×#386C2214yes—
31C2×#386C2214yes—
32C2×#385C2214yes—
33C2×#385C2214yes—
34C2×#382C2212yescomplemented
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2212yes—
38C2×#380C2211yescharacteristic
39C22×#95C2211yescharacteristic
40C22×#95C2212yes—
41#1445C2214yescomplemented
42#1445C2212yescomplemented
43#1443C2214yescomplemented
44#1443C2214yescomplemented
45#1442C2214yescomplemented
46#1442C2214yescomplemented
47#1442C2214yescomplemented
48#1440C2214yes—
49#1440C2214yes—
50#1440C2214yescomplemented
51#1440C2214yescomplemented
52#1439C2214yescomplemented
53#1439C2214yescomplemented
54#1438C2214yes—
55#1438C2214yes—
56#1438C2212yescomplemented
57#1437C2214yes—
58#1437C2212yes—
59#1437C2212yes—
60#1437C2212yescomplemented
61#1437C2212yescomplemented
62C4⋊#95C2212yescomplemented
63C4⋊#95C2212yescomplemented
64#1434C2214yescomplemented
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C26C2311yescharacteristic
70C23×D4C2312yes—
71C23×D4C2311yescharacteristic
72C23×D4C2314yes—
73C24×C4C2312yes—
74C24×C4C2312yes—
75C24×C4C2311yescharacteristic
76C24×C4C2314yes—
77C24×C4C2311yescharacteristic
78C2×#121—24no—
79C2×#121—24no—
80C2×#121—24no—
81C2×#120—28no—
82C2×#120—216no—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4—24no—
89C22×C22⋊C4—24no—
90C22×C22⋊C4—28no—
91C22×C22⋊C4—28no—
92C22×C22⋊C4—44no—
93C22×C22⋊C4—44no—
94C22×C22⋊C4—48no—
95C22×C22⋊C4D418yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C42—24no—
112C22×C42—44no—
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116#397—28no—
117#397—48no—
118#394—216no—
119#392—28no—
120#390—44no—
121#386—28no—
122#386—48no—
123#386—48no—
124#386—48no—
125#386—48no—
126#385—216no—
127#385—216no—
128#382—44no—
129#380—28no—
130#380—24no—
131#380—24no—
132#380—24no—
133C2×#95—24no—
134C2×#95—44no—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C25—22no—
157C25—22no—
158C25—24no—
159C25—42no—
160C25—42no—
161C25—42no—
162C25—42no—
163C25C2×D414yes—
164C25#4118yes—
165C25C2412yes—
166C25C2412yes—
167C25C2411yescharacteristic
168C22×D4—24no—
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—28no—
174C22×D4—28no—
175C22×D4—28no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—42no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4—42no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—28no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4#4114yes—
215C23×C4#4114yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226#121—416no—
227#120—432no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C22⋊C4—232no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
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—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—432no—
273C2×C22⋊C4—432no—
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⋊C4C2×D418yes—
287C2×C22⋊C4C2×D418yes—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—416no—
294C2×C42—42no—
295C2×C42—42no—
296#95—28no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—44no—
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—
316#95—44no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—28no—
329C24—28no—
330C24—216no—
331C24—216no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—48no—
342C24—48no—
343C24—48no—
344C24—48no—
345C24—48no—
346C24—48no—
347C24—82no—
348C24—82no—
349C24—84no—
350C24—84no—
351C24—82no—
352C24—82no—
353C24—84no—
354C24—84no—
355C24—84no—
356C24—84no—
357C24#124116yes—
358C24C22×D414yes—
359C24C22×D412yes—
360C24C2×#4118yes—
361C24C2×#4112yes—
362C24C2×#4112yes—
363C24C2×#4114yes—
364C24H(5,𝔽2)14yes—
365C24C2511yescharacteristic
366C2×D4—48no—
367C2×D4—416no—
368C2×D4—416no—
369C2×D4—48no—
370C2×D4—416no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—416no—
374C2×D4—416no—
375C2×D4—44no—
376C2×D4—82no—
377C2×D4—82no—
378C2×D4—84no—
379C2×D4—88no—
380C2×D4—88no—
381C2×D4—88no—
382C2×D4—88no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—216no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
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—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—28no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
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—24no—
437C22×C4—24no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—44no—
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—416no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—84no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4—88no—
457C22×C4—88no—
458C22×C4—84no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—84no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4—88no—
465C22×C4—88no—
466C22×C4#123116yes—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
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—
476C4⋊C4—44no—
477C4⋊C4—84no—
478C4⋊C4—84no—
479C4⋊C4—84no—
480C4⋊C4—88no—
481C22⋊C4—216no—
482C22⋊C4—216no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—416no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C22⋊C4—88no—
495C22⋊C4—88no—
496C42—28no—
497C42—48no—
498C42—44no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—232no—
526C23—432no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—44no—
531C23—42no—
532C23—42no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—42no—
545C23—44no—
546C23—42no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—88no—
559C23—816no—
560C23—816no—
561C23—816no—
562C23—816no—
563C23—816no—
564C23—88no—
565C23C2×#12314yes—
566C23C2×#12418yes—
567C23C2×#12414yes—
568C23#53514yes—
569C23#55318yes—
570C23#55318yes—
571C23C23×D411yescharacteristic
572C23C22×#4111yescharacteristic
573C23C22×#4111yescharacteristic
574C23C22×#4112yes—
575C23C2×H(5,𝔽2)12yes—
576C23C2×H(5,𝔽2)12yes—
577C23C2×H(5,𝔽2)12yes—
578C23#58512yes—
579C23#58512yes—
580D4—88no—
581D4—832no—
582D4—816no—
583C2×C4—22no—
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—24no—
595C2×C4—24no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—416no—
600C2×C4—432no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
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—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—88no—
646C2×C4—816no—
647C2×C4—832no—
648C2×C4C2×#12314yes—
649C2×C4C2×#12314yes—
650C22—24no—
651C22—28no—
652C22—28no—
653C22—28no—
654C22—28no—
655C22—28no—
656C22—28no—
657C22—24no—
658C22—216no—
659C22—28no—
660C22—44no—
661C22—48no—
662C22—48no—
663C22—48no—
664C22—48no—
665C22—48no—
666C22—44no—
667C22—832no—
668C22—82no—
669C22—84no—
670C22—84no—
671C22—84no—
672C22—84no—
673C22—84no—
674C22—84no—
675C22—82no—
676C22—82no—
677C22#250418yes—
678C22C22×#12311yescharacteristic
679C22C22×#12411yescharacteristic
680C22C2×#53511yescharacteristic
681C22#345912yes—
682C22#345912yes—
683C22C2×#55312yes—
684C22C2×#55312yes—
685C22#349011yescharacteristic
686C22#349012yes—
687C22#349111yescharacteristic
688C22#349112yes—
689C22#349412yes—
690C22#349612yes—
691C22#349612yes—
692C22#350512yes—
693C22#350811yescharacteristic
694C22#351011yescharacteristic
695C22#353412yes—
696C22#353712yes—
697C22#355812yes—
698C22#356012yes—
699C4—24no—
700C4—24no—
701C4—44no—
702C4—44no—
703C4—44no—
704C4—84no—
705C4—84no—
706C4—84no—
707C4—84no—
708C4—82no—
709C2—28no—
710C2—48no—
711C2—162no—
712C2C2×#250411yescharacteristic
713C2#3499312yes—
714C2#3536011yescharacteristic
715C2#3554111yescharacteristic
716C2#3673912yes—
717C2#3794012yes—
718C2#3794212yes—
719C2#3954812yes—
720C2#3955012yes—
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×#2315C2semidirect
3. #15554C2semidirect
4. #15544C2semidirect
5. #15391C2semidirect
6. #15377C2semidirect
7. #14735C2semidirect
8. #14698C2semidirect
9. #14682C2semidirect
10. #14347C2semidirect
11. #14271C2semidirect
12. #14134C2semidirect
13. C2×#1437C2semidirect
22. C2×#397C22semidirect
26. C2×#390C22semidirect
34. C2×#382C22semidirect
41. #1445C22semidirect
42. #1445C22semidirect
43. #1443C22semidirect
44. #1443C22semidirect
45. #1442C22semidirect
46. #1442C22semidirect
47. #1442C22semidirect
50. #1440C22semidirect
51. #1440C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
56. #1438C22semidirect
60. #1437C22semidirect
61. #1437C22semidirect
62. C4⋊#95C22semidirect
63. C4⋊#95C22semidirect
64. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26172
C2×#23152
#155542
#155442
#153912
#153772
#147352
#146982
#146822
#143472
#142712
#141342
C2×#14372