Finite Group Databasebeta

← #612923

Subgroups — #612923

Order 512.

5178 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2569C211yesmaximal, characteristic, complemented
2C23×#95C211yesmaximal, characteristic
3#15015C212yesmaximal, complemented
4#14920C212yesmaximal, complemented
5#14767C214yesmaximal, complemented
6#14766C214yesmaximal, complemented
7#14693C212yesmaximal
8#14680C212yesmaximal
9#14680C211yesmaximal, characteristic
10#14149C211yesmaximal, characteristic, complemented
11#14028C214yesmaximal, complemented
12#13980C214yesmaximal, complemented
13#13966C211yesmaximal, characteristic, complemented
14#13908C211yesmaximal, characteristic, complemented
15C4×#379C211yesmaximal, characteristic
16C25×C4C2212yes—
17C25×C4C2211yescharacteristic
18#2569—28no—
19C2×#400C2212yescomplemented
20C2×#398C2212yescomplemented
21C2×#387C2212yescomplemented
22C2×#387C2212yescomplemented
23C2×#385C2212yes—
24C2×#385C2214yes—
25C2×#385C2212yes—
26C2×#385C2214yes—
27C2×#380C2211yescharacteristic
28C2×#380C2212yes—
29C2×#380C2211yescharacteristic
30C2×#380C2214yes—
31C2×#380C2212yes—
32C2×#380C2214yes—
33C2×#380C2214yes—
34C2×#379C2212yes—
35C2×#379C2211yescharacteristic
36C2×#379C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2214yes—
40C22×#95—28no—
41C22×#95C2211yescharacteristic
42C22×#95C2212yes—
43C22×#95C2214yes—
44C22×#95C2212yes—
45C22×#95C2212yes—
46C22×#95C2211yescharacteristic
47#1445C2214yescomplemented
48#1445C2214yescomplemented
49#1444C2212yescomplemented
50#1443C2214yescomplemented
51#1443C2214yescomplemented
52#1442C2214yescomplemented
53#1442C2212yescomplemented
54#1442C2214yescomplemented
55#1442C2214yescomplemented
56#1441C2214yescomplemented
57#1441C2214yescomplemented
58#1441C2214yescomplemented
59#1436C2214yescomplemented
60#1436C2214yescomplemented
61#1434C2212yescomplemented
62#1434C2212yescomplemented
63C22⋊#95C2212yes—
64C22⋊#95C2211yescharacteristic
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2214yes—
70C22⋊#95C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2214yes—
77C26C2311yescharacteristic
78C24×C4—28no—
79C24×C4—24no—
80C24×C4—216no—
81C24×C4—24no—
82C24×C4—28no—
83C24×C4—48no—
84C24×C4—44no—
85C24×C4—48no—
86C24×C4Q818yes—
87C24×C4C2312yes—
88C24×C4C2314yes—
89C24×C4C2312yes—
90C24×C4C2312yes—
91C24×C4C2312yes—
92C24×C4C2314yes—
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2312yes—
115#400—28no—
116#398—28no—
117#387—28no—
118#387—28no—
119#385—28no—
120#385—216no—
121#385—48no—
122#385—44no—
123#380—24no—
124#380—28no—
125#380—216no—
126#380—24no—
127#380—48no—
128#380—44no—
129#380—48no—
130#379—24no—
131#379—28no—
132#379—24no—
133C4×C22⋊C4—24no—
134C4×C22⋊C4—24no—
135C4×C22⋊C4—48no—
136C2×#95—28no—
137C2×#95—24no—
138C2×#95—216no—
139C2×#95—28no—
140C2×#95—28no—
141C2×#95D418yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2311yescharacteristic
148C2×#95C2311yescharacteristic
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C25—22no—
173C25—28no—
174C25—22no—
175C25—28no—
176C25—22no—
177C25—22no—
178C25C2×Q814yes—
179C25#4114yes—
180C25#4114yes—
181C25C2411yescharacteristic
182C25C2412yes—
183C23×C4—264no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—28no—
188C23×C4—28no—
189C23×C4—22no—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4—24no—
193C23×C4—28no—
194C23×C4—232no—
195C23×C4—28no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—24no—
206C23×C4—28no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—432no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—432no—
221C23×C4—464no—
222C23×C4—432no—
223C23×C4—432no—
224C23×C4—432no—
225C23×C4—464no—
226C23×C4C2×D418yes—
227C23×C4C2×D414yes—
228C23×C4C2×Q818yes—
229C23×C4C2×Q818yes—
230C23×C4C2×Q814yes—
231C23×C4C2×Q814yes—
232C23×C4#41116yes—
233C23×C4#4114yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2414yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2412yes—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—44no—
298C2×C42—44no—
299#95—28no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—24no—
304#95—22no—
305#95—22no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95C2×D418yes—
327#95C2×D414yes—
328#95C2×D414yes—
329C24—216no—
330C24—216no—
331C24—232no—
332C24—216no—
333C24—216no—
334C24—232no—
335C24—28no—
336C24—216no—
337C24—216no—
338C24—28no—
339C24—28no—
340C24—28no—
341C24—232no—
342C24—28no—
343C24—232no—
344C24—28no—
345C24—22no—
346C24—22no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—22no—
352C24—22no—
353C24#125116yes—
354C24C22×D412yes—
355C24C22×Q812yes—
356C24C22×Q814yes—
357C24C2×#4112yes—
358C24C2×#4112yes—
359C24C2×#4112yes—
360C24C2×#4114yes—
361C24C2×#4114yes—
362C24C2×#4118yes—
363C24H(5,𝔽2)18yes—
364C24H(5,𝔽2)12yes—
365C242−1+412yes—
366C24C2511yescharacteristic
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—232no—
373C22×C4—216no—
374C22×C4—28no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—232no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—216no—
394C22×C4—216no—
395C22×C4—216no—
396C22×C4—232no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—4128no—
459C22×C4—4256no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—416no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—416no—
466C22×C4C22⋊Q8116yes—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D414yes—
471C22×C4C22×Q814yes—
472C22×C4C22×Q812yes—
473C22×C4C22×Q812yes—
474C22×C4C2×#4118yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4118yes—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C4⋊C4—88no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—88no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—48no—
489C22⋊C4—416no—
490C22⋊C4—44no—
491C22⋊C4—416no—
492C22⋊C4—416no—
493C22⋊C4—416no—
494C22⋊C4—44no—
495C22⋊C4—48no—
496C42—84no—
497C42—84no—
498C23—264no—
499C23—232no—
500C23—232no—
501C23—264no—
502C23—264no—
503C23—264no—
504C23—264no—
505C23—264no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—28no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—232no—
519C23—28no—
520C23—28no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—24no—
525C23—24no—
526C23—28no—
527C23—28no—
528C23—24no—
529C23—28no—
530C23—28no—
531C23—28no—
532C23—28no—
533C23—28no—
534C23—28no—
535C23C2×C22⋊Q814yes—
536C23C2×#12518yes—
537C23C2×#12514yes—
538C23#53318yes—
539C23#54318yes—
540C23#54414yes—
541C23#54614yes—
542C23#54814yes—
543C23#551116yes—
544C23#55214yes—
545C23#553116yes—
546C23#55314yes—
547C23C23×D411yescharacteristic
548C23C23×Q811yescharacteristic
549C23C22×#4111yescharacteristic
550C23C22×#4114yes—
551C23C22×#4111yescharacteristic
552C23C22×#4111yescharacteristic
553C23C2×H(5,𝔽2)14yes—
554C23C2×H(5,𝔽2)11yescharacteristic
555C23C2×2−1+411yescharacteristic
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—416no—
576C2×C4—432no—
577C2×C4—464no—
578C2×C4—432no—
579C2×C4—432no—
580C2×C4—432no—
581C2×C4—464no—
582C2×C4—416no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
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×C4C2×C22⋊Q814yes—
629C2×C4C2×C22⋊Q814yes—
630C2×C4#52518yes—
631C22—2128no—
632C22—24no—
633C22—28no—
634C22—28no—
635C22—28no—
636C22—28no—
637C22—216no—
638C22—216no—
639C22—28no—
640C22—28no—
641C22—24no—
642C22—28no—
643C22—28no—
644C22—216no—
645C22—28no—
646C22—28no—
647C22—28no—
648C22—216no—
649C22—28no—
650C22#244218yes—
651C22#2569116yescomplemented
652C22C22×C22⋊Q811yescharacteristic
653C22C2×#52512yes—
654C22C22×#12511yescharacteristic
655C22C2×#53312yes—
656C22#344412yes—
657C22C2×#54312yes—
658C22C2×#54411yescharacteristic
659C22C2×#54611yescharacteristic
660C22C2×#54811yescharacteristic
661C22C2×#55114yes—
662C22C2×#55211yescharacteristic
663C22C2×#55314yes—
664C22C2×#55311yescharacteristic
665C22#348412yes—
666C22#349014yes—
667C22#349012yes—
668C22#349214yes—
669C4—28no—
670C4—24no—
671C4—24no—
672C4—44no—
673C4—48no—
674C4—44no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—84no—
682C2—216no—
683C2—28no—
684C2C2×#244211yescharacteristic
685C2C2×#256912yes—
686C2#3498211yescharacteristic
687C2#3505212yes—
688C2#3615812yes—
689C2#3620612yes—
690C2#3658112yes—
691C2#3660712yes—
692C2#3662511yescharacteristic
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×#2569C2semidirect
3. #15015C2semidirect
4. #14920C2semidirect
5. #14767C2semidirect
6. #14766C2semidirect
10. #14149C2semidirect
11. #14028C2semidirect
12. #13980C2semidirect
13. #13966C2semidirect
14. #13908C2semidirect
19. C2×#400C22semidirect
20. C2×#398C22semidirect
21. C2×#387C22semidirect
22. C2×#387C22semidirect
47. #1445C22semidirect
48. #1445C22semidirect
49. #1444C22semidirect
50. #1443C22semidirect
51. #1443C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
59. #1436C22semidirect
60. #1436C22semidirect
61. #1434C22semidirect
62. #1434C22semidirect
651. C22#2569semidirect

Maximal subgroups

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

subgroupindex
C2×#25692
C23×#952
#150152
#149202
#147672
#147662
#146932
#146802
#141492
#140282
#139802
#139662
#139082
C4×#3792