Finite Group Databasebeta

← #918976

Subgroups — #918976

Order 512.

2944 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2482C211yesmaximal, characteristic
2C22×#399C211yesmaximal, characteristic
3#17168C212yesmaximal, complemented
4#17160C212yesmaximal, complemented
5#16830C212yesmaximal, complemented
6#16824C212yesmaximal, complemented
7#16591C214yesmaximal, complemented
8#15828C214yesmaximal, complemented
9#15073C212yesmaximal, complemented
10#15073C212yesmaximal, complemented
11#14918C212yesmaximal
12#14265C212yesmaximal, complemented
13#13870C212yesmaximal, complemented
14#13869C212yesmaximal
15#13869C211yesmaximal, characteristic
16C22×#117C2211yescharacteristic
17C23×C4⋊C4C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22#2482—28no—
23C2×#399—28no—
24C2×#399C2212yes—
25C2×#399C2211yescharacteristic
26C2×#399C2211yescharacteristic
27C2×#399C2212yes—
28C2×#399C2212yes—
29C2×#399C2212yes—
30C2×#397C2211yescharacteristic
31C2×#397C2211yescharacteristic
32C2×#388C2212yes—
33C2×#388C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#381C2211yescharacteristic
37C2×#381C2212yes—
38C2×#381C2212yes—
39C2×#381C2212yes—
40C2×#381C2212yes—
41C2×#381C2212yes—
42C2×#379C2212yes—
43C2×#379C2211yescharacteristic
44C22×#95C2211yescharacteristic
45#1445C2214yescomplemented
46#1443C2214yescomplemented
47#1442C2214yescomplemented
48#1442C2214yescomplemented
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1442C2214yescomplemented
52#1441C2214yescomplemented
53#1441C2214yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1441C2214yescomplemented
57#1441C2214yescomplemented
58#1441C2214yescomplemented
59#1440C2212yes—
60#1440C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1440C2212yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1436C2214yescomplemented
68#1436C2214yescomplemented
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C2×#117—24no—
84C2×#117—42no—
85C2×#117—42no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—28no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4—42no—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—28no—
103C22×C22⋊C4—42no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4—42no—
106C22×C22⋊C4—42no—
107C22×C22⋊C4—44no—
108C22×C22⋊C4D418yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C42C2312yes—
127#399—28no—
128#399—28no—
129#399—28no—
130#399—28no—
131#399—24no—
132#399—24no—
133#397—24no—
134#397—24no—
135#388—28no—
136#388—28no—
137#385—28no—
138#385—28no—
139#381—24no—
140#381—44no—
141#381—44no—
142#381—44no—
143#381—44no—
144#381—44no—
145#379—28no—
146#379—42no—
147C2×#95—24no—
148C2×#95C2311yescharacteristic
149C2×#95C2311yescharacteristic
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
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×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C25—24no—
175C25—24no—
176C25—22no—
177C25—24no—
178C25—22no—
179C25—42no—
180C25—44no—
181C25C2×D414yes—
182C25C2411yescharacteristic
183C25C2411yescharacteristic
184C25C2411yescharacteristic
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—42no—
205C23×C4—44no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4#4118yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2414yes—
223#117—416no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—432no—
246C2×C4⋊C4—416no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—416no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—432no—
288C2×C22⋊C4—416no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4C2×D414yes—
296C2×C22⋊C4C2×D414yes—
297C2×C22⋊C4C2×D414yes—
298C2×C22⋊C4C2×D414yes—
299C2×C42—24no—
300C2×C42—42no—
301C2×C42—42no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—22no—
306#95—22no—
307#95—24no—
308#95—24no—
309#95—28no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
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#95—44no—
327#95—44no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—22no—
332C24—21no—
333C24—21no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—28no—
338C24—22no—
339C24—21no—
340C24—21no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—28no—
346C24—216no—
347C24—44no—
348C24—48no—
349C24—416no—
350C24—48no—
351C24—48no—
352C24—416no—
353C24—48no—
354C24—44no—
355C24—44no—
356C24—88no—
357C24—84no—
358C24—88no—
359C24—84no—
360C24C22×D412yes—
361C24C22×D412yes—
362C24C22×D412yes—
363C24C2×#4114yes—
364C24H(5,𝔽2)12yes—
365C24H(5,𝔽2)12yes—
366C24H(5,𝔽2)14yes—
367C242−1+414yes—
368C24C2511yescharacteristic
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—216no—
372C22×C4—216no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
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—48no—
435C22×C4—416no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—416no—
441C22×C4—48no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—84no—
453C22×C4—88no—
454C22×C4—84no—
455C22×C4—88no—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C2×#4118yes—
461C4⋊C4—416no—
462C4⋊C4—416no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—84no—
468C4⋊C4—84no—
469C4⋊C4—88no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—416no—
475C22⋊C4—416no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—84no—
484C22⋊C4—84no—
485C22⋊C4—84no—
486C22⋊C4—84no—
487C42—48no—
488C23—22no—
489C23—22no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—24no—
494C23—28no—
495C23—24no—
496C23—24no—
497C23—28no—
498C23—24no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—24no—
505C23—24no—
506C23—28no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—416no—
513C23—432no—
514C23—42no—
515C23—41no—
516C23—44no—
517C23—41no—
518C23—42no—
519C23—44no—
520C23—42no—
521C23—42no—
522C23—416no—
523C23—41no—
524C23—44no—
525C23—44no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—88no—
534C23—816no—
535C23—816no—
536C23—816no—
537C23—832no—
538C23#53414yes—
539C23#53614yes—
540C23#54018yes—
541C23#54618yes—
542C23#55914yes—
543C23#56014yes—
544C23C23×D411yescharacteristic
545C23C22×#4112yes—
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×H(5,𝔽2)12yes—
549C23C2×2−1+412yes—
550C23#58512yes—
551C23#58514yes—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—416no—
564C2×C4—432no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—416no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
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—816no—
622C2×C4—816no—
623C22—28no—
624C22—24no—
625C22—28no—
626C22—24no—
627C22—28no—
628C22—28no—
629C22—216no—
630C22—24no—
631C22—28no—
632C22—44no—
633C22—44no—
634C22—42no—
635C22—42no—
636C22—44no—
637C22—48no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—44no—
642C22—44no—
643C22—832no—
644C22—82no—
645C22—82no—
646C22—84no—
647C22#270918yes—
648C22C2×#53411yescharacteristic
649C22C2×#53611yescharacteristic
650C22#344911yescharacteristic
651C22C2×#54012yes—
652C22C2×#54612yes—
653C22C2×#55911yescharacteristic
654C22#348512yes—
655C22#349114yes—
656C22#349514yes—
657C22C2×#56011yescharacteristic
658C22#353312yes—
659C22#353314yes—
660C22#353312yes—
661C22#353512yes—
662C22#353514yes—
663C22#353512yes—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—48no—
669C4—84no—
670C4—82no—
671C4—82no—
672C4—84no—
673C4—82no—
674C4—82no—
675C4—84no—
676C2—28no—
677C2—44no—
678C2—82no—
679C2#3695511yescharacteristic
680C2#3744711yescharacteristic
681C2#3783512yes—
682C2#3784212yes—
683C2#4142914yes—
684C2#4359812yes—
685C2C2×#270911yescharacteristic
686C2#5274212yes—
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
3. #17168C2semidirect
4. #17160C2semidirect
5. #16830C2semidirect
6. #16824C2semidirect
7. #16591C2semidirect
8. #15828C2semidirect
9. #15073C2semidirect
10. #15073C2semidirect
12. #14265C2semidirect
13. #13870C2semidirect
45. #1445C22semidirect
46. #1443C22semidirect
47. #1442C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
67. #1436C22semidirect
68. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24822
C22×#3992
#171682
#171602
#168302
#168242
#165912
#158282
#150732
#149182
#142652
#138702
#138692