Finite Group Databasebeta

← #765669

Subgroups — #765669

Order 512.

3094 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#399C211yesmaximal, characteristic
2C2×#2313C211yesmaximal, characteristic, complemented
3#15653C214yesmaximal, complemented
4#15652C214yesmaximal, complemented
5#15650C214yesmaximal, complemented
6#15649C214yesmaximal, complemented
7#14822C211yesmaximal, characteristic, complemented
8#14821C211yesmaximal, characteristic, complemented
9#14820C211yesmaximal, characteristic
10#14734C211yesmaximal, characteristic
11#14039C212yesmaximal, complemented
12#13991C212yesmaximal
13#13962C212yesmaximal, complemented
14#13868C212yesmaximal
15#13795C211yesmaximal, characteristic, complemented
16C22×#117C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C2×#400C2212yescomplemented
24C2×#399—28no—
25C2×#399C2214yes—
26C2×#399C2214yes—
27C2×#399C2212yes—
28C2×#398C2212yescomplemented
29C2×#397C2212yes—
30#2313—28no—
31C2×#388C2214yes—
32C2×#387C2212yescomplemented
33C2×#387C2212yescomplemented
34C2×#387C2212yescomplemented
35C2×#387C2212yescomplemented
36C2×#387C2214yescomplemented
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#381C2212yes—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#379C2212yes—
43C2×#379C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C22×#95C2211yescharacteristic
46#1442C2214yescomplemented
47#1442C2214yescomplemented
48#1441C2212yescomplemented
49#1441C2214yescomplemented
50#1441C2214yescomplemented
51#1441C2214yescomplemented
52#1441C2214yescomplemented
53#1441C2212yescomplemented
54#1440C2214yes—
55#1440C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2212yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2212yes—
62#1437C2214yes—
63#1437C2214yes—
64#1437C2214yes—
65#1436C2214yescomplemented
66C4⋊#95C2214yescomplemented
67C4⋊#95C2214yescomplemented
68C4⋊#95C2214yescomplemented
69#1434C2214yescomplemented
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2212yes—
76C22⋊#95C2211yescharacteristic
77C26C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×#117—28no—
85C2×#117—28no—
86C2×#117—44no—
87C22×C4⋊C4—28no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4—44no—
92C22×C4⋊C4D418yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C22⋊C4—24no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—28no—
109C22×C22⋊C4—44no—
110C22×C22⋊C4—44no—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C42C2314yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126#400—44no—
127#399—216no—
128#399—216no—
129#399—44no—
130#398—44no—
131#397—44no—
132#388—48no—
133#387—28no—
134#387—28no—
135#387—28no—
136#387—28no—
137#387—48no—
138#385—28no—
139#385—44no—
140#381—28no—
141#380—24no—
142#380—24no—
143#379—28no—
144#379—44no—
145C4×C22⋊C4—28no—
146C2×#95—24no—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C25—22no—
173C25—24no—
174C25—24no—
175C25—24no—
176C25—22no—
177C25—22no—
178C25—42no—
179C25—42no—
180C25—42no—
181C25C2412yes—
182C25C2411yescharacteristic
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4C2×D414yes—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4#4118yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222#117—432no—
223C2×C4⋊C4—232no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—416no—
249C2×C4⋊C4—416no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4C2×D418yes—
257C2×C4⋊C4C2×D418yes—
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—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—416no—
287C2×C22⋊C4—416no—
288C2×C22⋊C4—416no—
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⋊C4—44no—
296C2×C22⋊C4#4118yes—
297C2×C42—28no—
298C2×C42—28no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42—44no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—28no—
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—42no—
324#95—42no—
325#95—44no—
326#95—42no—
327#95—42no—
328#95—44no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—28no—
334C24—22no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—416no—
345C24—416no—
346C24—48no—
347C24—48no—
348C24—44no—
349C24—44no—
350C24—44no—
351C24—44no—
352C24—44no—
353C24—44no—
354C24—48no—
355C24—48no—
356C24—88no—
357C24—88no—
358C24—84no—
359C24—84no—
360C24C22×D412yes—
361C24C2×#4114yes—
362C24C2×#4114yes—
363C24H(5,𝔽2)12yes—
364C242−1+414yes—
365C242−1+412yes—
366C24C2511yescharacteristic
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—216no—
370C22×C4—216no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—416no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—88no—
452C22×C4—88no—
453C22×C4—88no—
454C22×C4#123116yes—
455C22×C4C22×D414yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D414yes—
459C22×C4C2×#4118yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4114yes—
462C22×C4H(5,𝔽2)14yes—
463C22×C4H(5,𝔽2)14yes—
464C4⋊C4—216no—
465C4⋊C4—216no—
466C4⋊C4—416no—
467C4⋊C4—416no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—416no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4—82no—
475C4⋊C4—88no—
476C22⋊C4—216no—
477C22⋊C4—416no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—416no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—416no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C42—416no—
489C42—84no—
490C42—84no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—28no—
505C23—28no—
506C23—24no—
507C23—24no—
508C23—416no—
509C23—416no—
510C23—416no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—832no—
530C23—832no—
531C23—88no—
532C23—88no—
533C23—88no—
534C23—88no—
535C23C2×#12314yes—
536C23#53214yes—
537C23#53714yes—
538C23#55118yes—
539C23#55218yes—
540C23C23×D411yescharacteristic
541C23C22×#4112yes—
542C23C22×#4112yes—
543C23C2×H(5,𝔽2)11yescharacteristic
544C23C2×H(5,𝔽2)12yes—
545C23C2×H(5,𝔽2)12yes—
546C23C2×2−1+411yescharacteristic
547C23C2×2−1+412yes—
548C23#58512yes—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—232no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—416no—
574C2×C4—416no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
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—44no—
622C2×C4—88no—
623C2×C4—88no—
624C2×C4—88no—
625C2×C4C2×#12318yes—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—24no—
630C22—24no—
631C22—216no—
632C22—24no—
633C22—216no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—48no—
638C22—48no—
639C22—832no—
640C22—82no—
641C22—82no—
642C22—84no—
643C22—82no—
644C22—82no—
645C22—84no—
646C22#249818yes—
647C22C22×#12311yescharacteristic
648C22C2×#53211yescharacteristic
649C22C2×#53711yescharacteristic
650C22C2×#55112yes—
651C22C2×#55212yes—
652C22#348411yescharacteristic
653C22#348411yescharacteristic
654C22#348611yescharacteristic
655C22#348711yescharacteristic
656C22#349514yes—
657C22#350814yes—
658C22#351314yes—
659C22#352014yes—
660C22#353412yes—
661C22#353512yes—
662C22#353612yes—
663C22#353812yes—
664C4—28no—
665C4—44no—
666C4—44no—
667C4—48no—
668C4—44no—
669C4—42no—
670C4—42no—
671C4—82no—
672C4—82no—
673C4—82no—
674C4—82no—
675C4—84no—
676C4—82no—
677C2—28no—
678C2—84no—
679C2C2×#249811yescharacteristic
680C2#3539611yescharacteristic
681C2#3539711yescharacteristic
682C2#3555411yescharacteristic
683C2#3567211yescharacteristic
684C2#3671911yescharacteristic
685C2#3672211yescharacteristic
686C2#4397514yes—
687C2#4402114yes—
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
2. C2×#2313C2semidirect
3. #15653C2semidirect
4. #15652C2semidirect
5. #15650C2semidirect
6. #15649C2semidirect
7. #14822C2semidirect
8. #14821C2semidirect
11. #14039C2semidirect
13. #13962C2semidirect
15. #13795C2semidirect
23. C2×#400C22semidirect
28. C2×#398C22semidirect
32. C2×#387C22semidirect
33. C2×#387C22semidirect
34. C2×#387C22semidirect
35. C2×#387C22semidirect
36. C2×#387C22semidirect
46. #1442C22semidirect
47. #1442C22semidirect
48. #1441C22semidirect
49. #1441C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
65. #1436C22semidirect
66. C4⋊#95C22semidirect
67. C4⋊#95C22semidirect
68. C4⋊#95C22semidirect
69. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3992
C2×#23132
#156532
#156522
#156502
#156492
#148222
#148212
#148202
#147342
#140392
#139912
#139622
#138682
#137952