Finite Group Databasebeta

← #759340

Subgroups — #759340

Order 512.

4362 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2442C212yesmaximal, complemented
2C22×#380C212yesmaximal
3#16304C214yesmaximal, complemented
4#15223C211yesmaximal, characteristic, complemented
5#15220C212yesmaximal, complemented
6#15216C211yesmaximal, characteristic, complemented
7#14767C214yesmaximal, complemented
8#14766C214yesmaximal, complemented
9#14130C212yesmaximal, complemented
10#13907C212yesmaximal, complemented
11#13881C212yesmaximal, complemented
12#13877C214yesmaximal
13C4×#379C211yesmaximal, characteristic
14C25×C4C2211yescharacteristic
15C22×#125C2211yescharacteristic, complemented
16C23×C4⋊C4C2212yes—
17C23×C4⋊C4C2212yes—
18C23×C22⋊C4C2214yes—
19#2442—216no—
20C2×#400C2214yescomplemented
21C2×#398C2214yescomplemented
22C2×#389C2212yescomplemented
23C2×#389C2212yescomplemented
24C2×#389C2212yescomplemented
25C2×#385C2214yes—
26C2×#385C2214yes—
27C2×#382C2212yescomplemented
28C2×#382C2211yescharacteristic, complemented
29C2×#380—216no—
30C2×#380C2214yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2214yes—
35C2×#380C2214yes—
36C2×#380C2214yes—
37C2×#380C2212yes—
38C2×#379C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C22×#95C2214yes—
44#1445C2214yescomplemented
45#1444C2214yescomplemented
46#1443C2214yescomplemented
47#1442C2214yescomplemented
48#1442C2214yescomplemented
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1441C2214yescomplemented
52#1441C2214yescomplemented
53#1441C2214yescomplemented
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58C4⋊#95C2211yescharacteristic, complemented
59C4⋊#95C2212yescomplemented
60C4⋊#95C2212yescomplemented
61C4⋊#95C2211yescharacteristic, complemented
62C22⋊#95C2214yes—
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C26C2311yescharacteristic
70C24×C4—28no—
71C24×C4—28no—
72C24×C4—42no—
73C24×C4—42no—
74C24×C4Q818yes—
75C24×C4C2311yescharacteristic
76C24×C4C2314yes—
77C24×C4C2314yes—
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C2×#125—28no—
82C2×#125—42no—
83C22×C4⋊C4—28no—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C22⋊C4—216no—
100C22×C22⋊C4—216no—
101C22×C22⋊C4—48no—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115#400—216no—
116#398—216no—
117#389—28no—
118#389—28no—
119#389—44no—
120#385—216no—
121#385—48no—
122#382—28no—
123#382—42no—
124#380—28no—
125#380—28no—
126#380—28no—
127#380—216no—
128#380—216no—
129#380—216no—
130#380—48no—
131#380D4116yes—
132#379—28no—
133C4×C22⋊C4—28no—
134C4×C22⋊C4—28no—
135C4×C22⋊C4—42no—
136C4×C22⋊C4—42no—
137C2×#95—216no—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C25—24no—
162C25—24no—
163C25—24no—
164C25—24no—
165C25—41no—
166C25—41no—
167C25—42no—
168C25C2×Q814yes—
169C25#4118yes—
170C25C2411yescharacteristic
171C25C2412yes—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—264no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—28no—
188C23×C4—28no—
189C23×C4—28no—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4—28no—
193C23×C4—28no—
194C23×C4—28no—
195C23×C4—416no—
196C23×C4—41no—
197C23×C4—41no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—416no—
201C23×C4—432no—
202C23×C4—416no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4C2×D418yes—
210C23×C4C2×Q814yes—
211C23×C4C2×Q818yes—
212C23×C4C2×Q814yes—
213C23×C4C2×Q818yes—
214C23×C4#4118yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226#125—416no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—432no—
246C2×C4⋊C4—432no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C2×D418yes—
258C2×C4⋊C4C2×D418yes—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—464no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4C2×D4116yes—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—24no—
307#95—24no—
308#95—44no—
309#95—44no—
310#95—41no—
311#95—42no—
312#95—41no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95C2×D4116yes—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—216no—
329C24—216no—
330C24—216no—
331C24—216no—
332C24—216no—
333C24—216no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—21no—
338C24—21no—
339C24—22no—
340C24—22no—
341C24—44no—
342C24—44no—
343C24—48no—
344C24—48no—
345C24—44no—
346C24—48no—
347C24—44no—
348C24—48no—
349C24—48no—
350C24—48no—
351C24—48no—
352C24—48no—
353C24—44no—
354C24—44no—
355C24#125116yescomplemented
356C24C22×D414yes—
357C24C22×Q812yes—
358C24C22×Q814yes—
359C24C2×#4114yes—
360C24C2×#4114yes—
361C24C2×#4114yes—
362C24C2×#4114yes—
363C24H(5,𝔽2)14yes—
364C242−1+414yes—
365C24C2511yescharacteristic
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—216no—
373C22×C4—216no—
374C22×C4—216no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—216no—
394C22×C4—216no—
395C22×C4—216no—
396C22×C4—28no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—44no—
444C22×C4—4128no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—44no—
458C22×C4—416no—
459C22×C4—416no—
460C22×C4—88no—
461C22×C4—88no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4—88no—
465C22×C4—88no—
466C22×C4C22⋊Q8132yes—
467C22×C4C22×D418yes—
468C22×C4C22×D418yes—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C22×C4C22×Q812yes—
472C22×C4C22×Q814yes—
473C22×C4C22×Q812yes—
474C22×C4C2×#4114yes—
475C22×C4C2×#4114yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C4H(5,𝔽2)12yes—
478C4⋊C4—216no—
479C4⋊C4—216no—
480C4⋊C4—416no—
481C4⋊C4—416no—
482C4⋊C4—416no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—88no—
486C4⋊C4—88no—
487C4⋊C4—88no—
488C22⋊C4—232no—
489C22⋊C4—416no—
490C22⋊C4—416no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—416no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C42—44no—
499C42—48no—
500C42—44no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—22no—
531C23—24no—
532C23—22no—
533C23—24no—
534C23—24no—
535C23—264no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—416no—
541C23—416no—
542C23—432no—
543C23—416no—
544C23—432no—
545C23—432no—
546C23—432no—
547C23—432no—
548C23—41no—
549C23—42no—
550C23—41no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—41no—
557C23—41no—
558C23C2×C22⋊Q818yes—
559C23C2×#12514yes—
560C23C2×#12518yes—
561C23#54418yes—
562C23#54618yes—
563C23#54818yes—
564C23#55218yes—
565C23#55318yes—
566C23C23×D412yes—
567C23C23×Q811yescharacteristic
568C23C22×#4112yes—
569C23C22×#4112yes—
570C23C2×H(5,𝔽2)11yescharacteristic
571C23C2×H(5,𝔽2)11yescharacteristic
572C23C2×H(5,𝔽2)12yes—
573C23C2×2−1+412yes—
574C23#58512yes—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—264no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—416no—
591C2×C4—432no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—416no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—832no—
647C2×C4—832no—
648C2×C4C2×C22⋊Q818yes—
649C2×C4C2×C22⋊Q818yes—
650C2×C4#54314yes—
651C2×C4D4214yes—
652C2×C4#54618yes—
653C2×C4#55114yes—
654C22—28no—
655C22—28no—
656C22—28no—
657C22—28no—
658C22—28no—
659C22—28no—
660C22—28no—
661C22—28no—
662C22—28no—
663C22—28no—
664C22—28no—
665C22—28no—
666C22—28no—
667C22—28no—
668C22—28no—
669C22—24no—
670C22—464no—
671C22—44no—
672C22—42no—
673C22—42no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22#2442116yescomplemented
679C22C22×C22⋊Q812yes—
680C22C22×#12511yescharacteristic
681C22C2×#54311yescharacteristic
682C22C2×#54412yes—
683C22#345811yescharacteristic
684C22C2×D4211yescharacteristic
685C22C2×#54612yes—
686C22C2×#54612yes—
687C22C2×#54812yes—
688C22C2×#55111yescharacteristic
689C22C2×#55212yes—
690C22C2×#55312yes—
691C22#348812yes—
692C22#348912yes—
693C22#349412yes—
694C22#349512yes—
695C22#350512yes—
696C22#350512yes—
697C22#353612yes—
698C22#353711yescharacteristic
699C22#353811yescharacteristic
700C4—28no—
701C4—42no—
702C4—48no—
703C4—48no—
704C4—82no—
705C4—82no—
706C4—84no—
707C4—82no—
708C4—84no—
709C4—82no—
710C4#243718yes—
711C2—216no—
712C2—44no—
713C2C2×#243711yescharacteristic
714C2C2×#244212yes—
715C2#3502011yescharacteristic
716C2#3525312yes—
717C2#3747711yescharacteristic
718C2#5899612yes—
719C2#5901312yes—
720C2#5901412yes—
721C2#5919712yes—
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×#2442C2semidirect
3. #16304C2semidirect
4. #15223C2semidirect
5. #15220C2semidirect
6. #15216C2semidirect
7. #14767C2semidirect
8. #14766C2semidirect
9. #14130C2semidirect
10. #13907C2semidirect
11. #13881C2semidirect
15. C22×#125C22semidirect
20. C2×#400C22semidirect
21. C2×#398C22semidirect
22. C2×#389C22semidirect
23. C2×#389C22semidirect
24. C2×#389C22semidirect
27. C2×#382C22semidirect
28. C2×#382C22semidirect
44. #1445C22semidirect
45. #1444C22semidirect
46. #1443C22semidirect
47. #1442C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1441C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
58. C4⋊#95C22semidirect
59. C4⋊#95C22semidirect
60. C4⋊#95C22semidirect
61. C4⋊#95C22semidirect
355. C24#125semidirect
678. C22#2442semidirect

Maximal subgroups

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

subgroupindex
C2×#24422
C22×#3802
#163042
#152232
#152202
#152162
#147672
#147662
#141302
#139072
#138812
#138772
C4×#3792