Finite Group Databasebeta

← #536262

Subgroups — #536262

Order 512.

3568 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#393C211yesmaximal, characteristic
2#14609C212yesmaximal, complemented
3#14596C212yesmaximal, complemented
4#14560C214yesmaximal, complemented
5#14243C214yesmaximal, complemented
6#14232C212yesmaximal, complemented
7#14213C212yesmaximal, complemented
8#13922C212yesmaximal, complemented
9#13874C212yesmaximal, complemented
10#13873C212yesmaximal
11#13869C211yesmaximal, characteristic
12#13868C211yesmaximal, characteristic
13#13799C212yesmaximal, complemented
14C22⋊C42C212yesmaximal
15#13796C212yesmaximal, complemented
16C24×Q8C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C2×#393C4116yescomplemented
22C2×#393C2211yescharacteristic
23C2×#393C2211yescharacteristic
24C2×#393C2212yes—
25C2×#393C2211yescharacteristic
26C2×#393C2211yescharacteristic
27C2×#393C2212yes—
28C2×#391C2214yescomplemented
29C2×#391C2214yescomplemented
30C2×#388C2214yes—
31C2×#381C2212yes—
32C2×#381C2211yescharacteristic
33C2×#380C2212yes—
34C2×#379C2212yes—
35C2×#379C2211yescharacteristic
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2214yes—
40C22×#95C2212yes—
41C22×#95C2211yescharacteristic
42#1446C2214yescomplemented
43#1446C2214yescomplemented
44#1443C2214yescomplemented
45#1442C2214yescomplemented
46#1440C2212yes—
47#1440C2214yes—
48#1440C2212yes—
49#1439C2212yes—
50#1439C2212yes—
51#1439C2214yes—
52#1437C2214yes—
53#1437C2212yes—
54#1437C2214yes—
55#1437C2212yes—
56#1437C2214yes—
57#1436C2214yescomplemented
58#1436C2214yescomplemented
59#1436C2214yescomplemented
60#1434C2214yescomplemented
61#1434C2214yescomplemented
62#1434C2214yescomplemented
63#1434C2214yescomplemented
64#1434C2214yescomplemented
65#1434C2214yescomplemented
66C22⋊#95C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C4×#95C2214yescomplemented
76C26C2311yescharacteristic
77C23×Q8—42no—
78C23×Q8—42no—
79C23×Q8—48no—
80C23×Q8C2×C418yescomplemented
81C23×Q8C2311yescharacteristic
82C23×Q8C2312yes—
83C23×Q8C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×#117—28no—
90C2×#117—42no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C22⋊C4—28no—
94C22×C22⋊C4—48no—
95C22×C22⋊C4—42no—
96C22×C22⋊C4—42no—
97C22×C22⋊C4—42no—
98C22×C22⋊C4C2×C4116yes—
99C22×C22⋊C4C2×C418yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2314yes—
119C22×C42C2314yes—
120C22×C42C2314yes—
121#393C2×C418yes—
122#393C2×C4116yes—
123#393C2×C418yes—
124#393C2×C418yescomplemented
125#393C2×C4116yescomplemented
126#393C2×C418yescomplemented
127#391—48no—
128#391—48no—
129#388—48no—
130#381—28no—
131#381—42no—
132#380—44no—
133#379—28no—
134#379—42no—
135C4×C22⋊C4—28no—
136C4×C22⋊C4—28no—
137C4×C22⋊C4—48no—
138C4×C22⋊C4—44no—
139C2×#95C2×C4116yescomplemented
140C2×#95C2×C418yescomplemented
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C25—24no—
169C25—24no—
170C25—24no—
171C25—22no—
172C25—22no—
173C25—44no—
174C25—42no—
175C25C22×C414yes—
176C25C2411yescharacteristic
177C25C2411yescharacteristic
178C25C2411yescharacteristic
179C22×Q8—28no—
180C22×Q8—28no—
181C22×Q8—416no—
182C22×Q8—41no—
183C22×Q8—42no—
184C22×Q8—44no—
185C22×Q8—41no—
186C22×Q8—42no—
187C22×Q8—41no—
188C22×Q8—41no—
189C22×Q8—44no—
190C22×Q8—432no—
191C22×Q8—432no—
192C22×Q8—432no—
193C22×Q8C22×C414yes—
194C22×Q8C22×C418yes—
195C22×Q8C22×C414yes—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—28no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—42no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—42no—
216C23×C4C22×C414yes—
217C23×C4C22×C418yes—
218C23×C4C22×C414yes—
219C23×C4C22×C418yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232#117—416no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
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—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—432no—
256C2×C22⋊C4—416no—
257C2×C22⋊C4—416no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4—41no—
266C2×C22⋊C4—41no—
267C2×C22⋊C4—41no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4C22×C418yes—
277C2×C22⋊C4C22×C418yes—
278C2×C22⋊C4C22×C418yes—
279C2×C22⋊C4C22×C414yes—
280C2×C22⋊C4C22×C414yes—
281C2×C22⋊C4C22×C418yes—
282C2×C22⋊C4C22×C414yes—
283C2×C22⋊C4C22×C414yes—
284C2×C22⋊C4#4118yes—
285C2×C22⋊C4#4118yes—
286C2×C42—28no—
287C2×C42—28no—
288C2×C42—28no—
289C2×C42—28no—
290C2×C42—42no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—44no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—42no—
309#95—42no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95C22×C418yes—
318#95C22×C414yes—
319#95C22×C418yes—
320#95C22×C414yes—
321#95C22×C418yes—
322#95C22×C418yes—
323#95C22×C418yes—
324C24—24no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—21no—
331C24—21no—
332C24—22no—
333C24—28no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—21no—
338C24—21no—
339C24—22no—
340C24—22no—
341C24—216no—
342C24—28no—
343C24—48no—
344C24—44no—
345C24—416no—
346C24—416no—
347C24—48no—
348C24—48no—
349C24—44no—
350C24—44no—
351C24—48no—
352C24—88no—
353C24—88no—
354C24—84no—
355C24—84no—
356C24C23×C412yes—
357C24C23×C412yes—
358C24C23×C412yes—
359C24C2×#4114yes—
360C24C2×#4114yes—
361C24H(5,𝔽2)12yes—
362C24H(5,𝔽2)12yes—
363C242−1+414yes—
364C24C2511yescharacteristic
365C2×Q8—216no—
366C2×Q8—216no—
367C2×Q8—44no—
368C2×Q8—48no—
369C2×Q8—48no—
370C2×Q8—44no—
371C2×Q8—48no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—416no—
375C2×Q8—416no—
376C2×Q8—4128no—
377C2×Q8—48no—
378C2×Q8—48no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—216no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—216no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—21no—
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—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—216no—
425C22×C4—216no—
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—24no—
439C22×C4—24no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—416no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—48no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4—88no—
456C22×C4—84no—
457C22×C4—84no—
458C22×C4C23×C412yes—
459C22×C4C23×C412yes—
460C22×C4C23×C414yes—
461C22×C4C23×C414yes—
462C22×C4C23×C414yes—
463C22×C4C23×C412yes—
464C22×C4C23×C412yes—
465C22×C4C23×C414yes—
466C22×C4C2×#4118yes—
467C22×C4C2×#4118yes—
468C22×C4H(5,𝔽2)18yes—
469C4⋊C4—48no—
470C4⋊C4—88no—
471C22⋊C4—216no—
472C22⋊C4—216no—
473C22⋊C4—216no—
474C22⋊C4—216no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—48no—
482C22⋊C4—416no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—88no—
486C22⋊C4—84no—
487C22⋊C4—84no—
488C42—48no—
489C42—88no—
490C42—88no—
491C42—88no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—28no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—28no—
508C23—28no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—416no—
517C23—432no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—44no—
522C23—44no—
523C23—42no—
524C23—42no—
525C23—416no—
526C23—41no—
527C23—41no—
528C23—42no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—88no—
538C23—816no—
539C23—816no—
540C23—832no—
541C23—816no—
542C23C4×#4118yes—
543C23#51814yes—
544C23#51914yes—
545C23#55114yes—
546C23#55114yes—
547C23#55518yes—
548C23C24×C411yescharacteristic
549C23C22×#4112yes—
550C23C22×#4112yes—
551C23C2×H(5,𝔽2)11yescharacteristic
552C23C2×H(5,𝔽2)11yescharacteristic
553C23C2×H(5,𝔽2)14yes—
554C23C2×2−1+412yes—
555C23#58512yes—
556Q8—416no—
557Q8—432no—
558Q8—416no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—432no—
587C2×C4—416no—
588C2×C4—416no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
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—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—816no—
628C2×C4—816no—
629C2×C4#51818yes—
630C22—24no—
631C22—24no—
632C22—28no—
633C22—216no—
634C22—28no—
635C22—28no—
636C22—28no—
637C22—24no—
638C22—28no—
639C22—42no—
640C22—42no—
641C22—44no—
642C22—48no—
643C22—44no—
644C22—44no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—44no—
649C22—44no—
650C22—832no—
651C22—82no—
652C22—84no—
653C22—82no—
654C22#237218yes—
655C22C2×C4×#4112yes—
656C22C2×#51811yescharacteristic
657C22C2×#51812yes—
658C22C2×#51911yescharacteristic
659C22#342811yescharacteristic
660C22C2×#55111yescharacteristic
661C22C2×#55111yescharacteristic
662C22C2×#55512yes—
663C22#351414yes—
664C22#351414yes—
665C22#352014yes—
666C22#352014yes—
667C22#353212yes—
668C22#353212yes—
669C22#353512yes—
670C22#353512yes—
671C4—28no—
672C4—44no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—48no—
678C4—48no—
679C4—44no—
680C4—44no—
681C4—84no—
682C4—84no—
683C2—28no—
684C2—44no—
685C2—82no—
686C2C2×#237211yescharacteristic
687C2#3424012yes—
688C2#3424512yes—
689C2#3435611yescharacteristic
690C2#3435911yescharacteristic
691C2#4364312yes—
692C2#4364514yes—
693C2#4366712yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #14609C2semidirect
3. #14596C2semidirect
4. #14560C2semidirect
5. #14243C2semidirect
6. #14232C2semidirect
7. #14213C2semidirect
8. #13922C2semidirect
9. #13874C2semidirect
13. #13799C2semidirect
15. #13796C2semidirect
21. C2×#393C4semidirect
28. C2×#391C22semidirect
29. C2×#391C22semidirect
42. #1446C22semidirect
43. #1446C22semidirect
44. #1443C22semidirect
45. #1442C22semidirect
57. #1436C22semidirect
58. #1436C22semidirect
59. #1436C22semidirect
60. #1434C22semidirect
61. #1434C22semidirect
62. #1434C22semidirect
63. #1434C22semidirect
64. #1434C22semidirect
65. #1434C22semidirect
75. C4×#95C22semidirect
80. C23×Q8C2×C4semidirect
124. #393C2×C4semidirect
125. #393C2×C4semidirect
126. #393C2×C4semidirect
139. C2×#95C2×C4semidirect
140. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3932
#146092
#145962
#145602
#142432
#142322
#142132
#139222
#138742
#138732
#138692
#138682
#137992
C22⋊C422
#137962