Finite Group Databasebeta

← #4381116

Subgroups — #4381116

Order 512.

3634 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2561C211yesmaximal, characteristic, complemented
2C2×#2404C211yesmaximal, characteristic
3C2×#2616C211yesmaximal, characteristic, complemented
4C2×#2312C211yesmaximal, characteristic, complemented
5#16255C214yesmaximal, complemented
6#15687C214yesmaximal, complemented
7#15634C214yesmaximal, complemented
8#15632C214yesmaximal, complemented
9#15613C214yesmaximal, complemented
10#14838C214yesmaximal, complemented
11#14660C211yesmaximal, characteristic, complemented
12#14297C211yesmaximal, characteristic, complemented
13#13937C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×C22⋊Q8C2212yescomplemented
17C22×#120C2212yes—
18C23×C22⋊C4C2211yescharacteristic
19#2616—28no—
20#2561—28no—
21#2404—28no—
22C2×#397C2212yes—
23C2×#397C2214yes—
24C2×#395C2212yescomplemented
25C2×#394C2214yes—
26C2×#394C2214yes—
27C2×#394C2212yescomplemented
28C2×#393C2212yes—
29C2×#393C2212yes—
30C2×#392C2212yes—
31#2312—28no—
32C2×#391C2212yescomplemented
33C2×#390C2212yescomplemented
34C2×#388C2214yes—
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#386C2214yes—
38C2×#386C2212yes—
39C2×#381C2212yes—
40C2×#381C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2212yescomplemented
44C2×#379C2212yes—
45C2×#379C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48#1446C2212yescomplemented
49#1445C2212yescomplemented
50#1443C2214yescomplemented
51#1442C2214yescomplemented
52#1441C2214yescomplemented
53#1441C2214yescomplemented
54#1440C2214yes—
55#1440C2214yes—
56#1440C2214yescomplemented
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yescomplemented
61#1438C2214yes—
62#1438C2214yes—
63#1438C2214yes—
64#1438C2214yes—
65#1437C2214yescomplemented
66#1437C2214yescomplemented
67#1436C2214yescomplemented
68C4⋊#95C2214yescomplemented
69#1434C2214yescomplemented
70C22⋊#95C2214yescomplemented
71C26C2311yescharacteristic
72C22×#41—24no—
73C22×#41—24no—
74C22×#41—24no—
75C22×#41—44no—
76C22×#41—44no—
77C23×Q8C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C23×D4C2312yes—
81C23×D4C2314yes—
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—44no—
90C2×C22⋊Q8—44no—
91C2×#120—28no—
92C2×#120—44no—
93C2×#120—44no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4D418yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C42C2314yes—
121C22×C42C2312yes—
122#397—28no—
123#397—48no—
124#395—28no—
125#394—216no—
126#394—28no—
127#394—48no—
128#393—28no—
129#393—28no—
130#392—28no—
131#391—44no—
132#390—44no—
133#388—216no—
134#388—216no—
135#388—216no—
136#386—28no—
137#386—216no—
138#381—28no—
139#381—24no—
140#380—24no—
141#380—28no—
142#380D418yescomplemented
143#379—28no—
144#379—28no—
145C4×C22⋊C4—24no—
146C4×C22⋊C4—28no—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C25—22no—
167C25—22no—
168C25—42no—
169C25—42no—
170C25—42no—
171C25—42no—
172C25—44no—
173C25C2×D414yes—
174C25C2412yes—
175C25C2412yes—
176C25C2411yescharacteristic
177C2×#41—432no—
178C2×#41—432no—
179C2×#41—416no—
180C2×#41—416no—
181C2×#41—416no—
182C22×Q8—22no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—28no—
198C22×D4—28no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—24no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—44no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—44no—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4#4114yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2414yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C22⋊Q8—432no—
238C22⋊Q8—416no—
239#120—432no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C2×D418yes—
258C2×C22⋊C4—232no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
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—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4C2×D418yes—
307C2×C22⋊C4C2×D418yes—
308C2×C22⋊C4C2×D418yes—
309C2×C42—28no—
310C2×C42—24no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42—44no—
314C2×C42—44no—
315#95—28no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—28no—
324#95—28no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95C2×D418yes—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—22no—
358C24—22no—
359C24—28no—
360C24—48no—
361C24—48no—
362C24—48no—
363C24—44no—
364C24—44no—
365C24—48no—
366C24—84no—
367C24—82no—
368C24—82no—
369C24—84no—
370C24—84no—
371C24—88no—
372C24—88no—
373C24—88no—
374C24—88no—
375C24C22×D414yes—
376C24C22×D412yes—
377C24C22×D412yes—
378C24C2×#4112yes—
379C24H(5,𝔽2)14yes—
380C24C2511yescharacteristic
381#41—4128no—
382C2×Q8—28no—
383C2×Q8—48no—
384C2×Q8—48no—
385C2×Q8—44no—
386C2×Q8—48no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—84no—
390C2×Q8—82no—
391C2×Q8—82no—
392C2×D4—216no—
393C2×D4—28no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—44no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—416no—
404C2×D4—416no—
405C2×D4—44no—
406C2×D4—88no—
407C2×D4—88no—
408C2×D4—88no—
409C2×D4—88no—
410C2×D4—88no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—216no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4#121116yes—
478C22×C4C22×D414yes—
479C22×C4C22×D418yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)14yes—
487C4⋊C4—48no—
488C4⋊C4—416no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—84no—
492C4⋊C4—88no—
493C22⋊C4—216no—
494C22⋊C4—216no—
495C22⋊C4—48no—
496C22⋊C4—416no—
497C22⋊C4—416no—
498C22⋊C4—416no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C42—84no—
511C42—88no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—22no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—28no—
528C23—24no—
529C23—28no—
530C23—24no—
531C23—28no—
532C23—28no—
533C23—416no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—42no—
539C23—44no—
540C23—44no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—42no—
559C23—42no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—88no—
575C23—88no—
576C23—832no—
577C23—816no—
578C23—816no—
579C23—816no—
580C23—816no—
581C23C2×#12114yes—
582C23C23×D411yescharacteristic
583C23C23×D411yescharacteristic
584C23C22×#4111yescharacteristic
585C23C2×H(5,𝔽2)12yes—
586C23C2×H(5,𝔽2)12yes—
587C23#58512yes—
588C23#58512yes—
589C23#58514yes—
590Q8—416no—
591Q8—88no—
592D4—416no—
593D4—432no—
594D4—832no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—416no—
612C2×C4—432no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—88no—
656C2×C4C2×#12114yes—
657C2×C4C2×#12114yes—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—48no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—48no—
673C22—48no—
674C22—48no—
675C22—48no—
676C22—832no—
677C22—82no—
678C22—82no—
679C22—82no—
680C22—84no—
681C22—82no—
682C22—84no—
683C22—84no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—84no—
689C22C22×#12111yescharacteristic
690C22#344912yes—
691C22#348412yes—
692C22#348412yes—
693C22#348512yes—
694C22#348511yescharacteristic
695C22#348711yescharacteristic
696C22#349011yescharacteristic
697C22#349011yescharacteristic
698C22#349312yes—
699C22#350012yes—
700C22#351012yes—
701C22#354514yes—
702C22#355114yes—
703C22#356014yes—
704C22#356214yes—
705C4—24no—
706C4—24no—
707C4—44no—
708C4—44no—
709C4—48no—
710C4—82no—
711C4—82no—
712C4—84no—
713C4—84no—
714C4—84no—
715C2—44no—
716C2—84no—
717C2—84no—
718C2#3515011yescharacteristic
719C2#3534811yescharacteristic
720C2#3561711yescharacteristic
721C2#4141514yes—
722C2#4642712yes—
723C2#4643012yes—
724C2#4959612yes—
725C2#4959712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2404C2
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×#2561C2semidirect
3. C2×#2616C2semidirect
4. C2×#2312C2semidirect
5. #16255C2semidirect
6. #15687C2semidirect
7. #15634C2semidirect
8. #15632C2semidirect
9. #15613C2semidirect
10. #14838C2semidirect
11. #14660C2semidirect
12. #14297C2semidirect
13. #13937C2semidirect
16. C22×C22⋊Q8C22semidirect
24. C2×#395C22semidirect
27. C2×#394C22semidirect
32. C2×#391C22semidirect
33. C2×#390C22semidirect
43. C2×#380C22semidirect
48. #1446C22semidirect
49. #1445C22semidirect
50. #1443C22semidirect
51. #1442C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
56. #1440C22semidirect
60. #1439C22semidirect
65. #1437C22semidirect
66. #1437C22semidirect
67. #1436C22semidirect
68. C4⋊#95C22semidirect
69. #1434C22semidirect
70. C22⋊#95C22semidirect
142. #380D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25612
C2×#24042
C2×#26162
C2×#23122
#162552
#156872
#156342
#156322
#156132
#148382
#146602
#142972
#139372