Finite Group Databasebeta

← #4389626

Subgroups — #4389626

Order 512.

3572 conjugacy classes of subgroups.

The subgroup lattice has 764 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2431C211yesmaximal, characteristic
2C2×#2616C211yesmaximal, characteristic, complemented
3C2×#2639C211yesmaximal, characteristic, complemented
4C2×#2332C211yesmaximal, characteristic, complemented
5#16802C212yesmaximal, complemented
6#16505C212yesmaximal, complemented
7#15764C214yesmaximal, complemented
8#15419C214yesmaximal, complemented
9#15403C214yesmaximal, complemented
10#15280C214yesmaximal, complemented
11#15164C212yesmaximal, complemented
12#15127C212yesmaximal, complemented
13#14659C211yesmaximal, characteristic, complemented
14#13965C211yesmaximal, characteristic, complemented
15#13956C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2212yes—
21C22×#121C2211yescharacteristic
22#2639—28no—
23#2616—28no—
24#2431—28no—
25C2×#400C2212yes—
26C2×#397C2212yes—
27C2×#396C2214yes—
28C2×#395C2212yes—
29C2×#394C2212yes—
30C2×#394C2212yes—
31C2×#394C2214yes—
32C2×#394C2214yes—
33C2×#393C2212yes—
34C2×#392C2214yes—
35#2332—28no—
36C2×#391C2212yes—
37C2×#391C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#390C2214yes—
40C2×#390C2214yes—
41C2×#388C2212yes—
42C2×#388C2212yes—
43C2×#387C2212yes—
44C2×#387C2212yes—
45C2×#386C2214yes—
46C2×#386C2214yes—
47C2×#386C2212yes—
48C2×#386C2212yes—
49C2×#385C2214yes—
50C2×#381C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×#380C2212yes—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55C22×#95C2211yescharacteristic
56#1446C2212yes—
57#1445C2212yes—
58#1444C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2214yes—
70#1438C2212yes—
71#1438C2212yes—
72#1438C2212yes—
73#1438C2212yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77#1434C2214yes—
78C22⋊#95C2214yes—
79C22×#41—24no—
80C22×#41—24no—
81C22×#41—24no—
82C22×#41—44no—
83C22×#41—44no—
84C23×Q8C2311yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×D4C2312yes—
87C23×D4C2311yescharacteristic
88C23×D4C2314yes—
89C23×D4C2312yes—
90C23×D4C2311yescharacteristic
91C24×C4C2312yes—
92C24×C4C2312yes—
93C24×C4C2311yescharacteristic
94C2×#127—24no—
95C2×#127—44no—
96C2×#124—24no—
97C2×#124—44no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8—44no—
102C2×C22⋊Q8—44no—
103C2×C22⋊Q8D418yescomplemented
104C2×#121—24no—
105C2×#121—24no—
106C2×#121—24no—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#400—28no—
137#397—28no—
138#396—48no—
139#395—28no—
140#394—28no—
141#394—28no—
142#394—216no—
143#394—48no—
144#393—28no—
145#392—216no—
146#391—24no—
147#391—28no—
148#390—24no—
149#390—48no—
150#390—48no—
151#388—28no—
152#388—28no—
153#387—28no—
154#387—28no—
155#386—216no—
156#386—216no—
157#386—44no—
158#386—44no—
159#385—48no—
160#381—24no—
161#380—24no—
162#380—28no—
163C4×C4⋊C4—28no—
164C4×C22⋊C4D418yes—
165C2×#95—24no—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C25C2412yes—
187C25C2412yes—
188C25C2411yescharacteristic
189C2×#41—432no—
190C2×#41—416no—
191C2×#41—432no—
192C2×#41—416no—
193C2×#41—416no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—24no—
200C22×Q8—42no—
201C22×Q8—42no—
202C22×Q8—42no—
203C22×Q8C2×D414yes—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—28no—
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—28no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—24no—
221C22×D4—28no—
222C22×D4—28no—
223C22×D4—42no—
224C22×D4—42no—
225C22×D4—42no—
226C22×D4—42no—
227C22×D4—44no—
228C22×D4—42no—
229C22×D4—44no—
230C22×D4—44no—
231C22×D4—44no—
232C22×D4—44no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—24no—
238C23×C4—24no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4—42no—
243C23×C4—44no—
244C23×C4—44no—
245C23×C4—42no—
246C23×C4—42no—
247C23×C4C2×D414yes—
248C23×C4C2×D414yes—
249C23×C4#4114yes—
250C23×C4C2414yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2414yes—
254C23×C4C2412yes—
255C23×C4C2412yes—
256C23×C4C2412yes—
257C23×C4C2412yes—
258C23×C4C2412yes—
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C23×C4C2412yes—
262#127—416no—
263#124—416no—
264C22⋊Q8—232no—
265C22⋊Q8—432no—
266#121—416no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D418yes—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4C2×D418yes—
337C2×C22⋊C4C2×D418yes—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—42no—
343C2×C42—42no—
344C2×C42—44no—
345C2×C42C2×D414yes—
346C2×C42C2×D414yes—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—28no—
352#95—24no—
353#95—24no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—28no—
361#95—28no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95C2×D418yes—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—24no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—24no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—24no—
380C24—24no—
381C24—24no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—24no—
386C24—24no—
387C24—24no—
388C24—24no—
389C24—24no—
390C24—24no—
391C24C22×D412yes—
392C24C22×D412yes—
393C24C2×#4112yes—
394C24H(5,𝔽2)14yes—
395C24C2511yescharacteristic
396#41—4128no—
397C2×Q8—28no—
398C2×Q8—28no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×Q8—48no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×Q8—84no—
407C2×D4—28no—
408C2×D4—216no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—44no—
414C2×D4—416no—
415C2×D4—416no—
416C2×D4—416no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—48no—
422C2×D4—48no—
423C2×D4—416no—
424C2×D4—48no—
425C2×D4—44no—
426C2×D4—44no—
427C2×D4—416no—
428C2×D4—84no—
429C2×D4—84no—
430C2×D4—88no—
431C2×D4—88no—
432C2×D4—88no—
433C2×D4—88no—
434C2×D4—84no—
435C2×D4—84no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—216no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—84no—
504C22×C4—84no—
505C22×C4—88no—
506C22×C4—88no—
507C22×C4#121116yes—
508C22×C4C22×D414yes—
509C22×C4C22×D414yes—
510C22×C4C22×D414yes—
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D414yes—
516C22×C4C2×#4112yes—
517C22×C4C2×#4112yes—
518C22×C4H(5,𝔽2)14yes—
519C22×C4H(5,𝔽2)14yes—
520C22×C42−1+414yes—
521C4⋊C4—28no—
522C4⋊C4—216no—
523C4⋊C4—416no—
524C4⋊C4—48no—
525C4⋊C4—44no—
526C4⋊C4—48no—
527C4⋊C4—88no—
528C4⋊C4—84no—
529C4⋊C4—88no—
530C22⋊C4—216no—
531C22⋊C4—216no—
532C22⋊C4—416no—
533C22⋊C4—416no—
534C22⋊C4—48no—
535C22⋊C4—88no—
536C22⋊C4—88no—
537C22⋊C4—88no—
538C22⋊C4—88no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C22⋊C4—84no—
542C22⋊C4—88no—
543C22⋊C4—88no—
544C22⋊C4—88no—
545C22⋊C4—88no—
546C42—44no—
547C42—44no—
548C42—48no—
549C42—84no—
550C23—24no—
551C23—22no—
552C23—22no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—22no—
557C23—22no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—28no—
565C23—28no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—42no—
571C23—44no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—42no—
589C23—44no—
590C23—42no—
591C23—44no—
592C23—42no—
593C23—44no—
594C23—44no—
595C23—42no—
596C23—42no—
597C23—44no—
598C23—42no—
599C23—42no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23C2×#12114yes—
610C23C23×D411yescharacteristic
611C23C23×D411yescharacteristic
612C23C22×#4111yescharacteristic
613C23C2×H(5,𝔽2)12yes—
614C23C2×H(5,𝔽2)12yes—
615C23C2×H(5,𝔽2)12yes—
616C23C2×2−1+412yes—
617C23#58514yes—
618Q8—416no—
619Q8—88no—
620D4—416no—
621D4—432no—
622D4—832no—
623D4—88no—
624D4—816no—
625C2×C4—24no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—28no—
636C2×C4—28no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—28no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—416no—
649C2×C4—432no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—816no—
693C2×C4C2×#12114yes—
694C2×C4C2×#12114yes—
695C2×C4#53718yes—
696C22—24no—
697C22—24no—
698C22—24no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—48no—
703C22—44no—
704C22—44no—
705C22—44no—
706C22—48no—
707C22—44no—
708C22—48no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—84no—
713C22—84no—
714C22—82no—
715C22—82no—
716C22—84no—
717C22—84no—
718C22—84no—
719C22—82no—
720C22—84no—
721C22—84no—
722C22—84no—
723C22—82no—
724C22C22×#12111yescharacteristic
725C22C2×#53712yes—
726C22#348412yes—
727C22#348411yescharacteristic
728C22#348411yescharacteristic
729C22#348712yes—
730C22#349012yes—
731C22#349811yescharacteristic
732C22#349911yescharacteristic
733C22#350012yes—
734C22#351012yes—
735C22#351112yes—
736C22#353414yes—
737C22#353514yes—
738C22#354614yes—
739C22#355414yes—
740C4—24no—
741C4—24no—
742C4—28no—
743C4—44no—
744C4—44no—
745C4—44no—
746C4—44no—
747C4—82no—
748C4—84no—
749C4—84no—
750C4—82no—
751C4—82no—
752C2—44no—
753C2—84no—
754C2—162no—
755C2#3527311yescharacteristic
756C2#3534811yescharacteristic
757C2#3567711yescharacteristic
758C2#4049212yes—
759C2#4242912yes—
760C2#4636112yes—
761C2#4642512yes—
762C2#4801314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2431C2
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×#2616C2semidirect
3. C2×#2639C2semidirect
4. C2×#2332C2semidirect
5. #16802C2semidirect
6. #16505C2semidirect
7. #15764C2semidirect
8. #15419C2semidirect
9. #15403C2semidirect
10. #15280C2semidirect
11. #15164C2semidirect
12. #15127C2semidirect
13. #14659C2semidirect
14. #13965C2semidirect
15. #13956C2semidirect
103. C2×C22⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24312
C2×#26162
C2×#26392
C2×#23322
#168022
#165052
#157642
#154192
#154032
#152802
#151642
#151272
#146592
#139652
#139562