Finite Group Databasebeta

← #647271

Subgroups — #647271

Order 512.

3078 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic, complemented
2C22×#388C211yesmaximal, characteristic
3#17045C214yesmaximal, complemented
4#16982C212yesmaximal, complemented
5#16803C212yesmaximal, complemented
6#15092C212yesmaximal, complemented
7#15077C214yesmaximal, complemented
8#14936C211yesmaximal, characteristic, complemented
9#14934C211yesmaximal, characteristic, complemented
10#14918C212yesmaximal
11#14734C211yesmaximal, characteristic
12#14717C211yesmaximal, characteristic
13#14683C211yesmaximal, characteristic, complemented
14#14305C212yesmaximal, complemented
15#14301C212yesmaximal, complemented
16#13992C212yesmaximal, complemented
17#13991C212yesmaximal
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic, complemented
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C42C2211yescharacteristic
25C2×#399C2212yes—
26C2×#393C2212yes—
27C4×C22⋊Q8—28no—
28C2×#391C2211yescharacteristic, complemented
29C2×#389C2211yescharacteristic, complemented
30C2×#388—28no—
31C2×#388C2212yes—
32C2×#388C2212yes—
33C2×#388C2212yes—
34C2×#388C2212yes—
35C2×#388C2214yes—
36C2×#387C2212yescomplemented
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#383C2212yescomplemented
41C2×#381C2212yes—
42C2×#381C2212yes—
43C2×#380C2211yescharacteristic
44C2×#379C2212yes—
45C2×#379C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic, complemented
47C2×C4×C4⋊C4C2212yescomplemented
48C2×C4×C22⋊C4C2212yes—
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51#1446C2212yescomplemented
52#1445C2214yescomplemented
53#1445C2212yescomplemented
54#1444C2212yescomplemented
55#1444C2214yescomplemented
56#1444C2214yescomplemented
57#1443C2214yescomplemented
58#1442C2212yescomplemented
59#1442C2214yescomplemented
60#1442C2214yescomplemented
61#1441C2214yescomplemented
62#1441C2214yescomplemented
63#1440C2214yes—
64#1440C2214yes—
65#1440C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2212yes—
69#1439C2212yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1437C2214yes—
74#1436C2214yescomplemented
75#1434C2212yescomplemented
76#1434C2212yescomplemented
77#1434C2212yescomplemented
78#1434C2214yescomplemented
79C22⋊#95C2212yes—
80C22⋊#95C2211yescharacteristic
81C22⋊#95C2212yes—
82C22⋊#95C2211yescharacteristic
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C26C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—28no—
95C2×C4×Q8—24no—
96C2×C4×Q8—44no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—28no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—44no—
114C22×C22⋊C4—44no—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C42—44no—
130C22×C42D418yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#399—28no—
138#393—28no—
139#391—24no—
140#389—24no—
141#388—28no—
142#388—28no—
143#388—28no—
144#388—28no—
145#388—48no—
146#387—28no—
147#385—28no—
148#385—44no—
149#385—44no—
150#383—44no—
151#381—28no—
152#381—44no—
153#380—24no—
154#379—28no—
155#379—44no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—24no—
158C4×C22⋊C4—28no—
159C2×#95—24no—
160C2×#95—24no—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C25—22no—
189C25—24no—
190C25—24no—
191C25—24no—
192C25—22no—
193C25—22no—
194C25—42no—
195C25—42no—
196C25—42no—
197C25C2412yes—
198C25C2411yescharacteristic
199C22×Q8—22no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—42no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C22⋊Q8—232no—
245C4×Q8—416no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—416no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C22⋊C4—232no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—416no—
301C2×C22⋊C4—416no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4#4118yes—
310C2×C22⋊C4#4118yes—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—28no—
316C2×C42—416no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42—44no—
320C2×C42C2×D414yes—
321C2×C42C2×D414yes—
322C2×C42C2×D414yes—
323C2×C42C2×D414yes—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—42no—
346#95—44no—
347#95—42no—
348#95—44no—
349#95—44no—
350#95—44no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—28no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—24no—
366C24—416no—
367C24—48no—
368C24—48no—
369C24—416no—
370C24—44no—
371C24—44no—
372C24—44no—
373C24—44no—
374C24—44no—
375C24—44no—
376C24—48no—
377C24—48no—
378C24—88no—
379C24—84no—
380C24—84no—
381C24—88no—
382C24C22×D412yes—
383C24C2×#4114yes—
384C24C2×#4114yes—
385C24H(5,𝔽2)14yes—
386C242−1+412yes—
387C242−1+412yes—
388C24C2511yescharacteristic
389C2×Q8—28no—
390C2×Q8—416no—
391C2×Q8—44no—
392C2×Q8—44no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—416no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—416no—
471C22×C4—44no—
472C22×C4—416no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—88no—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C2×#4114yes—
490C22×C4C2×#4114yes—
491C22×C4C2×#4114yes—
492C22×C4C2×#4114yes—
493C22×C4H(5,𝔽2)14yes—
494C4⋊C4—28no—
495C4⋊C4—216no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—88no—
503C4⋊C4—88no—
504C22⋊C4—216no—
505C22⋊C4—216no—
506C22⋊C4—216no—
507C22⋊C4—416no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C22⋊C4—88no—
518C42—28no—
519C42—28no—
520C42—44no—
521C42—44no—
522C42—48no—
523C42—48no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—28no—
529C23—24no—
530C23—24no—
531C23—28no—
532C23—28no—
533C23—24no—
534C23—24no—
535C23—28no—
536C23—24no—
537C23—28no—
538C23—24no—
539C23—24no—
540C23—28no—
541C23—416no—
542C23—416no—
543C23—416no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—42no—
554C23—42no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—832no—
563C23—832no—
564C23—88no—
565C23—88no—
566C23—88no—
567C23—88no—
568C23#52514yes—
569C23#53514yes—
570C23#53614yes—
571C23#54218yes—
572C23#54218yes—
573C23C23×D411yescharacteristic
574C23C22×#4112yes—
575C23C22×#4112yes—
576C23C2×H(5,𝔽2)12yes—
577C23C2×H(5,𝔽2)12yes—
578C23C2×2−1+411yescharacteristic
579C23C2×2−1+411yescharacteristic
580C23#58512yes—
581C23#58512yes—
582Q8—416no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—416no—
609C2×C4—416no—
610C2×C4—416no—
611C2×C4—416no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
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—44no—
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—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—88no—
661C2×C4—88no—
662C2×C4#52514yes—
663C2×C4#53414yes—
664C22—24no—
665C22—24no—
666C22—24no—
667C22—216no—
668C22—24no—
669C22—216no—
670C22—24no—
671C22—24no—
672C22—48no—
673C22—44no—
674C22—48no—
675C22—44no—
676C22—44no—
677C22—832no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—84no—
683C22—84no—
684C22#263218yes—
685C22C2×#52511yescharacteristic
686C22C2×#52511yescharacteristic
687C22C2×#53411yescharacteristic
688C22C2×#53511yescharacteristic
689C22C2×#53611yescharacteristic
690C22#344911yescharacteristic
691C22#344911yescharacteristic
692C22C2×#54212yes—
693C22C2×#54212yes—
694C22#348114yes—
695C22#349512yes—
696C22#349512yes—
697C22#351314yes—
698C22#352114yes—
699C22#353512yes—
700C22#353512yes—
701C22#354214yes—
702C4—24no—
703C4—24no—
704C4—42no—
705C4—42no—
706C4—42no—
707C4—44no—
708C4—42no—
709C4—44no—
710C4—44no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—84no—
715C4—84no—
716C4—82no—
717C2—28no—
718C2—84no—
719C2C2×#263211yescharacteristic
720C2#3495611yescharacteristic
721C2#3514311yescharacteristic
722C2#3523311yescharacteristic
723C2#3532511yescharacteristic
724C2#3743711yescharacteristic
725C2#3744111yescharacteristic
726C2#3851414yes—
727C2#3854714yes—
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×C4×C22⋊Q8C2semidirect
3. #17045C2semidirect
4. #16982C2semidirect
5. #16803C2semidirect
6. #15092C2semidirect
7. #15077C2semidirect
8. #14936C2semidirect
9. #14934C2semidirect
13. #14683C2semidirect
14. #14305C2semidirect
15. #14301C2semidirect
16. #13992C2semidirect
19. C22×C4×Q8C22semidirect
28. C2×#391C22semidirect
29. C2×#389C22semidirect
36. C2×#387C22semidirect
40. C2×#383C22semidirect
46. C2×C4×C4⋊C4C22semidirect
47. C2×C4×C4⋊C4C22semidirect
51. #1446C22semidirect
52. #1445C22semidirect
53. #1445C22semidirect
54. #1444C22semidirect
55. #1444C22semidirect
56. #1444C22semidirect
57. #1443C22semidirect
58. #1442C22semidirect
59. #1442C22semidirect
60. #1442C22semidirect
61. #1441C22semidirect
62. #1441C22semidirect
74. #1436C22semidirect
75. #1434C22semidirect
76. #1434C22semidirect
77. #1434C22semidirect
78. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
C22×#3882
#170452
#169822
#168032
#150922
#150772
#149362
#149342
#149182
#147342
#147172
#146832
#143052
#143012
#139922
#139912