Finite Group Databasebeta

← #722645

Subgroups — #722645

Order 512.

3678 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2434C211yesmaximal, characteristic, complemented
2C2×#2626C211yesmaximal, characteristic
3C22×#393C211yesmaximal, characteristic
4C2×#2368C211yesmaximal, characteristic, complemented
5#16296C214yesmaximal, complemented
6#15855C214yesmaximal, complemented
7#15322C214yesmaximal, complemented
8#14955C212yesmaximal, complemented
9#14789C214yesmaximal, complemented
10#14726C212yesmaximal
11#14149C212yesmaximal, complemented
12#13991C212yesmaximal
13#13922C211yesmaximal, characteristic, complemented
14#13908C211yesmaximal, characteristic, complemented
15#13775C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C4⋊Q8C2212yescomplemented
18C22×C22⋊Q8C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21#2626—28no—
22#2434—28no—
23C2×#399C2212yes—
24C2×#396C2214yes—
25C2×#396C2214yes—
26C2×#395C2214yescomplemented
27C2×#393—28no—
28C2×#393C2212yes—
29C2×#393C2212yes—
30C2×#393C2212yes—
31C2×#393C2214yes—
32#2368—28no—
33C2×#391C2212yescomplemented
34C2×#391C2212yescomplemented
35C2×#391C2214yescomplemented
36C2×#388C2214yes—
37C2×#387C2212yescomplemented
38C2×#387C2212yescomplemented
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#380C2212yes—
44C2×#380C2211yescharacteristic
45C2×#379C2211yescharacteristic
46C2×#379C2212yes—
47C2×#379C2211yescharacteristic
48C2×C4×C4⋊C4C2212yescomplemented
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C22×#95C2212yes—
53C22×#95C2211yescharacteristic
54#1446C2214yescomplemented
55#1444C2212yescomplemented
56#1442C2212yescomplemented
57#1442C2214yescomplemented
58#1442C2214yescomplemented
59#1441C2214yescomplemented
60#1441C2214yescomplemented
61#1441C2214yescomplemented
62#1441C2214yescomplemented
63#1440C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1437C2214yes—
69#1437C2214yes—
70#1437C2214yes—
71#1436C2214yescomplemented
72#1436C2214yescomplemented
73#1436C2214yescomplemented
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2211yescharacteristic
79C26C2311yescharacteristic
80C23×Q8—24no—
81C23×Q8—24no—
82C23×Q8—24no—
83C23×Q8—48no—
84C23×Q8C2312yes—
85C23×Q8C2312yes—
86C23×Q8C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—48no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8D418yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—28no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—48no—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128#399—28no—
129#396—216no—
130#396—48no—
131#395—216no—
132#393—28no—
133#393—28no—
134#393—28no—
135#393—216no—
136#391—28no—
137#391—28no—
138#391—48no—
139#388—48no—
140#387—28no—
141#387—28no—
142#385—28no—
143#385—28no—
144#385—28no—
145#385—28no—
146#380—24no—
147#380—28no—
148#379—28no—
149#379—24no—
150#379Q818yescomplemented
151C4×C4⋊C4—28no—
152C4×C22⋊C4—28no—
153C4×C22⋊C4—24no—
154C4×C22⋊C4—24no—
155C2×#95—28no—
156C2×#95D418yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2311yescharacteristic
180C2×#95C2311yescharacteristic
181C25—22no—
182C25—22no—
183C25—22no—
184C25—22no—
185C25—22no—
186C25—22no—
187C25—44no—
188C25—44no—
189C25C2×Q814yes—
190C25C2412yes—
191C25C2411yescharacteristic
192C22×Q8—24no—
193C22×Q8—22no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—22no—
199C22×Q8—24no—
200C22×Q8—28no—
201C22×Q8—28no—
202C22×Q8—44no—
203C22×Q8—44no—
204C22×Q8—44no—
205C22×Q8—432no—
206C22×Q8—432no—
207C22×Q8—432no—
208C22×Q8—416no—
209C22×Q8C2×D414yes—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—28no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4C2×D418yes—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4#4114yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C4⋊Q8—432no—
242C22⋊Q8—232no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D418yes—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
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—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—432no—
299C2×C22⋊C4—416no—
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×Q814yes—
308C2×C22⋊C4C2×Q814yes—
309C2×C22⋊C4C2×Q8116yescomplemented
310C2×C22⋊C4#4114yes—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42—44no—
318C2×C42—44no—
319#95—28no—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95C2×D414yes—
341#95C2×D414yes—
342#95C2×D418yes—
343C24—24no—
344C24—24no—
345C24—28no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24—24no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—28no—
357C24—28no—
358C24—416no—
359C24—48no—
360C24—48no—
361C24—416no—
362C24—44no—
363C24—44no—
364C24—44no—
365C24—44no—
366C24—84no—
367C24—84no—
368C24—84no—
369C24—84no—
370C24—88no—
371C24—88no—
372C24C22×D412yes—
373C24C22×D412yes—
374C24C22×Q814yes—
375C24C22×Q812yes—
376C24C2×#4112yes—
377C24C2×#4112yes—
378C242−1+412yes—
379C242−1+412yes—
380C24C2511yescharacteristic
381C2×Q8—216no—
382C2×Q8—216no—
383C2×Q8—28no—
384C2×Q8—48no—
385C2×Q8—416no—
386C2×Q8—416no—
387C2×Q8—416no—
388C2×Q8—416no—
389C2×Q8—48no—
390C2×Q8—48no—
391C2×Q8—4128no—
392C2×Q8—48no—
393C2×Q8—48no—
394C2×Q8—84no—
395C2×Q8—84no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—216no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
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—24no—
433C22×C4—216no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—44no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4—84no—
465C22×C4—84no—
466C22×C4#121116yes—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×Q814yes—
476C22×C4C22×Q814yes—
477C22×C4C2×#4114yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4H(5,𝔽2)18yes—
481C4⋊C4—28no—
482C4⋊C4—216no—
483C4⋊C4—416no—
484C4⋊C4—416no—
485C4⋊C4—48no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C4⋊C4—88no—
489C4⋊C4—88no—
490C22⋊C4—216no—
491C22⋊C4—28no—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—416no—
495C22⋊C4—416no—
496C22⋊C4—416no—
497C22⋊C4—44no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C42—48no—
505C42—48no—
506C42—84no—
507C42—84no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—24no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—28no—
522C23—28no—
523C23—432no—
524C23—42no—
525C23—44no—
526C23—416no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—816no—
544C23—816no—
545C23—88no—
546C23—816no—
547C23—816no—
548C23—816no—
549C23—88no—
550C23C2×#12114yes—
551C23#53314yes—
552C23#54114yes—
553C23#54714yes—
554C23#54814yes—
555C23D4×Q814yes—
556C23D4×Q814yes—
557C23C23×D411yescharacteristic
558C23C23×D411yescharacteristic
559C23C23×Q811yescharacteristic
560C23C22×#4111yescharacteristic
561C23C22×#4111yescharacteristic
562C23C2×H(5,𝔽2)14yes—
563C23C2×2−1+411yescharacteristic
564C23C2×2−1+411yescharacteristic
565C23#58514yes—
566Q8—432no—
567Q8—432no—
568Q8—88no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—416no—
595C2×C4—432no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
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—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—816no—
637C2×C4—88no—
638C2×C4C2×#12114yes—
639C2×C4C2×#12114yes—
640C22—24no—
641C22—28no—
642C22—28no—
643C22—28no—
644C22—28no—
645C22—28no—
646C22—28no—
647C22—24no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—48no—
652C22—48no—
653C22—832no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—84no—
658C22—84no—
659C22—82no—
660C22#242918yes—
661C22C22×#12111yescharacteristic
662C22C2×#53311yescharacteristic
663C22#344912yes—
664C22C2×#54111yescharacteristic
665C22C2×#54711yescharacteristic
666C22C2×#54811yescharacteristic
667C22C2×D4×Q811yescharacteristic
668C22C2×D4×Q811yescharacteristic
669C22#348412yes—
670C22#348412yes—
671C22#349012yes—
672C22#350012yes—
673C22#350114yes—
674C22#351012yes—
675C22#351414yes—
676C22#354114yes—
677C22#356114yes—
678C4—24no—
679C4—24no—
680C4—48no—
681C4—44no—
682C4—44no—
683C4—44no—
684C4—44no—
685C4—44no—
686C4—44no—
687C4—44no—
688C4—84no—
689C4—82no—
690C4—82no—
691C2—28no—
692C2—84no—
693C2C2×#242911yescharacteristic
694C2#3532711yescharacteristic
695C2#3534811yescharacteristic
696C2#3570512yes—
697C2#3804412yes—
698C2#3958812yes—
699C2#4054112yes—
700C2#4300312yes—
701C2#4389512yes—
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×#2434C2semidirect
4. C2×#2368C2semidirect
5. #16296C2semidirect
6. #15855C2semidirect
7. #15322C2semidirect
8. #14955C2semidirect
9. #14789C2semidirect
11. #14149C2semidirect
13. #13922C2semidirect
14. #13908C2semidirect
17. C22×C4⋊Q8C22semidirect
26. C2×#395C22semidirect
33. C2×#391C22semidirect
34. C2×#391C22semidirect
35. C2×#391C22semidirect
37. C2×#387C22semidirect
38. C2×#387C22semidirect
48. C2×C4×C4⋊C4C22semidirect
54. #1446C22semidirect
55. #1444C22semidirect
56. #1442C22semidirect
57. #1442C22semidirect
58. #1442C22semidirect
59. #1441C22semidirect
60. #1441C22semidirect
61. #1441C22semidirect
62. #1441C22semidirect
71. #1436C22semidirect
72. #1436C22semidirect
73. #1436C22semidirect
150. #379Q8semidirect
309. C2×C22⋊C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24342
C2×#26262
C22×#3932
C2×#23682
#162962
#158552
#153222
#149552
#147892
#147262
#141492
#139912
#139222
#139082
#137752