Finite Group Databasebeta

← #625113

Subgroups — #625113

Order 512.

3022 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2329C211yesmaximal, characteristic, complemented
2C22×#388C211yesmaximal, characteristic
3#15977C214yesmaximal, complemented
4#15970C214yesmaximal, complemented
5#15459C214yesmaximal, complemented
6#15424C214yesmaximal, complemented
7#15014C211yesmaximal, characteristic, complemented
8#14914C211yesmaximal, characteristic, complemented
9#14820C211yesmaximal, characteristic
10#14717C211yesmaximal, characteristic
11#14695C212yesmaximal, complemented
12#14349C211yesmaximal, characteristic, complemented
13#14262C212yesmaximal
14#14039C212yesmaximal, complemented
15#13991C212yesmaximal
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22C2×#397C2212yes—
23C2×#393C2212yes—
24#2329—28no—
25C2×#391C2212yescomplemented
26C2×#389C2212yescomplemented
27C2×#388—28no—
28C2×#388C2214yes—
29C2×#388C2214yes—
30C2×#388C2214yes—
31C2×#387C2212yescomplemented
32C2×#387C2212yescomplemented
33C2×#387C2214yescomplemented
34C2×#387C2214yescomplemented
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#380C2211yescharacteristic
39C2×#379C2212yes—
40C2×#379C2212yes—
41C2×C4×C22⋊C4C2212yes—
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44#1446C2212yescomplemented
45#1445C2212yescomplemented
46#1444C2212yescomplemented
47#1444C2212yescomplemented
48#1443C2214yescomplemented
49#1443C2214yescomplemented
50#1442C2214yescomplemented
51#1442C2214yescomplemented
52#1441C2214yescomplemented
53#1441C2214yescomplemented
54#1440C2214yes—
55#1440C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2212yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2212yes—
64#1437C2214yes—
65#1436C2214yescomplemented
66#1436C2214yescomplemented
67#1436C2214yescomplemented
68#1434C2214yescomplemented
69C22⋊#95C2211yescharacteristic
70C22⋊#95C2212yes—
71C22⋊#95C2211yescharacteristic
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—28no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—44no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C42—24no—
118C22×C42—44no—
119C22×C42C2314yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122#397—44no—
123#393—44no—
124#391—28no—
125#389—28no—
126#388—216no—
127#388—216no—
128#388—48no—
129#387—28no—
130#387—28no—
131#387—48no—
132#387—48no—
133#385—28no—
134#385—28no—
135#385—28no—
136#380—24no—
137#379—28no—
138#379—28no—
139C4×C22⋊C4—28no—
140C2×#95—24no—
141C2×#95D418yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C25—22no—
170C25—22no—
171C25—24no—
172C25—24no—
173C25—42no—
174C25—42no—
175C25—42no—
176C25—42no—
177C25#4114yes—
178C25C2412yes—
179C25C2411yescharacteristic
180C22×Q8—22no—
181C22×Q8—24no—
182C22×Q8—42no—
183C22×Q8—42no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4#4114yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C22⋊Q8—232no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—416no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C22⋊C4—232no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—416no—
278C2×C22⋊C4—416no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4#4118yes—
286C2×C42—28no—
287C2×C42—28no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—44no—
291C2×C42—416no—
292C2×C42—44no—
293C2×C42—44no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—44no—
307#95—44no—
308#95—42no—
309#95—44no—
310#95—42no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—42no—
318#95—42no—
319#95C2×D418yes—
320#95C2×D418yes—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—28no—
326C24—22no—
327C24—22no—
328C24—24no—
329C24—24no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24—22no—
335C24—22no—
336C24—216no—
337C24—48no—
338C24—48no—
339C24—44no—
340C24—44no—
341C24—48no—
342C24—44no—
343C24—44no—
344C24—48no—
345C24—44no—
346C24—44no—
347C24—48no—
348C24—88no—
349C24—88no—
350C24—88no—
351C24—88no—
352C24C22×D412yes—
353C24C2×#4114yes—
354C24C2×#4112yes—
355C24C2×#4114yes—
356C24C2×#4112yes—
357C242−1+414yes—
358C24C2511yescharacteristic
359C2×Q8—28no—
360C2×Q8—82no—
361C2×Q8—82no—
362C2×Q8—88no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
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—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—48no—
429C22×C4—44no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—88no—
445C22×C4—88no—
446C22×C4C22×D412yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4114yes—
453C22×C4H(5,𝔽2)14yes—
454C22×C4H(5,𝔽2)14yes—
455C4⋊C4—28no—
456C4⋊C4—216no—
457C4⋊C4—416no—
458C4⋊C4—416no—
459C4⋊C4—416no—
460C4⋊C4—84no—
461C4⋊C4—82no—
462C4⋊C4—84no—
463C4⋊C4—82no—
464C22⋊C4—232no—
465C22⋊C4—216no—
466C22⋊C4—416no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—416no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C42—416no—
477C42—84no—
478C42—84no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—24no—
492C23—28no—
493C23—28no—
494C23—24no—
495C23—24no—
496C23—416no—
497C23—416no—
498C23—416no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—832no—
518C23—88no—
519C23—88no—
520C23—88no—
521C23—832no—
522C23—88no—
523C23#52514yes—
524C23#52514yes—
525C23#53614yes—
526C23#55218yes—
527C23#55618yes—
528C23C23×D411yescharacteristic
529C23C22×#4111yescharacteristic
530C23C22×#4111yescharacteristic
531C23C22×#4112yes—
532C23C2×H(5,𝔽2)12yes—
533C23C2×H(5,𝔽2)12yes—
534C23C2×2−1+412yes—
535C23#58512yes—
536C23#58512yes—
537Q8—88no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—416no—
561C2×C4—416no—
562C2×C4—416no—
563C2×C4—416no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
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—44no—
607C2×C4—44no—
608C2×C4—88no—
609C2×C4—88no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—216no—
614C22—216no—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—44no—
619C22—48no—
620C22—48no—
621C22—44no—
622C22—44no—
623C22—832no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—84no—
628C22—84no—
629C22—82no—
630C22#259918yes—
631C22C2×#52511yescharacteristic
632C22C2×#52511yescharacteristic
633C22C2×#53611yescharacteristic
634C22#345912yes—
635C22#345912yes—
636C22C2×#55212yes—
637C22C2×#55612yes—
638C22#348411yescharacteristic
639C22#348411yescharacteristic
640C22#348511yescharacteristic
641C22#348711yescharacteristic
642C22#350512yes—
643C22#350612yes—
644C22#351314yes—
645C22#351414yes—
646C22#353614yes—
647C22#356414yes—
648C4—44no—
649C4—44no—
650C4—44no—
651C4—44no—
652C4—42no—
653C4—42no—
654C4—44no—
655C4—44no—
656C4—44no—
657C4—84no—
658C4—84no—
659C4—82no—
660C4—82no—
661C2—28no—
662C2—84no—
663C2C2×#259911yescharacteristic
664C2#3498611yescharacteristic
665C2#3498711yescharacteristic
666C2#3535611yescharacteristic
667C2#3545111yescharacteristic
668C2#3671811yescharacteristic
669C2#3696211yescharacteristic
670C2#3825814yes—
671C2#3928914yes—
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×#2329C2semidirect
3. #15977C2semidirect
4. #15970C2semidirect
5. #15459C2semidirect
6. #15424C2semidirect
7. #15014C2semidirect
8. #14914C2semidirect
11. #14695C2semidirect
12. #14349C2semidirect
14. #14039C2semidirect
25. C2×#391C22semidirect
26. C2×#389C22semidirect
31. C2×#387C22semidirect
32. C2×#387C22semidirect
33. C2×#387C22semidirect
34. C2×#387C22semidirect
44. #1446C22semidirect
45. #1445C22semidirect
46. #1444C22semidirect
47. #1444C22semidirect
48. #1443C22semidirect
49. #1443C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
68. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#23292
C22×#3882
#159772
#159702
#154592
#154242
#150142
#149142
#148202
#147172
#146952
#143492
#142622
#140392
#139912