Finite Group Databasebeta

← #641786

Subgroups — #641786

Order 512.

4006 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#385C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic
3#16892C214yesmaximal, complemented
4#16400C214yesmaximal, complemented
5#15092C212yesmaximal, complemented
6#14936C212yesmaximal, complemented
7#14920C214yesmaximal, complemented
8#14843C212yesmaximal, complemented
9#14822C212yesmaximal, complemented
10#14680C211yesmaximal, characteristic
11#14339C212yesmaximal, complemented
12#14275C212yesmaximal, complemented
13#14127C212yesmaximal
14#13869C212yesmaximal
15C25×C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23C2×#400C2214yescomplemented
24C2×#385—28no—
25C2×#385C2212yes—
26C2×#385C2212yes—
27C2×#385C2212yes—
28C2×#385C2212yes—
29C2×#385C2214yes—
30C2×#385C2214yes—
31C2×#381C2214yes—
32C2×#380—28no—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#379C2212yes—
41C22×#95C2212yes—
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44#1445C2214yescomplemented
45#1445C2218yescomplemented
46#1444C2214yescomplemented
47#1444C2214yescomplemented
48#1442C2218yescomplemented
49#1442C2214yescomplemented
50#1441C2218yescomplemented
51#1441C2214yescomplemented
52#1441C2214yescomplemented
53#1441C2218yescomplemented
54#1441C2214yescomplemented
55#1440C2218yes—
56#1439C2218yes—
57C22⋊#95C2214yes—
58C22⋊#95C2214yes—
59C22⋊#95C2214yes—
60C22⋊#95C2214yes—
61C22⋊#95C2214yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C26C2311yescharacteristic
66C24×C4—28no—
67C24×C4—44no—
68C24×C4—44no—
69C24×C4D418yes—
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C22×C4⋊C4—24no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2318yes—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—28no—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—48no—
105C22×C22⋊C4—44no—
106C22×C22⋊C4Q818yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116#400—48no—
117#385—28no—
118#385—28no—
119#385—28no—
120#385—28no—
121#385—48no—
122#385—48no—
123#381—48no—
124#380—28no—
125#380—28no—
126#380—28no—
127#380—28no—
128#380—28no—
129#380—44no—
130#380—44no—
131#379—28no—
132C2×#95—28no—
133C2×#95—24no—
134C2×#95—24no—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2318yes—
144C2×#95C2318yes—
145C2×#95C2318yes—
146C2×#95C2314yes—
147C2×#95C2318yes—
148C2×#95C2318yes—
149C2×#95C2318yes—
150C2×#95C2318yes—
151C25—24no—
152C25—24no—
153C25—24no—
154C25—24no—
155C25—42no—
156C25—42no—
157C25C2×D414yes—
158C25C2×Q814yes—
159C25#4114yes—
160C25C2411yescharacteristic
161C25C2411yescharacteristic
162C25C2411yescharacteristic
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—232no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—28no—
186C23×C4—24no—
187C23×C4—28no—
188C23×C4—416no—
189C23×C4—42no—
190C23×C4—416no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—42no—
207C23×C4—464no—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2×Q814yes—
215C23×C4C2×Q814yes—
216C23×C4#4118yes—
217C23×C4C2418yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—416no—
246C2×C4⋊C4—416no—
247C2×C4⋊C4—416no—
248C2×C4⋊C4—416no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—48no—
253C2×C4⋊C4—48no—
254C2×C22⋊C4—232no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—416no—
277C2×C22⋊C4—432no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4C2×Q818yes—
281C2×C22⋊C4C2×Q818yes—
282#95—28no—
283#95—28no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—28no—
289#95—28no—
290#95—48no—
291#95—44no—
292#95—48no—
293#95—48no—
294#95—48no—
295#95—48no—
296#95—48no—
297#95—48no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—28no—
307C24—28no—
308C24—28no—
309C24—28no—
310C24—216no—
311C24—216no—
312C24—216no—
313C24—216no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—44no—
321C24—44no—
322C24—48no—
323C24—48no—
324C24—416no—
325C24—416no—
326C24—416no—
327C24—416no—
328C24C22⋊Q8116yes—
329C24C22×D412yes—
330C24C22×D412yes—
331C24C22×D412yes—
332C24C22×Q812yes—
333C24C22×Q812yes—
334C24C22×Q812yes—
335C24C2×#4112yes—
336C24C2×#4112yes—
337C24C2×#4112yes—
338C24C2×#4114yes—
339C24H(5,𝔽2)14yes—
340C24H(5,𝔽2)14yes—
341C242−1+414yes—
342C24C2511yescharacteristic
343C22×C4—232no—
344C22×C4—28no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—216no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—24no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—28no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—24no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—416no—
407C22×C4—48no—
408C22×C4—416no—
409C22×C4—416no—
410C22×C4—416no—
411C22×C4—4128no—
412C22×C4—416no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—48no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—48no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—44no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—48no—
429C22×C4—44no—
430C22×C4—48no—
431C22×C4—88no—
432C22×C4—88no—
433C22×C4—816no—
434C22×C4—816no—
435C22×C4#123116yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×Q814yes—
441C22×C4C22×Q814yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4114yes—
444C22×C4H(5,𝔽2)14yes—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—88no—
454C4⋊C4—816no—
455C22⋊C4—216no—
456C22⋊C4—216no—
457C22⋊C4—48no—
458C22⋊C4—416no—
459C22⋊C4—48no—
460C22⋊C4—416no—
461C22⋊C4—416no—
462C22⋊C4—416no—
463C22⋊C4—88no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—232no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—232no—
502C23—416no—
503C23—416no—
504C23—464no—
505C23—416no—
506C23—416no—
507C23—416no—
508C23—464no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23C2×C22⋊Q814yes—
514C23C2×C22⋊Q814yes—
515C23C2×C22⋊Q814yes—
516C23C2×#12314yes—
517C23#53414yes—
518C23#53514yes—
519C23#53614yes—
520C23#53918yes—
521C23#54418yes—
522C23#55318yes—
523C23#55418yes—
524C23C23×D411yescharacteristic
525C23C23×Q811yescharacteristic
526C23C22×#4111yescharacteristic
527C23C22×#4112yes—
528C23C2×H(5,𝔽2)12yes—
529C23C2×H(5,𝔽2)12yes—
530C23C2×H(5,𝔽2)12yes—
531C23C2×2−1+412yes—
532C23#58512yes—
533C2×C4—232no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—416no—
551C2×C4—416no—
552C2×C4—416no—
553C2×C4—416no—
554C2×C4—416no—
555C2×C4—416no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—416no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—48no—
588C2×C4—48no—
589C2×C4—816no—
590C2×C4—88no—
591C2×C4—816no—
592C2×C4—88no—
593C2×C4C2×#12314yes—
594C2×C4C2×#12314yes—
595C2×C4#53414yes—
596C22—216no—
597C22—216no—
598C22—24no—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—24no—
609C22—216no—
610C22—216no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—464no—
616C22—48no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—48no—
622C22#248718yes—
623C22#262718yes—
624C22C22×C22⋊Q811yescharacteristic
625C22C22×#12311yescharacteristic
626C22C2×#53411yescharacteristic
627C22C2×#53411yescharacteristic
628C22C2×#53511yescharacteristic
629C22C2×#53611yescharacteristic
630C22#344911yescharacteristic
631C22C2×#53912yes—
632C22C2×#54412yes—
633C22C2×#55312yes—
634C22C2×#55412yes—
635C22#349014yes—
636C22#349514yes—
637C22#350112yes—
638C22#350512yes—
639C22#353414yes—
640C22#353614yes—
641C4—24no—
642C4—24no—
643C4—24no—
644C4—42no—
645C4—44no—
646C4—44no—
647C4—88no—
648C4—82no—
649C4—84no—
650C4—84no—
651C4—82no—
652C2—28no—
653C2—28no—
654C2—44no—
655C2C2×#262711yescharacteristic
656C2C2×#248711yescharacteristic
657C2#3499211yescharacteristic
658C2#3501011yescharacteristic
659C2#3531111yescharacteristic
660C2#3744011yescharacteristic
661C2#3745111yescharacteristic
662C2#3787314yes—
663C2#3789014yes—
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
3. #16892C2semidirect
4. #16400C2semidirect
5. #15092C2semidirect
6. #14936C2semidirect
7. #14920C2semidirect
8. #14843C2semidirect
9. #14822C2semidirect
11. #14339C2semidirect
12. #14275C2semidirect
23. C2×#400C22semidirect
44. #1445C22semidirect
45. #1445C22semidirect
46. #1444C22semidirect
47. #1444C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1441C22semidirect
51. #1441C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3852
C22×#3802
#168922
#164002
#150922
#149362
#149202
#148432
#148222
#146802
#143392
#142752
#141272
#138692