Finite Group Databasebeta

← #636505

Subgroups — #636505

Order 512.

3642 conjugacy classes of subgroups.

The subgroup lattice has 671 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2569C211yesmaximal, characteristic, complemented
2C22×#380C211yesmaximal, characteristic
3#16304C214yesmaximal, complemented
4#16304C214yesmaximal, complemented
5#14822C212yesmaximal, complemented
6#14775C214yesmaximal, complemented
7#14774C214yesmaximal, complemented
8#14728C212yesmaximal, complemented
9#14680C211yesmaximal, characteristic
10#14338C212yesmaximal
11#14262C212yesmaximal
12#14149C211yesmaximal, characteristic, complemented
13#13966C211yesmaximal, characteristic, complemented
14#13908C211yesmaximal, characteristic, complemented
15C4×#379C211yesmaximal, characteristic
16C25×C4C2211yescharacteristic
17C23×C4⋊C4C2212yes—
18C23×C22⋊C4C2212yes—
19#2569—28no—
20C2×#400C2212yescomplemented
21C2×#400C2214yescomplemented
22C2×#398C2212yescomplemented
23C2×#398C2214yescomplemented
24C2×#387C2212yescomplemented
25C2×#387C2212yescomplemented
26C2×#385C2212yes—
27C2×#385C2212yes—
28C2×#380—28no—
29C2×#380C2212yes—
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2214yes—
35C2×#380C2212yes—
36C2×#380C2214yes—
37C2×#379C2211yescharacteristic
38C2×#379C2212yes—
39C2×#379C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2214yes—
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45#1444C2212yescomplemented
46#1444C2214yescomplemented
47#1442C2212yescomplemented
48#1442C2214yescomplemented
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1442C2214yescomplemented
52#1441C2214yescomplemented
53#1441C2214yescomplemented
54#1441C2214yescomplemented
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1434C2212yescomplemented
66#1434C2212yescomplemented
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2211yescharacteristic
77C26C2311yescharacteristic
78C24×C4—24no—
79C24×C4—24no—
80C24×C4—48no—
81C24×C4Q818yes—
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2312yes—
85C24×C4C2312yes—
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C22×C4⋊C4—28no—
89C22×C4⋊C4—48no—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C22⋊C4—28no—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—44no—
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117#400—28no—
118#400—48no—
119#398—28no—
120#398—48no—
121#387—28no—
122#387—28no—
123#385—28no—
124#385—44no—
125#380—24no—
126#380—28no—
127#380—24no—
128#380—216no—
129#380—28no—
130#380—28no—
131#380—48no—
132#380—44no—
133#379—24no—
134#379—24no—
135#379—28no—
136C4×C22⋊C4—24no—
137C4×C22⋊C4—24no—
138C4×C22⋊C4—48no—
139C2×#95—24no—
140C2×#95D418yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
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×#95C2314yes—
166C2×#95C2311yescharacteristic
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C25—22no—
170C25—22no—
171C25—22no—
172C25—22no—
173C25—44no—
174C25—44no—
175C25C2×Q814yes—
176C25#4114yes—
177C25#4114yes—
178C25C2412yes—
179C25C2411yescharacteristic
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—28no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—232no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—432no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—432no—
204C23×C4—432no—
205C23×C4C2×D418yes—
206C23×C4C2×D414yes—
207C23×C4C2×Q814yes—
208C23×C4C2×Q818yes—
209C23×C4C2×Q818yes—
210C23×C4C2×Q814yes—
211C23×C4#4114yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—432no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—432no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—44no—
290C2×C42—44no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—22no—
295#95—22no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95C2×D418yes—
317#95C2×D414yes—
318#95C2×D414yes—
319C24—24no—
320C24—24no—
321C24—28no—
322C24—28no—
323C24—28no—
324C24—28no—
325C24—28no—
326C24—28no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—22no—
333C24—24no—
334C24—24no—
335C24—48no—
336C24—48no—
337C24—416no—
338C24—416no—
339C24—48no—
340C24—48no—
341C24—48no—
342C24—48no—
343C24—416no—
344C24—416no—
345C24#125116yes—
346C24C22×D412yes—
347C24C22×Q814yes—
348C24C22×Q812yes—
349C24C2×#4114yes—
350C24C2×#4114yes—
351C24C2×#4112yes—
352C24C2×#4112yes—
353C24C2×#4112yes—
354C24H(5,𝔽2)12yes—
355C242−1+412yes—
356C24C2511yescharacteristic
357C22×C4—216no—
358C22×C4—216no—
359C22×C4—28no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—216no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—216no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
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—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—416no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—4128no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4C22⋊Q8116yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×Q812yes—
448C22×C4C22×Q812yes—
449C22×C4C22×Q814yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4H(5,𝔽2)18yes—
453C4⋊C4—416no—
454C4⋊C4—416no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—84no—
458C4⋊C4—84no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C4⋊C4—88no—
462C4⋊C4—88no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—416no—
466C22⋊C4—416no—
467C22⋊C4—44no—
468C22⋊C4—44no—
469C22⋊C4—48no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—48no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C42—84no—
476C42—84no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—232no—
496C23—24no—
497C23—24no—
498C23—28no—
499C23—28no—
500C23—28no—
501C23—28no—
502C23—432no—
503C23—416no—
504C23—432no—
505C23—432no—
506C23—432no—
507C23—432no—
508C23—416no—
509C23—432no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23C2×C22⋊Q814yes—
519C23C2×#12518yes—
520C23C2×#12514yes—
521C23#54414yes—
522C23#54614yes—
523C23#54814yes—
524C23#55214yes—
525C23#55314yes—
526C23C23×D411yescharacteristic
527C23C23×Q811yescharacteristic
528C23C22×#4111yescharacteristic
529C23C22×#4111yescharacteristic
530C23C22×#4111yescharacteristic
531C23C2×H(5,𝔽2)14yes—
532C23C2×H(5,𝔽2)11yescharacteristic
533C23C2×2−1+411yescharacteristic
534C23#58514yes—
535C2×C4—232no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—416no—
553C2×C4—432no—
554C2×C4—432no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—816no—
604C2×C4—816no—
605C2×C4C2×C22⋊Q814yes—
606C2×C4C2×C22⋊Q814yes—
607C2×C4#54318yes—
608C22—28no—
609C22—28no—
610C22—28no—
611C22—28no—
612C22—24no—
613C22—28no—
614C22—24no—
615C22—28no—
616C22—28no—
617C22—28no—
618C22—28no—
619C22—464no—
620C22—48no—
621C22—48no—
622C22—48no—
623C22—48no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22#244218yes—
629C22C22×C22⋊Q811yescharacteristic
630C22C22×#12511yescharacteristic
631C22#344912yes—
632C22#345512yes—
633C22C2×#54312yes—
634C22C2×#54411yescharacteristic
635C22#345812yes—
636C22C2×#54611yescharacteristic
637C22C2×#54811yescharacteristic
638C22C2×#55211yescharacteristic
639C22C2×#55311yescharacteristic
640C22#348412yes—
641C22#349012yes—
642C22#350514yes—
643C22#350514yes—
644C22#354114yes—
645C22#354214yes—
646C4—28no—
647C4—24no—
648C4—24no—
649C4—44no—
650C4—44no—
651C4—82no—
652C4—84no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—84no—
659C2—28no—
660C2—48no—
661C2#3463411yescharacteristic
662C2C2×#244211yescharacteristic
663C2#3498211yescharacteristic
664C2#3502012yes—
665C2#3747612yes—
666C2#3946012yes—
667C2#3947212yes—
668C2#3971612yes—
669C2#3972812yes—
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×#2569C2semidirect
3. #16304C2semidirect
4. #16304C2semidirect
5. #14822C2semidirect
6. #14775C2semidirect
7. #14774C2semidirect
8. #14728C2semidirect
12. #14149C2semidirect
13. #13966C2semidirect
14. #13908C2semidirect
20. C2×#400C22semidirect
21. C2×#400C22semidirect
22. C2×#398C22semidirect
23. C2×#398C22semidirect
24. C2×#387C22semidirect
25. C2×#387C22semidirect
45. #1444C22semidirect
46. #1444C22semidirect
47. #1442C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1434C22semidirect
66. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25692
C22×#3802
#163042
#148222
#147752
#147742
#147282
#146802
#143382
#142622
#141492
#139662
#139082
C4×#3792