Finite Group Databasebeta

← #4344917

Subgroups — #4344917

Order 512.

2838 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic, complemented
2C2×#2328C211yesmaximal, characteristic
3#16895C212yesmaximal, complemented
4#16689C212yesmaximal, complemented
5#16599C212yesmaximal, complemented
6#16398C212yesmaximal, complemented
7#15961C214yesmaximal, complemented
8#15327C214yesmaximal, complemented
9#15121C212yesmaximal, complemented
10#14836C212yesmaximal, complemented
11#14807C212yesmaximal, complemented
12#14777C212yesmaximal, complemented
13#14419C212yesmaximal
14#14401C212yesmaximal, complemented
15#13960C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2487—28no—
21C2×#400C2212yes—
22C2×#399C2212yes—
23C2×#399C2212yes—
24C2×#399C2212yes—
25#2328—28no—
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#389C2212yes—
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2214yes—
36C2×#386C2214yes—
37C2×#385C2212yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#385C2214yes—
41C2×#382C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2214yes—
46C2×#376C2211yescharacteristic
47C22×#95C2211yescharacteristic
48#1445C2212yes—
49#1445C2212yes—
50#1445C2214yes—
51#1444C2212yes—
52#1444C2214yes—
53#1444C2214yes—
54#1443C2214yes—
55#1442C2212yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2212yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2212yes—
63#1439C2214yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2214yes—
69#1438C2214yes—
70#1438C2214yes—
71#1438C2214yes—
72#1436C2214yes—
73#1434C2214yes—
74#1434C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×D4C2311yescharacteristic
78C23×D4C2312yes—
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2312yes—
86C2×#127—24no—
87C2×#127—24no—
88C2×#127Q818yes—
89C2×C4×D4—24no—
90C2×C4×D4—42no—
91C2×C4×D4—42no—
92C2×C4×D4—42no—
93C2×C4×D4—42no—
94C2×C4×D4D418yes—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—42no—
97C22×C4⋊C4—42no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#400—28no—
127#399—28no—
128#399—44no—
129#399—44no—
130#390—24no—
131#390—24no—
132#389—28no—
133#386—28no—
134#386—28no—
135#386—28no—
136#386—28no—
137#386—28no—
138#386—28no—
139#386—48no—
140#386—48no—
141#385—28no—
142#385—48no—
143#385—48no—
144#385—48no—
145#382—24no—
146#380—28no—
147#380—24no—
148#380—48no—
149#380—42no—
150#376—24no—
151C2×#95—42no—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C25C2412yes—
175C25C2411yescharacteristic
176C25C2411yescharacteristic
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—41no—
184C22×D4—41no—
185C22×D4—41no—
186C22×D4—41no—
187C22×D4—41no—
188C22×D4—41no—
189C22×D4—41no—
190C22×D4—41no—
191C22×D4—44no—
192C22×D4—44no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4C2×D414yes—
196C22×D4C2×Q818yes—
197C22×D4C2×Q818yes—
198C22×D4C2×Q814yes—
199C22×D4C2×Q814yes—
200C22×D4#4114yes—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—24no—
209C23×C4—44no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236#127—232no—
237C4×D4—416no—
238C4×D4—416no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—416no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4C2×D414yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
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—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—44no—
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⋊C4—44no—
307C2×C22⋊C4C2×D414yes—
308C2×C22⋊C4C2×D414yes—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—41no—
315C2×C42—41no—
316C2×C42C2×D414yes—
317C2×C42C2×Q814yes—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—24no—
330#95—28no—
331#95—28no—
332#95—42no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340C24—24no—
341C24—21no—
342C24—24no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—21no—
348C24—21no—
349C24—24no—
350C24—21no—
351C24—24no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24—24no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—24no—
362C24C22×D412yes—
363C24C22×D412yes—
364C24C22×Q814yes—
365C24C22×Q812yes—
366C24C22×Q812yes—
367C24C2×#4112yes—
368C24C2×#4112yes—
369C24C2511yescharacteristic
370C2×D4—216no—
371C2×D4—216no—
372C2×D4—28no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—416no—
386C2×D4—416no—
387C2×D4—84no—
388C2×D4—84no—
389C2×D4—84no—
390C2×D4—84no—
391C2×D4—88no—
392C2×D4—88no—
393C2×D4C22⋊Q8116yes—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
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—48no—
458C22×C4—48no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—84no—
462C22×C4—84no—
463C22×C4—88no—
464C22×C4—84no—
465C22×C4—88no—
466C22×C4—84no—
467C22×C4—88no—
468C22×C4—88no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4—88no—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×Q814yes—
478C22×C4C22×Q812yes—
479C22×C4C2×#4112yes—
480C22×C4H(5,𝔽2)18yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C4H(5,𝔽2)12yes—
483C4⋊C4—44no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—88no—
489C4⋊C4—88no—
490C4⋊C4—88no—
491C4⋊C4—84no—
492C4⋊C4—88no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—416no—
496C22⋊C4—416no—
497C22⋊C4—48no—
498C22⋊C4—416no—
499C22⋊C4—48no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C22⋊C4—88no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C22⋊C4—88no—
506C22⋊C4—84no—
507C22⋊C4—88no—
508C42—28no—
509C42—28no—
510C42—44no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—28no—
524C23—28no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—28no—
530C23—28no—
531C23—24no—
532C23—24no—
533C23—28no—
534C23—28no—
535C23—24no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23—42no—
541C23—44no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23C2×C22⋊Q814yes—
567C23C2×C22⋊Q814yes—
568C23C23×D411yescharacteristic
569C23C23×Q811yescharacteristic
570C23C22×#4111yescharacteristic
571C23C2×H(5,𝔽2)11yescharacteristic
572C23C2×H(5,𝔽2)11yescharacteristic
573C23C2×H(5,𝔽2)14yes—
574C23#58511yescharacteristic
575C23#58511yescharacteristic
576C23#58514yes—
577D4—416no—
578D4—416no—
579D4—816no—
580D4—816no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—416no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
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—88no—
647C2×C4—816no—
648C2×C4—816no—
649C2×C4—816no—
650C2×C4C2×C22⋊Q814yes—
651C2×C4#53414yes—
652C22—24no—
653C22—24no—
654C22—28no—
655C22—28no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—42no—
661C22—42no—
662C22—44no—
663C22—44no—
664C22—42no—
665C22—42no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—48no—
671C22—44no—
672C22—48no—
673C22—44no—
674C22—44no—
675C22—84no—
676C22—82no—
677C22—84no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22C22×C22⋊Q811yescharacteristic
682C22C2×#53411yescharacteristic
683C22#344911yescharacteristic
684C22#348512yes—
685C22#349812yes—
686C22#350111yescharacteristic
687C22#350114yes—
688C22#350111yescharacteristic
689C22#350511yescharacteristic
690C22#350511yescharacteristic
691C22#351314yes—
692C22#353312yes—
693C22#353314yes—
694C22#353312yes—
695C22#353712yes—
696C22#353914yes—
697C22#354912yes—
698C4—24no—
699C4—24no—
700C4—48no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4—44no—
705C4—84no—
706C4—82no—
707C4—84no—
708C4—82no—
709C4—84no—
710C2—44no—
711C2—44no—
712C2—84no—
713C2#3501011yescharacteristic
714C2#3583612yes—
715C2#3695511yescharacteristic
716C2#3729711yescharacteristic
717C2#4195412yes—
718C2#4753512yes—
719C2#4781812yes—
720C2#5200312yes—
721C2#5203912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2487C2semidirect
3. #16895C2semidirect
4. #16689C2semidirect
5. #16599C2semidirect
6. #16398C2semidirect
7. #15961C2semidirect
8. #15327C2semidirect
9. #15121C2semidirect
10. #14836C2semidirect
11. #14807C2semidirect
12. #14777C2semidirect
14. #14401C2semidirect
15. #13960C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C2×#23282
#168952
#166892
#165992
#163982
#159612
#153272
#151212
#148362
#148072
#147772
#144192
#144012
#139602