Finite Group Databasebeta

← #4866559

Subgroups — #4866559

Order 512.

3180 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2535C211yesmaximal, characteristic
2C2×#2409C211yesmaximal, characteristic, complemented
3#17201C212yesmaximal, complemented
4#17105C212yesmaximal, complemented
5#16807C212yesmaximal, complemented
6#16694C212yesmaximal, complemented
7#16520C214yesmaximal, complemented
8#16470C212yesmaximal, complemented
9#16288C212yesmaximal, complemented
10#16265C212yesmaximal, complemented
11#16263C212yesmaximal, complemented
12#15312C214yesmaximal, complemented
13#14873C212yesmaximal, complemented
14#14865C212yesmaximal, complemented
15#14344C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×#121C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2535—28no—
22#2409—28no—
23C2×#397C2211yescharacteristic
24C2×#397C2211yescharacteristic
25C2×#397C2214yes—
26C2×#396C2214yes—
27C2×#395C2212yes—
28C2×#394C2214yes—
29C2×#394C2214yes—
30C2×#393C2212yes—
31C2×#393C2212yes—
32C2×#393C2212yes—
33C2×#393C2214yes—
34C2×#392C2211yescharacteristic
35C2×#392C2211yescharacteristic
36C2×#391C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#386C2214yes—
41C2×#385C2212yes—
42C2×#382C2211yescharacteristic
43C2×#381C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47#1446C2214yes—
48#1446C2212yes—
49#1446C2212yes—
50#1445C2214yes—
51#1444C2212yes—
52#1444C2214yes—
53#1444C2212yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2212yes—
59#1441C2214yes—
60#1441C2214yes—
61#1440C2212yes—
62#1440C2214yes—
63#1439C2212yes—
64#1439C2212yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2212yes—
70#1438C2214yes—
71#1438C2214yes—
72#1434C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C23×Q8—24no—
78C23×Q8—42no—
79C23×Q8—42no—
80C23×Q8—48no—
81C23×Q8C2312yes—
82C23×Q8C2312yes—
83C23×D4C2312yes—
84C23×D4C2311yescharacteristic
85C23×D4C2311yescharacteristic
86C23×D4C2312yes—
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—48no—
93C2×#121—28no—
94C2×#121—48no—
95C2×C4×D4—24no—
96C2×C4×D4—42no—
97C2×C4×D4—42no—
98C2×C4×D4—42no—
99C2×C4×D4—42no—
100C2×C4×D4D418yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#397—24no—
127#397—24no—
128#397—48no—
129#396—48no—
130#395—28no—
131#394—48no—
132#394—48no—
133#393—28no—
134#393—44no—
135#393—44no—
136#393—48no—
137#392—24no—
138#392—24no—
139#391—28no—
140#386—28no—
141#386—28no—
142#386—28no—
143#386—48no—
144#385—28no—
145#382—24no—
146#381—42no—
147#380—24no—
148#380—28no—
149#380—42no—
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C25C2411yescharacteristic
175C25C2412yes—
176C25C2411yescharacteristic
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—28no—
180C22×Q8—28no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—44no—
184C22×Q8—44no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×Q8—416no—
188C22×Q8—432no—
189C22×Q8—432no—
190C22×Q8—432no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—28no—
197C22×D4—24no—
198C22×D4—22no—
199C22×D4—24no—
200C22×D4—41no—
201C22×D4—41no—
202C22×D4—41no—
203C22×D4—41no—
204C22×D4—42no—
205C22×D4—44no—
206C22×D4—42no—
207C22×D4—42no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4C2×D414yes—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—28no—
218C23×C4—28no—
219C23×C4—44no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—44no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C22⋊Q8—432no—
242#121—432no—
243C4×D4—416no—
244C4×D4—416no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
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×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
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⋊C4C2×D414yes—
307C2×C22⋊C4C2×D414yes—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42C2×D414yes—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—22no—
316#95—22no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—44no—
325#95—42no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335C24—21no—
336C24—21no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—21no—
348C24—21no—
349C24—24no—
350C24—22no—
351C24—24no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24C22×D412yes—
359C24C22×D412yes—
360C24H(5,𝔽2)14yes—
361C242−1+414yes—
362C24C2511yescharacteristic
363C2×Q8—216no—
364C2×Q8—416no—
365C2×Q8—416no—
366C2×Q8—416no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×Q8—48no—
370C2×Q8—416no—
371C2×Q8—4128no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×D4—28no—
375C2×D4—28no—
376C2×D4—44no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—48no—
383C2×D4—44no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—416no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—84no—
390C2×D4—84no—
391C2×D4—88no—
392C2×D4—88no—
393C2×D4—88no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
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—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—416no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—416no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—84no—
465C22×C4—84no—
466C22×C4—88no—
467C22×C4—88no—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4H(5,𝔽2)18yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C42−1+414yes—
477C4⋊C4—44no—
478C4⋊C4—416no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—84no—
483C4⋊C4—88no—
484C4⋊C4—88no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—416no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—84no—
493C22⋊C4—84no—
494C22⋊C4—88no—
495C22⋊C4—88no—
496C22⋊C4—88no—
497C22⋊C4—88no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C22⋊C4—88no—
501C42—28no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—28no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—28no—
517C23—24no—
518C23—24no—
519C23—28no—
520C23—42no—
521C23—42no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23—41no—
544C23—41no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23#53514yes—
557C23C23×D411yescharacteristic
558C23C2×H(5,𝔽2)11yescharacteristic
559C23C2×H(5,𝔽2)11yescharacteristic
560C23C2×H(5,𝔽2)12yes—
561C23C2×H(5,𝔽2)14yes—
562C23C2×2−1+412yes—
563C23C2×2−1+412yes—
564C23#58512yes—
565Q8—432no—
566Q8—432no—
567D4—416no—
568D4—416no—
569D4—816no—
570D4—816no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—416no—
588C2×C4—432no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
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—88no—
637C2×C4—816no—
638C2×C4#53414yes—
639C2×C4#53614yes—
640C22—24no—
641C22—28no—
642C22—48no—
643C22—44no—
644C22—44no—
645C22—42no—
646C22—44no—
647C22—42no—
648C22—48no—
649C22—48no—
650C22—42no—
651C22—42no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—42no—
658C22—84no—
659C22—84no—
660C22—82no—
661C22—84no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22C2×#53411yescharacteristic
668C22C2×#53511yescharacteristic
669C22C2×#53611yescharacteristic
670C22#348412yes—
671C22#348912yes—
672C22#353212yes—
673C22#353212yes—
674C22#353612yes—
675C22#353614yes—
676C22#353612yes—
677C22#353811yescharacteristic
678C22#354011yescharacteristic
679C22#354112yes—
680C22#354714yes—
681C22#355214yes—
682C22#355814yes—
683C4—24no—
684C4—24no—
685C4—42no—
686C4—48no—
687C4—42no—
688C4—42no—
689C4—84no—
690C4—84no—
691C4—82no—
692C4—84no—
693C4—82no—
694C4—82no—
695C2—44no—
696C2—84no—
697C2—82no—
698C2#3670211yescharacteristic
699C2#3737411yescharacteristic
700C2#3745111yescharacteristic
701C2#4059312yes—
702C2#4248512yes—
703C2#5356312yes—
704C2#5383212yes—
705C2#5889714yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2535C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2409C2semidirect
3. #17201C2semidirect
4. #17105C2semidirect
5. #16807C2semidirect
6. #16694C2semidirect
7. #16520C2semidirect
8. #16470C2semidirect
9. #16288C2semidirect
10. #16265C2semidirect
11. #16263C2semidirect
12. #15312C2semidirect
13. #14873C2semidirect
14. #14865C2semidirect
15. #14344C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25352
C2×#24092
#172012
#171052
#168072
#166942
#165202
#164702
#162882
#162652
#162632
#153122
#148732
#148652
#143442