Finite Group Databasebeta

← #4867136

Subgroups — #4867136

Order 512.

3172 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2543C211yesmaximal, characteristic
2C2×#2409C211yesmaximal, characteristic, complemented
3#17143C212yesmaximal, complemented
4#17084C212yesmaximal, complemented
5#16807C212yesmaximal, complemented
6#16665C212yesmaximal, complemented
7#16566C212yesmaximal, complemented
8#16462C212yesmaximal, complemented
9#16320C214yesmaximal, complemented
10#16282C212yesmaximal, complemented
11#16253C212yesmaximal, complemented
12#15277C214yesmaximal, complemented
13#14914C212yesmaximal, complemented
14#14873C212yesmaximal, complemented
15#14345C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×#121C2211yescharacteristic
19C22×#121C2212yes—
20C22×C4×D4C2211yescharacteristic
21#2543—28no—
22#2409—28no—
23C2×#398C2211yescharacteristic
24C2×#397C2211yescharacteristic
25C2×#397C2214yes—
26C2×#395C2212yes—
27C2×#394C2211yescharacteristic
28C2×#394C2214yes—
29C2×#394C2214yes—
30C2×#393C2212yes—
31C2×#393C2212yes—
32C2×#393C2212yes—
33C2×#393C2214yes—
34C2×#392C2211yescharacteristic
35C2×#391C2212yes—
36C2×#388C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#386C2214yes—
41C2×#385C2212yes—
42C2×#385C2214yes—
43C2×#383C2211yescharacteristic
44C2×#382C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C22×#95C2211yescharacteristic
47#1446C2212yes—
48#1446C2212yes—
49#1446C2214yes—
50#1445C2212yes—
51#1445C2214yes—
52#1444C2212yes—
53#1444C2212yes—
54#1443C2214yes—
55#1442C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1440C2214yes—
62#1439C2212yes—
63#1439C2212yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1438C2212yes—
71#1438C2214yes—
72#1438C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
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×D4C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—48no—
93C2×#121—28no—
94C2×#121—24no—
95C2×#121—42no—
96C2×#121—42no—
97C2×#121—48no—
98C2×C4×D4—42no—
99C2×C4×D4—42no—
100C2×C4×D4D418yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#398—24no—
126#397—24no—
127#397—48no—
128#395—28no—
129#394—24no—
130#394—48no—
131#394—48no—
132#393—28no—
133#393—44no—
134#393—44no—
135#393—48no—
136#392—24no—
137#391—28no—
138#388—28no—
139#386—28no—
140#386—28no—
141#386—28no—
142#386—48no—
143#385—28no—
144#385—48no—
145#383—42no—
146#382—42no—
147#380—24no—
148C2×#95—24no—
149C2×#95C2311yescharacteristic
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2311yescharacteristic
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C25C2412yes—
175C25C2411yescharacteristic
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—28no—
196C22×D4—28no—
197C22×D4—22no—
198C22×D4—24no—
199C22×D4—41no—
200C22×D4—41no—
201C22×D4—41no—
202C22×D4—41no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4—44no—
206C22×D4—42no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4C2×D414yes—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—28no—
217C23×C4—24no—
218C23×C4—44no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—42no—
225C23×C4—44no—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C22⋊Q8—432no—
241#121—416no—
242#121—432no—
243C4×D4—416no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
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⋊C4C2×D414yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
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⋊C4C2×D414yes—
306C2×C22⋊C4C2×D414yes—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42C2×D414yes—
310#95—24no—
311#95—22no—
312#95—22no—
313#95—24no—
314#95—28no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—42no—
323#95—42no—
324#95—42no—
325#95—44no—
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—24no—
336C24—21no—
337C24—21no—
338C24—24no—
339C24—21no—
340C24—21no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—24no—
353C24—22no—
354C24—24no—
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—416no—
379C2×D4—44no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—84no—
387C2×D4—84no—
388C2×D4—88no—
389C2×D4—88no—
390C2×D4—88no—
391C2×D4—84no—
392C2×D4—84no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—22no—
421C22×C4—24no—
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—22no—
434C22×C4—22no—
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—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—416no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—416no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—84no—
463C22×C4—84no—
464C22×C4—88no—
465C22×C4—84no—
466C22×C4—88no—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4H(5,𝔽2)18yes—
473C22×C4H(5,𝔽2)12yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C42−1+414yes—
476C4⋊C4—44no—
477C4⋊C4—416no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—88no—
482C4⋊C4—84no—
483C4⋊C4—88no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—48no—
487C22⋊C4—416no—
488C22⋊C4—48no—
489C22⋊C4—416no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—84no—
494C22⋊C4—88no—
495C22⋊C4—88no—
496C22⋊C4—88no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C42—28no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—42no—
523C23—42no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—44no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23#53514yes—
554C23#53614yes—
555C23C23×D411yescharacteristic
556C23C2×H(5,𝔽2)11yescharacteristic
557C23C2×H(5,𝔽2)11yescharacteristic
558C23C2×H(5,𝔽2)12yes—
559C23C2×H(5,𝔽2)14yes—
560C23C2×2−1+412yes—
561C23C2×2−1+412yes—
562C23#58512yes—
563Q8—432no—
564Q8—432no—
565D4—416no—
566D4—816no—
567D4—88no—
568D4—816no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—416no—
587C2×C4—432no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
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—88no—
636C2×C4—816no—
637C2×C4#53414yes—
638C22—24no—
639C22—24no—
640C22—28no—
641C22—28no—
642C22—42no—
643C22—42no—
644C22—48no—
645C22—48no—
646C22—42no—
647C22—42no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—82no—
655C22—84no—
656C22—84no—
657C22—82no—
658C22—82no—
659C22—84no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—84no—
665C22C2×#53411yescharacteristic
666C22C2×#53511yescharacteristic
667C22C2×#53611yescharacteristic
668C22#348412yes—
669C22#348912yes—
670C22#353212yes—
671C22#353212yes—
672C22#353612yes—
673C22#353614yes—
674C22#353612yes—
675C22#354114yes—
676C22#354712yes—
677C22#355011yescharacteristic
678C22#355111yescharacteristic
679C22#355214yes—
680C22#355814yes—
681C4—24no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—48no—
686C4—42no—
687C4—44no—
688C4—84no—
689C4—82no—
690C4—84no—
691C4—84no—
692C4—82no—
693C2—44no—
694C2—44no—
695C2—162no—
696C2#3671411yescharacteristic
697C2#3737611yescharacteristic
698C2#3745111yescharacteristic
699C2#4059112yes—
700C2#4059312yes—
701C2#5366912yes—
702C2#5387612yes—
703C2#5889714yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2543C2
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. #17143C2semidirect
4. #17084C2semidirect
5. #16807C2semidirect
6. #16665C2semidirect
7. #16566C2semidirect
8. #16462C2semidirect
9. #16320C2semidirect
10. #16282C2semidirect
11. #16253C2semidirect
12. #15277C2semidirect
13. #14914C2semidirect
14. #14873C2semidirect
15. #14345C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25432
C2×#24092
#171432
#170842
#168072
#166652
#165662
#164622
#163202
#162822
#162532
#152772
#149142
#148732
#143452