Finite Group Databasebeta

← #4462555

Subgroups — #4462555

Order 512.

2822 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic, complemented
2C22×#396C211yesmaximal, characteristic
3#17195C214yesmaximal, complemented
4#17188C214yesmaximal, complemented
5#17119C214yesmaximal, complemented
6#17100C214yesmaximal, complemented
7#15073C212yesmaximal, complemented
8#14936C211yesmaximal, characteristic, complemented
9#14934C211yesmaximal, characteristic, complemented
10#14918C212yesmaximal
11#14918C212yesmaximal
12#14786C212yesmaximal, complemented
13#14734C211yesmaximal, characteristic
14#14717C211yesmaximal, characteristic
15#14342C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23#2627—28no—
24C2×#400C2212yescomplemented
25C2×#399C2212yes—
26C2×#398C2212yescomplemented
27C2×#398C2212yescomplemented
28C2×#398C2214yescomplemented
29C2×#396—28no—
30C2×#396C2214yes—
31C2×#396C2214yes—
32C2×#395C2212yescomplemented
33C2×#393C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#381C2212yes—
38C2×#381C2212yes—
39C2×#381C2212yes—
40C2×#381C2212yes—
41C2×#380C2211yescharacteristic
42C2×#379C2212yes—
43C2×#379C2212yes—
44C22×#95C2211yescharacteristic
45#1446C2212yescomplemented
46#1445C2214yescomplemented
47#1445C2214yescomplemented
48#1445C2214yescomplemented
49#1445C2214yescomplemented
50#1445C2212yescomplemented
51#1444C2214yescomplemented
52#1444C2212yescomplemented
53#1442C2212yescomplemented
54#1442C2214yescomplemented
55#1442C2214yescomplemented
56#1442C2214yescomplemented
57#1441C2214yescomplemented
58#1441C2214yescomplemented
59#1441C2214yescomplemented
60#1440C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—44no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4—44no—
92C22×C4⋊C4D418yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—44no—
109C22×C22⋊C4—44no—
110C22×C22⋊C4—44no—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123#400—28no—
124#399—44no—
125#398—28no—
126#398—28no—
127#398—48no—
128#396—216no—
129#396—216no—
130#395—28no—
131#393—44no—
132#385—28no—
133#385—44no—
134#385—44no—
135#381—28no—
136#381—44no—
137#381—44no—
138#381—44no—
139#380—24no—
140#379—44no—
141#379—44no—
142C2×#95—24no—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C25—22no—
171C25—24no—
172C25—24no—
173C25—24no—
174C25—22no—
175C25—22no—
176C25—42no—
177C25—42no—
178C25—42no—
179C25C2412yes—
180C25C2411yescharacteristic
181C22×Q8—22no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8—42no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C22⋊Q8—416no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—416no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—416no—
247C2×C4⋊C4—416no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4C2×D418yes—
258C2×C4⋊C4C2×D418yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—416no—
286C2×C22⋊C4—416no—
287C2×C22⋊C4—416no—
288C2×C22⋊C4—44no—
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—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—42no—
309#95—44no—
310#95—44no—
311#95—42no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—42no—
322#95—44no—
323#95—44no—
324#95—42no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—28no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—24no—
334C24—24no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—24no—
339C24—24no—
340C24—416no—
341C24—416no—
342C24—48no—
343C24—48no—
344C24—44no—
345C24—44no—
346C24—44no—
347C24—44no—
348C24—44no—
349C24—44no—
350C24—48no—
351C24—48no—
352C24—88no—
353C24—88no—
354C24—84no—
355C24—84no—
356C24C22×D412yes—
357C24H(5,𝔽2)14yes—
358C24H(5,𝔽2)14yes—
359C242−1+414yes—
360C242−1+412yes—
361C242−1+412yes—
362C24C2511yescharacteristic
363C2×Q8—44no—
364C2×Q8—82no—
365C2×Q8—82no—
366C2×Q8—88no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—28no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—416no—
442C22×C4—44no—
443C22×C4—416no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—88no—
456C22×C4—88no—
457C22×C4C22×D414yes—
458C22×C4C22×D412yes—
459C22×C4C22×D414yes—
460C22×C4C22×D412yes—
461C22×C4H(5,𝔽2)14yes—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—416no—
465C4⋊C4—416no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C4⋊C4—82no—
469C4⋊C4—84no—
470C4⋊C4—82no—
471C4⋊C4—84no—
472C4⋊C4—88no—
473C4⋊C4—88no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—88no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—28no—
491C23—28no—
492C23—28no—
493C23—24no—
494C23—24no—
495C23—28no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—28no—
501C23—24no—
502C23—24no—
503C23—416no—
504C23—416no—
505C23—416no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—832no—
525C23—832no—
526C23—88no—
527C23—88no—
528C23—88no—
529C23—88no—
530C23#53414yes—
531C23#53514yes—
532C23#53614yes—
533C23#56218yes—
534C23#56218yes—
535C23C23×D411yescharacteristic
536C23C2×H(5,𝔽2)12yes—
537C23C2×H(5,𝔽2)12yes—
538C23C2×H(5,𝔽2)12yes—
539C23C2×2−1+411yescharacteristic
540C23C2×2−1+411yescharacteristic
541C23C2×2−1+412yes—
542C23#58512yes—
543C23#58512yes—
544Q8—88no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—28no—
559C2×C4—416no—
560C2×C4—416no—
561C2×C4—416no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
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—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—88no—
618C2×C4—88no—
619C2×C4—88no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—216no—
626C22—216no—
627C22—24no—
628C22—44no—
629C22—44no—
630C22—48no—
631C22—48no—
632C22—44no—
633C22—832no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—84no—
639C22—84no—
640C22#281618yes—
641C22C2×#53411yescharacteristic
642C22C2×#53511yescharacteristic
643C22C2×#53611yescharacteristic
644C22#348411yescharacteristic
645C22#348511yescharacteristic
646C22#348611yescharacteristic
647C22#348711yescharacteristic
648C22C2×#56212yes—
649C22C2×#56212yes—
650C22#353414yes—
651C22#353514yes—
652C22#353514yes—
653C22#353812yes—
654C22#353812yes—
655C22#354214yes—
656C22#356412yes—
657C22#356412yes—
658C4—44no—
659C4—44no—
660C4—44no—
661C4—42no—
662C4—42no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—84no—
669C4—84no—
670C4—82no—
671C2—28no—
672C2—84no—
673C2C2×#281611yescharacteristic
674C2#3670811yescharacteristic
675C2#3671511yescharacteristic
676C2#3696211yescharacteristic
677C2#3711811yescharacteristic
678C2#3718511yescharacteristic
679C2#3718611yescharacteristic
680C2#5234814yes—
681C2#5237414yes—
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×#2627C2semidirect
3. #17195C2semidirect
4. #17188C2semidirect
5. #17119C2semidirect
6. #17100C2semidirect
7. #15073C2semidirect
8. #14936C2semidirect
9. #14934C2semidirect
12. #14786C2semidirect
15. #14342C2semidirect
24. C2×#400C22semidirect
26. C2×#398C22semidirect
27. C2×#398C22semidirect
28. C2×#398C22semidirect
32. C2×#395C22semidirect
45. #1446C22semidirect
46. #1445C22semidirect
47. #1445C22semidirect
48. #1445C22semidirect
49. #1445C22semidirect
50. #1445C22semidirect
51. #1444C22semidirect
52. #1444C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
56. #1442C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
C22×#3962
#171952
#171882
#171192
#171002
#150732
#149362
#149342
#149182
#147862
#147342
#147172
#143422