Finite Group Databasebeta

← #920681

Subgroups — #920681

Order 512.

2844 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2C2×#2398C211yesmaximal, characteristic
3#17201C212yesmaximal, complemented
4#17198C212yesmaximal, complemented
5#16874C214yesmaximal, complemented
6#16432C214yesmaximal
7#15579C212yesmaximal, complemented
8#15568C212yesmaximal, complemented
9#15201C212yesmaximal, complemented
10#15183C212yesmaximal, complemented
11#15179C212yesmaximal, complemented
12#14261C212yesmaximal, complemented
13#14138C212yesmaximal, complemented
14#13922C212yesmaximal, complemented
15#13871C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117C2211yescharacteristic
21#2409—28no—
22#2398—28no—
23C2×#400C2212yes—
24C2×#397C2211yescharacteristic
25C2×#397C2211yescharacteristic
26C2×#395C2212yes—
27C2×#394C2212yes—
28C2×#393C2211yescharacteristic
29C2×#393C2212yes—
30C2×#393C2211yescharacteristic
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#389C2212yes—
37C2×#386C2212yes—
38C2×#385C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2212yes—
42C2×C4×C22⋊C4C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C2×#376C2211yescharacteristic
45#1446C2214yes—
46#1446C2214yes—
47#1446C2214yes—
48#1445C2214yes—
49#1443C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1440C2212yes—
60#1440C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1438C2212yes—
67#1437C2212yes—
68#1436C2212yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C23×Q8—28no—
78C23×Q8—24no—
79C23×Q8—42no—
80C23×Q8—42no—
81C23×Q8—44no—
82C23×Q8C2312yes—
83C23×Q8C2312yes—
84C23×Q8C2311yescharacteristic
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—44no—
92C2×C4×Q8—24no—
93C2×C4×Q8—42no—
94C2×C4×Q8—42no—
95C2×C4×D4—42no—
96C2×C4×D4—42no—
97C2×C4×D4D418yes—
98C2×#117—44no—
99C2×#117—42no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123#400—28no—
124#397—24no—
125#397—24no—
126#395—28no—
127#394—28no—
128#393—24no—
129#393—24no—
130#393—28no—
131#391—28no—
132#391—28no—
133#391—44no—
134#391—44no—
135#391—44no—
136#389—44no—
137#386—28no—
138#385—28no—
139#380—24no—
140#380—44no—
141#380—42no—
142C4×C22⋊C4—28no—
143C4×C22⋊C4—28no—
144#376—24no—
145C2×#95C2312yes—
146C2×#95C2311yescharacteristic
147C2×#95C2311yescharacteristic
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C25C2411yescharacteristic
175C25C2411yescharacteristic
176C22×Q8—24no—
177C22×Q8—22no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—28no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—41no—
187C22×Q8—41no—
188C22×Q8—44no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×Q8—432no—
193C22×Q8—416no—
194C22×Q8—416no—
195C22×Q8—432no—
196C22×Q8—416no—
197C22×Q8#4118yes—
198C22×D4—24no—
199C22×D4—41no—
200C22×D4—41no—
201C22×D4—42no—
202C22×D4C2×D414yes—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C22⋊Q8—432no—
232C4×Q8—416no—
233C4×D4—416no—
234#117—416no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
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⋊C4C2×D414yes—
259C2×C22⋊C4—28no—
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—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4C2×D414yes—
288C2×C22⋊C4C2×D414yes—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—42no—
298C2×C42—44no—
299C2×C42C2×D414yes—
300#95—22no—
301#95—22no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—42no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329C24—22no—
330C24—22no—
331C24—21no—
332C24—21no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—21no—
339C24—21no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24C22×D412yes—
345C24C22×D412yes—
346C242−1+414yes—
347C24C2511yescharacteristic
348C2×Q8—216no—
349C2×Q8—216no—
350C2×Q8—216no—
351C2×Q8—48no—
352C2×Q8—44no—
353C2×Q8—416no—
354C2×Q8—48no—
355C2×Q8—44no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—416no—
359C2×Q8—44no—
360C2×Q8—4128no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—48no—
364C2×Q8—48no—
365C2×Q8—48no—
366C2×D4—28no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—48no—
370C2×D4—48no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—216no—
402C22×C4—216no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—48no—
446C22×C4—416no—
447C22×C4—416no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C2×#4118yes—
462C22×C4C2×#4114yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C4H(5,𝔽2)14yes—
466C22×C42−1+414yes—
467C4⋊C4—44no—
468C4⋊C4—416no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—88no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—416no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—88no—
485C22⋊C4—84no—
486C42—28no—
487C42—48no—
488C42—48no—
489C42—48no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—24no—
505C23—42no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23#53614yes—
532C23C23×D411yescharacteristic
533C23C22×#4112yes—
534C23C2×H(5,𝔽2)11yescharacteristic
535C23C2×H(5,𝔽2)11yescharacteristic
536C23C2×H(5,𝔽2)12yes—
537C23C2×2−1+412yes—
538C23C2×2−1+412yes—
539C23#58512yes—
540C23#58512yes—
541Q8—416no—
542Q8—432no—
543Q8—432no—
544D4—416no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—432no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—416no—
581C2×C4—416no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4#53414yes—
625C2×C4#53818yes—
626C2×C4#54618yes—
627C2×C4#55114yes—
628C2×C4H(3,𝔽4)14yes—
629C22—24no—
630C22—28no—
631C22—44no—
632C22—44no—
633C22—42no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—44no—
641C22—44no—
642C22—42no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22C2×#53411yescharacteristic
648C22C2×#53611yescharacteristic
649C22#344911yescharacteristic
650C22C2×#53812yes—
651C22C2×#54612yes—
652C22C2×#55111yescharacteristic
653C22#348912yes—
654C22#351612yes—
655C22#351812yes—
656C22#352312yes—
657C22#352412yes—
658C22C2×H(3,𝔽4)11yescharacteristic
659C22#353811yescharacteristic
660C22#354011yescharacteristic
661C22#354112yes—
662C22#354112yes—
663C22#354112yes—
664C22#355212yes—
665C22#355412yes—
666C22#356112yes—
667C22#356212yes—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—48no—
672C4—42no—
673C4—44no—
674C4—82no—
675C4—84no—
676C4—82no—
677C4—82no—
678C4—84no—
679C4—84no—
680C4#268418yes—
681C2—44no—
682C2—82no—
683C2#3737411yescharacteristic
684C2#3743811yescharacteristic
685C2#3802712yes—
686C2#3803512yes—
687C2#4285112yes—
688C2#4286912yes—
689C2#4528212yes—
690C2C2×#268411yescharacteristic
691C2#5314412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17201C2semidirect
4. #17198C2semidirect
5. #16874C2semidirect
7. #15579C2semidirect
8. #15568C2semidirect
9. #15201C2semidirect
10. #15183C2semidirect
11. #15179C2semidirect
12. #14261C2semidirect
13. #14138C2semidirect
14. #13922C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C2×#23982
#172012
#171982
#168742
#164322
#155792
#155682
#152012
#151832
#151792
#142612
#141382
#139222
#138712