Finite Group Databasebeta

← #922739

Subgroups — #922739

Order 512.

2556 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17235C212yesmaximal, complemented
2#17121C212yesmaximal, complemented
3#16838C214yesmaximal, complemented
4#16548C214yesmaximal, complemented
5#16217C212yesmaximal, complemented
6#16047C212yesmaximal, complemented
7#15150C211yesmaximal, characteristic, complemented
8#15135C211yesmaximal, characteristic, complemented
9#15132C212yesmaximal, complemented
10#15131C212yesmaximal, complemented
11#14337C212yesmaximal
12#13993C212yesmaximal, complemented
13#13973C212yesmaximal, complemented
14#13876C212yesmaximal, complemented
15#13851C211yesmaximal, characteristic
16C23×#41C2211yescharacteristic
17C2×#399C2212yes—
18C2×#397C2214yes—
19C2×#397C2212yes—
20C2×#397C2212yes—
21C2×#396C2214yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#388C2214yes—
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C22⋊C4C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic
38#1446C2214yes—
39#1446C2211yescharacteristic
40#1446C2211yescharacteristic
41#1445C2212yes—
42#1445C2212yes—
43#1445C2212yes—
44#1444C2212yes—
45#1443C2214yes—
46#1443C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1440C2214yes—
53#1439C2214yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2214yes—
59#1438C2214yes—
60#1438C2214yes—
61#1438C2212yes—
62#1438C2211yescharacteristic
63#1438C2212yes—
64#1438C2211yescharacteristic
65#1437C2214yes—
66#1437C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2211yescharacteristic
74#1434C2214yes—
75C22×#41—28no—
76C22×#41—24no—
77C22×#41—42no—
78C22×#41—44no—
79C22×#41—42no—
80C23×Q8C2311yescharacteristic
81C23×D4C2312yes—
82C23×D4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2311yescharacteristic
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2311yescharacteristic
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C42C2311yescharacteristic
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2314yes—
112#399—44no—
113#397—44no—
114#397—48no—
115#397—44no—
116#396—48no—
117#389—44no—
118#389—44no—
119#388—48no—
120#384—42no—
121#384—42no—
122#380—28no—
123#380—28no—
124#380—44no—
125#380—44no—
126#380—44no—
127#380—44no—
128C4×C4⋊C4—44no—
129C4×C22⋊C4—28no—
130C4×C22⋊C4—44no—
131C4×C22⋊C4—42no—
132C4×C22⋊C4—42no—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2311yescharacteristic
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2311yescharacteristic
157C2×#95C2311yescharacteristic
158C2×#95C2311yescharacteristic
159C2×#95C2314yes—
160C25C2412yes—
161C25C2411yescharacteristic
162C2×#41—416no—
163C2×#41—432no—
164C2×#41—432no—
165C2×#41—416no—
166C2×#41—416no—
167C22×Q8—22no—
168C22×Q8—24no—
169C22×Q8—41no—
170C22×Q8—42no—
171C22×Q8—41no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—22no—
176C22×D4—24no—
177C22×D4—42no—
178C22×D4—42no—
179C22×D4—42no—
180C22×D4—41no—
181C22×D4—42no—
182C22×D4—41no—
183C22×D4—42no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4#4118yes—
195C23×C4#4114yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42—44no—
287#95—24no—
288#95—24no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—24no—
294#95—28no—
295#95—41no—
296#95—42no—
297#95—42no—
298#95—41no—
299#95—41no—
300#95—41no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—42no—
311#95—42no—
312#95—44no—
313#95—42no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—21no—
320C24—21no—
321C24—21no—
322C24—22no—
323C24—21no—
324C24—22no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—21no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24C2×#4112yes—
339C24C2×#4114yes—
340C24C2511yescharacteristic
341#41—4128no—
342C2×Q8—44no—
343C2×Q8—48no—
344C2×Q8—44no—
345C2×Q8—48no—
346C2×Q8—44no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—44no—
351C2×D4—48no—
352C2×D4—44no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—44no—
358C2×D4—48no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—216no—
392C22×C4—28no—
393C22×C4—216no—
394C22×C4—216no—
395C22×C4—216no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
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—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4C2×#4112yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4112yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C4H(5,𝔽2)12yes—
444C22×C4H(5,𝔽2)12yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C4H(5,𝔽2)14yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C42−1+414yes—
449C22×C42−1+412yes—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—84no—
455C4⋊C4—84no—
456C4⋊C4—84no—
457C4⋊C4—88no—
458C4⋊C4—84no—
459C4⋊C4—88no—
460C22⋊C4—216no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C42—48no—
474C42—44no—
475C42—44no—
476C42—88no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—22no—
487C23—24no—
488C23—22no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23—42no—
508C23—41no—
509C23—42no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23—41no—
531C23—42no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23C22×#4111yescharacteristic
536C23C22×#4112yes—
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×H(5,𝔽2)11yescharacteristic
539C23C2×H(5,𝔽2)12yes—
540C23C2×H(5,𝔽2)11yescharacteristic
541C23C2×H(5,𝔽2)11yescharacteristic
542C23C2×H(5,𝔽2)11yescharacteristic
543C23C2×2−1+412yes—
544C23C2×2−1+411yescharacteristic
545C23#58512yes—
546Q8—416no—
547D4—416no—
548D4—432no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—432no—
589C2×C4—416no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4#53818yes—
626C2×C4#53814yes—
627C2×C4#55114yes—
628C2×C4#55118yes—
629C2×C4#56014yes—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—42no—
637C22—44no—
638C22—42no—
639C22—44no—
640C22—42no—
641C22—44no—
642C22—81no—
643C22—82no—
644C22—81no—
645C22—81no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—81no—
655C22—82no—
656C22C2×#53811yescharacteristic
657C22C2×#53812yes—
658C22#346112yes—
659C22C2×#55111yescharacteristic
660C22C2×#55112yes—
661C22#349212yes—
662C22#349412yes—
663C22#350712yes—
664C22#350811yescharacteristic
665C22#351112yes—
666C22#351111yescharacteristic
667C22C2×#56011yescharacteristic
668C22#353611yescharacteristic
669C22#353711yescharacteristic
670C22#353812yes—
671C22#354112yes—
672C22#354212yes—
673C22#354612yes—
674C22#354812yes—
675C22#354811yescharacteristic
676C22#354911yescharacteristic
677C22#355011yescharacteristic
678C22#355111yescharacteristic
679C4—42no—
680C4—42no—
681C4—42no—
682C4—42no—
683C4—44no—
684C4—44no—
685C4—42no—
686C4—84no—
687C4—82no—
688C4—84no—
689C4—84no—
690C4—84no—
691C4#273518yes—
692C2—84no—
693C2—82no—
694C2#3731611yescharacteristic
695C2#3751411yescharacteristic
696C2#3797012yes—
697C2#3956112yes—
698C2#4244912yes—
699C2#4292312yes—
700C2#4511511yescharacteristic
701C2#4533011yescharacteristic
702C2C2×#273511yescharacteristic
703C2#5296611yescharacteristic
704C2#5300711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17235C2semidirect
2. #17121C2semidirect
3. #16838C2semidirect
4. #16548C2semidirect
5. #16217C2semidirect
6. #16047C2semidirect
7. #15150C2semidirect
8. #15135C2semidirect
9. #15132C2semidirect
10. #15131C2semidirect
12. #13993C2semidirect
13. #13973C2semidirect
14. #13876C2semidirect

Maximal subgroups

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

subgroupindex
#172352
#171212
#168382
#165482
#162172
#160472
#151502
#151352
#151322
#151312
#143372
#139932
#139732
#138762
#138512