Finite Group Databasebeta

← #923335

Subgroups — #923335

Order 512.

2620 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17281C212yesmaximal, complemented
2#17159C212yesmaximal, complemented
3#17049C214yesmaximal, complemented
4#16544C214yesmaximal, complemented
5#16185C212yesmaximal, complemented
6#15569C212yesmaximal, complemented
7#15195C211yesmaximal, characteristic, complemented
8#15132C212yesmaximal, complemented
9#15125C211yesmaximal, characteristic, complemented
10#15125C212yesmaximal, complemented
11#14337C212yesmaximal
12#13994C212yesmaximal, complemented
13#13973C212yesmaximal, complemented
14#13876C212yesmaximal, complemented
15#13851C211yesmaximal, characteristic
16C23×#41C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C2×#397C2214yes—
19C2×#393C2212yes—
20C2×#393C2214yes—
21C2×#389C2212yes—
22C2×#388C2214yes—
23C2×#388C2214yes—
24C2×#384C2211yescharacteristic
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C22⋊C4C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36#1446C2214yes—
37#1446C2214yes—
38#1446C2212yes—
39#1446C2211yescharacteristic
40#1446C2211yescharacteristic
41#1443C2214yes—
42#1442C2212yes—
43#1442C2212yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2212yes—
49#1442C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2212yes—
56#1439C2212yes—
57#1439C2212yes—
58#1438C2214yes—
59#1438C2214yes—
60#1438C2214yes—
61#1438C2211yescharacteristic
62#1438C2212yes—
63#1438C2212yes—
64#1438C2211yescharacteristic
65#1437C2214yes—
66#1436C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2211yescharacteristic
73#1434C2214yes—
74#1434C2214yes—
75C22×#41—28no—
76C22×#41—24no—
77C22×#41—42no—
78C22×#41—44no—
79C22×#41—42no—
80C23×Q8C2311yescharacteristic
81C23×Q8C2312yes—
82C23×D4C2311yescharacteristic
83C23×D4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C2×C4⋊Q8—42no—
87C2×C4⋊Q8—44no—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2311yescharacteristic
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2314yes—
113#397—48no—
114#393—48no—
115#393—44no—
116#389—44no—
117#388—48no—
118#388—48no—
119#384—42no—
120#380—28no—
121#380—28no—
122#380—44no—
123#380—44no—
124#380—44no—
125#380—44no—
126C4×C4⋊C4—44no—
127C4×C22⋊C4—28no—
128C4×C22⋊C4—42no—
129C4×C22⋊C4—44no—
130C4×C22⋊C4—42no—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2311yescharacteristic
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2311yescharacteristic
154C2×#95C2311yescharacteristic
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2314yes—
159C25C2412yes—
160C25C2411yescharacteristic
161C2×#41—416no—
162C2×#41—432no—
163C2×#41—432no—
164C2×#41—416no—
165C2×#41—416no—
166C22×Q8—22no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—41no—
170C22×Q8—42no—
171C22×Q8—41no—
172C22×Q8—42no—
173C22×Q8—44no—
174C22×Q8—44no—
175C22×Q8—42no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—22no—
180C22×D4—24no—
181C22×D4—42no—
182C22×D4—42no—
183C22×D4—42no—
184C22×D4—41no—
185C22×D4—42no—
186C22×D4—41no—
187C22×D4—42no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4#4118yes—
199C23×C4#4114yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C4⋊Q8—416no—
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—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
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×C42—24no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—28no—
282C2×C42—44no—
283C2×C42—44no—
284#95—24no—
285#95—24no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—44no—
295#95—42no—
296#95—41no—
297#95—41no—
298#95—41no—
299#95—42no—
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—42no—
310#95—44no—
311#95—44no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—21no—
320C24—21no—
321C24—22no—
322C24—21no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24C2×#4112yes—
337C24C2×#4114yes—
338C24C2511yescharacteristic
339#41—4128no—
340C2×Q8—44no—
341C2×Q8—48no—
342C2×Q8—44no—
343C2×Q8—48no—
344C2×Q8—44no—
345C2×Q8—48no—
346C2×Q8—84no—
347C2×Q8—88no—
348C2×Q8—88no—
349C2×Q8—84no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—44no—
354C2×D4—48no—
355C2×D4—44no—
356C2×D4—48no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—44no—
361C2×D4—48no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—216no—
395C22×C4—28no—
396C22×C4—216no—
397C22×C4—216no—
398C22×C4—216no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4C2×#4112yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4112yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C4H(5,𝔽2)14yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C4H(5,𝔽2)14yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C42−1+412yes—
451C22×C42−1+412yes—
452C22×C42−1+412yes—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—84no—
459C4⋊C4—84no—
460C4⋊C4—88no—
461C22⋊C4—216no—
462C22⋊C4—44no—
463C22⋊C4—48no—
464C22⋊C4—44no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—416no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C42—48no—
475C42—44no—
476C42—44no—
477C42—88no—
478C23—22no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—41no—
536C23C22×#4111yescharacteristic
537C23C22×#4112yes—
538C23C2×H(5,𝔽2)11yescharacteristic
539C23C2×H(5,𝔽2)11yescharacteristic
540C23C2×H(5,𝔽2)12yes—
541C23C2×H(5,𝔽2)12yes—
542C23C2×H(5,𝔽2)11yescharacteristic
543C23C2×2−1+411yescharacteristic
544C23C2×2−1+411yescharacteristic
545C23C2×2−1+411yescharacteristic
546C23#58512yes—
547Q8—416no—
548Q8—816no—
549D4—416no—
550D4—432no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—432no—
591C2×C4—416no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4#53818yes—
628C2×C4#53814yes—
629C2×C4#55114yes—
630C2×C4#55118yes—
631C2×C4H(3,𝔽4)14yes—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—42no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—42no—
642C22—42no—
643C22—44no—
644C22—81no—
645C22—82no—
646C22—82no—
647C22—81no—
648C22—81no—
649C22—82no—
650C22—81no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22C2×#53811yescharacteristic
659C22C2×#53812yes—
660C22#346012yes—
661C22C2×#55111yescharacteristic
662C22C2×#55112yes—
663C22#349212yes—
664C22#349412yes—
665C22#350712yes—
666C22#350911yescharacteristic
667C22#351111yescharacteristic
668C22#351112yes—
669C22C2×H(3,𝔽4)11yescharacteristic
670C22#353611yescharacteristic
671C22#353612yes—
672C22#353811yescharacteristic
673C22#354112yes—
674C22#354212yes—
675C22#354712yes—
676C22#354811yescharacteristic
677C22#354912yes—
678C22#354911yescharacteristic
679C22#355011yescharacteristic
680C22#355111yescharacteristic
681C4—42no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—44no—
687C4—44no—
688C4—84no—
689C4—84no—
690C4—84no—
691C4—82no—
692C4—84no—
693C4#273618yes—
694C2—84no—
695C2—82no—
696C2#3732411yescharacteristic
697C2#3751311yescharacteristic
698C2#3796912yes—
699C2#3956312yes—
700C2#4244612yes—
701C2#4292312yes—
702C2#4491411yescharacteristic
703C2#4511511yescharacteristic
704C2C2×#273611yescharacteristic
705C2#5300511yescharacteristic
706C2#5301511yescharacteristic
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. #17281C2semidirect
2. #17159C2semidirect
3. #17049C2semidirect
4. #16544C2semidirect
5. #16185C2semidirect
6. #15569C2semidirect
7. #15195C2semidirect
8. #15132C2semidirect
9. #15125C2semidirect
10. #15125C2semidirect
12. #13994C2semidirect
13. #13973C2semidirect
14. #13876C2semidirect

Maximal subgroups

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

subgroupindex
#172812
#171592
#170492
#165442
#161852
#155692
#151952
#151322
#151252
#143372
#139942
#139732
#138762
#138512