Finite Group Databasebeta

← #663909

Subgroups — #663909

Order 512.

3642 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic, complemented
2C2×#2432C211yesmaximal, characteristic
3C2×#2339C211yesmaximal, characteristic
4C2×#2297C211yesmaximal, characteristic
5#15538C214yesmaximal, complemented
6#15383C214yesmaximal, complemented
7#15188C212yesmaximal, complemented
8#15186C212yesmaximal, complemented
9#14965C214yesmaximal, complemented
10#14138C211yesmaximal, characteristic, complemented
11#14129C211yesmaximal, characteristic, complemented
12#14047C214yesmaximal, complemented
13#14027C212yesmaximal, complemented
14#13920C212yesmaximal, complemented
15C4×#381C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×D4C2212yes—
21C22×#117C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2855—28no—
24#2432—28no—
25C2×#394C2214yes—
26#2339—28no—
27#2297—28no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2211yescharacteristic
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#390C2211yescharacteristic
34C2×#390C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#387C2214yes—
38C2×#386C2212yes—
39C2×#386C2214yes—
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#385C2214yes—
43C2×#381C2212yes—
44C2×#381C2212yes—
45C2×#381C2212yes—
46C2×#380C2214yes—
47C2×#380C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2214yes—
51C22×#95C2212yes—
52#1445C2214yes—
53#1443C2214yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1440C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1436C2214yes—
67#1436C2212yes—
68#1436C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2212yes—
72#1434C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C4×#95C2212yes—
78C4×#95C2212yes—
79C22×#41—28no—
80C22×#41—24no—
81C22×#41—24no—
82C22×#41—44no—
83C22×#41D418yes—
84C23×Q8C2311yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×D4C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C24×C4C2314yes—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—44no—
95C2×#124—24no—
96C2×#124—28no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C4×D4—28no—
101C2×C4×D4—28no—
102C2×C4×D4—44no—
103C2×#117—24no—
104C2×#117—44no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C42—24no—
123C22×C42—24no—
124C22×C42—24no—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131#394—216no—
132#391—24no—
133#391—28no—
134#391—28no—
135#391—28no—
136#391—44no—
137#390—24no—
138#390—28no—
139#389—28no—
140#389—44no—
141#387—216no—
142#386—28no—
143#386—28no—
144#386—216no—
145#385—216no—
146#385—28no—
147#381—28no—
148#381—28no—
149#381—28no—
150#380—216no—
151#380D418yes—
152C4×C22⋊C4—24no—
153C4×C22⋊C4—24no—
154C4×C22⋊C4—216no—
155C2×#95—28no—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2311yescharacteristic
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C25C2411yescharacteristic
184C25C2412yes—
185C2×#41—232no—
186C2×#41—232no—
187C2×#41—416no—
188C2×#41—432no—
189C2×#41—432no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—24no—
195C22×Q8—22no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8C2×D414yes—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—28no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—24no—
206C22×D4—44no—
207C22×D4C2×D418yes—
208C22×D4C2×D414yes—
209C22×D4#4118yes—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—28no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—28no—
224C23×C4—28no—
225C23×C4—28no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4C2×D418yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4#4114yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2414yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C4⋊Q8—416no—
251#124—416no—
252C22⋊Q8—416no—
253C4×D4—432no—
254#117—416no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4C2×D418yes—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4C2×D418yes—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—44no—
319C2×C42—416no—
320#95—22no—
321#95—28no—
322#95—22no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—42no—
338#95—42no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95C2×D418yes—
346#95#4118yescomplemented
347C24—22no—
348C24—22no—
349C24—24no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—24no—
354C24—24no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—22no—
359C24—24no—
360C24C22×D412yes—
361C24C22×D412yes—
362C24C22×D414yes—
363C24C2×#4112yes—
364C24C2×#4118yes—
365C24C2511yescharacteristic
366#41—4128no—
367C2×Q8—28no—
368C2×Q8—28no—
369C2×Q8—28no—
370C2×Q8—44no—
371C2×Q8—44no—
372C2×Q8—44no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—48no—
377C2×D4—216no—
378C2×D4—216no—
379C2×D4—28no—
380C2×D4—28no—
381C2×D4—216no—
382C2×D4—216no—
383C2×D4—48no—
384C2×D4—416no—
385C2×D4—416no—
386C2×D4#123116yes—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—216no—
392C22×C4—216no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—216no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—216no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—48no—
463C22×C4—416no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—416no—
469C22×C4—416no—
470C22×C4—416no—
471C22×C4—416no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—88no—
477C22×C4—88no—
478C22×C4—88no—
479C22×C4—88no—
480C22×C4#121116yes—
481C22×C4C22×D414yes—
482C22×C4C22×D418yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C2×#4114yes—
488C22×C4C2×#4114yes—
489C22×C4C2×#4114yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4114yes—
493C22×C4H(5,𝔽2)14yes—
494C22×C42−1+414yes—
495C4⋊C4—216no—
496C4⋊C4—416no—
497C4⋊C4—416no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—88no—
502C22⋊C4—48no—
503C22⋊C4—416no—
504C22⋊C4—48no—
505C22⋊C4—416no—
506C22⋊C4—48no—
507C22⋊C4—416no—
508C22⋊C4—416no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—416no—
512C42—48no—
513C42—48no—
514C42—48no—
515C42—44no—
516C42—44no—
517C42—48no—
518C23—24no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—22no—
524C23—28no—
525C23—24no—
526C23—28no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—28no—
531C23—28no—
532C23—28no—
533C23—22no—
534C23—22no—
535C23—28no—
536C23—24no—
537C23—24no—
538C23—28no—
539C23—28no—
540C23—42no—
541C23—42no—
542C23—44no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—44no—
548C23—42no—
549C23—44no—
550C23—44no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23C2×#12114yes—
559C23C2×#12318yes—
560C23C23×D411yescharacteristic
561C23C23×D411yescharacteristic
562C23C22×#4111yescharacteristic
563C23C22×#4112yes—
564C23C22×#4112yes—
565C23C2×H(5,𝔽2)12yes—
566C23C2×2−1+412yes—
567C23#58512yes—
568C23#58512yes—
569Q8—416no—
570Q8—88no—
571D4—232no—
572D4—432no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—232no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—28no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—432no—
597C2×C4—416no—
598C2×C4—416no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—832no—
653C2×C4C2×#12114yes—
654C2×C4C2×#12114yes—
655C2×C4C2×#12314yes—
656C2×C4#53414yes—
657C2×C4#54618yes—
658C2×C4#55118yes—
659C22—24no—
660C22—24no—
661C22—28no—
662C22—216no—
663C22—24no—
664C22—44no—
665C22—48no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—48no—
670C22—44no—
671C22—44no—
672C22—48no—
673C22—48no—
674C22—44no—
675C22—48no—
676C22—44no—
677C22C22×#12111yescharacteristic
678C22C22×#12311yescharacteristic
679C22#344411yescharacteristic
680C22C2×#53411yescharacteristic
681C22#345511yescharacteristic
682C22C2×#54612yes—
683C22D4×#4112yes—
684C22D4×#4112yes—
685C22C2×#55112yes—
686C22#347512yes—
687C22#348212yes—
688C22#348511yescharacteristic
689C22#348611yescharacteristic
690C22#348912yes—
691C22#349512yes—
692C22#351112yes—
693C22#351412yes—
694C22#351512yes—
695C22#352112yes—
696C22#355412yes—
697C22#356212yes—
698C4—24no—
699C4—44no—
700C4—44no—
701C4—44no—
702C4—44no—
703C4—48no—
704C4—44no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—84no—
709C4—82no—
710C4#241218yes—
711C2—44no—
712C2—48no—
713C2C2×#241211yescharacteristic
714C2#3474311yescharacteristic
715C2#3563311yescharacteristic
716C2#3621712yes—
717C2#3639012yes—
718C2#3806012yes—
719C2#3986412yes—
720C2#5909812yes—
721C2#5910312yes—
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. C2×#2855C2semidirect
5. #15538C2semidirect
6. #15383C2semidirect
7. #15188C2semidirect
8. #15186C2semidirect
9. #14965C2semidirect
10. #14138C2semidirect
11. #14129C2semidirect
12. #14047C2semidirect
13. #14027C2semidirect
14. #13920C2semidirect
15. C4×#381C2semidirect
346. #95#41semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×#24322
C2×#23392
C2×#22972
#155382
#153832
#151882
#151862
#149652
#141382
#141292
#140472
#140272
#139202
C4×#3812