Finite Group Databasebeta

← #789607

Subgroups — #789607

Order 512.

2822 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic, complemented
2C2×Q8×C22⋊C4C211yesmaximal, characteristic
3#16694C212yesmaximal, complemented
4#16068C214yesmaximal, complemented
5#15432C214yesmaximal
6#15199C211yesmaximal, characteristic, complemented
7#15192C211yesmaximal, characteristic, complemented
8#15135C211yesmaximal, characteristic
9#15104C212yesmaximal, complemented
10#15017C212yesmaximal
11#14225C212yesmaximal
12#14142C212yesmaximal, complemented
13#13976C212yesmaximal, complemented
14#13931C211yesmaximal, characteristic, complemented
15#13922C212yesmaximal
16#13876C212yesmaximal
17C4×#397C211yesmaximal, characteristic
18C24×Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#399C2212yes—
22C2×#397C2211yescharacteristic
23#2355—28no—
24Q8×C22⋊C4—28no—
25C2×#391C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#389C2212yes—
34C2×#388C2214yes—
35C2×#387C2214yes—
36C2×#384C2211yescharacteristic
37C2×#384C2212yes—
38C2×#382C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48#1446C2214yes—
49#1446C2214yes—
50#1445C2211yescharacteristic
51#1445C2212yes—
52#1445C2212yes—
53#1445C2214yes—
54#1444C2212yes—
55#1444C2214yes—
56#1444C2212yes—
57#1444C2214yes—
58#1444C2212yes—
59#1444C2214yes—
60#1443C2212yes—
61#1443C2214yes—
62#1443C2214yes—
63#1442C2211yescharacteristic
64#1442C2214yes—
65#1442C2214yes—
66#1442C2212yes—
67#1440C2212yes—
68#1439C2212yes—
69#1439C2211yescharacteristic
70#1439C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78#1434C2212yes—
79#1434C2212yes—
80#1434C2214yes—
81C22⋊#95C2214yes—
82C22⋊#95C2214yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8—24no—
86C23×Q8—24no—
87C23×Q8—28no—
88C23×Q8—24no—
89C23×Q8—44no—
90C23×Q8C2311yescharacteristic
91C23×Q8C2312yes—
92C23×Q8C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C24×C4C2312yes—
95C2×C4×Q8—28no—
96C2×C4×Q8—24no—
97C2×C4×Q8—44no—
98C2×C4×Q8D418yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2314yes—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2314yes—
125#399—44no—
126#397Q818yescomplemented
127#391—28no—
128#391—28no—
129#391—28no—
130#391—24no—
131#391—28no—
132#391—28no—
133#391—24no—
134#391—44no—
135#389—44no—
136#388—48no—
137#387—216no—
138#384—24no—
139#384—44no—
140#382—28no—
141#380—24no—
142#380—28no—
143#380—44no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—28no—
147C4×C22⋊C4—28no—
148C4×C22⋊C4—28no—
149C4×C22⋊C4—24no—
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C25C2411yescharacteristic
185C22×Q8—232no—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—22no—
194C22×Q8—28no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—44no—
198C22×Q8—432no—
199C22×Q8—432no—
200C22×Q8—416no—
201C22×Q8—416no—
202C22×Q8C2×D414yes—
203C22×Q8#4114yes—
204C22×Q8#4118yes—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—44no—
214C23×C4#4114yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C4×Q8—232no—
232C4×Q8—416no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
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—24no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×D418yes—
265C2×C4⋊C4C2×Q814yescomplemented
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4C2×Q818yes—
283C2×C22⋊C4C2×Q814yes—
284C2×C42—28no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42C2×D418yes—
300C2×C42C2×D414yes—
301#95—24no—
302#95—22no—
303#95—22no—
304#95—24no—
305#95—28no—
306#95—24no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—24no—
319#95—22no—
320#95—24no—
321#95—28no—
322#95—24no—
323#95—42no—
324#95—42no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—42no—
333#95C2×Q818yescomplemented
334#95C2×Q814yescomplemented
335C24—21no—
336C24—21no—
337C24—22no—
338C24—22no—
339C24—21no—
340C24—22no—
341C24—21no—
342C24—22no—
343C24C22×Q812yes—
344C24C2×#4112yes—
345C24H(5,𝔽2)12yes—
346C24C2511yescharacteristic
347C2×Q8—216no—
348C2×Q8—28no—
349C2×Q8—28no—
350C2×Q8—28no—
351C2×Q8—216no—
352C2×Q8—216no—
353C2×Q8—216no—
354C2×Q8—416no—
355C2×Q8—416no—
356C2×Q8—48no—
357C2×Q8—4128no—
358C2×Q8—48no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8—48no—
362C2×Q8#123116yes—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—216no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—28no—
408C22×C4—216no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4C22×D414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×Q812yes—
453C22×C4C22×Q814yes—
454C22×C4C22×Q814yes—
455C22×C4C22×Q812yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4118yes—
459C22×C4C2×#4114yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4H(5,𝔽2)14yes—
463C22×C42−1+414yes—
464C22×C42−1+412yes—
465C4⋊C4—216no—
466C4⋊C4—28no—
467C4⋊C4—28no—
468C4⋊C4—44no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—84no—
475C4⋊C4—88no—
476C4⋊C4—88no—
477C4⋊C4—88no—
478C22⋊C4—216no—
479C22⋊C4—28no—
480C22⋊C4—48no—
481C22⋊C4—44no—
482C22⋊C4—48no—
483C22⋊C4—88no—
484C42—216no—
485C42—28no—
486C42—48no—
487C42—48no—
488C42—44no—
489C42—48no—
490C42—88no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—24no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23#55414yes—
514C23C23×D411yescharacteristic
515C23C23×Q811yescharacteristic
516C23C22×#4112yes—
517C23C22×#4111yescharacteristic
518C23C22×#4111yescharacteristic
519C23C2×H(5,𝔽2)12yes—
520C23C2×H(5,𝔽2)11yescharacteristic
521C23C2×2−1+411yescharacteristic
522C23C2×2−1+412yes—
523C23#58512yes—
524C23#58511yescharacteristic
525Q8—232no—
526Q8—432no—
527Q8—416no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—432no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—416no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
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—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4C2×#12318yes—
616C2×C4C2×#12314yes—
617C2×C4#52514yes—
618C2×C4#53514yes—
619C2×C4D4×Q814yes—
620C2×C4#55118yes—
621C2×C4#55118yes—
622C22—28no—
623C22—24no—
624C22—44no—
625C22—44no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—44no—
632C22—42no—
633C22C22×#12311yescharacteristic
634C22C2×#52511yescharacteristic
635C22C2×#53511yescharacteristic
636C22C2×D4×Q811yescharacteristic
637C22C2×#55112yes—
638C22C2×#55112yes—
639C22C2×#55411yescharacteristic
640C22Q8×#4111yescharacteristic
641C22Q8×#4112yes—
642C22#348711yescharacteristic
643C22#348811yescharacteristic
644C22#349512yes—
645C22#349911yescharacteristic
646C22#350312yes—
647C22#350711yescharacteristic
648C22#351612yes—
649C22#351612yes—
650C22#351812yes—
651C22#352212yes—
652C22#352211yescharacteristic
653C22#355212yes—
654C22#355812yes—
655C22#356212yes—
656C4—24no—
657C4—28no—
658C4—44no—
659C4—44no—
660C4—44no—
661C4—42no—
662C4—44no—
663C4—42no—
664C4—42no—
665C4—44no—
666C4—81no—
667C4—82no—
668C4—84no—
669C4—82no—
670C4—84no—
671C4#248218yes—
672C2—44no—
673C2C2×#248211yescharacteristic
674C2#3599611yescharacteristic
675C2#3603011yescharacteristic
676C2Q8×#12311yescharacteristic
677C2#3654411yescharacteristic
678C2#3961611yescharacteristic
679C2#4289011yescharacteristic
680C2#4511712yes—
681C2#4529212yes—
682C2#5927412yes—
683C2#5947712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2355C2semidirect
3. #16694C2semidirect
4. #16068C2semidirect
6. #15199C2semidirect
7. #15192C2semidirect
9. #15104C2semidirect
12. #14142C2semidirect
13. #13976C2semidirect
14. #13931C2semidirect
126. #397Q8semidirect
265. C2×C4⋊C4C2×Q8semidirect
333. #95C2×Q8semidirect
334. #95C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23552
C2×Q8×C22⋊C42
#166942
#160682
#154322
#151992
#151922
#151352
#151042
#150172
#142252
#141422
#139762
#139312
#139222
#138762
C4×#3972