Finite Group Databasebeta

← #613650

Subgroups — #613650

Order 512.

7018 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4×D4C211yesmaximal, characteristic
2C2×#2580C212yesmaximal, complemented
3C2×C4×C22⋊Q8C211yesmaximal, characteristic, complemented
4#15203C214yesmaximal, complemented
5#14865C212yesmaximal, complemented
6#14807C212yesmaximal, complemented
7#14734C212yesmaximal, complemented
8#14717C212yesmaximal, complemented
9#14693C214yesmaximal, complemented
10#14682C211yesmaximal, characteristic, complemented
11#14130C212yesmaximal, complemented
12#14018C214yesmaximal, complemented
13#13941C212yesmaximal, complemented
14#13902C212yesmaximal
15C24×D4C2211yescharacteristic
16C25×C4C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic, complemented
19C22×C4×D4—28no—
20C22×C4×D4—216no—
21C22×C4×D4C2212yes—
22C22×C4×D4C2212yes—
23C22×C4×D4C2212yes—
24C22×C4×D4C2212yes—
25C23×C4⋊C4C2211yescharacteristic
26C23×C22⋊C4C2212yes—
27C23×C42C2211yescharacteristic
28#2580—216no—
29C2×#399C2214yescomplemented
30C2×#393C2214yescomplemented
31C4×C22⋊Q8—28no—
32C2×#391C2211yescharacteristic, complemented
33C2×#390C2214yes—
34C2×#390C2214yes—
35C2×#389C2214yescomplemented
36C2×#389C2211yescharacteristic, complemented
37C2×#387C2212yescomplemented
38C2×#385C2212yes—
39C2×#385C2214yes—
40C2×#385C2218yescomplemented
41C2×#385C2218yescomplemented
42C2×#382C2212yescomplemented
43C2×#380C2212yes—
44C2×#380C2211yescharacteristic
45C2×#379C2214yes—
46C2×#379C2214yes—
47C2×C4×C4⋊C4C2211yescharacteristic, complemented
48C2×C4×C4⋊C4C2212yescomplemented
49C2×C4×C22⋊C4C2212yes—
50#1444C2214yescomplemented
51#1442C2214yescomplemented
52#1439C2214yescomplemented
53#1439C2218yescomplemented
54#1439C2218yescomplemented
55#1439C2214yescomplemented
56#1438C2214yes—
57#1438C2214yes—
58C4⋊#95C2214yescomplemented
59C4⋊#95C2212yescomplemented
60#1434C2212yescomplemented
61#1434C2212yescomplemented
62C22⋊#95C2214yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C22⋊#95C2218yescomplemented
66C22⋊#95C2218yescomplemented
67C26C2312yes—
68C23×Q8C2311yescharacteristic
69C23×D4—28no—
70C23×D4—28no—
71C23×D4—24no—
72C23×D4—24no—
73C23×D4Q818yescomplemented
74C23×D4C2312yes—
75C23×D4C2312yes—
76C24×C4—216no—
77C24×C4—28no—
78C24×C4—48no—
79C24×C4—48no—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2314yes—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2314yes—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—28no—
88C2×C4×Q8—28no—
89C2×C4×Q8—24no—
90C2×C4×D4—2128no—
91C2×C4×D4—28no—
92C2×C4×D4—216no—
93C2×C4×D4—28no—
94C2×C4×D4—216no—
95C2×C4×D4—28no—
96C2×C4×D4—28no—
97C2×C4×D4—28no—
98C2×C4×D4—28no—
99C2×C4×D4—28no—
100C2×C4×D4D4116yes—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2318yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—28no—
114C22×C22⋊C4—216no—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2318yes—
119C22×C22⋊C4C2318yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2318yes—
123C22×C22⋊C4C2318yes—
124C22×C42—28no—
125C22×C42D418yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#399—216no—
133#393—216no—
134#391—24no—
135#390—216no—
136#390—216no—
137#389—216no—
138#389—24no—
139#387—28no—
140#385—232no—
141#385—28no—
142#385—416no—
143#385—48no—
144#382—28no—
145#380—24no—
146#380—28no—
147#379—216no—
148#379—48no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—24no—
151C4×C22⋊C4—28no—
152C2×#95C2314yes—
153C2×#95C2318yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2318yes—
163C2×#95C2314yes—
164C2×#95C2318yes—
165C25—28no—
166C25—24no—
167C25—28no—
168C25—24no—
169C25—28no—
170C25—28no—
171C25—44no—
172C25—44no—
173C25C2×Q818yes—
174C25C2414yes—
175C25C2411yescharacteristic
176C22×Q8—22no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—42no—
180C22×D4—232no—
181C22×D4—232no—
182C22×D4—264no—
183C22×D4—28no—
184C22×D4—28no—
185C22×D4—264no—
186C22×D4—232no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4C2×D418yes—
194C22×D4C2×Q818yes—
195C22×D4C2×Q818yes—
196C22×D4#4118yes—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—264no—
203C23×C4—24no—
204C23×C4—28no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—28no—
208C23×C4—28no—
209C23×C4—28no—
210C23×C4—28no—
211C23×C4—28no—
212C23×C4—28no—
213C23×C4—216no—
214C23×C4—48no—
215C23×C4—48no—
216C23×C4—48no—
217C23×C4—44no—
218C23×C4—42no—
219C23×C4—4128no—
220C23×C4—432no—
221C23×C4—432no—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D4116yes—
226C23×C4C2×Q814yes—
227C23×C4#4114yes—
228C23×C4C2412yes—
229C23×C4C2418yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C22⋊Q8—232no—
241C4×Q8—416no—
242C4×D4—264no—
243C4×D4—264no—
244C4×D4—264no—
245C4×D4—264no—
246C2×C4⋊C4—232no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—216no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—48no—
269C2×C4⋊C4—48no—
270C2×C4⋊C4C2×D418yes—
271C2×C22⋊C4—264no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—216no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—216no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—216no—
288C2×C22⋊C4—216no—
289C2×C22⋊C4—216no—
290C2×C22⋊C4—216no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4—48no—
297C2×C22⋊C4C2×D4116yes—
298C2×C22⋊C4#4118yes—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—232no—
303C2×C42—24no—
304C2×C42—28no—
305C2×C42—24no—
306C2×C42—28no—
307C2×C42C2×D414yes—
308C2×C42C2×D414yes—
309C2×C42C2×D414yes—
310C2×C42C2×D414yes—
311C2×C42C2×D418yes—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—216no—
320#95—28no—
321#95—28no—
322#95—44no—
323#95—48no—
324#95—48no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—216no—
330C24—216no—
331C24—232no—
332C24—232no—
333C24—216no—
334C24—24no—
335C24—24no—
336C24—28no—
337C24—24no—
338C24—24no—
339C24—28no—
340C24—28no—
341C24—28no—
342C24—232no—
343C24—416no—
344C24—432no—
345C24—432no—
346C24—432no—
347C24—432no—
348C24—48no—
349C24—48no—
350C24—416no—
351C24—48no—
352C24—48no—
353C24—416no—
354C24C22×D418yes—
355C24C22×D412yes—
356C24C22×Q818yes—
357C24C22×Q812yes—
358C24C2×#4118yes—
359C24C2×#4112yes—
360C24C2×#4114yes—
361C24H(5,𝔽2)14yes—
362C24C2511yescharacteristic
363C2×Q8—28no—
364C2×Q8—416no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×D4—216no—
368C2×D4—216no—
369C2×D4—232no—
370C2×D4—2256no—
371C2×D4—232no—
372C2×D4—216no—
373C2×D4—216no—
374C2×D4—216no—
375C2×D4—216no—
376C2×D4—216no—
377C2×D4—216no—
378C2×D4—216no—
379C2×D4C22⋊Q8132yescomplemented
380C22×C4—232no—
381C22×C4—28no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—216no—
393C22×C4—28no—
394C22×C4—216no—
395C22×C4—28no—
396C22×C4—216no—
397C22×C4—232no—
398C22×C4—28no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—28no—
415C22×C4—232no—
416C22×C4—22no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—22no—
426C22×C4—28no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—28no—
432C22×C4—24no—
433C22×C4—432no—
434C22×C4—44no—
435C22×C4—432no—
436C22×C4—416no—
437C22×C4—432no—
438C22×C4—432no—
439C22×C4—4256no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—432no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—416no—
447C22×C4C22⋊Q8116yes—
448C22×C4C22×D412yes—
449C22×C4C22×D418yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D414yes—
458C22×C4C22×Q812yes—
459C22×C4C22×Q812yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4114yes—
464C22×C4C2×#4112yes—
465C22×C4H(5,𝔽2)14yes—
466C4⋊C4—216no—
467C4⋊C4—28no—
468C4⋊C4—216no—
469C4⋊C4—216no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—416no—
474C4⋊C4—48no—
475C4⋊C4—816no—
476C22⋊C4—232no—
477C22⋊C4—216no—
478C22⋊C4—232no—
479C22⋊C4—432no—
480C22⋊C4—432no—
481C22⋊C4—48no—
482C22⋊C4—416no—
483C22⋊C4—816no—
484C22⋊C4—816no—
485C42—28no—
486C42—28no—
487C42—28no—
488C42—48no—
489C42—48no—
490C42#120116yes—
491C23—22no—
492C23—24no—
493C23—22no—
494C23—24no—
495C23—22no—
496C23—22no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—264no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—216no—
511C23—216no—
512C23—216no—
513C23—216no—
514C23—216no—
515C23—216no—
516C23—4128no—
517C23—432no—
518C23—4128no—
519C23—432no—
520C23—432no—
521C23—432no—
522C23—432no—
523C23—432no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—48no—
529C23—48no—
530C23—48no—
531C23—48no—
532C23—48no—
533C23—48no—
534C23C2×C22⋊Q8116yes—
535C23C2×C22⋊Q814yes—
536C23#52514yes—
537C23#53514yes—
538C23#55418yes—
539C23#55918yes—
540C23C23×D411yescharacteristic
541C23C23×D412yes—
542C23C23×Q811yescharacteristic
543C23C22×#4111yescharacteristic
544C23C22×#4112yes—
545C23C22×#4112yes—
546C23C2×H(5,𝔽2)12yes—
547C23C2×H(5,𝔽2)12yes—
548C23#58512yes—
549Q8—416no—
550D4—264no—
551D4—264no—
552C2×C4—232no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—232no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—216no—
576C2×C4—416no—
577C2×C4—464no—
578C2×C4—464no—
579C2×C4—432no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—48no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—48no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—44no—
601C2×C4—48no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—44no—
607C2×C4C2×#12014yes—
608C2×C4C2×#12014yes—
609C2×C4C2×#12014yes—
610C2×C4C2×C22⋊Q814yes—
611C2×C4C2×C22⋊Q818yes—
612C2×C4C2×C22⋊Q814yes—
613C2×C4#52514yes—
614C2×C4#52518yes—
615C2×C4#53514yes—
616C22—24no—
617C22—216no—
618C22—24no—
619C22—216no—
620C22—24no—
621C22—24no—
622C22—24no—
623C22—24no—
624C22—216no—
625C22—232no—
626C22—4128no—
627C22—48no—
628C22—416no—
629C22—416no—
630C22—48no—
631C22—416no—
632C22—48no—
633C22—48no—
634C22—48no—
635C22—48no—
636C22#258018yescomplemented
637C22C22×#12011yescharacteristic
638C22C22×C22⋊Q811yescharacteristic
639C22C22×C22⋊Q812yes—
640C22C2×#52512yes—
641C22C2×#52511yescharacteristic
642C22C2×#52511yescharacteristic
643C22C2×#53511yescharacteristic
644C22C2×#53511yescharacteristic
645C22#344911yescharacteristic
646C22D4×#4114yes—
647C22C2×#55412yes—
648C22C2×#55912yes—
649C22#348114yes—
650C22#348414yes—
651C22#350112yes—
652C22#350612yes—
653C22#351314yes—
654C4—24no—
655C4—24no—
656C4—24no—
657C4—24no—
658C4—44no—
659C4—48no—
660C4—42no—
661C4—44no—
662C4—88no—
663C4—82no—
664C4—82no—
665C4#284818yes—
666C2—28no—
667C2—416no—
668C2#3473611yescharacteristic
669C2C2×#258011yescharacteristic
670C2#3496311yescharacteristic
671C2#3517211yescharacteristic
672C2#3522211yescharacteristic
673C2#3523211yescharacteristic
674C2#3621514yes—
675C2#3644114yes—
676C2C2×#284811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2580C2semidirect
3. C2×C4×C22⋊Q8C2semidirect
4. #15203C2semidirect
5. #14865C2semidirect
6. #14807C2semidirect
7. #14734C2semidirect
8. #14717C2semidirect
9. #14693C2semidirect
10. #14682C2semidirect
11. #14130C2semidirect
12. #14018C2semidirect
13. #13941C2semidirect
18. C22×C4×Q8C22semidirect
29. C2×#399C22semidirect
30. C2×#393C22semidirect
32. C2×#391C22semidirect
35. C2×#389C22semidirect
36. C2×#389C22semidirect
37. C2×#387C22semidirect
40. C2×#385C22semidirect
41. C2×#385C22semidirect
42. C2×#382C22semidirect
47. C2×C4×C4⋊C4C22semidirect
48. C2×C4×C4⋊C4C22semidirect
50. #1444C22semidirect
51. #1442C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
54. #1439C22semidirect
55. #1439C22semidirect
58. C4⋊#95C22semidirect
59. C4⋊#95C22semidirect
60. #1434C22semidirect
61. #1434C22semidirect
65. C22⋊#95C22semidirect
66. C22⋊#95C22semidirect
73. C23×D4Q8semidirect
379. C2×D4C22⋊Q8semidirect
636. C22#2580semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D42
C2×#25802
C2×C4×C22⋊Q82
#152032
#148652
#148072
#147342
#147172
#146932
#146822
#141302
#140182
#139412
#139022