Finite Group Databasebeta

← #4414092

Subgroups — #4414092

Order 512.

5556 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
1C23×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2816C212yesmaximal, complemented
3C2×#2847C211yesmaximal, characteristic
4#16600C214yesmaximal, complemented
5#16365C212yesmaximal, complemented
6#16263C214yesmaximal, complemented
7#16253C212yesmaximal, complemented
8#14873C212yesmaximal, complemented
9#14821C212yesmaximal, complemented
10#14777C212yesmaximal, complemented
11#14719C212yesmaximal, complemented
12#14718C214yesmaximal, complemented
13#14681C211yesmaximal, characteristic, complemented
14#14660C212yesmaximal, complemented
15C24×Q8C2211yescharacteristic
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8—28no—
18C22×C22⋊Q8—216no—
19C22×C22⋊Q8C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×C22⋊Q8C2212yes—
23C22×#121C2211yescharacteristic
24C22×#121C2211yescharacteristic
25C22×#121C2211yescharacteristic
26C22×#120C2212yes—
27C23×C4⋊C4C2212yes—
28C23×C4⋊C4C2211yescharacteristic
29C23×C22⋊C4C2212yes—
30#2847—28no—
31#2816—216no—
32C2×#399C2214yescomplemented
33C2×#398C2214yescomplemented
34C2×#394C2218yes—
35C2×#394C2214yescomplemented
36C2×#393C2214yescomplemented
37C2×#393C2214yescomplemented
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#386C2214yes—
42C2×#386C2214yes—
43C2×#385C2214yes—
44C2×#381C2212yes—
45C2×#381C2212yes—
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#379C2212yes—
50#1446C2214yescomplemented
51#1446C2214yescomplemented
52#1445C2214yescomplemented
53#1444C2214yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1441C2218yescomplemented
57#1440C2218yes—
58#1439C2218yes—
59#1439C2214yescomplemented
60#1439C2218yescomplemented
61#1438C2218yes—
62C22⋊#95C2212yes—
63C22⋊#95C2214yes—
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C26C2311yescharacteristic
68C23×Q8—28no—
69C23×Q8—24no—
70C23×Q8—44no—
71C23×Q8—44no—
72C23×Q8C2312yes—
73C23×Q8C2312yes—
74C23×D4C2312yes—
75C23×D4C2312yes—
76C23×D4C2311yescharacteristic
77C23×D4C2312yes—
78C24×C4—28no—
79C24×C4—44no—
80C24×C4—44no—
81C24×C4D418yes—
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—2128no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—216no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—216no—
100C2×C22⋊Q8D4116yescomplemented
101C2×#121—28no—
102C2×#121—24no—
103C2×#121—24no—
104C2×#121—24no—
105C2×#121—44no—
106C2×#121—44no—
107C2×#120—28no—
108C2×#120—48no—
109C22×C4⋊C4—24no—
110C22×C4⋊C4—28no—
111C22×C4⋊C4—216no—
112C22×C4⋊C4—28no—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2318yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—28no—
122C22×C22⋊C4—216no—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2318yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2318yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2314yes—
136#399—216no—
137#398—216no—
138#394—216no—
139#394—232no—
140#393—216no—
141#393—48no—
142#386—28no—
143#386—28no—
144#386—28no—
145#386—48no—
146#386—48no—
147#385—48no—
148#381—28no—
149#381—44no—
150#380—24no—
151#380—24no—
152#380—24no—
153#379—44no—
154C2×#95C2318yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2318yes—
162C2×#95C2314yes—
163C2×#95C2318yes—
164C2×#95C2318yes—
165C2×#95C2318yes—
166C25—24no—
167C25—24no—
168C25—24no—
169C25—24no—
170C25—24no—
171C25—42no—
172C25—42no—
173C25C2×D414yes—
174C25C2412yes—
175C25C2411yescharacteristic
176C25C2411yescharacteristic
177C25C2411yescharacteristic
178C22×Q8—232no—
179C22×Q8—28no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—28no—
184C22×Q8—44no—
185C22×Q8—44no—
186C22×Q8—44no—
187C22×Q8—464no—
188C22×Q8—416no—
189C22×Q8—416no—
190C22×Q8C2×D418yes—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—28no—
195C22×D4—28no—
196C22×D4—28no—
197C22×D4—28no—
198C22×D4—28no—
199C22×D4—22no—
200C22×D4—24no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4—44no—
206C22×D4—44no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—232no—
215C23×C4—28no—
216C23×C4—28no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—28no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—416no—
226C23×C4—416no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4—464no—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2×D414yes—
240C23×C4C2×D414yes—
241C23×C4C2×D414yes—
242C23×C4C2×D414yes—
243C23×C4C2×D418yes—
244C23×C4C2418yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2414yes—
249C23×C4C2414yes—
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C22⋊Q8—264no—
255C22⋊Q8—264no—
256C22⋊Q8—264no—
257C22⋊Q8—264no—
258#121—416no—
259#121—416no—
260#121—416no—
261#120—432no—
262C2×C4⋊C4—232no—
263C2×C4⋊C4—264no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—216no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—48no—
281C2×C4⋊C4—48no—
282C2×C4⋊C4C2×D4116yes—
283C2×C4⋊C4C2×D418yes—
284C2×C22⋊C4—264no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—216no—
305C2×C22⋊C4—216no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—48no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—48no—
315C2×C22⋊C4—48no—
316C2×C22⋊C4—48no—
317C2×C22⋊C4C2×D4116yes—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—28no—
323#95—216no—
324#95—28no—
325#95—44no—
326#95—48no—
327#95—48no—
328#95—48no—
329#95—48no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—232no—
335C24—28no—
336C24—28no—
337C24—28no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—28no—
351C24—28no—
352C24—24no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—44no—
361C24—44no—
362C24—416no—
363C24—416no—
364C24—416no—
365C24—416no—
366C24—416no—
367C24—416no—
368C24C22×D412yes—
369C24C22×D414yes—
370C24C22×D412yes—
371C24C22×D412yes—
372C24H(5,𝔽2)14yes—
373C24H(5,𝔽2)14yes—
374C242−1+414yes—
375C242−1+414yes—
376C24C2511yescharacteristic
377C2×Q8—216no—
378C2×Q8—216no—
379C2×Q8—416no—
380C2×Q8—432no—
381C2×Q8—4128no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—48no—
385C2×Q8—416no—
386C2×Q8—48no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—432no—
399C2×D4—44no—
400C2×D4—88no—
401C2×D4—816no—
402C2×D4—816no—
403C22×C4—232no—
404C22×C4—28no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—216no—
414C22×C4—216no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—28no—
447C22×C4—24no—
448C22×C4—28no—
449C22×C4—28no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—28no—
453C22×C4—24no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—416no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—416no—
466C22×C4—416no—
467C22×C4—432no—
468C22×C4—4128no—
469C22×C4—416no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4#120116yes—
475C22×C4C22×D418yes—
476C22×C4C22×D412yes—
477C22×C4C22×D418yes—
478C22×C4C22×D412yes—
479C22×C4C22×D414yes—
480C22×C4C22×D414yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4H(5,𝔽2)14yes—
484C4⋊C4—216no—
485C4⋊C4—232no—
486C4⋊C4—216no—
487C4⋊C4—232no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—816no—
492C22⋊C4—232no—
493C22⋊C4—232no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—416no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—816no—
502C22⋊C4—816no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—232no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—28no—
532C23—28no—
533C23—28no—
534C23—416no—
535C23—416no—
536C23—416no—
537C23—416no—
538C23—416no—
539C23—4128no—
540C23—416no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—48no—
557C23—48no—
558C23—48no—
559C23—48no—
560C23—48no—
561C23—48no—
562C23—48no—
563C23C2×#12014yes—
564C23#53414yes—
565C23#53514yes—
566C23#53614yes—
567C23#53614yes—
568C23#562116yes—
569C23C23×D411yescharacteristic
570C23C23×D412yes—
571C23C2×H(5,𝔽2)12yes—
572C23C2×H(5,𝔽2)12yes—
573C23C2×H(5,𝔽2)12yes—
574C23C2×2−1+412yes—
575C23C2×2−1+412yes—
576C23#58512yes—
577Q8—432no—
578Q8—432no—
579D4—88no—
580D4—816no—
581D4—816no—
582D4—816no—
583C2×C4—232no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—216no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—416no—
602C2×C4—416no—
603C2×C4—416no—
604C2×C4—416no—
605C2×C4—416no—
606C2×C4—432no—
607C2×C4—432no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—48no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—48no—
626C2×C4—48no—
627C2×C4—48no—
628C2×C4—48no—
629C2×C4—48no—
630C2×C4—48no—
631C2×C4—48no—
632C2×C4—48no—
633C2×C4—48no—
634C2×C4—48no—
635C2×C4C2×#12014yes—
636C2×C4C2×#12014yes—
637C2×C4#53414yes—
638C22—24no—
639C22—24no—
640C22—24no—
641C22—232no—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—464no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—416no—
659C22—416no—
660C22—42no—
661C22—42no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—816no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22#281618yescomplemented
671C22C22×#12011yescharacteristic
672C22C2×#53411yescharacteristic
673C22C2×#53411yescharacteristic
674C22C2×#53511yescharacteristic
675C22C2×#53611yescharacteristic
676C22C2×#53611yescharacteristic
677C22#344911yescharacteristic
678C22#348414yes—
679C22#348614yes—
680C22#348814yes—
681C22C2×#56214yes—
682C22#353514yes—
683C22#353614yes—
684C22#354114yes—
685C4—24no—
686C4—24no—
687C4—24no—
688C4—42no—
689C4—48no—
690C4—48no—
691C4—82no—
692C4—82no—
693C4—88no—
694C4—82no—
695C2—28no—
696C2—44no—
697C2—44no—
698C2—162no—
699C2#3498011yescharacteristic
700C2#3524211yescharacteristic
701C2#3525011yescharacteristic
702C2C2×#281611yescharacteristic
703C2#3743811yescharacteristic
704C2#3744711yescharacteristic
705C2#3745111yescharacteristic
706C2#4544518yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2847C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊Q8C2semidirect
2. C2×#2816C2semidirect
4. #16600C2semidirect
5. #16365C2semidirect
6. #16263C2semidirect
7. #16253C2semidirect
8. #14873C2semidirect
9. #14821C2semidirect
10. #14777C2semidirect
11. #14719C2semidirect
12. #14718C2semidirect
13. #14681C2semidirect
14. #14660C2semidirect
32. C2×#399C22semidirect
33. C2×#398C22semidirect
35. C2×#394C22semidirect
36. C2×#393C22semidirect
37. C2×#393C22semidirect
50. #1446C22semidirect
51. #1446C22semidirect
52. #1445C22semidirect
53. #1444C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
100. C2×C22⋊Q8D4semidirect
670. C22#2816semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#28162
C2×#28472
#166002
#163652
#162632
#162532
#148732
#148212
#147772
#147192
#147182
#146812
#146602