Finite Group Databasebeta

← #641716

Subgroups — #641716

Order 512.

5510 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊Q8C211yesmaximal, characteristic
2C2×#2627C212yesmaximal, complemented
3C22×#385C211yesmaximal, characteristic
4#16601C214yesmaximal, complemented
5#16264C214yesmaximal, complemented
6#14934C212yesmaximal, complemented
7#14874C212yesmaximal, complemented
8#14843C212yesmaximal, complemented
9#14728C214yesmaximal, complemented
10#14720C212yesmaximal, complemented
11#14680C211yesmaximal, characteristic
12#14281C212yesmaximal, complemented
13#14127C212yesmaximal
14#13869C212yesmaximal
15C24×Q8C2211yescharacteristic
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8—28no—
18C22×C22⋊Q8—216no—
19C22×C22⋊Q8C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×C22⋊Q8C2212yes—
23C23×C4⋊C4C2212yes—
24C23×C4⋊C4C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26C23×C4⋊C4C2211yescharacteristic
27C23×C22⋊C4C2212yes—
28C23×C22⋊C4C2211yescharacteristic
29C23×C22⋊C4C2211yescharacteristic
30#2627—216no—
31C2×#400C2214yescomplemented
32C2×#398C2218yescomplemented
33C2×#395C2214yescomplemented
34C2×#395C2214yescomplemented
35C2×#385—28no—
36C2×#385C2212yes—
37C2×#385C2214yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#385C2214yes—
42C2×#381C2214yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×#379C2212yes—
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49#1446C2214yescomplemented
50#1446C2214yescomplemented
51#1445C2214yescomplemented
52#1444C2218yescomplemented
53#1442C2218yescomplemented
54#1442C2214yescomplemented
55#1441C2214yescomplemented
56#1441C2218yescomplemented
57#1440C2218yes—
58#1439C2218yes—
59C22⋊#95C2214yes—
60C22⋊#95C2212yes—
61C22⋊#95C2214yes—
62C22⋊#95C2214yes—
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C26C2311yescharacteristic
68C23×Q8—28no—
69C23×Q8—24no—
70C23×Q8—44no—
71C23×Q8—44no—
72C23×Q8C2312yes—
73C23×Q8C2312yes—
74C24×C4—28no—
75C24×C4—44no—
76C24×C4—44no—
77C24×C4D418yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
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⋊Q8D4116yes—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—216no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—44no—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2318yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4—28no—
120C22×C22⋊C4—24no—
121C22×C22⋊C4—24no—
122C22×C22⋊C4—216no—
123C22×C22⋊C4—24no—
124C22×C22⋊C4—44no—
125C22×C22⋊C4Q818yescomplemented
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135#400—216no—
136#398—232no—
137#395—216no—
138#395—48no—
139#385—28no—
140#385—28no—
141#385—28no—
142#385—28no—
143#385—216no—
144#385—48no—
145#381—216no—
146#380—28no—
147#380—44no—
148#380—44no—
149#379—28no—
150C2×#95—24no—
151C2×#95—24no—
152C2×#95C2318yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2318yes—
156C2×#95C2318yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C2×#95C2318yes—
164C2×#95C2314yes—
165C2×#95C2318yes—
166C25—24no—
167C25—24no—
168C25—24no—
169C25—24no—
170C25—42no—
171C25—42no—
172C25C2×D414yes—
173C25C2×Q814yes—
174C25#4114yes—
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—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—232no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—416no—
219C23×C4—42no—
220C23×C4—416no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—464no—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2×D414yes—
240C23×C4C2×D414yes—
241C23×C4C2×D414yes—
242C23×C4C2×D418yes—
243C23×C4C2×Q814yescomplemented
244C23×C4C2×Q814yescomplemented
245C23×C4C2418yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2414yes—
249C23×C4C2414yes—
250C23×C4C2411yescharacteristic
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C22⋊Q8—264no—
257C22⋊Q8—264no—
258C22⋊Q8—264no—
259C22⋊Q8—264no—
260C2×C4⋊C4—232no—
261C2×C4⋊C4—264no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—216no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—416no—
282C2×C4⋊C4—416no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—48no—
286C2×C4⋊C4—48no—
287C2×C4⋊C4C2×D4116yes—
288C2×C4⋊C4C2×D418yes—
289C2×C22⋊C4—264no—
290C2×C22⋊C4—232no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—416no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4C2×D4116yes—
315C2×C22⋊C4C2×Q818yes—
316C2×C22⋊C4C2×Q818yescomplemented
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—216no—
323#95—216no—
324#95—216no—
325#95—28no—
326#95—48no—
327#95—48no—
328#95—44no—
329#95—48no—
330#95—48no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—28no—
340C24—28no—
341C24—28no—
342C24—28no—
343C24—216no—
344C24—216no—
345C24—216no—
346C24—216no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—44no—
354C24—44no—
355C24—48no—
356C24—48no—
357C24—416no—
358C24—416no—
359C24—416no—
360C24—416no—
361C24C22⋊Q8116yes—
362C24C22×D412yes—
363C24C22×D412yes—
364C24C22×D414yes—
365C24C22×D412yes—
366C24C22×Q812yes—
367C24C22×Q812yes—
368C24C22×Q812yes—
369C24C2×#4112yes—
370C24C2×#4112yes—
371C24C2×#4112yes—
372C24H(5,𝔽2)14yes—
373C242−1+414yes—
374C242−1+414yes—
375C24C2511yescharacteristic
376C2×Q8—216no—
377C2×Q8—216no—
378C2×Q8—416no—
379C2×Q8—432no—
380C2×Q8—4128no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—416no—
385C2×Q8—48no—
386C22×C4—232no—
387C22×C4—28no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—216no—
397C22×C4—216no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—216no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—24no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—22no—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—24no—
440C22×C4—28no—
441C22×C4—24no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—416no—
450C22×C4—48no—
451C22×C4—416no—
452C22×C4—432no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—432no—
456C22×C4—4128no—
457C22×C4—416no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—44no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—88no—
475C22×C4—88no—
476C22×C4#120116yes—
477C22×C4C22×D418yes—
478C22×C4C22×D418yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D414yes—
483C22×C4C22×D414yes—
484C22×C4C22×D412yes—
485C22×C4C22×Q814yes—
486C22×C4C22×Q814yes—
487C22×C4H(5,𝔽2)14yes—
488C4⋊C4—216no—
489C4⋊C4—232no—
490C4⋊C4—216no—
491C4⋊C4—232no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—88no—
497C4⋊C4—816no—
498C22⋊C4—232no—
499C22⋊C4—232no—
500C22⋊C4—216no—
501C22⋊C4—216no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—416no—
505C22⋊C4—416no—
506C22⋊C4—88no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—28no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—232no—
532C23—22no—
533C23—232no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—416no—
546C23—416no—
547C23—464no—
548C23—416no—
549C23—464no—
550C23—416no—
551C23—416no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23C2×#12014yes—
557C23C2×C22⋊Q814yes—
558C23C2×C22⋊Q814yes—
559C23C2×C22⋊Q814yes—
560C23#53414yes—
561C23#53614yes—
562C23#53614yes—
563C23#53918yes—
564C23#54418yes—
565C23#54818yes—
566C23D4×Q818yes—
567C23C23×D411yescharacteristic
568C23C23×D412yes—
569C23C23×Q811yescharacteristic
570C23C22×#4111yescharacteristic
571C23C2×H(5,𝔽2)12yes—
572C23C2×H(5,𝔽2)12yes—
573C23C2×2−1+412yes—
574C23C2×2−1+412yes—
575C23#58512yes—
576Q8—432no—
577Q8—432no—
578C2×C4—232no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—28no—
596C2×C4—216no—
597C2×C4—216no—
598C2×C4—28no—
599C2×C4—416no—
600C2×C4—416no—
601C2×C4—416no—
602C2×C4—416no—
603C2×C4—416no—
604C2×C4—416no—
605C2×C4—432no—
606C2×C4—432no—
607C2×C4—42no—
608C2×C4—416no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—48no—
619C2×C4—48no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
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×C4—48no—
636C2×C4—88no—
637C2×C4—88no—
638C2×C4C2×#12014yes—
639C2×C4C2×#12014yes—
640C2×C4#53414yes—
641C22—216no—
642C22—216no—
643C22—24no—
644C22—24no—
645C22—216no—
646C22—216no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—24no—
657C22—24no—
658C22—24no—
659C22—24no—
660C22—464no—
661C22—48no—
662C22—42no—
663C22—48no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22#240918yes—
668C22#262718yescomplemented
669C22C22×#12011yescharacteristic
670C22C22×C22⋊Q811yescharacteristic
671C22C2×#53411yescharacteristic
672C22C2×#53411yescharacteristic
673C22C2×#53611yescharacteristic
674C22C2×#53611yescharacteristic
675C22#344911yescharacteristic
676C22C2×#53912yes—
677C22C2×#54412yes—
678C22C2×#54812yes—
679C22C2×D4×Q812yes—
680C22#348414yes—
681C22#348914yes—
682C22#350112yes—
683C22#350512yes—
684C22#353514yes—
685C22#353814yes—
686C4—24no—
687C4—24no—
688C4—24no—
689C4—44no—
690C4—42no—
691C4—48no—
692C4—44no—
693C4—48no—
694C4—82no—
695C4—88no—
696C4—82no—
697C2—28no—
698C2—28no—
699C2—44no—
700C2C2×#262711yescharacteristic
701C2C2×#240911yescharacteristic
702C2#3498011yescharacteristic
703C2#3501011yescharacteristic
704C2#3530511yescharacteristic
705C2#3744111yescharacteristic
706C2#3745311yescharacteristic
707C2#3768514yes—
708C2#3770214yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2627C2semidirect
4. #16601C2semidirect
5. #16264C2semidirect
6. #14934C2semidirect
7. #14874C2semidirect
8. #14843C2semidirect
9. #14728C2semidirect
10. #14720C2semidirect
12. #14281C2semidirect
31. C2×#400C22semidirect
32. C2×#398C22semidirect
33. C2×#395C22semidirect
34. C2×#395C22semidirect
49. #1446C22semidirect
50. #1446C22semidirect
51. #1445C22semidirect
52. #1444C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
125. C22×C22⋊C4Q8semidirect
243. C23×C4C2×Q8semidirect
244. C23×C4C2×Q8semidirect
316. C2×C22⋊C4C2×Q8semidirect
668. C22#2627semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊Q82
C2×#26272
C22×#3852
#166012
#162642
#149342
#148742
#148432
#147282
#147202
#146802
#142812
#141272
#138692