Finite Group Databasebeta

← #759987

Subgroups — #759987

Order 512.

3826 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#533C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic, complemented
3C2×#2567C211yesmaximal, characteristic, complemented
4C2×C4×C22⋊Q8C211yesmaximal, characteristic
5#16601C214yesmaximal
6#15238C212yesmaximal, complemented
7#15228C212yesmaximal, complemented
8#15179C214yesmaximal
9#14876C214yesmaximal, complemented
10#14161C211yesmaximal, characteristic, complemented
11#14155C211yesmaximal, characteristic, complemented
12#14138C212yesmaximal
13#14017C214yesmaximal, complemented
14#13879C212yesmaximal
15C4×#380C211yesmaximal, characteristic
16C2×#533—28no—
17C2×#533—28no—
18C2×#533—28no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#125C2212yes—
22C22×C22⋊Q8C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C4×Q8C2212yes—
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2212yes—
28C22×#117C2212yes—
29C23×C42C2211yescharacteristic
30#2568—28no—
31#2567—28no—
32C2×#400C2214yes—
33C2×#395C2214yes—
34C4×C22⋊Q8—28no—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#391C2214yes—
38C2×#391C2214yes—
39C2×#391C2211yescharacteristic
40C2×#389C2214yes—
41C2×#389C2211yescharacteristic
42C2×#387C2212yes—
43C2×#385C2212yes—
44C2×#384C2212yes—
45C2×#382C2212yes—
46C2×#380C2211yescharacteristic
47C2×#380C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2212yes—
54#1446C2214yes—
55#1445C2214yes—
56#1444C2218yes—
57#1442C2218yes—
58#1442C2218yes—
59#1441C2218yes—
60#1440C2218yes—
61#1439C2218yes—
62#1436C2218yes—
63#1436C2218yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2214yes—
69C4⋊#95C2212yes—
70#1434C2212yes—
71C22⋊#95C2212yes—
72C4×#95C2212yes—
73C23×Q8C2312yes—
74C23×Q8C2311yescharacteristic
75C23×Q8C2312yes—
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79#533—2128no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8D418yes—
85C2×C4⋊Q8D418yes—
86C2×#125—28no—
87C2×#125—28no—
88C2×#125—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8Q818yescomplemented
98C2×C4×Q8—28no—
99C2×C4×Q8—28no—
100C2×C4×Q8—28no—
101C2×C4×Q8—28no—
102C2×C4×Q8—28no—
103C2×C4×Q8—28no—
104C2×C4×Q8—24no—
105C2×C4×Q8—24no—
106C2×C4×Q8—24no—
107C2×#117—28no—
108C2×#117—28no—
109C2×#117—28no—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2318yes—
124C22×C4⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C42—24no—
130C22×C42—24no—
131C22×C42D418yes—
132C22×C42C2314yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#400—216no—
142#395—216no—
143#391—216no—
144#391—28no—
145#391—28no—
146#391—216no—
147#391—24no—
148#389—216no—
149#389—24no—
150#387—28no—
151#385—28no—
152#384—28no—
153#382—28no—
154#380—24no—
155#380—28no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4—24no—
160C4×C22⋊C4—28no—
161C4×C22⋊C4—28no—
162C2×#95C2312yes—
163C2×#95C2318yes—
164C2×#95C2312yes—
165C2×#95C2318yes—
166C2×#95C2318yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2318yes—
170C2×#95C2318yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2318yes—
178C2×#95C2318yes—
179C25C2411yescharacteristic
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—28no—
189C22×Q8—28no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—44no—
193C22×Q8—44no—
194C22×Q8C2×D418yes—
195C22×Q8C2×D418yes—
196C22×Q8C2×Q814yes—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4C2×Q814yes—
213C23×C4#4114yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C4⋊Q8—232no—
229C4⋊Q8—232no—
230#125—264no—
231C22⋊Q8—264no—
232C22⋊Q8—264no—
233C22⋊Q8—232no—
234C4×Q8—264no—
235C4×Q8—264no—
236C4×Q8—416no—
237#117—264no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—216no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—48no—
274C2×C4⋊C4—48no—
275C2×C4⋊C4C2×D418yes—
276C2×C4⋊C4C2×D418yes—
277C2×C4⋊C4C2×D418yes—
278C2×C4⋊C4C2×D418yes—
279C2×C4⋊C4C2×Q814yes—
280C2×C4⋊C4C2×Q818yes—
281C2×C4⋊C4#4118yes—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4C2×Q818yes—
293C2×C42—28no—
294C2×C42—28no—
295C2×C42—28no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—232no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—28no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42C2×D414yes—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—216no—
328#95—216no—
329#95—216no—
330#95—28no—
331#95—28no—
332#95—216no—
333#95—216no—
334#95—48no—
335#95—44no—
336#95—48no—
337C24—21no—
338C24—21no—
339C24—22no—
340C24—21no—
341C24—21no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24C22×D412yes—
346C24C22×Q812yes—
347C24C2×#4112yes—
348C24C2511yescharacteristic
349C2×Q8—216no—
350C2×Q8—216no—
351C2×Q8—28no—
352C2×Q8—232no—
353C2×Q8—28no—
354C2×Q8—28no—
355C2×Q8—416no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—416no—
359C2×Q8—44no—
360C2×Q8—48no—
361C2×Q8—44no—
362C2×Q8—48no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—416no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—28no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4C22⋊Q8116yes—
438C22×C4C22×D418yes—
439C22×C4C22×D412yes—
440C22×C4C22×D418yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×Q814yes—
453C22×C4C22×Q814yes—
454C22×C4C22×Q812yes—
455C22×C4C22×Q812yes—
456C22×C4C2×#4118yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4114yes—
460C22×C42−1+414yes—
461C22×C42−1+412yes—
462C22×C42−1+412yes—
463C4⋊C4—232no—
464C4⋊C4—232no—
465C4⋊C4—216no—
466C4⋊C4—216no—
467C4⋊C4—216no—
468C4⋊C4—216no—
469C4⋊C4—28no—
470C4⋊C4—216no—
471C4⋊C4—216no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—816no—
478C22⋊C4—232no—
479C22⋊C4—216no—
480C22⋊C4—416no—
481C22⋊C4—48no—
482C42—232no—
483C42—28no—
484C42—28no—
485C42—28no—
486C42—48no—
487C42—48no—
488C42—48no—
489C42—48no—
490C42#120116yes—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23C2×C22⋊Q814yes—
514C23C23×D411yescharacteristic
515C23C23×D411yescharacteristic
516C23C23×D411yescharacteristic
517C23C23×Q811yescharacteristic
518C23C22×#4111yescharacteristic
519C23C22×#4112yes—
520C23C2×2−1+412yes—
521C23C2×2−1+411yescharacteristic
522C23C2×2−1+411yescharacteristic
523C23#58512yes—
524C23#58512yes—
525Q8—432no—
526Q8—416no—
527Q8—432no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—232no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—216no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—216no—
556C2×C4—28no—
557C2×C4—416no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—48no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—48no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—416no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4C2×#12014yes—
598C2×C4C2×#12014yes—
599C2×C4C2×#12014yes—
600C2×C4C2×C22⋊Q814yes—
601C2×C4C2×C22⋊Q814yes—
602C2×C4#53214yes—
603C2×C4#53614yes—
604C2×C4#53614yes—
605C2×C4#54718yes—
606C2×C4#54814yes—
607C2×C4#54814yes—
608C2×C4#54818yes—
609C2×C4D4×Q814yes—
610C2×C4D4×Q814yes—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22C22×#12011yescharacteristic
623C22C22×C22⋊Q811yescharacteristic
624C22C2×#53211yescharacteristic
625C22C2×#53611yescharacteristic
626C22C2×#53611yescharacteristic
627C22#344911yescharacteristic
628C22#344911yescharacteristic
629C22C2×#54712yes—
630C22C2×#54811yescharacteristic
631C22C2×#54811yescharacteristic
632C22C2×#54812yes—
633C22C2×D4×Q811yescharacteristic
634C22C2×D4×Q811yescharacteristic
635C22#347512yes—
636C22Q8×#4112yes—
637C22#348512yes—
638C22#348712yes—
639C22#350212yes—
640C22#350712yes—
641C22#351512yes—
642C22#351812yes—
643C22#355512yes—
644C22#355512yes—
645C4—24no—
646C4—24no—
647C4—42no—
648C4—48no—
649C4—44no—
650C4—44no—
651C4—42no—
652C4—48no—
653C4—48no—
654C4—82no—
655C4—82no—
656C4—84no—
657C4#240918yes—
658C4#243218yes—
659C2—44no—
660C2C2×#240911yescharacteristic
661C2C2×#243211yescharacteristic
662C2#3494711yescharacteristic
663C2#3514911yescharacteristic
664C2#3517911yescharacteristic
665C2#3520111yescharacteristic
666C2#3763511yescharacteristic
667C2#5906512yes—
668C2#5906712yes—
669C2#5907912yes—
670C2#5908112yes—
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
2. C2×#2568C2semidirect
3. C2×#2567C2semidirect
6. #15238C2semidirect
7. #15228C2semidirect
9. #14876C2semidirect
10. #14161C2semidirect
11. #14155C2semidirect
13. #14017C2semidirect
97. C2×C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5332
C2×#25682
C2×#25672
C2×C4×C22⋊Q82
#166012
#152382
#152282
#151792
#148762
#141612
#141552
#141382
#140172
#138792
C4×#3802