Finite Group Databasebeta

← #719646

Subgroups — #719646

Order 512.

5038 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Q8C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic, complemented
3C2×#2536C211yesmaximal, characteristic
4C2×#2434C212yesmaximal, complemented
5C2×#2327C211yesmaximal, characteristic, complemented
6#16507C214yesmaximal, complemented
7#15354C218yesmaximal, complemented
8#15236C212yesmaximal, complemented
9#15178C212yesmaximal, complemented
10#14801C214yesmaximal
11#14274C211yesmaximal, characteristic, complemented
12#13942C211yesmaximal, characteristic, complemented
13#13922C212yesmaximal, complemented
14#13847C211yesmaximal, characteristic
15C24×Q8C2212yes—
16C2×D4×Q8—28no—
17C2×D4×Q8—28no—
18C2×D4×Q8—28no—
19C22×C4⋊Q8C2212yes—
20C22×C4⋊Q8C2212yes—
21C22×C4⋊Q8C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C22×C22⋊Q8C2212yes—
24C22×C22⋊Q8C2214yes—
25C22×C22⋊Q8C2212yes—
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2211yescharacteristic
28C22×C4×D4C2212yes—
29C22×C4×D4C2211yescharacteristic
30#2546—28no—
31#2536—28no—
32#2434—216no—
33C2×#398C2212yes—
34C2×#398C2214yes—
35C2×#395C2218yes—
36C2×#394C2212yes—
37#2327—28no—
38C2×#391C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#391C2212yes—
41C2×#391C2212yes—
42C2×#391C2212yes—
43C2×#391C2212yes—
44C2×#389C2214yes—
45C2×#388C2214yes—
46C2×#387C2214yes—
47C2×#387C2214yes—
48C2×#385C2218yes—
49C2×#382C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2212yes—
55C2×#376C2211yescharacteristic
56#1446C2218yes—
57#1445C2212yes—
58#1445C2214yes—
59#1445C2214yes—
60#1444C2218yes—
61#1442C2218yes—
62#1441C2214yes—
63#1440C2218yes—
64#1439C2214yes—
65#1439C2218yes—
66#1438C2212yes—
67C4⋊#95C2212yes—
68#1434C2214yes—
69C22⋊#95C2218yes—
70C23×Q8—28no—
71C23×Q8—28no—
72C23×Q8—28no—
73C23×Q8—28no—
74C23×Q8—28no—
75C23×Q8—44no—
76C23×Q8—44no—
77C23×Q8C2312yes—
78C23×Q8C2311yescharacteristic
79C23×Q8C2314yes—
80C23×Q8C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2314yes—
84D4×Q8—2128no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—28no—
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—44no—
95C2×C4⋊Q8D418yes—
96C2×C4⋊Q8D418yescomplemented
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8—216no—
102C2×C22⋊Q8—28no—
103C2×C22⋊Q8—216no—
104C2×C22⋊Q8—216no—
105C2×C22⋊Q8—44no—
106C2×C4×Q8—24no—
107C2×C4×Q8—24no—
108C2×C4×Q8—24no—
109C2×C4×Q8—24no—
110C2×C4×Q8—24no—
111C2×C4×Q8D418yes—
112C2×C4×D4—28no—
113C2×C4×D4—28no—
114C2×C4×D4—28no—
115C2×C4×D4—24no—
116C2×C4×D4—24no—
117C2×C4×D4D418yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2318yes—
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2312yes—
137C22×C42C2312yes—
138C22×C42C2312yes—
139C22×C42C2311yescharacteristic
140C22×C42C2312yes—
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143#398—28no—
144#398—216no—
145#395—232no—
146#394—28no—
147#391—24no—
148#391—28no—
149#391—28no—
150#391—24no—
151#391—28no—
152#391—44no—
153#389—216no—
154#388—216no—
155#387—216no—
156#387—216no—
157#385—232no—
158#382—24no—
159#382—24no—
160#380—28no—
161C4×C4⋊C4—24no—
162C4×C22⋊C4—28no—
163C4×C22⋊C4—44no—
164#376—24no—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2318yes—
168C2×#95C2318yes—
169C2×#95C2312yes—
170C2×#95C2318yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2318yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2318yes—
180C2×#95C2314yes—
181C25C2412yes—
182C22×Q8—264no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—28no—
189C22×Q8—28no—
190C22×Q8—28no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—28no—
195C22×Q8—22no—
196C22×Q8—24no—
197C22×Q8—28no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—44no—
205C22×Q8—44no—
206C22×Q8—432no—
207C22×Q8—432no—
208C22×Q8—432no—
209C22×Q8—432no—
210C22×Q8—432no—
211C22×Q8—432no—
212C22×Q8C2×D414yes—
213C22×Q8C2×D418yes—
214C22×Q8C2×D414yes—
215C22×Q8C2×D414yes—
216C22×Q8#4114yes—
217C22×D4—28no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4C2×D414yes—
221C23×C4—28no—
222C23×C4—28no—
223C23×C4—28no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—28no—
231C23×C4—28no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4C2×D418yes—
235C23×C4C2414yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2414yes—
243C23×C4C2414yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C4⋊Q8—264no—
249C4⋊Q8—232no—
250C4⋊Q8—232no—
251C4⋊Q8—432no—
252C22⋊Q8—264no—
253C22⋊Q8—2128no—
254C22⋊Q8—432no—
255C4×Q8—232no—
256C4×Q8—416no—
257C4×D4—264no—
258C4×D4—232no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—216no—
287C2×C4⋊C4—216no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—48no—
291C2×C4⋊C4C2×D418yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D414yes—
294C2×C4⋊C4C2×D418yes—
295C2×C4⋊C4C2×D418yes—
296C2×C4⋊C4C2×D4116yes—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—216no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4C2×D418yes—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42C2×D418yes—
328C2×C42C2×D414yes—
329C2×C42C2×D414yes—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—24no—
339#95—24no—
340#95—216no—
341#95—216no—
342#95—28no—
343#95—28no—
344#95—216no—
345#95—216no—
346#95—44no—
347#95—48no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24C22×D414yes—
358C242−1+418yes—
359C24C2511yescharacteristic
360C2×Q8—232no—
361C2×Q8—28no—
362C2×Q8—216no—
363C2×Q8—28no—
364C2×Q8—28no—
365C2×Q8—28no—
366C2×Q8—28no—
367C2×Q8—28no—
368C2×Q8—28no—
369C2×Q8—216no—
370C2×Q8—28no—
371C2×Q8—28no—
372C2×Q8—216no—
373C2×Q8—416no—
374C2×Q8—416no—
375C2×Q8—416no—
376C2×Q8—416no—
377C2×Q8—416no—
378C2×Q8—4256no—
379C2×Q8—44no—
380C2×Q8—416no—
381C2×Q8—48no—
382C2×Q8—44no—
383C2×Q8—48no—
384C2×Q8—48no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—48no—
388C2×Q8#121116yes—
389C2×D4—216no—
390C2×D4—216no—
391C2×D4—216no—
392C2×D4—28no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—232no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—216no—
432C22×C4—216no—
433C22×C4—216no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—28no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—416no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4—416no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4C22×D414yes—
459C22×C4C22×D414yes—
460C22×C4C22×D418yes—
461C22×C4C22×D414yes—
462C22×C4C22×D414yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C2×#4114yes—
476C22×C4C2×#4112yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C42−1+418yes—
480C4⋊C4—216no—
481C4⋊C4—232no—
482C4⋊C4—216no—
483C4⋊C4—28no—
484C4⋊C4—216no—
485C4⋊C4—28no—
486C4⋊C4—216no—
487C4⋊C4—232no—
488C4⋊C4—416no—
489C4⋊C4—432no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C22⋊C4—232no—
494C22⋊C4—232no—
495C22⋊C4—216no—
496C22⋊C4—416no—
497C22⋊C4—416no—
498C22⋊C4—416no—
499C42—216no—
500C42—28no—
501C42—48no—
502C42—44no—
503C42—48no—
504C42#120116yes—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—24no—
513C23—24no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23#53618yes—
528C23C23×D411yescharacteristic
529C23C23×D411yescharacteristic
530C23C23×D411yescharacteristic
531C23C23×D411yescharacteristic
532C23C22×#4111yescharacteristic
533C23C2×H(5,𝔽2)11yescharacteristic
534C23C2×H(5,𝔽2)11yescharacteristic
535C23C2×2−1+414yes—
536C23C2×2−1+414yes—
537Q8—232no—
538Q8—464no—
539Q8—416no—
540Q8—432no—
541D4—232no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—464no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—432no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—48no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—48no—
607C2×C4C2×#12014yes—
608C2×C4C2×#12014yes—
609C2×C4C2×#12014yes—
610C2×C4C2×#12118yes—
611C2×C4C2×#12114yes—
612C2×C4#53514yes—
613C2×C4D4214yes—
614C2×C4D4214yes—
615C2×C4#54614yes—
616C2×C4#54614yes—
617C22—28no—
618C22—28no—
619C22—28no—
620C22—48no—
621C22—48no—
622C22—44no—
623C22—44no—
624C22—48no—
625C22—44no—
626C22—48no—
627C22—44no—
628C22C22×#12011yescharacteristic
629C22C22×#12111yescharacteristic
630C22C2×#53511yescharacteristic
631C22C2×#53612yes—
632C22C2×D4211yescharacteristic
633C22C2×D4211yescharacteristic
634C22C2×#54611yescharacteristic
635C22C2×#54611yescharacteristic
636C22#348612yes—
637C22#348612yes—
638C22#348612yes—
639C22#348614yes—
640C22#349912yes—
641C22#349912yes—
642C22#352314yes—
643C22#353814yes—
644C22#355214yes—
645C4—24no—
646C4—28no—
647C4—24no—
648C4—44no—
649C4—48no—
650C4—44no—
651C4—48no—
652C4—44no—
653C4—44no—
654C4—82no—
655C4—84no—
656C4—82no—
657C4#240818yes—
658C2—48no—
659C2C2×#240811yescharacteristic
660C2#3524211yescharacteristic
661C2#3566611yescharacteristic
662C2#3570012yes—
663C2#3767512yes—
664C2#3808112yes—
665C2#4279212yes—
666C2#4533612yes—
667C2#4541812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2546C2semidirect
4. C2×#2434C2semidirect
5. C2×#2327C2semidirect
6. #16507C2semidirect
7. #15354C2semidirect
8. #15236C2semidirect
9. #15178C2semidirect
11. #14274C2semidirect
12. #13942C2semidirect
13. #13922C2semidirect
96. C2×C4⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C22×D4×Q82
C2×#25462
C2×#25362
C2×#24342
C2×#23272
#165072
#153542
#152362
#151782
#148012
#142742
#139422
#139222
#138472