Finite Group Databasebeta

← #657584

Subgroups — #657584

Order 512.

4102 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic
2C2×#2503C211yesmaximal, characteristic, complemented
3C2×#2486C211yesmaximal, characteristic
4C22×#382C211yesmaximal, characteristic, complemented
5#15881C214yesmaximal, complemented
6#15670C214yesmaximal, complemented
7#15236C212yesmaximal, complemented
8#15230C212yesmaximal, complemented
9#14934C214yesmaximal, complemented
10#14702C211yesmaximal, characteristic, complemented
11#13978C212yesmaximal, complemented
12#13968C214yesmaximal
13#13924C211yesmaximal, characteristic, complemented
14#13881C212yesmaximal, complemented
15C4×#386C211yesmaximal, characteristic
16C2×#554—28no—
17C2×#554—28no—
18C2×#554—28no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×#125C2212yes—
21C22×#123C2212yes—
22C22×C22⋊Q8C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C22⋊Q8C2211yescharacteristic
26C22×C4×Q8C2211yescharacteristic
27C22×C4×D4C2212yes—
28C22×C4×D4C2211yescharacteristic
29C23×C4⋊C4C2211yescharacteristic
30C23×C4⋊C4C2211yescharacteristic
31C23×C4⋊C4C2211yescharacteristic
32C23×C42C2211yescharacteristic
33#2503—28no—
34#2486—28no—
35C2×#400C2214yes—
36C2×#398C2212yes—
37C2×#397C2212yes—
38C2×#391C2212yes—
39C2×#391C2212yes—
40C2×#389C2212yes—
41C2×#386C2212yes—
42C2×#382—28no—
43C2×#382C2214yes—
44C2×#382C2212yes—
45C2×#382C2212yes—
46C2×#382C2212yes—
47C2×#381C2212yes—
48C2×#381C2214yes—
49C2×#380C2212yes—
50C2×#380C2211yescharacteristic
51C2×C4×C22⋊C4C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53C22×#95C2214yes—
54#1446C2218yes—
55#1445C2214yes—
56#1442C2218yes—
57#1442C2218yes—
58#1442C2218yes—
59#1441C2218yes—
60#1440C2218yes—
61#1437C2212yes—
62#1436C2212yes—
63#1436C2214yes—
64#1436C2218yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C22⋊#95C2218yes—
69C22⋊#95C2212yes—
70C22⋊#95C2214yes—
71C22⋊#95C2214yes—
72C4×#95C2212yes—
73C23×Q8C2311yescharacteristic
74C23×Q8C2312yes—
75C23×D4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82#554—2128no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—24no—
86C2×#125—28no—
87C2×#125—28no—
88C2×#125—28no—
89C2×#123—28no—
90C2×#123—216no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8—28no—
102C2×C22⋊Q8D418yes—
103C2×C4×Q8—24no—
104C2×C4×Q8—24no—
105C2×C4×Q8—24no—
106C2×C4×D4—28no—
107C2×C4×D4—28no—
108C2×C4×D4—24no—
109C2×C4×D4—24no—
110C2×C4×D4—28no—
111C2×C4×D4D418yes—
112C22×C4⋊C4—24no—
113C22×C4⋊C4—24no—
114C22×C4⋊C4—24no—
115C22×C4⋊C4—24no—
116C22×C4⋊C4—24no—
117C22×C4⋊C4—24no—
118C22×C4⋊C4—24no—
119C22×C4⋊C4—24no—
120C22×C4⋊C4D418yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2314yes—
140C22×C22⋊C4C2311yescharacteristic
141C22×C22⋊C4C2312yes—
142C22×C22⋊C4C2311yescharacteristic
143C22×C42—24no—
144C22×C42—28no—
145C22×C42C2312yes—
146C22×C42C2312yes—
147C22×C42C2312yes—
148C22×C42C2311yescharacteristic
149C22×C42C2312yes—
150#400—216no—
151#398—28no—
152#397—28no—
153#391—28no—
154#391—28no—
155#389—28no—
156#386—28no—
157#382—28no—
158#382—28no—
159#382—216no—
160#382—28no—
161#381—28no—
162#381—216no—
163#380—28no—
164#380—24no—
165C4×C22⋊C4—24no—
166C4×C22⋊C4—28no—
167C2×#95—216no—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2318yes—
173C2×#95C2318yes—
174C2×#95C2314yes—
175C2×#95C2318yes—
176C2×#95C2314yes—
177C2×#95C2318yes—
178C2×#95C2318yes—
179C2×#95C2314yes—
180C2×#95C2318yes—
181C2×#95C2318yes—
182C2×#95C2318yes—
183C25C2411yescharacteristic
184C25C2411yescharacteristic
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—48no—
193C22×Q8C2×D414yes—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—24no—
197C22×D4C2×D414yes—
198C22×D4#4118yes—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—28no—
219C23×C4—44no—
220C23×C4—48no—
221C23×C4—48no—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2×D418yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C4⋊Q8—232no—
241#125—264no—
242#123—432no—
243C22⋊Q8—232no—
244C22⋊Q8—264no—
245C22⋊Q8—264no—
246C22⋊Q8—232no—
247C4×Q8—232no—
248C4×D4—264no—
249C4×D4—232no—
250C2×C4⋊C4—232no—
251C2×C4⋊C4—232no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—216no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—416no—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×D418yes—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D418yes—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—216no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4C2×D418yes—
314C2×C22⋊C4C2×D414yes—
315C2×C22⋊C4C2×D414yes—
316C2×C42—232no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—28no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42C2×D414yes—
330C2×C42#4118yes—
331#95—216no—
332#95—216no—
333#95—24no—
334#95—28no—
335#95—24no—
336#95—216no—
337#95—28no—
338#95—216no—
339#95—28no—
340#95—28no—
341#95—216no—
342#95—48no—
343#95—44no—
344#95—48no—
345#95—48no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—21no—
350C24—21no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24C22×D412yes—
356C24C22×D412yes—
357C24C22×D412yes—
358C24C22×D412yes—
359C24C2×#4114yes—
360C24C2×#4114yes—
361C242−1+414yes—
362C24C2511yescharacteristic
363C2×Q8—216no—
364C2×Q8—28no—
365C2×Q8—216no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—48no—
369C2×Q8—88no—
370C2×Q8—88no—
371C2×Q8—88no—
372C2×D4—28no—
373C2×D4—28no—
374C2×D4—216no—
375C2×D4—216no—
376C2×D4#124116yes—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—216no—
385C22×C4—216no—
386C22×C4—216no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—24no—
413C22×C4—216no—
414C22×C4—216no—
415C22×C4—28no—
416C22×C4—216no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—28no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—416no—
435C22×C4—416no—
436C22×C4—416no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4—88no—
452C22×C4#120116yes—
453C22×C4C22×D414yes—
454C22×C4C22×D414yes—
455C22×C4C22×D412yes—
456C22×C4C22×D414yes—
457C22×C4C22×D412yes—
458C22×C4C22×D414yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C2×#4118yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4114yes—
469C22×C42−1+414yes—
470C4⋊C4—216no—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—28no—
475C4⋊C4—216no—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—216no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—416no—
483C4⋊C4—48no—
484C22⋊C4—216no—
485C22⋊C4—216no—
486C22⋊C4—28no—
487C22⋊C4—28no—
488C22⋊C4—416no—
489C22⋊C4—48no—
490C22⋊C4—416no—
491C22⋊C4—48no—
492C42—216no—
493C42—216no—
494C42—28no—
495C42—216no—
496C42—48no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—24no—
501C23—28no—
502C23—24no—
503C23—28no—
504C23—24no—
505C23—24no—
506C23—28no—
507C23—28no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—24no—
515C23—28no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—42no—
522C23—44no—
523C23C2×#12014yes—
524C23C2×#12414yes—
525C23C2×#12414yes—
526C23#53614yes—
527C23#54818yes—
528C23#54818yes—
529C23C23×D411yescharacteristic
530C23C23×D411yescharacteristic
531C23C23×D411yescharacteristic
532C23C22×#4112yes—
533C23C22×#4112yes—
534C23C2×2−1+412yes—
535C23C2×2−1+412yes—
536C23#58514yes—
537Q8—416no—
538Q8—816no—
539D4—232no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—232no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—432no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—416no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—816no—
602C2×C4—816no—
603C2×C4—816no—
604C2×C4C2×#12014yes—
605C2×C4C2×#12014yes—
606C2×C4#52514yes—
607C2×C4C2×#12414yes—
608C2×C4#53614yes—
609C2×C4#54718yes—
610C2×C4#54718yes—
611C22—24no—
612C22—28no—
613C22—28no—
614C22—24no—
615C22—216no—
616C22—28no—
617C22—28no—
618C22—28no—
619C22—24no—
620C22—42no—
621C22—44no—
622C22—42no—
623C22—48no—
624C22—44no—
625C22—44no—
626C22#243618yes—
627C22C22×#12011yescharacteristic
628C22C2×#52511yescharacteristic
629C22C22×#12411yescharacteristic
630C22C2×#53611yescharacteristic
631C22C2×#53611yescharacteristic
632C22C2×#54712yes—
633C22C2×#54712yes—
634C22C2×#54812yes—
635C22C2×#54812yes—
636C22#347514yes—
637C22#348512yes—
638C22#349514yes—
639C22#350012yes—
640C22#350012yes—
641C22#351514yes—
642C22#356314yes—
643C4—24no—
644C4—24no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—42no—
649C4—48no—
650C4—42no—
651C4—44no—
652C4—84no—
653C4—84no—
654C4—84no—
655C4#240618yes—
656C2—28no—
657C2—44no—
658C2C2×#243611yescharacteristic
659C2C2×#240611yescharacteristic
660C2#3561611yescharacteristic
661C2#3616412yes—
662C2#3651612yes—
663C2#3786512yes—
664C2#3912212yes—
665C2#5931714yes—
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×#2503C2semidirect
4. C22×#382C2semidirect
5. #15881C2semidirect
6. #15670C2semidirect
7. #15236C2semidirect
8. #15230C2semidirect
9. #14934C2semidirect
10. #14702C2semidirect
11. #13978C2semidirect
13. #13924C2semidirect
14. #13881C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#25032
C2×#24862
C22×#3822
#158812
#156702
#152362
#152302
#149342
#147022
#139782
#139682
#139242
#138812
C4×#3862