Finite Group Databasebeta

← #657581

Subgroups — #657581

Order 512.

3968 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2C2×#2502C211yesmaximal, characteristic, complemented
3C2×#2487C211yesmaximal, characteristic
4C22×#382C211yesmaximal, characteristic, complemented
5#15927C214yesmaximal, complemented
6#15881C214yesmaximal, complemented
7#15236C212yesmaximal, complemented
8#15228C212yesmaximal, complemented
9#14912C214yesmaximal
10#14702C211yesmaximal, characteristic, complemented
11#14128C214yesmaximal, complemented
12#13978C212yesmaximal, complemented
13#13904C211yesmaximal, characteristic, complemented
14#13879C212yesmaximal, complemented
15C4×#386C211yesmaximal, characteristic
16C2×#552—28no—
17C2×#552—28no—
18C2×#552—28no—
19C22×#126C2212yes—
20C22×#125C2212yes—
21C22×#125C2211yescharacteristic
22C22×#123C2212yes—
23C22×C22⋊Q8C2211yescharacteristic
24C22×C22⋊Q8C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26C22×C4×D4C2211yescharacteristic
27C22×#117C2212yes—
28C22×#117C2211yescharacteristic
29C23×C4⋊C4C2211yescharacteristic
30C23×C4⋊C4C2211yescharacteristic
31C23×C42C2211yescharacteristic
32#2502—28no—
33#2487—28no—
34C2×#400C2212yes—
35C2×#399C2212yes—
36C2×#398C2214yes—
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#389C2212yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#385C2212yes—
43C2×#385C2214yes—
44C2×#384C2212yes—
45C2×#382—28no—
46C2×#382C2214yes—
47C2×#382C2212yes—
48C2×#382C2212yes—
49C2×#381C2214yes—
50C2×#380C2212yes—
51C2×#380C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54C22×#95C2214yes—
55#1445C2214yes—
56#1444C2218yes—
57#1442C2218yes—
58#1442C2218yes—
59#1441C2218yes—
60#1440C2218yes—
61#1439C2218yes—
62#1437C2212yes—
63#1436C2212yes—
64#1436C2218yes—
65#1436C2214yes—
66#1436C2218yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2214yes—
72C4×#95C2212yes—
73C23×Q8C2311yescharacteristic
74C23×D4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80#552—2128no—
81C2×#126—28no—
82C2×#126—28no—
83C2×#126—28no—
84C2×#125—28no—
85C2×#125—28no—
86C2×#125—28no—
87C2×#125—24no—
88C2×#125—24no—
89C2×#125—24no—
90C2×#123—28no—
91C2×#123—28no—
92C2×#123—28no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8D418yes—
99C2×C4×Q8—24no—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4D418yes—
105C2×#117—24no—
106C2×#117—28no—
107C2×#117—24no—
108C2×#117—24no—
109C2×#117—28no—
110C2×#117—28no—
111C22×C4⋊C4—24no—
112C22×C4⋊C4—24no—
113C22×C4⋊C4—24no—
114C22×C4⋊C4—24no—
115C22×C4⋊C4—24no—
116C22×C4⋊C4D418yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2318yes—
131C22×C4⋊C4C2314yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2314yes—
136C22×C22⋊C4C2311yescharacteristic
137C22×C22⋊C4C2311yescharacteristic
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2314yes—
141C22×C42—24no—
142C22×C42—28no—
143C22×C42C2312yes—
144C22×C42C2312yes—
145C22×C42C2312yes—
146C22×C42C2312yes—
147C22×C42C2311yescharacteristic
148C22×C42C2312yes—
149#400—28no—
150#399—28no—
151#398—216no—
152#391—28no—
153#391—28no—
154#389—28no—
155#386—28no—
156#386—28no—
157#385—28no—
158#385—216no—
159#384—28no—
160#382—28no—
161#382—28no—
162#382—216no—
163#381—216no—
164#380—28no—
165#380—24no—
166C4×C22⋊C4—24no—
167C4×C22⋊C4—28no—
168C2×#95—216no—
169C2×#95C2312yes—
170C2×#95C2318yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2318yes—
177C2×#95C2318yes—
178C2×#95C2318yes—
179C2×#95C2318yes—
180C2×#95C2318yes—
181C2×#95C2314yes—
182C2×#95C2318yes—
183C25C2411yescharacteristic
184C25C2411yescharacteristic
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8C2×D414yes—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4C2×D414yes—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—28no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—44no—
213C23×C4—48no—
214C23×C4C2×D418yes—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232#126—264no—
233#125—264no—
234#125—232no—
235#123—264no—
236C22⋊Q8—232no—
237C22⋊Q8—232no—
238C4×Q8—232no—
239C4×D4—232no—
240#117—232no—
241#117—264no—
242C2×C4⋊C4—232no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
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—216no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—416no—
279C2×C4⋊C4—48no—
280C2×C4⋊C4—48no—
281C2×C4⋊C4—48no—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D418yes—
285C2×C4⋊C4C2×D414yes—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D418yes—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
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—44no—
308C2×C22⋊C4—48no—
309C2×C22⋊C4C2×D414yes—
310C2×C22⋊C4C2×D418yes—
311C2×C22⋊C4C2×D414yes—
312C2×C22⋊C4#4118yes—
313C2×C42—232no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
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—28no—
339#95—216no—
340#95—216no—
341#95—28no—
342#95—216no—
343#95—48no—
344#95—44no—
345C24—22no—
346C24—22no—
347C24—24no—
348C24—21no—
349C24—21no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24C22×D412yes—
356C24C22×D412yes—
357C24C22×D412yes—
358C24C22×D412yes—
359C24C2×#4114yes—
360C242−1+414yes—
361C24C2511yescharacteristic
362C2×Q8—28no—
363C2×Q8—216no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—48no—
367C2×D4—28no—
368C2×D4—28no—
369C2×D4—28no—
370C2×D4—48no—
371C2×D4—48no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—216no—
380C22×C4—232no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—28no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—216no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—28no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—416no—
432C22×C4—416no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—416no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—416no—
442C22×C4—48no—
443C22×C4—88no—
444C22×C4—88no—
445C22×C4#120116yes—
446C22×C4C22×D414yes—
447C22×C4C22×D412yes—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C22×D414yes—
451C22×C4C22×D412yes—
452C22×C4C22×D414yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C2×#4118yes—
460C22×C4C2×#4118yes—
461C22×C4C2×#4114yes—
462C22×C42−1+414yes—
463C4⋊C4—216no—
464C4⋊C4—216no—
465C4⋊C4—216no—
466C4⋊C4—216no—
467C4⋊C4—28no—
468C4⋊C4—28no—
469C4⋊C4—216no—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—216no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—816no—
477C4⋊C4—88no—
478C22⋊C4—216no—
479C22⋊C4—216no—
480C22⋊C4—28no—
481C22⋊C4—216no—
482C22⋊C4—216no—
483C22⋊C4—28no—
484C22⋊C4—416no—
485C22⋊C4—48no—
486C22⋊C4—88no—
487C42—216no—
488C42—216no—
489C42—216no—
490C42—28no—
491C42—216no—
492C42—48no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—28no—
498C23—24no—
499C23—28no—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—28no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—42no—
516C23—44no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23C2×#12014yes—
521C23#53214yes—
522C23#53614yes—
523C23#54718yes—
524C23#54818yes—
525C23C23×D411yescharacteristic
526C23C23×D411yescharacteristic
527C23C23×D411yescharacteristic
528C23C22×#4112yes—
529C23C22×#4112yes—
530C23C2×2−1+412yes—
531C23C2×2−1+412yes—
532C23#58514yes—
533Q8—416no—
534D4—416no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—232no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—432no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—432no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—416no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—48no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—816no—
601C2×C4C2×#12014yes—
602C2×C4C2×#12014yes—
603C2×C4#52514yes—
604C2×C4#53614yes—
605C2×C4#54718yes—
606C2×C4#54718yes—
607C22—28no—
608C22—28no—
609C22—28no—
610C22—24no—
611C22—24no—
612C22—216no—
613C22—28no—
614C22—44no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—44no—
619C22—82no—
620C22—82no—
621C22—84no—
622C22#243218yes—
623C22C22×#12011yescharacteristic
624C22C2×#52511yescharacteristic
625C22C2×#53211yescharacteristic
626C22C2×#53611yescharacteristic
627C22C2×#53611yescharacteristic
628C22C2×#54712yes—
629C22C2×#54712yes—
630C22C2×#54712yes—
631C22C2×#54812yes—
632C22#347514yes—
633C22#348712yes—
634C22#349514yes—
635C22#350012yes—
636C22#350012yes—
637C22#351514yes—
638C22#356314yes—
639C4—24no—
640C4—48no—
641C4—42no—
642C4—44no—
643C4—44no—
644C4—44no—
645C4—42no—
646C4—42no—
647C4—44no—
648C4—48no—
649C4—84no—
650C4—84no—
651C4#240618yes—
652C2—28no—
653C2—82no—
654C2C2×#243211yescharacteristic
655C2C2×#240611yescharacteristic
656C2#3567611yescharacteristic
657C2#3618412yes—
658C2#3642712yes—
659C2#3786512yes—
660C2#3891912yes—
661C2#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×#2502C2semidirect
4. C22×#382C2semidirect
5. #15927C2semidirect
6. #15881C2semidirect
7. #15236C2semidirect
8. #15228C2semidirect
10. #14702C2semidirect
11. #14128C2semidirect
12. #13978C2semidirect
13. #13904C2semidirect
14. #13879C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5522
C2×#25022
C2×#24872
C22×#3822
#159272
#158812
#152362
#152282
#149122
#147022
#141282
#139782
#139042
#138792
C4×#3862