Finite Group Databasebeta

← #556568

Subgroups — #556568

Order 512.

2730 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3C22×#387C211yesmaximal, characteristic
4#14631C212yesmaximal, complemented
5#14626C212yesmaximal, complemented
6#14308C212yesmaximal, complemented
7#14302C212yesmaximal, complemented
8#14280C214yesmaximal, complemented
9#14265C212yesmaximal
10#14265C212yesmaximal
11#14159C211yesmaximal, characteristic
12#14081C214yesmaximal, complemented
13#14040C212yesmaximal
14#14039C212yesmaximal
15#13966C211yesmaximal, characteristic
16#13923C211yesmaximal, characteristic
17#13753C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23#2583C4116yescomplemented
24C2×#398C2212yes—
25C2×#395C2212yes—
26C4×C22⋊Q8—28no—
27C2×#391C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#389C2214yes—
30C2×#387—28no—
31C2×#387C2212yes—
32C2×#387C2212yes—
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#387C2212yes—
36C2×#387C2214yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#380C2211yescharacteristic
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×#376C2212yes—
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51#1446C2212yes—
52#1445C2214yes—
53#1445C2212yes—
54#1445C2214yes—
55#1445C2214yes—
56#1443C2214yes—
57#1443C2214yes—
58#1443C2214yes—
59#1442C2214yes—
60#1442C2212yes—
61#1442C2212yes—
62#1442C2214yes—
63#1442C2214yes—
64#1441C2214yes—
65#1441C2212yes—
66#1441C2214yes—
67#1441C2212yes—
68#1441C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1434C2214yes—
74#1434C2214yes—
75#1434C2212yes—
76#1434C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8C2×C418yes—
97C2×C4×Q8—24no—
98C2×C4×Q8—44no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—44no—
103C22×C4⋊C4C2×C418yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C42—44no—
121C22×C42D418yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#398C2×C4116yescomplemented
131#395C2×C4116yescomplemented
132#391—24no—
133#389—24no—
134#389C2×C4132yescomplemented
135#387—28no—
136#387—28no—
137#387—28no—
138#387—28no—
139#387—28no—
140#387—48no—
141#385—28no—
142#385C2×C4116yes—
143#385C2×C4116yes—
144#380—24no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—24no—
147C4×C22⋊C4—28no—
148C4×C22⋊C4—24no—
149C4×C22⋊C4C2×C418yes—
150#376—44no—
151C2×#95—24no—
152C2×#95—24no—
153C2×#95—24no—
154C2×#95—24no—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C25C2411yescharacteristic
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8C22×C414yes—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4C22×C414yes—
219C23×C4C22×C414yes—
220C23×C4C22×C414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4#4114yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C22⋊Q8—232no—
242C4×Q8—416no—
243C2×C4⋊C4—232no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—416no—
269C2×C4⋊C4C22×C418yes—
270C2×C4⋊C4C22×C418yes—
271C2×C4⋊C4C22×C418yes—
272C2×C4⋊C4C22×C414yes—
273C2×C4⋊C4C22×C4116yes—
274C2×C4⋊C4C22×C4116yes—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4C22×C418yes—
285C2×C22⋊C4C22×C418yes—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—28no—
292C2×C42—416no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—44no—
299C2×C42—42no—
300C2×C42C22×C418yes—
301C2×C42C2×D414yes—
302C2×C42C2×D414yes—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—28no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95C22×C418yes—
336#95C22×C4116yes—
337#95C22×C4116yes—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24C23×C412yes—
343C24C22×D412yes—
344C24C2×#4112yes—
345C242−1+414yes—
346C242−1+414yes—
347C24C2511yescharacteristic
348C2×Q8—28no—
349C2×Q8—48no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×Q8—48no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—28no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—48no—
431C22×C4—416no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—416no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4C4×D4116yes—
448C22×C4C23×C418yes—
449C22×C4C23×C414yes—
450C22×C4C23×C414yes—
451C22×C4C23×C414yes—
452C22×C4C23×C414yes—
453C22×C4C23×C412yes—
454C22×C4C23×C412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4H(5,𝔽2)14yes—
464C4⋊C4—216no—
465C4⋊C4—216no—
466C4⋊C4—28no—
467C4⋊C4—216no—
468C4⋊C4—416no—
469C4⋊C4—416no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C22⋊C4—216no—
477C22⋊C4—216no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C42—28no—
481C42—28no—
482C42—44no—
483C42—44no—
484C42—48no—
485C42—48no—
486C42—48no—
487C42—48no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—24no—
496C23—28no—
497C23—28no—
498C23—28no—
499C23—28no—
500C23C2×C4×D414yes—
501C23#51918yes—
502C23#53614yes—
503C23#53614yes—
504C23#54118yes—
505C23C24×C411yescharacteristic
506C23C23×D411yescharacteristic
507C23C22×#4111yescharacteristic
508C23C2×H(5,𝔽2)12yes—
509C23C2×2−1+412yes—
510C23C2×2−1+412yes—
511C23#58512yes—
512C23#58512yes—
513C23#58512yes—
514Q8—416no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—416no—
548C2×C4—416no—
549C2×C4—416no—
550C2×C4—416no—
551C2×C4—416no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—48no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—88no—
598C2×C4—88no—
599C2×C4C2×C4×D414yes—
600C2×C4C2×C4×D414yes—
601C2×C4#53414yes—
602C22—24no—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—216no—
607C22—24no—
608C22—216no—
609C22—24no—
610C22#232718yes—
611C22C22×C4×D411yescharacteristic
612C22C2×#51912yes—
613C22#342812yes—
614C22C4×H(5,𝔽2)12yes—
615C22C2×#53411yescharacteristic
616C22C2×#53611yescharacteristic
617C22C2×#53611yescharacteristic
618C22#344911yescharacteristic
619C22#344911yescharacteristic
620C22#344911yescharacteristic
621C22C2×#54112yes—
622C22#345512yes—
623C22#345912yes—
624C22#353514yes—
625C22#354114yes—
626C22#355314yes—
627C22#356414yes—
628C4—24no—
629C4—24no—
630C4—24no—
631C4—44no—
632C4—44no—
633C4—44no—
634C4—44no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—48no—
639C4—82no—
640C4—82no—
641C4—84no—
642C4—82no—
643C4—82no—
644C2—28no—
645C2#3406511yescharacteristic
646C2C2×#232711yescharacteristic
647C2C4×#53411yescharacteristic
648C2#3448614yes—
649C2#3743811yescharacteristic
650C2#3744111yescharacteristic
651C2#3748814yes—
652C2#3754811yescharacteristic
653C2#3760711yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #14631C2semidirect
5. #14626C2semidirect
6. #14308C2semidirect
7. #14302C2semidirect
8. #14280C2semidirect
12. #14081C2semidirect
23. #2583C4semidirect
130. #398C2×C4semidirect
131. #395C2×C4semidirect
134. #389C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
C2×C4×C22⋊Q82
C22×#3872
#146312
#146262
#143082
#143022
#142802
#142652
#141592
#140812
#140402
#140392
#139662
#139232
#137532