Finite Group Databasebeta

← #704577

Subgroups — #704577

Order 512.

2850 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2484C211yesmaximal, characteristic
2C22×#398C211yesmaximal, characteristic
3C2×#2344C211yesmaximal, characteristic
4C2×#2317C211yesmaximal, characteristic
5#17061C212yesmaximal, complemented
6#16705C212yesmaximal, complemented
7#15874C214yesmaximal, complemented
8#15347C214yesmaximal, complemented
9#15151C212yesmaximal, complemented
10#15085C212yesmaximal, complemented
11#14919C212yesmaximal
12#14728C212yesmaximal
13#14342C211yesmaximal, characteristic
14#14043C212yesmaximal
15#13962C212yesmaximal
16#13962C211yesmaximal, characteristic
17#13958C211yesmaximal, characteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23C23×C4⋊C4C2212yes—
24#2484—28no—
25C2×#400C2211yescharacteristic
26C2×#400C2211yescharacteristic
27C2×#398—28no—
28C2×#398C2212yes—
29C2×#398C2211yescharacteristic
30C2×#398C2212yes—
31C2×#398C2212yes—
32C2×#398C2211yescharacteristic
33C2×#398C2212yes—
34C2×#398C2214yes—
35#2344—28no—
36#2317—28no—
37C2×#389C2214yes—
38C2×#389C2212yes—
39C2×#389C2214yes—
40C2×#387C2212yes—
41C2×#387C2212yes—
42C2×#387C2212yes—
43C2×#387C2212yes—
44C2×#385C2212yes—
45C2×#385C2212yes—
46C2×#385C2212yes—
47C2×#385C2212yes—
48C2×#384C2212yes—
49C2×#382C2212yes—
50C2×#381C2211yescharacteristic
51C2×#381C2211yescharacteristic
52C2×#381C2211yescharacteristic
53C2×#381C2212yes—
54C2×#380C2211yescharacteristic
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C22⋊C4C2211yescharacteristic
57C22×#95C2212yes—
58C22×#95C2211yescharacteristic
59#1445C2214yes—
60#1445C2214yes—
61#1445C2212yes—
62#1445C2212yes—
63#1445C2212yes—
64#1445C2212yes—
65#1444C2214yes—
66#1444C2214yes—
67#1443C2214yes—
68#1443C2214yes—
69#1442C2214yes—
70#1442C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2214yes—
75#1441C2214yes—
76#1436C2214yes—
77C4⋊#95C2214yes—
78C4⋊#95C2212yes—
79#1434C2214yes—
80#1434C2214yes—
81#1434C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C22⋊#95C2212yes—
86C22⋊#95C2212yes—
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117—24no—
95C2×#117—24no—
96C2×#117—28no—
97C2×#117—28no—
98C2×#117D418yes—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—48no—
106C22×C4⋊C4D418yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2314yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135#400—24no—
136#400—24no—
137#398—28no—
138#398—28no—
139#398—28no—
140#398—28no—
141#398—24no—
142#398—24no—
143#398—48no—
144#389—216no—
145#389—28no—
146#389—48no—
147#387—28no—
148#387—28no—
149#387—28no—
150#387—28no—
151#385—28no—
152#385—28no—
153#385—28no—
154#385—28no—
155#384—28no—
156#382—28no—
157#381—24no—
158#381—24no—
159#381—28no—
160#381—24no—
161#380—24no—
162C4×C4⋊C4—28no—
163C4×C22⋊C4—24no—
164C2×#95—24no—
165C2×#95—28no—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2311yescharacteristic
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C25C2411yescharacteristic
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—28no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—44no—
216C23×C4—42no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4#4114yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2414yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240#117—232no—
241#117—416no—
242#117—416no—
243C2×C4⋊C4—232no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—432no—
286C2×C4⋊C4—416no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D414yes—
294C2×C4⋊C4C2×D414yes—
295C2×C4⋊C4C2×D418yes—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4C2×D414yes—
308C2×C22⋊C4C2×D414yes—
309C2×C22⋊C4#4118yes—
310C2×C42—28no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—28no—
317C2×C42—24no—
318C2×C42—42no—
319C2×C42C2×D418yes—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—22no—
336#95—22no—
337#95—28no—
338#95—28no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352C24—22no—
353C24—24no—
354C24—22no—
355C24C22×D412yes—
356C24C22×D412yes—
357C24C2×#4112yes—
358C24C2×#4114yes—
359C242−1+414yes—
360C24C2511yescharacteristic
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—216no—
364C22×C4—216no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—216no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—216no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—44no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—88no—
453C22×C4—88no—
454C22×C4#121116yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D414yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D412yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4118yes—
468C22×C4H(5,𝔽2)14yes—
469C22×C42−1+418yes—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—216no—
475C4⋊C4—48no—
476C4⋊C4—416no—
477C4⋊C4—48no—
478C4⋊C4—416no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—416no—
486C4⋊C4—88no—
487C4⋊C4—88no—
488C22⋊C4—28no—
489C22⋊C4—48no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—48no—
493C22⋊C4#123116yes—
494C42—216no—
495C42—48no—
496C42—48no—
497C42—48no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23C2×#12114yes—
510C23C2×#12314yes—
511C23#53614yes—
512C23#54718yes—
513C23#55218yes—
514C23C23×D411yescharacteristic
515C23C23×D411yescharacteristic
516C23C22×#4111yescharacteristic
517C23C22×#4112yes—
518C23C2×H(5,𝔽2)12yes—
519C23C2×2−1+412yes—
520C23C2×2−1+414yes—
521C23#58512yes—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—232no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—416no—
553C2×C4—432no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—416no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—816no—
607C2×C4C2×#12114yes—
608C2×C4C2×#12114yes—
609C2×C4C2×#12318yes—
610C2×C4#53514yes—
611C22—28no—
612C22—24no—
613C22—28no—
614C22—28no—
615C22—24no—
616C22—216no—
617C22—28no—
618C22#248618yes—
619C22C22×#12111yescharacteristic
620C22C22×#12311yescharacteristic
621C22C2×#53511yescharacteristic
622C22C2×#53611yescharacteristic
623C22#344911yescharacteristic
624C22C2×#54712yes—
625C22C2×#55212yes—
626C22#348612yes—
627C22#348812yes—
628C22#349412yes—
629C22#349912yes—
630C22#350912yes—
631C22#351514yes—
632C22#352014yes—
633C22#353814yes—
634C22#356314yes—
635C4—28no—
636C4—24no—
637C4—24no—
638C4—24no—
639C4—48no—
640C4—44no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—82no—
645C4—82no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—84no—
650C4—82no—
651C2—28no—
652C2C2×#248611yescharacteristic
653C2#3525211yescharacteristic
654C2#3566711yescharacteristic
655C2#3599812yes—
656C2#3804512yes—
657C2#3891712yes—
658C2#4193912yes—
659C2#4397714yes—
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
5. #17061C2semidirect
6. #16705C2semidirect
7. #15874C2semidirect
8. #15347C2semidirect
9. #15151C2semidirect
10. #15085C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24842
C22×#3982
C2×#23442
C2×#23172
#170612
#167052
#158742
#153472
#151512
#150852
#149192
#147282
#143422
#140432
#139622
#139582