Finite Group Databasebeta

← #614570

Subgroups — #614570

Order 512.

2930 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic
3#15099C214yesmaximal, complemented
4#15098C212yesmaximal, complemented
5#15097C212yesmaximal, complemented
6#15014C211yesmaximal, characteristic
7#14914C211yesmaximal, characteristic
8#14821C211yesmaximal, characteristic
9#14720C211yesmaximal, characteristic
10#14705C212yesmaximal
11#14705C212yesmaximal
12#14241C212yesmaximal, complemented
13#14226C212yesmaximal, complemented
14#14083C214yesmaximal, complemented
15#14044C212yesmaximal
16#14043C212yesmaximal
17C4×#381C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22C23×C42C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2855—28no—
25C2×#398C2212yes—
26C2×#395C2212yes—
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#389C2214yes—
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#387C2212yes—
33C2×#387C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#381C2212yes—
40C2×#381C2212yes—
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4—28no—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50#1446C2212yes—
51#1445C2212yes—
52#1444C2214yes—
53#1444C2212yes—
54#1444C2212yes—
55#1444C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2212yes—
61#1441C2212yes—
62#1441C2214yes—
63#1441C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2214yes—
75#1434C2212yes—
76#1434C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2211yescharacteristic
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2211yescharacteristic
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—44no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C42—24no—
118C22×C42—24no—
119C22×C42—24no—
120C22×C42—24no—
121C22×C42—44no—
122C22×C42—44no—
123C22×C42D418yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2314yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#398—28no—
134#395—28no—
135#391—24no—
136#391—24no—
137#389—216no—
138#389—24no—
139#389—24no—
140#387—28no—
141#387—28no—
142#385—28no—
143#385—28no—
144#385—28no—
145#385—44no—
146#385—44no—
147#381—28no—
148#381Q8116yescomplemented
149#380—24no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—28no—
154C4×C22⋊C4—44no—
155C2×#95—24no—
156C2×#95—24no—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C25C2411yescharacteristic
188C22×Q8—22no—
189C22×Q8—24no—
190C22×Q8—42no—
191C22×Q8—42no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2×Q818yes—
221C23×C4#4114yes—
222C23×C4#4114yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C22⋊Q8—416no—
240C2×C4⋊C4—232no—
241C2×C4⋊C4—232no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4#4118yes—
278C2×C4⋊C4#4118yes—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4C2×Q818yes—
290C2×C22⋊C4C2×Q818yes—
291C2×C42—232no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—28no—
301C2×C42—416no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42—44no—
307C2×C42—44no—
308C2×C42—416no—
309C2×C42—44no—
310C2×C42—44no—
311C2×C42C2×D414yes—
312C2×C42C2×D414yes—
313C2×C42C2×D414yes—
314C2×C42C2×D414yes—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—28no—
332#95—24no—
333#95—28no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—42no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95C2×Q8116yes—
344#95C2×Q8116yes—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24C22×D412yes—
350C24C22×Q814yes—
351C24C2×#4112yes—
352C24C2×#4112yes—
353C24H(5,𝔽2)14yes—
354C24C2511yescharacteristic
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—88no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—216no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—216no—
366C22×C4—216no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—216no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4C4⋊Q8116yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×Q818yes—
461C22×C4C22×Q818yes—
462C22×C4C22×Q814yes—
463C22×C4C22×Q814yes—
464C22×C4C2×#4114yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4114yes—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C4⋊C4—216no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—416no—
483C4⋊C4—88no—
484C4⋊C4—88no—
485C4⋊C4—88no—
486C22⋊C4—216no—
487C22⋊C4—216no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C42—216no—
491C42—216no—
492C42—44no—
493C42—44no—
494C42—44no—
495C42—44no—
496C42—48no—
497C42—48no—
498C42—88no—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—22no—
504C23—22no—
505C23—28no—
506C23—28no—
507C23—24no—
508C23—24no—
509C23—28no—
510C23—28no—
511C23#52514yes—
512C23C2×C4⋊Q814yes—
513C23#53414yes—
514C23#55418yes—
515C23#55418yes—
516C23C23×D411yescharacteristic
517C23C23×Q812yes—
518C23C22×#4111yescharacteristic
519C23C22×#4112yes—
520C23C22×#4112yes—
521C23C22×#4111yescharacteristic
522C23C2×H(5,𝔽2)12yes—
523C23#58512yes—
524C23#58512yes—
525Q8—88no—
526C2×C4—232no—
527C2×C4—232no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—416no—
567C2×C4—416no—
568C2×C4—416no—
569C2×C4—416no—
570C2×C4—416no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4#52514yes—
611C2×C4#52514yes—
612C2×C4#52918yes—
613C2×C4C2×C4⋊Q814yes—
614C2×C4C2×C4⋊Q814yes—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—216no—
620C22—216no—
621C22—24no—
622C22—24no—
623C22#258318yes—
624C22C2×#52511yescharacteristic
625C22C2×#52511yescharacteristic
626C22C2×#52511yescharacteristic
627C22C2×#52912yes—
628C22C22×C4⋊Q811yescharacteristic
629C22#344412yes—
630C22C2×#53411yescharacteristic
631C22#344911yescharacteristic
632C22#344911yescharacteristic
633C22C2×#55412yes—
634C22C2×#55412yes—
635C22Q8×#4114yes—
636C22Q8×#4114yes—
637C22#349112yes—
638C22#349312yes—
639C22#351314yes—
640C22#351314yes—
641C4—28no—
642C4—24no—
643C4—24no—
644C4—24no—
645C4—24no—
646C4—44no—
647C4—48no—
648C4—48no—
649C4—42no—
650C4—44no—
651C4—42no—
652C4—44no—
653C4—82no—
654C4—82no—
655C4—82no—
656C4—82no—
657C2—28no—
658C2C2×#258311yescharacteristic
659C2#3495311yescharacteristic
660C2#3495311yescharacteristic
661C2#3506211yescharacteristic
662C2#3506311yescharacteristic
663C2#3511811yescharacteristic
664C2#3655314yes—
665C2#3656914yes—
666C2#3662111yescharacteristic
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
3. #15099C2semidirect
4. #15098C2semidirect
5. #15097C2semidirect
12. #14241C2semidirect
13. #14226C2semidirect
14. #14083C2semidirect
148. #381Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C22×C4×C4⋊C42
#150992
#150982
#150972
#150142
#149142
#148212
#147202
#147052
#142412
#142262
#140832
#140442
#140432
C4×#3812