Finite Group Databasebeta

← #769971

Subgroups — #769971

Order 512.

3330 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C211yesmaximal, characteristic
2C2×#2434C211yesmaximal, characteristic
3C22×#395C211yesmaximal, characteristic
4C2×#2329C211yesmaximal, characteristic
5#15740C214yesmaximal, complemented
6#15623C214yesmaximal, complemented
7#15360C214yesmaximal, complemented
8#15318C214yesmaximal, complemented
9#14955C212yesmaximal
10#14786C212yesmaximal
11#14149C212yesmaximal
12#14130C212yesmaximal
13#13923C211yesmaximal, characteristic
14#13922C211yesmaximal, characteristic
15#13778C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C4⋊Q8C2212yes—
18C22×C22⋊Q8C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2583—28no—
23#2434—28no—
24C2×#398C2212yes—
25C2×#398C2214yes—
26C2×#395—28no—
27C2×#395C2214yes—
28C2×#395C2214yes—
29C2×#395C2212yes—
30C2×#395C2214yes—
31#2329—28no—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#389C2214yes—
37C2×#389C2212yes—
38C2×#389C2214yes—
39C2×#387C2212yes—
40C2×#387C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2212yes—
44C2×#385C2212yes—
45C2×#385C2212yes—
46C2×#380C2211yescharacteristic
47C2×#380C2212yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51C22×#95C2211yescharacteristic
52C22×#95C2211yescharacteristic
53C22×#95C2211yescharacteristic
54#1446C2214yes—
55#1445C2214yes—
56#1444C2214yes—
57#1444C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2212yes—
62#1441C2214yes—
63#1441C2212yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73C4⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×Q8—24no—
80C23×Q8—24no—
81C23×Q8—24no—
82C23×Q8—44no—
83C23×Q8—44no—
84C23×Q8C2312yes—
85C23×Q8C2312yes—
86C23×Q8C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—44no—
96C2×C4⋊Q8—44no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8D418yes—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—44no—
106C22×C4⋊C4—44no—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130#398—28no—
131#398—48no—
132#395—28no—
133#395—216no—
134#395—216no—
135#395—216no—
136#391—28no—
137#391—28no—
138#391—44no—
139#391—44no—
140#389—216no—
141#389—28no—
142#389—48no—
143#387—28no—
144#387—28no—
145#385—28no—
146#385—28no—
147#385—28no—
148#385—28no—
149#385—28no—
150#380—24no—
151#380Q8116yescomplemented
152C4×C22⋊C4—24no—
153C4×C22⋊C4—28no—
154C4×C22⋊C4—24no—
155C2×#95—24no—
156C2×#95—24no—
157C2×#95D418yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C25C2411yescharacteristic
185C22×Q8—24no—
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—22no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—42no—
198C22×Q8—42no—
199C22×Q8—42no—
200C22×Q8—42no—
201C22×Q8—42no—
202C22×Q8—42no—
203C22×Q8—416no—
204C22×Q8—416no—
205C22×Q8—432no—
206C22×Q8—432no—
207C22×Q8—416no—
208C22×Q8C2×D414yes—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2×D414yes—
240C23×C4C2×Q818yes—
241C23×C4#4114yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2414yes—
249C23×C4C2414yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C23×C4C2412yes—
256C4⋊Q8—432no—
257C22⋊Q8—232no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—416no—
291C2×C4⋊C4—416no—
292C2×C4⋊C4—416no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4C2×D418yes—
300C2×C4⋊C4C2×D414yes—
301C2×C4⋊C4C2×Q8116yescomplemented
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4C2×D418yes—
311C2×C22⋊C4C2×Q818yes—
312C2×C22⋊C4C2×Q818yes—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42—44no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42—44no—
323#95—28no—
324#95—28no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95C2×D418yes—
347#95C2×D418yes—
348#95C2×Q8116yescomplemented
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24C22×D412yes—
354C24C22×D412yes—
355C24C22×Q814yes—
356C24C2×#4112yes—
357C242−1+414yes—
358C24C2511yescharacteristic
359C2×Q8—216no—
360C2×Q8—216no—
361C2×Q8—28no—
362C2×Q8—48no—
363C2×Q8—48no—
364C2×Q8—48no—
365C2×Q8—48no—
366C2×Q8—48no—
367C2×Q8—48no—
368C2×Q8—48no—
369C2×Q8—48no—
370C2×Q8—48no—
371C2×Q8—48no—
372C2×Q8—48no—
373C2×Q8—4128no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—82no—
377C2×Q8—82no—
378C2×Q8—84no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
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—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—416no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—44no—
459C22×C4—416no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4#121116yes—
471C22×C4C4⋊Q8116yescomplemented
472C22×C4C22×D414yes—
473C22×C4C22×D414yes—
474C22×C4C22×D414yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D414yes—
480C22×C4C22×Q818yes—
481C22×C4C22×Q818yes—
482C22×C4C22×Q814yes—
483C22×C4C22×Q814yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)14yes—
487C22×C4H(5,𝔽2)14yes—
488C22×C42−1+414yes—
489C4⋊C4—28no—
490C4⋊C4—216no—
491C4⋊C4—416no—
492C4⋊C4—416no—
493C4⋊C4—416no—
494C4⋊C4—416no—
495C4⋊C4—416no—
496C4⋊C4—416no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—84no—
502C4⋊C4—88no—
503C4⋊C4—84no—
504C22⋊C4—216no—
505C22⋊C4—216no—
506C22⋊C4—216no—
507C22⋊C4—48no—
508C42—48no—
509C42—84no—
510C42—84no—
511C42—84no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—28no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23C2×#12114yes—
525C23C2×C4⋊Q814yes—
526C23#53614yes—
527C23D4×Q818yes—
528C23#55718yes—
529C23C23×D411yescharacteristic
530C23C23×D411yescharacteristic
531C23C23×Q812yes—
532C23C22×#4111yescharacteristic
533C23C2×H(5,𝔽2)12yes—
534C23C2×H(5,𝔽2)12yes—
535C23C2×2−1+412yes—
536C23C2×2−1+412yes—
537C23#58512yes—
538Q8—432no—
539Q8—432no—
540Q8—88no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—416no—
576C2×C4—416no—
577C2×C4—416no—
578C2×C4—416no—
579C2×C4—416no—
580C2×C4—416no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—48no—
620C2×C4—48no—
621C2×C4—88no—
622C2×C4C2×#12114yes—
623C2×C4C2×#12114yes—
624C2×C4C2×C4⋊Q818yes—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—216no—
629C22—24no—
630C22—24no—
631C22—216no—
632C22—24no—
633C22#254618yes—
634C22C22×#12111yescharacteristic
635C22C22×C4⋊Q811yescharacteristic
636C22C2×#53611yescharacteristic
637C22C2×D4×Q812yes—
638C22C2×#55712yes—
639C22#348411yescharacteristic
640C22#348412yes—
641C22#348411yescharacteristic
642C22#348511yescharacteristic
643C22#348611yescharacteristic
644C22#348912yes—
645C22#349012yes—
646C22#349512yes—
647C22#350114yes—
648C22#350314yes—
649C22#354114yes—
650C22#355214yes—
651C4—24no—
652C4—24no—
653C4—28no—
654C4—44no—
655C4—44no—
656C4—44no—
657C4—44no—
658C4—44no—
659C4—48no—
660C4—48no—
661C4—44no—
662C4—82no—
663C4—82no—
664C4—82no—
665C2—28no—
666C2C2×#254611yescharacteristic
667C2#3534811yescharacteristic
668C2#3539111yescharacteristic
669C2#3557611yescharacteristic
670C2#3563311yescharacteristic
671C2#3670511yescharacteristic
672C2#3723311yescharacteristic
673C2#4310014yes—
674C2#4312514yes—
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. #15740C2semidirect
6. #15623C2semidirect
7. #15360C2semidirect
8. #15318C2semidirect
151. #380Q8semidirect
301. C2×C4⋊C4C2×Q8semidirect
348. #95C2×Q8semidirect
471. C22×C4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
C2×#24342
C22×#3952
C2×#23292
#157402
#156232
#153602
#153182
#149552
#147862
#141492
#141302
#139232
#139222
#137782