Finite Group Databasebeta

← #818879

Subgroups — #818879

Order 512.

2694 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2367C211yesmaximal, characteristic
3#16892C212yesmaximal, complemented
4#15770C214yesmaximal, complemented
5#15586C214yesmaximal, complemented
6#15193C211yesmaximal, characteristic, complemented
7#15191C211yesmaximal, characteristic, complemented
8#15128C211yesmaximal, characteristic, complemented
9#15063C212yesmaximal, complemented
10#14826C212yesmaximal, complemented
11#14027C212yesmaximal
12#13982C212yesmaximal, complemented
13#13978C212yesmaximal, complemented
14#13920C212yesmaximal
15#13908C212yesmaximal, complemented
16C4×#400C211yesmaximal, characteristic, complemented
17D4×#95C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2212yes—
22C22×C4×D4C2211yescharacteristic
23#2563—28no—
24C2×#400C2211yescharacteristic
25C2×#395C2212yes—
26C2×#393C2214yes—
27#2367—28no—
28C2×#391C2211yescharacteristic
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#390C2211yescharacteristic
32C2×#390C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#388C2212yes—
36C2×#388C2212yes—
37C2×#385C2212yes—
38C2×#382C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2214yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46C22×#95C2212yes—
47#1446C2212yes—
48#1446C2214yes—
49#1445C2211yescharacteristic
50#1445C2214yes—
51#1445C2212yes—
52#1445C2214yes—
53#1444C2212yes—
54#1444C2212yes—
55#1443C2212yes—
56#1443C2214yes—
57#1442C2211yescharacteristic
58#1442C2212yes—
59#1442C2212yes—
60#1442C2212yes—
61#1442C2214yes—
62#1442C2214yes—
63#1441C2212yes—
64#1441C2211yescharacteristic
65#1440C2214yes—
66#1440C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76#1434C2214yes—
77C22⋊#95C2214yes—
78C22⋊#95C2214yes—
79C22⋊#95C2214yes—
80C22⋊#95C2214yes—
81C22⋊#95C2212yes—
82C4×#95C2211yescharacteristic
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2314yes—
89C24×C4C2312yes—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—44no—
92C2×#124—24no—
93C2×#124—24no—
94C2×#124Q818yescomplemented
95C2×C4×Q8—24no—
96C2×C4×Q8—44no—
97C2×C4×D4—28no—
98C2×C4×D4—28no—
99C2×C4×D4—44no—
100C2×C4×D4—44no—
101C2×C4×D4D418yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2312yes—
129#400—24no—
130#395—44no—
131#393—216no—
132#391—28no—
133#391—24no—
134#391—44no—
135#390—28no—
136#390—24no—
137#389—28no—
138#389—44no—
139#388—28no—
140#388—28no—
141#385—28no—
142#382—28no—
143#380—28no—
144#380—28no—
145#380—48no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—24no—
149C4×C22⋊C4—28no—
150C2×#95—28no—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2311yescharacteristic
184C25C2412yes—
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—44no—
191C22×Q8—44no—
192C22×Q8—42no—
193C22×Q8—42no—
194C22×Q8C2×Q814yes—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4C2×D414yes—
198C22×D4C2×Q814yes—
199C22×D4#4114yes—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4C2×D418yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C4⋊Q8—416no—
229#124—232no—
230C4×Q8—416no—
231C4×D4—416no—
232C4×D4—432no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4#4118yes—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4C2×D418yes—
284C2×C22⋊C4C2×Q818yes—
285C2×C22⋊C4C2×Q818yes—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42C2×D414yes—
302C2×C42C2×Q814yes—
303#95—22no—
304#95—24no—
305#95—24no—
306#95—22no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—28no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—22no—
320#95—44no—
321#95—42no—
322#95—42no—
323#95—42no—
324#95—44no—
325#95—42no—
326#95—42no—
327#95—44no—
328#95—42no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—42no—
335#95#4114yes—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24C22×D414yes—
345C24C22×Q814yes—
346C24C2×#4114yes—
347C24C2511yescharacteristic
348C2×Q8—28no—
349C2×Q8—216no—
350C2×Q8—48no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8—48no—
354C2×Q8—84no—
355C2×Q8—84no—
356C2×Q8—88no—
357C2×Q8—88no—
358C2×D4—28no—
359C2×D4—28no—
360C2×D4—216no—
361C2×D4—216no—
362C2×D4C22⋊Q8116yes—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—216no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4—88no—
452C22×C4—88no—
453C22×C4C22×D414yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×Q812yes—
458C22×C4C22×Q812yes—
459C22×C4C22×Q814yes—
460C22×C4C22×Q812yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#4118yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4114yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C4H(5,𝔽2)14yes—
468C22×C42−1+412yes—
469C4⋊C4—28no—
470C4⋊C4—216no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—88no—
480C4⋊C4—88no—
481C22⋊C4—216no—
482C22⋊C4—216no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—416no—
486C22⋊C4—84no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C42—28no—
490C42—28no—
491C42—48no—
492C42—48no—
493C42—44no—
494C42—48no—
495C42—84no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—28no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—28no—
508C23—24no—
509C23—28no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23C2×C22⋊Q818yes—
519C23C23×D411yescharacteristic
520C23C23×Q811yescharacteristic
521C23C22×#4111yescharacteristic
522C23C22×#4112yes—
523C23C22×#4111yescharacteristic
524C23C2×H(5,𝔽2)12yes—
525C23C2×H(5,𝔽2)11yescharacteristic
526C23C2×2−1+411yescharacteristic
527C23#58512yes—
528C23#58511yescharacteristic
529C23#58512yes—
530Q8—416no—
531Q8—816no—
532D4—232no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—816no—
622C2×C4—832no—
623C2×C4C2×C22⋊Q814yes—
624C2×C4#53214yes—
625C2×C4#53414yes—
626C2×C4#54714yes—
627C2×C4#55118yes—
628C2×C4#55418yes—
629C22—28no—
630C22—28no—
631C22—28no—
632C22—44no—
633C22—48no—
634C22—44no—
635C22—48no—
636C22—48no—
637C22—48no—
638C22—44no—
639C22—44no—
640C22C22×C22⋊Q811yescharacteristic
641C22C2×#53211yescharacteristic
642C22C2×#53411yescharacteristic
643C22C2×#54711yescharacteristic
644C22C2×#55112yes—
645C22C2×#55412yes—
646C22#347512yes—
647C22Q8×#4111yescharacteristic
648C22#348111yescharacteristic
649C22#348511yescharacteristic
650C22#348811yescharacteristic
651C22#350011yescharacteristic
652C22#350311yescharacteristic
653C22#350412yes—
654C22#351412yes—
655C22#351412yes—
656C22#351511yescharacteristic
657C22#351812yes—
658C22#352112yes—
659C22#352412yes—
660C22#355812yes—
661C22#356112yes—
662C22#356212yes—
663C4—24no—
664C4—44no—
665C4—44no—
666C4—48no—
667C4—44no—
668C4—42no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—81no—
673C4—82no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—84no—
678C4#256418yes—
679C2—48no—
680C2C2×#256411yescharacteristic
681C2#3603111yescharacteristic
682C2#3605111yescharacteristic
683C2#3618511yescharacteristic
684C2#3627511yescharacteristic
685C2#3891811yescharacteristic
686C2#4297911yescharacteristic
687C2#4519212yes—
688C2#4521312yes—
689C2#5946212yes—
690C2#5948412yes—
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
3. #16892C2semidirect
4. #15770C2semidirect
5. #15586C2semidirect
6. #15193C2semidirect
7. #15191C2semidirect
8. #15128C2semidirect
9. #15063C2semidirect
10. #14826C2semidirect
12. #13982C2semidirect
13. #13978C2semidirect
15. #13908C2semidirect
16. C4×#400C2semidirect
94. C2×#124Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#23672
#168922
#157702
#155862
#151932
#151912
#151282
#150632
#148262
#140272
#139822
#139782
#139202
#139082
C4×#4002
D4×#952