Finite Group Databasebeta

← #4344496

Subgroups — #4344496

Order 512.

2950 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic, complemented
2C2×#2321C211yesmaximal, characteristic
3#16511C212yesmaximal, complemented
4#16398C212yesmaximal, complemented
5#16318C212yesmaximal, complemented
6#16297C212yesmaximal, complemented
7#16084C214yesmaximal, complemented
8#15476C214yesmaximal, complemented
9#15100C212yesmaximal, complemented
10#15063C212yesmaximal, complemented
11#14836C212yesmaximal, complemented
12#14734C212yesmaximal, complemented
13#14401C212yesmaximal, complemented
14#14357C212yesmaximal
15#13960C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic, complemented
17C22×#120C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20#2563—28no—
21C2×#399C2212yes—
22C2×#399C2212yes—
23C2×#395C2214yescomplemented
24C2×#393C2212yes—
25C2×#393C2214yes—
26C2×#393C2212yescomplemented
27#2321—28no—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#388C2212yes—
31C2×#388C2212yes—
32C2×#386C2212yes—
33C2×#386C2211yescharacteristic
34C2×#386C2212yes—
35C2×#386C2214yes—
36C2×#385C2211yescharacteristic
37C2×#385C2214yes—
38C2×#382C2212yescomplemented
39C2×#380C2212yes—
40C2×#380C2212yescomplemented
41C2×#380C2211yescharacteristic, complemented
42C2×#380C2214yescomplemented
43C2×#379C2211yescharacteristic
44C2×#379C2214yes—
45C2×C4×C4⋊C4C2211yescharacteristic, complemented
46C2×C4×C22⋊C4C2211yescharacteristic
47C22×#95C2211yescharacteristic, complemented
48#1445C2212yescomplemented
49#1445C2212yescomplemented
50#1445C2214yescomplemented
51#1444C2214yescomplemented
52#1444C2212yescomplemented
53#1442C2212yescomplemented
54#1442C2214yescomplemented
55#1441C2214yescomplemented
56#1441C2212yescomplemented
57#1440C2214yes—
58#1439C2212yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2212yes—
66#1439C2212yes—
67#1439C2214yes—
68#1439C2214yescomplemented
69#1438C2214yes—
70#1437C2214yes—
71#1436C2214yescomplemented
72#1436C2214yescomplemented
73#1434C2214yescomplemented
74#1434C2212yescomplemented
75C22⋊#95C2212yes—
76C22⋊#95C2212yescomplemented
77C26C2311yescharacteristic
78C23×Q8C2312yes—
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—42no—
87C2×C4⋊Q8—42no—
88C2×#120—24no—
89C2×#120—24no—
90C2×#120Q818yescomplemented
91C2×C4×D4—42no—
92C2×C4×D4—42no—
93C2×C4×D4D418yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—24no—
104C22×C22⋊C4—42no—
105C22×C22⋊C4—42no—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2312yes—
127#399—44no—
128#399—44no—
129#395—48no—
130#393—28no—
131#393—28no—
132#393—48no—
133#390—28no—
134#390—28no—
135#388—28no—
136#388—28no—
137#386—24no—
138#386—28no—
139#386—28no—
140#386—48no—
141#385—24no—
142#385—48no—
143#382—28no—
144#380—28no—
145#380—28no—
146#380—48no—
147#380—42no—
148#379—24no—
149#379—48no—
150C4×C4⋊C4—24no—
151C4×C22⋊C4—24no—
152C2×#95—42no—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C25—22no—
175C25—22no—
176C25—41no—
177C25—41no—
178C25—41no—
179C25—41no—
180C25—44no—
181C25—44no—
182C25C2×Q814yes—
183C25C2412yes—
184C25C2411yescharacteristic
185C25C2411yescharacteristic
186C22×Q8—24no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8—44no—
190C22×Q8—44no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—41no—
194C22×D4—41no—
195C22×D4—41no—
196C22×D4—41no—
197C22×D4—44no—
198C22×D4—44no—
199C22×D4C2×D414yes—
200C22×D4C2×Q818yes—
201C22×D4C2×Q814yes—
202C22×D4#4114yes—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—28no—
210C23×C4—41no—
211C23×C4—41no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—44no—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C4⋊Q8—416no—
233#120—232no—
234C4×D4—416no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4C2×D414yes—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—28no—
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—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—416no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4C2×D414yes—
308C2×C22⋊C4C2×D414yes—
309C2×C22⋊C4C2×Q814yes—
310C2×C22⋊C4C2×Q818yes—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—44no—
318C2×C42C2×D414yes—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—24no—
330#95—28no—
331#95—28no—
332#95—42no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340C24—24no—
341C24—21no—
342C24—24no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—24no—
348C24—21no—
349C24—21no—
350C24—21no—
351C24—24no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24—24no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—24no—
362C24—28no—
363C24—44no—
364C24—44no—
365C24—416no—
366C24—44no—
367C24—44no—
368C24—44no—
369C24—44no—
370C24—84no—
371C24—84no—
372C24—88no—
373C24—84no—
374C24—84no—
375C24—84no—
376C24—88no—
377C24—84no—
378C24—88no—
379C24C22×D412yes—
380C24C22×D412yes—
381C24C22×Q814yes—
382C24C22×Q812yes—
383C24C22×Q812yes—
384C24C2×#4112yes—
385C24C2×#4112yes—
386C24C2511yescharacteristic
387C2×Q8—48no—
388C2×Q8—88no—
389C2×Q8—88no—
390C2×Q8—84no—
391C2×Q8—84no—
392C2×D4—216no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—416no—
400C2×D4—84no—
401C2×D4—84no—
402C2×D4—88no—
403C2×D4C22⋊Q8116yes—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
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—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—416no—
465C22×C4—44no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4—88no—
471C22×C4—84no—
472C22×C4—84no—
473C22×C4—84no—
474C22×C4—84no—
475C22×C4—84no—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×Q814yes—
482C22×C4C22×Q812yes—
483C22×C4C2×#4112yes—
484C22×C4H(5,𝔽2)18yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C4H(5,𝔽2)12yes—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—84no—
492C4⋊C4—88no—
493C4⋊C4—88no—
494C4⋊C4—84no—
495C4⋊C4—88no—
496C22⋊C4—28no—
497C22⋊C4—216no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—416no—
501C22⋊C4—48no—
502C22⋊C4—44no—
503C22⋊C4—48no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C22⋊C4—84no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C22⋊C4—84no—
512C22⋊C4—88no—
513C22⋊C4—88no—
514C42—28no—
515C42—48no—
516C42—84no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—28no—
530C23—28no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—28no—
536C23—28no—
537C23—24no—
538C23—28no—
539C23—24no—
540C23—24no—
541C23—28no—
542C23—416no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23—42no—
552C23—42no—
553C23—44no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—816no—
574C23—816no—
575C23—816no—
576C23—816no—
577C23—816no—
578C23—88no—
579C23—816no—
580C23—88no—
581C23C2×C22⋊Q814yes—
582C23C2×C22⋊Q814yes—
583C23C23×D411yescharacteristic
584C23C23×Q811yescharacteristic
585C23C22×#4111yescharacteristic
586C23C2×H(5,𝔽2)11yescharacteristic
587C23C2×H(5,𝔽2)11yescharacteristic
588C23C2×H(5,𝔽2)14yes—
589C23#58511yescharacteristic
590C23#58511yescharacteristic
591C23#58514yes—
592Q8—816no—
593D4—416no—
594D4—816no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—28no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—88no—
660C2×C4—88no—
661C2×C4—816no—
662C2×C4—88no—
663C2×C4C2×C22⋊Q814yes—
664C2×C4#53414yes—
665C22—24no—
666C22—24no—
667C22—28no—
668C22—28no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—42no—
674C22—42no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—42no—
682C22—42no—
683C22—44no—
684C22—48no—
685C22—48no—
686C22—44no—
687C22—44no—
688C22—832no—
689C22—84no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—84no—
694C22—82no—
695C22C22×C22⋊Q811yescharacteristic
696C22C2×#53411yescharacteristic
697C22#344911yescharacteristic
698C22#348512yes—
699C22#349812yes—
700C22#350111yescharacteristic
701C22#350111yescharacteristic
702C22#350114yes—
703C22#350511yescharacteristic
704C22#350511yescharacteristic
705C22#351314yes—
706C22#353712yes—
707C22#354514yes—
708C22#354512yes—
709C22#354512yes—
710C22#354912yes—
711C22#355114yes—
712C4—24no—
713C4—24no—
714C4—48no—
715C4—42no—
716C4—44no—
717C4—42no—
718C4—42no—
719C4—84no—
720C4—82no—
721C4—82no—
722C4—84no—
723C4—84no—
724C2—44no—
725C2—44no—
726C2—84no—
727C2#3501011yescharacteristic
728C2#3583612yes—
729C2#3711911yescharacteristic
730C2#3729711yescharacteristic
731C2#4138112yes—
732C2#4684212yes—
733C2#4760412yes—
734C2#4883912yes—
735C2#4977112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2563C2semidirect
3. #16511C2semidirect
4. #16398C2semidirect
5. #16318C2semidirect
6. #16297C2semidirect
7. #16084C2semidirect
8. #15476C2semidirect
9. #15100C2semidirect
10. #15063C2semidirect
11. #14836C2semidirect
12. #14734C2semidirect
13. #14401C2semidirect
15. #13960C2semidirect
16. C22×C4⋊Q8C22semidirect
23. C2×#395C22semidirect
26. C2×#393C22semidirect
38. C2×#382C22semidirect
40. C2×#380C22semidirect
41. C2×#380C22semidirect
42. C2×#380C22semidirect
45. C2×C4×C4⋊C4C22semidirect
47. C22×#95C22semidirect
48. #1445C22semidirect
49. #1445C22semidirect
50. #1445C22semidirect
51. #1444C22semidirect
52. #1444C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
68. #1439C22semidirect
71. #1436C22semidirect
72. #1436C22semidirect
73. #1434C22semidirect
74. #1434C22semidirect
76. C22⋊#95C22semidirect
90. C2×#120Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#23212
#165112
#163982
#163182
#162972
#160842
#154762
#151002
#150632
#148362
#147342
#144012
#143572
#139602