Finite Group Databasebeta

← #4310267

Subgroups — #4310267

Order 512.

3330 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C211yesmaximal, characteristic
2C2×#2280C211yesmaximal, characteristic
3#17058C211yesmaximal, characteristic, complemented
4#17057C211yesmaximal, characteristic, complemented
5#16698C212yesmaximal, complemented
6#16394C212yesmaximal, complemented
7#16393C212yesmaximal, complemented
8#16391C214yesmaximal
9#16366C211yesmaximal, characteristic, complemented
10#16365C211yesmaximal, characteristic, complemented
11#16364C212yesmaximal, complemented
12#15205C214yesmaximal, complemented
13#14906C212yesmaximal, complemented
14#14892C212yesmaximal, complemented
15#14681C211yesmaximal, characteristic
16#14295C212yesmaximal, complemented
17#14292C212yesmaximal, complemented
18C25×C4C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C22×#117C2211yescharacteristic
24#2848—28no—
25C2×#400C2212yescomplemented
26C2×#398C2212yescomplemented
27C2×#397C2214yes—
28C2×#385C2212yes—
29C2×#385C2212yes—
30C2×#385C2212yes—
31#2280—28no—
32C2×#381C2212yes—
33C2×#381C2212yes—
34C2×#381C2212yes—
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#379C2212yes—
43C2×#379C2212yes—
44C2×#379C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47#1446C2212yescomplemented
48#1446C2212yescomplemented
49#1445C2212yescomplemented
50#1445C2214yescomplemented
51#1445C2212yescomplemented
52#1445C2212yescomplemented
53#1445C2212yescomplemented
54#1444C2212yescomplemented
55#1444C2212yescomplemented
56#1444C2212yescomplemented
57#1443C2214yescomplemented
58#1442C2212yescomplemented
59#1442C2214yescomplemented
60#1442C2212yescomplemented
61#1442C2212yescomplemented
62#1441C2212yescomplemented
63#1441C2212yescomplemented
64#1441C2214yescomplemented
65#1440C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1440C2214yes—
70#1440C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1436C2214yescomplemented
77#1436C2214yescomplemented
78#1436C2214yescomplemented
79#1436C2214yescomplemented
80C22⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2211yescharacteristic
84C22⋊#95C2211yescharacteristic
85C26C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C24×C4—28no—
88C24×C4—44no—
89C24×C4—44no—
90C24×C4D418yes—
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C24×C4C2311yescharacteristic
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8—44no—
102C2×C22⋊Q8—44no—
103C2×#117—24no—
104C2×#117—24no—
105C2×#117—44no—
106C2×#117—44no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137#400—44no—
138#398—44no—
139#397—48no—
140#385—28no—
141#385—28no—
142#385—28no—
143#381—28no—
144#381—28no—
145#381—44no—
146#380—24no—
147#380—24no—
148#380—24no—
149#380—24no—
150#380—24no—
151#380—24no—
152#380—24no—
153#379—28no—
154#379—28no—
155#379—44no—
156C4×C22⋊C4—28no—
157C4×C22⋊C4—28no—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C25—24no—
186C25—24no—
187C25—24no—
188C25—42no—
189C25—41no—
190C25—41no—
191C25—42no—
192C25—42no—
193C25C2×D414yes—
194C25C2411yescharacteristic
195C25C2411yescharacteristic
196C25C2411yescharacteristic
197C22×Q8—22no—
198C22×Q8—24no—
199C22×Q8—42no—
200C22×Q8—42no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—232no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—416no—
213C23×C4—416no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—464no—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4#4118yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C22⋊Q8—416no—
244C22⋊Q8—416no—
245C22⋊Q8—416no—
246#117—416no—
247#117—416no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
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—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—44no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—44no—
319C2×C42—44no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—21no—
358C24—21no—
359C24—22no—
360C24—22no—
361C24—21no—
362C24—21no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—21no—
367C24—21no—
368C24—28no—
369C24—28no—
370C24—28no—
371C24—44no—
372C24—44no—
373C24—44no—
374C24—44no—
375C24—416no—
376C24—416no—
377C24—48no—
378C24—48no—
379C24—48no—
380C24—48no—
381C24—416no—
382C24—416no—
383C24—48no—
384C24—48no—
385C24C22×D412yes—
386C24C22×D412yes—
387C24C22×D412yes—
388C24C2×#4114yes—
389C24H(5,𝔽2)14yes—
390C242−1+414yes—
391C24C2511yescharacteristic
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—88no—
396C22×C4—232no—
397C22×C4—28no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
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—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—416no—
480C22×C4—416no—
481C22×C4—416no—
482C22×C4—416no—
483C22×C4—4128no—
484C22×C4—416no—
485C22×C4#123116yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C2×#4114yes—
491C22×C4C2×#4114yes—
492C22×C4H(5,𝔽2)14yes—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—44no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—88no—
504C4⋊C4—88no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—88no—
518C22⋊C4—88no—
519C22⋊C4—84no—
520C22⋊C4—84no—
521C42—48no—
522C42—48no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—24no—
542C23—24no—
543C23—28no—
544C23—28no—
545C23—28no—
546C23—28no—
547C23—28no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—28no—
553C23—416no—
554C23—416no—
555C23—416no—
556C23—416no—
557C23—416no—
558C23—416no—
559C23—416no—
560C23—464no—
561C23—464no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—44no—
579C23—44no—
580C23C2×#12314yes—
581C23#53414yes—
582C23#53614yes—
583C23C23×D411yescharacteristic
584C23C22×#4112yes—
585C23C2×H(5,𝔽2)12yes—
586C23C2×H(5,𝔽2)12yes—
587C23C2×2−1+412yes—
588C23#58512yes—
589C23#58511yescharacteristic
590C23#58511yescharacteristic
591C23#58512yes—
592Q8—88no—
593C2×C4—232no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—416no—
607C2×C4—416no—
608C2×C4—416no—
609C2×C4—416no—
610C2×C4—416no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
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—42no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4C2×#12314yes—
669C2×C4C2×#12314yes—
670C2×C4#53414yes—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—464no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—42no—
691C22—48no—
692C22—48no—
693C22—48no—
694C22—48no—
695C22—48no—
696C22—48no—
697C22—42no—
698C22—42no—
699C22C22×#12311yescharacteristic
700C22C2×#53411yescharacteristic
701C22C2×#53411yescharacteristic
702C22C2×#53611yescharacteristic
703C22#344911yescharacteristic
704C22#344911yescharacteristic
705C22#344911yescharacteristic
706C22#345512yes—
707C22#345512yes—
708C22#349014yes—
709C22#349514yes—
710C22#353314yes—
711C22#353512yes—
712C22#353512yes—
713C22#353912yes—
714C22#354012yes—
715C22#354114yes—
716C4—24no—
717C4—24no—
718C4—24no—
719C4—42no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—84no—
728C4—84no—
729C4—82no—
730C2—44no—
731C2—44no—
732C2—44no—
733C2#3463511yescharacteristic
734C2#3499211yescharacteristic
735C2#3531111yescharacteristic
736C2#3673314yes—
737C2#3673614yes—
738C2#3744511yescharacteristic
739C2#3744711yescharacteristic
740C2#3744911yescharacteristic
741C2#3745411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17058C2semidirect
4. #17057C2semidirect
5. #16698C2semidirect
6. #16394C2semidirect
7. #16393C2semidirect
9. #16366C2semidirect
10. #16365C2semidirect
11. #16364C2semidirect
12. #15205C2semidirect
13. #14906C2semidirect
14. #14892C2semidirect
16. #14295C2semidirect
17. #14292C2semidirect
25. C2×#400C22semidirect
26. C2×#398C22semidirect
47. #1446C22semidirect
48. #1446C22semidirect
49. #1445C22semidirect
50. #1445C22semidirect
51. #1445C22semidirect
52. #1445C22semidirect
53. #1445C22semidirect
54. #1444C22semidirect
55. #1444C22semidirect
56. #1444C22semidirect
57. #1443C22semidirect
58. #1442C22semidirect
59. #1442C22semidirect
60. #1442C22semidirect
61. #1442C22semidirect
62. #1441C22semidirect
63. #1441C22semidirect
64. #1441C22semidirect
76. #1436C22semidirect
77. #1436C22semidirect
78. #1436C22semidirect
79. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C2×#22802
#170582
#170572
#166982
#163942
#163932
#163912
#163662
#163652
#163642
#152052
#149062
#148922
#146812
#142952
#142922