Finite Group Databasebeta

← #4308842

Subgroups — #4308842

Order 512.

2938 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2646C211yesmaximal, characteristic, complemented
2C22×#388C211yesmaximal, characteristic
3#16478C214yesmaximal, complemented
4#16477C214yesmaximal, complemented
5#16330C214yesmaximal, complemented
6#16326C214yesmaximal, complemented
7#14920C212yesmaximal, complemented
8#14728C212yesmaximal, complemented
9#14726C212yesmaximal
10#14726C212yesmaximal
11#14697C211yesmaximal, characteristic, complemented
12#14697C211yesmaximal, characteristic, complemented
13#14353C211yesmaximal, characteristic, complemented
14#14338C211yesmaximal, characteristic
15#14338C211yesmaximal, characteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2646—28no—
22C2×#400C2212yescomplemented
23C2×#400C2212yescomplemented
24C2×#398C2212yescomplemented
25C2×#398C2212yescomplemented
26C2×#398C2214yescomplemented
27C2×#396C2214yes—
28C2×#396C2214yes—
29C2×#389C2212yescomplemented
30C2×#389C2212yescomplemented
31C2×#388—28no—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#380C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×#379C2212yes—
45C2×#379C2212yes—
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48#1445C2214yescomplemented
49#1444C2214yescomplemented
50#1444C2214yescomplemented
51#1444C2214yescomplemented
52#1442C2214yescomplemented
53#1442C2214yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1441C2214yescomplemented
57#1439C2212yes—
58#1439C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2212yes—
62#1439C2212yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1436C2214yescomplemented
70#1436C2214yescomplemented
71C4⋊#95C2212yescomplemented
72C4⋊#95C2212yescomplemented
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C22×C4⋊C4—24no—
85C22×C4⋊C4—44no—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—44no—
106C22×C22⋊C4—44no—
107C22×C22⋊C4Q818yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C42—24no—
121C22×C42—44no—
122C22×C42C2312yes—
123C22×C42C2312yes—
124#400—28no—
125#400—28no—
126#398—28no—
127#398—28no—
128#398—48no—
129#396—48no—
130#396—48no—
131#389—44no—
132#389—44no—
133#388—216no—
134#388—216no—
135#385—28no—
136#385—28no—
137#385—44no—
138#385—44no—
139#380—24no—
140#380—24no—
141#380—24no—
142#380—24no—
143#380—44no—
144#380—44no—
145#379—44no—
146#379—44no—
147C2×#95—24no—
148C2×#95D418yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C25—22no—
173C25—22no—
174C25—22no—
175C25—22no—
176C25—22no—
177C25—22no—
178C25—42no—
179C25—42no—
180C25—42no—
181C25—42no—
182C25C2×Q814yes—
183C25C2412yes—
184C25C2411yescharacteristic
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4C2×Q814yes—
213C23×C4C2×Q814yes—
214C23×C4#4114yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—416no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C22⋊C4—232no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
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—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—416no—
290C2×C22⋊C4—416no—
291C2×C22⋊C4—44no—
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—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4C2×Q818yes—
304C2×C22⋊C4C2×Q818yes—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42—416no—
312#95—28no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—28no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95C2×D418yes—
334#95C2×D418yes—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—28no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—24no—
350C24—28no—
351C24—28no—
352C24—48no—
353C24—48no—
354C24—44no—
355C24—44no—
356C24—48no—
357C24—48no—
358C24—44no—
359C24—44no—
360C24—44no—
361C24—44no—
362C24—44no—
363C24—44no—
364C24—84no—
365C24—84no—
366C24—84no—
367C24—84no—
368C24—84no—
369C24—84no—
370C24—84no—
371C24—84no—
372C24C22×D412yes—
373C24C22×Q814yes—
374C24C22×Q812yes—
375C24C2×#4112yes—
376C24H(5,𝔽2)12yes—
377C24H(5,𝔽2)12yes—
378C242−1+412yes—
379C242−1+412yes—
380C24C2511yescharacteristic
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
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—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—48no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4C22⋊Q8116yes—
473C22×C4C22×D414yes—
474C22×C4C22×D414yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×Q814yes—
478C22×C4C22×Q812yes—
479C22×C4C22×Q812yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4H(5,𝔽2)14yes—
483C22×C4H(5,𝔽2)14yes—
484C4⋊C4—416no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—82no—
488C4⋊C4—82no—
489C4⋊C4—82no—
490C4⋊C4—82no—
491C4⋊C4—84no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C4⋊C4—88no—
496C4⋊C4—88no—
497C4⋊C4—88no—
498C4⋊C4—88no—
499C22⋊C4—216no—
500C22⋊C4—216no—
501C22⋊C4—416no—
502C22⋊C4—416no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C42—48no—
512C42—48no—
513C23—24no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—28no—
533C23—28no—
534C23—416no—
535C23—416no—
536C23—416no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—816no—
556C23—816no—
557C23—816no—
558C23—816no—
559C23—88no—
560C23—88no—
561C23—88no—
562C23—88no—
563C23C2×C22⋊Q814yes—
564C23#53714yes—
565C23#53714yes—
566C23#54014yes—
567C23#54114yes—
568C23#54414yes—
569C23#54414yes—
570C23C23×D411yescharacteristic
571C23C23×Q811yescharacteristic
572C23C22×#4111yescharacteristic
573C23C2×H(5,𝔽2)11yescharacteristic
574C23C2×H(5,𝔽2)11yescharacteristic
575C23C2×H(5,𝔽2)12yes—
576C23C2×H(5,𝔽2)12yes—
577C23C2×2−1+411yescharacteristic
578C23C2×2−1+411yescharacteristic
579C23#58512yes—
580C23#58512yes—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—416no—
604C2×C4—416no—
605C2×C4—416no—
606C2×C4—416no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
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—88no—
657C2×C4—88no—
658C2×C4C2×C22⋊Q814yes—
659C2×C4C2×C22⋊Q814yes—
660C22—28no—
661C22—28no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—24no—
666C22—24no—
667C22—28no—
668C22—28no—
669C22—24no—
670C22—44no—
671C22—44no—
672C22—48no—
673C22—48no—
674C22—44no—
675C22—832no—
676C22—84no—
677C22—84no—
678C22—82no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22#262918yes—
683C22C22×C22⋊Q811yescharacteristic
684C22C2×#53711yescharacteristic
685C22C2×#53711yescharacteristic
686C22C2×#54011yescharacteristic
687C22C2×#54111yescharacteristic
688C22C2×#54411yescharacteristic
689C22C2×#54411yescharacteristic
690C22#345912yes—
691C22#346012yes—
692C22#348411yescharacteristic
693C22#348411yescharacteristic
694C22#348511yescharacteristic
695C22#348711yescharacteristic
696C22#350112yes—
697C22#350112yes—
698C22#353412yes—
699C22#353512yes—
700C22#353612yes—
701C22#353712yes—
702C22#355812yes—
703C22#356012yes—
704C22#356112yes—
705C22#356412yes—
706C4—24no—
707C4—24no—
708C4—44no—
709C4—44no—
710C4—44no—
711C4—44no—
712C4—44no—
713C4—84no—
714C4—84no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C2—28no—
720C2—84no—
721C2C2×#262911yescharacteristic
722C2#3498511yescharacteristic
723C2#3517411yescharacteristic
724C2#3670611yescharacteristic
725C2#3672211yescharacteristic
726C2#3674912yes—
727C2#3675012yes—
728C2#3695711yescharacteristic
729C2#3697111yescharacteristic
730C2#3745912yes—
731C2#3746012yes—
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
1. C2×#2646C2semidirect
3. #16478C2semidirect
4. #16477C2semidirect
5. #16330C2semidirect
6. #16326C2semidirect
7. #14920C2semidirect
8. #14728C2semidirect
11. #14697C2semidirect
12. #14697C2semidirect
13. #14353C2semidirect
22. C2×#400C22semidirect
23. C2×#400C22semidirect
24. C2×#398C22semidirect
25. C2×#398C22semidirect
26. C2×#398C22semidirect
29. C2×#389C22semidirect
30. C2×#389C22semidirect
48. #1445C22semidirect
49. #1444C22semidirect
50. #1444C22semidirect
51. #1444C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
69. #1436C22semidirect
70. #1436C22semidirect
71. C4⋊#95C22semidirect
72. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26462
C22×#3882
#164782
#164772
#163302
#163262
#149202
#147282
#147262
#146972
#143532
#143382