Finite Group Databasebeta

← #4868589

Subgroups — #4868589

Order 512.

2858 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2534C211yesmaximal, characteristic
2C2×#2486C211yesmaximal, characteristic, complemented
3#17187C214yesmaximal, complemented
4#17058C212yesmaximal, complemented
5#16884C212yesmaximal, complemented
6#16676C212yesmaximal, complemented
7#16675C212yesmaximal, complemented
8#16661C214yesmaximal, complemented
9#16564C212yesmaximal, complemented
10#16391C212yesmaximal, complemented
11#15949C212yesmaximal, complemented
12#15948C212yesmaximal
13#14777C212yesmaximal, complemented
14#14734C212yesmaximal, complemented
15#14345C211yesmaximal, characteristic, complemented
16C22×#123C2211yescharacteristic
17C22×#123C2212yes—
18C22×#120C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic, complemented
21C23×C22⋊C4C2211yescharacteristic
22#2534—28no—
23#2486—28no—
24C2×#399C2212yes—
25C2×#399C2212yes—
26C2×#398C2212yescomplemented
27C2×#397C2212yes—
28C2×#396C2212yes—
29C2×#394C2212yes—
30C2×#389C2212yescomplemented
31C2×#388C2212yes—
32C2×#386C2212yes—
33C2×#386C2214yes—
34C2×#385C2214yes—
35C2×#385C2214yescomplemented
36C2×#383C2211yescharacteristic, complemented
37C2×#382C2211yescharacteristic, complemented
38C2×#381C2211yescharacteristic
39C2×#381C2212yescomplemented
40C2×#380C2211yescharacteristic
41C2×#380C2212yes—
42C2×#380C2211yescharacteristic, complemented
43C2×#379C2211yescharacteristic
44C2×#379C2214yes—
45#1445C2212yescomplemented
46#1445C2214yescomplemented
47#1445C2212yescomplemented
48#1445C2212yescomplemented
49#1444C2214yescomplemented
50#1444C2214yescomplemented
51#1444C2214yescomplemented
52#1443C2214yescomplemented
53#1442C2214yescomplemented
54#1442C2214yescomplemented
55#1442C2212yescomplemented
56#1442C2212yescomplemented
57#1441C2214yescomplemented
58#1440C2214yes—
59#1440C2214yes—
60#1440C2214yes—
61#1440C2214yes—
62#1440C2214yescomplemented
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2212yes—
71#1439C2212yes—
72#1439C2212yescomplemented
73#1438C2212yes—
74#1438C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yescomplemented
77C26C2311yescharacteristic
78C23×D4C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×#123—28no—
85C2×#123—24no—
86C2×#123—48no—
87C2×#123—44no—
88C2×#120—24no—
89C2×#120—44no—
90C2×C4×D4—42no—
91C2×C4×D4—42no—
92C2×C4×D4D418yes—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—42no—
95C22×C4⋊C4—42no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4—24no—
107C22×C22⋊C4—42no—
108C22×C22⋊C4—42no—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#399—44no—
129#399—44no—
130#398—28no—
131#397—28no—
132#396—28no—
133#394—28no—
134#389—28no—
135#388—28no—
136#386—28no—
137#386—48no—
138#385—48no—
139#385—48no—
140#383—42no—
141#382—42no—
142#381—28no—
143#381—24no—
144#380—28no—
145#380—24no—
146#380—24no—
147#379—24no—
148#379—48no—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C25—22no—
173C25—24no—
174C25—24no—
175C25—41no—
176C25—41no—
177C25—42no—
178C25—42no—
179C25—44no—
180C25C2412yes—
181C25C2411yescharacteristic
182C25C2411yescharacteristic
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—28no—
188C22×D4—41no—
189C22×D4—41no—
190C22×D4—42no—
191C22×D4—44no—
192C22×D4C2×D414yes—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—44no—
202C23×C4—41no—
203C23×C4—41no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4—42no—
208C23×C4—44no—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223#123—416no—
224#123—432no—
225#120—416no—
226C4×D4—416no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—416no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
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—44no—
253C2×C4⋊C4C2×D414yes—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—416no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
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⋊C4C2×D414yes—
302C2×C22⋊C4C2×D414yes—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42C2×D414yes—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—42no—
321#95—42no—
322#95—42no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329C24—22no—
330C24—22no—
331C24—21no—
332C24—21no—
333C24—21no—
334C24—21no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—24no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—22no—
347C24—24no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24—48no—
352C24—44no—
353C24—416no—
354C24—44no—
355C24—48no—
356C24—44no—
357C24—44no—
358C24—44no—
359C24—84no—
360C24—88no—
361C24—84no—
362C24—84no—
363C24—88no—
364C24—84no—
365C24—84no—
366C24—88no—
367C24C22×D412yes—
368C24C22×D412yes—
369C24H(5,𝔽2)14yes—
370C24H(5,𝔽2)14yes—
371C242−1+414yes—
372C24C2511yescharacteristic
373C2×D4—28no—
374C2×D4—28no—
375C2×D4—44no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—416no—
380C2×D4—48no—
381C2×D4—84no—
382C2×D4—88no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
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—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—44no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—48no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4—88no—
456C22×C4—88no—
457C22×C4—84no—
458C22×C4—84no—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C4H(5,𝔽2)12yes—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—416no—
471C4⋊C4—48no—
472C4⋊C4—84no—
473C4⋊C4—84no—
474C4⋊C4—88no—
475C4⋊C4—88no—
476C22⋊C4—44no—
477C22⋊C4—44no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—416no—
482C22⋊C4—416no—
483C22⋊C4—44no—
484C22⋊C4—48no—
485C22⋊C4—44no—
486C22⋊C4—416no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—84no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C42—28no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—28no—
513C23—28no—
514C23—24no—
515C23—24no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—42no—
529C23—42no—
530C23—416no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—42no—
548C23—42no—
549C23—816no—
550C23—88no—
551C23—816no—
552C23—816no—
553C23—832no—
554C23—88no—
555C23—816no—
556C23#53514yes—
557C23#53514yes—
558C23C23×D411yescharacteristic
559C23C2×H(5,𝔽2)11yescharacteristic
560C23C2×H(5,𝔽2)11yescharacteristic
561C23C2×H(5,𝔽2)12yes—
562C23C2×H(5,𝔽2)12yes—
563C23C2×2−1+412yes—
564C23#58514yes—
565C23#58512yes—
566D4—416no—
567D4—816no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—416no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—88no—
634C2×C4—816no—
635C2×C4—88no—
636C2×C4#53414yes—
637C22—24no—
638C22—24no—
639C22—28no—
640C22—28no—
641C22—44no—
642C22—44no—
643C22—42no—
644C22—42no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—48no—
650C22—48no—
651C22—42no—
652C22—42no—
653C22—48no—
654C22—44no—
655C22—44no—
656C22—48no—
657C22—44no—
658C22—832no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—84no—
664C22—84no—
665C22C2×#53411yescharacteristic
666C22C2×#53511yescharacteristic
667C22C2×#53511yescharacteristic
668C22#348512yes—
669C22#348912yes—
670C22#353312yes—
671C22#353312yes—
672C22#353314yes—
673C22#353314yes—
674C22#353514yes—
675C22#353514yes—
676C22#353712yes—
677C22#353712yes—
678C22#353811yescharacteristic
679C22#354011yescharacteristic
680C22#354112yes—
681C4—24no—
682C4—42no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—82no—
687C4—84no—
688C4—82no—
689C4—82no—
690C4—84no—
691C4—84no—
692C4—84no—
693C2—44no—
694C2—44no—
695C2—84no—
696C2#3696511yescharacteristic
697C2#3737411yescharacteristic
698C2#3745011yescharacteristic
699C2#4140712yes—
700C2#4140912yes—
701C2#5577914yes—
702C2#5831912yes—
703C2#5837512yes—
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
2. C2×#2486C2semidirect
3. #17187C2semidirect
4. #17058C2semidirect
5. #16884C2semidirect
6. #16676C2semidirect
7. #16675C2semidirect
8. #16661C2semidirect
9. #16564C2semidirect
10. #16391C2semidirect
11. #15949C2semidirect
13. #14777C2semidirect
14. #14734C2semidirect
15. #14345C2semidirect
20. C23×C4⋊C4C22semidirect
26. C2×#398C22semidirect
30. C2×#389C22semidirect
35. C2×#385C22semidirect
36. C2×#383C22semidirect
37. C2×#382C22semidirect
39. C2×#381C22semidirect
42. C2×#380C22semidirect
45. #1445C22semidirect
46. #1445C22semidirect
47. #1445C22semidirect
48. #1445C22semidirect
49. #1444C22semidirect
50. #1444C22semidirect
51. #1444C22semidirect
52. #1443C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
56. #1442C22semidirect
57. #1441C22semidirect
62. #1440C22semidirect
72. #1439C22semidirect
76. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25342
C2×#24862
#171872
#170582
#168842
#166762
#166752
#166612
#165642
#163912
#159492
#159482
#147772
#147342
#143452