Finite Group Databasebeta

← #4867700

Subgroups — #4867700

Order 512.

2972 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic, complemented
2C2×#2534C211yesmaximal, characteristic
3#17092C214yesmaximal, complemented
4#16668C212yesmaximal, complemented
5#16602C212yesmaximal, complemented
6#16515C214yesmaximal, complemented
7#16365C212yesmaximal, complemented
8#16298C212yesmaximal, complemented
9#16265C212yesmaximal, complemented
10#16263C212yesmaximal, complemented
11#15983C212yesmaximal, complemented
12#15281C212yesmaximal
13#15079C212yesmaximal, complemented
14#14717C212yesmaximal, complemented
15#14344C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic, complemented
17C22×#123C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C22⋊Q8C2212yescomplemented
20C22×#120C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23#2542—28no—
24#2534—28no—
25C2×#399C2212yes—
26C2×#399C2212yescomplemented
27C2×#398C2214yescomplemented
28C2×#396C2212yes—
29C2×#396C2214yes—
30C2×#394C2212yes—
31C2×#394C2214yes—
32C2×#393C2212yes—
33C2×#393C2212yes—
34C2×#390C2212yes—
35C2×#388C2212yes—
36C2×#388C2214yes—
37C2×#387C2214yescomplemented
38C2×#386C2212yes—
39C2×#384C2212yescomplemented
40C2×#381C2211yescharacteristic
41C2×#381C2211yescharacteristic, complemented
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic, complemented
44C2×#379C2211yescharacteristic
45C2×#379C2214yes—
46C2×C4×C4⋊C4C2211yescharacteristic, complemented
47#1446C2212yescomplemented
48#1446C2212yescomplemented
49#1445C2214yescomplemented
50#1444C2212yescomplemented
51#1444C2212yescomplemented
52#1442C2214yescomplemented
53#1441C2212yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1441C2214yescomplemented
57#1441C2214yescomplemented
58#1440C2212yescomplemented
59#1440C2214yescomplemented
60#1439C2212yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2212yes—
66#1439C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yescomplemented
72#1438C2212yes—
73#1438C2214yes—
74#1434C2212yescomplemented
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C26C2311yescharacteristic
78C23×Q8C2312yes—
79C23×D4C2311yescharacteristic
80C23×D4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—42no—
86C2×C4⋊Q8—42no—
87C2×#123—24no—
88C2×#123—44no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—48no—
92C2×C22⋊Q8—48no—
93C2×#120—24no—
94C2×#120—44no—
95C2×C4×D4—42no—
96C2×C4×D4—42no—
97C2×C4×D4D418yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4—24no—
110C22×C22⋊C4—42no—
111C22×C22⋊C4—42no—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#399—28no—
134#399—28no—
135#398—48no—
136#396—28no—
137#396—48no—
138#394—28no—
139#394—48no—
140#393—44no—
141#393—44no—
142#390—28no—
143#388—28no—
144#388—48no—
145#387—48no—
146#386—28no—
147#384—28no—
148#381—24no—
149#381—42no—
150#380—24no—
151#380—42no—
152#379—24no—
153#379—48no—
154C4×C4⋊C4—24no—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C25—22no—
176C25—24no—
177C25—24no—
178C25—41no—
179C25—41no—
180C25—42no—
181C25—42no—
182C25—44no—
183C25C2411yescharacteristic
184C25C2412yes—
185C25C2411yescharacteristic
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—44no—
190C22×Q8—44no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—41no—
196C22×D4—41no—
197C22×D4—44no—
198C22×D4—42no—
199C22×D4—44no—
200C22×D4C2×D414yes—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—41no—
208C23×C4—41no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—42no—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C4⋊Q8—416no—
227#123—416no—
228C22⋊Q8—432no—
229C22⋊Q8—432no—
230#120—416no—
231C4×D4—416no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D414yes—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
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—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—41no—
285C2×C22⋊C4—41no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—416no—
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—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4C2×D414yes—
311C2×C22⋊C4C2×D414yes—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—28no—
316C2×C42C2×D414yes—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—44no—
325#95—42no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337C24—24no—
338C24—22no—
339C24—21no—
340C24—21no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—22no—
347C24—24no—
348C24—24no—
349C24—24no—
350C24—21no—
351C24—21no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—48no—
361C24—44no—
362C24—416no—
363C24—44no—
364C24—48no—
365C24—44no—
366C24—44no—
367C24—44no—
368C24—84no—
369C24—84no—
370C24—84no—
371C24—88no—
372C24—88no—
373C24—88no—
374C24—84no—
375C24—84no—
376C24C22×D412yes—
377C24C22×D412yes—
378C242−1+414yes—
379C242−1+414yes—
380C24C2511yescharacteristic
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—88no—
385C2×Q8—88no—
386C2×D4—28no—
387C2×D4—28no—
388C2×D4—44no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—84no—
394C2×D4—88no—
395C2×D4—88no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
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—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—84no—
462C22×C4—84no—
463C22×C4—84no—
464C22×C4—84no—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4H(5,𝔽2)18yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C4H(5,𝔽2)14yes—
474C4⋊C4—216no—
475C4⋊C4—44no—
476C4⋊C4—416no—
477C4⋊C4—416no—
478C4⋊C4—416no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—84no—
483C4⋊C4—88no—
484C4⋊C4—84no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—48no—
488C22⋊C4—416no—
489C22⋊C4—416no—
490C22⋊C4—416no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—84no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C42—28no—
505C42—48no—
506C23—22no—
507C23—22no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—28no—
513C23—28no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—28no—
521C23—24no—
522C23—28no—
523C23—24no—
524C23—44no—
525C23—41no—
526C23—41no—
527C23—42no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—416no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—41no—
546C23—44no—
547C23—41no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—42no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—816no—
562C23—88no—
563C23—816no—
564C23—832no—
565C23—816no—
566C23—816no—
567C23—88no—
568C23#53614yes—
569C23C23×D411yescharacteristic
570C23C2×H(5,𝔽2)11yescharacteristic
571C23C2×H(5,𝔽2)11yescharacteristic
572C23C2×H(5,𝔽2)12yes—
573C23C2×H(5,𝔽2)14yes—
574C23C2×2−1+412yes—
575C23C2×2−1+412yes—
576C23#58512yes—
577Q8—816no—
578D4—416no—
579D4—816no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
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—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—88no—
644C2×C4—88no—
645C2×C4—88no—
646C2×C4#53414yes—
647C2×C4#53514yes—
648C22—24no—
649C22—28no—
650C22—44no—
651C22—44no—
652C22—42no—
653C22—48no—
654C22—44no—
655C22—42no—
656C22—42no—
657C22—48no—
658C22—42no—
659C22—44no—
660C22—44no—
661C22—48no—
662C22—44no—
663C22—44no—
664C22—42no—
665C22—44no—
666C22—832no—
667C22—82no—
668C22—82no—
669C22—84no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—84no—
674C22—84no—
675C22—82no—
676C22C2×#53411yescharacteristic
677C22C2×#53511yescharacteristic
678C22C2×#53611yescharacteristic
679C22#348412yes—
680C22#348912yes—
681C22#353612yes—
682C22#353612yes—
683C22#353614yes—
684C22#353811yescharacteristic
685C22#353814yes—
686C22#354011yescharacteristic
687C22#354112yes—
688C22#354214yes—
689C22#354312yes—
690C22#354312yes—
691C22#356114yes—
692C4—24no—
693C4—24no—
694C4—42no—
695C4—42no—
696C4—48no—
697C4—42no—
698C4—82no—
699C4—84no—
700C4—84no—
701C4—84no—
702C4—82no—
703C4—82no—
704C2—44no—
705C2—82no—
706C2—84no—
707C2#3723211yescharacteristic
708C2#3737411yescharacteristic
709C2#3745111yescharacteristic
710C2#4059512yes—
711C2#4247812yes—
712C2#5344114yes—
713C2#5896312yes—
714C2#5898212yes—
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×#2542C2semidirect
3. #17092C2semidirect
4. #16668C2semidirect
5. #16602C2semidirect
6. #16515C2semidirect
7. #16365C2semidirect
8. #16298C2semidirect
9. #16265C2semidirect
10. #16263C2semidirect
11. #15983C2semidirect
13. #15079C2semidirect
14. #14717C2semidirect
15. #14344C2semidirect
16. C22×C4⋊Q8C22semidirect
19. C22×C22⋊Q8C22semidirect
26. C2×#399C22semidirect
27. C2×#398C22semidirect
37. C2×#387C22semidirect
39. C2×#384C22semidirect
41. C2×#381C22semidirect
43. C2×#380C22semidirect
46. C2×C4×C4⋊C4C22semidirect
47. #1446C22semidirect
48. #1446C22semidirect
49. #1445C22semidirect
50. #1444C22semidirect
51. #1444C22semidirect
52. #1442C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
58. #1440C22semidirect
59. #1440C22semidirect
71. #1439C22semidirect
74. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#25342
#170922
#166682
#166022
#165152
#163652
#162982
#162652
#162632
#159832
#152812
#150792
#147172
#143442