Finite Group Databasebeta

← #4397637

Subgroups — #4397637

Order 512.

3442 conjugacy classes of subgroups.

The subgroup lattice has 740 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2608C211yesmaximal, characteristic, complemented
2C2×#2435C211yesmaximal, characteristic
3#16888C212yesmaximal, complemented
4#16558C212yesmaximal, complemented
5#15639C214yesmaximal, complemented
6#15348C214yesmaximal, complemented
7#15297C214yesmaximal, complemented
8#15297C214yesmaximal, complemented
9#14982C212yesmaximal, complemented
10#14932C212yesmaximal, complemented
11#14660C211yesmaximal, characteristic, complemented
12#14335C211yesmaximal, characteristic
13#14292C211yesmaximal, characteristic, complemented
14#13902C211yesmaximal, characteristic
15#13780C211yesmaximal, characteristic, complemented
16C24×D4C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20#2608—28no—
21#2435—28no—
22C2×#397C2214yescomplemented
23C2×#397C2214yescomplemented
24C2×#396C2212yes—
25C2×#396C2212yes—
26C2×#394C2214yes—
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#389C2212yescomplemented
32C2×#389C2212yescomplemented
33C2×#388C2214yescomplemented
34C2×#388C2214yescomplemented
35C2×#387C2212yescomplemented
36C2×#387C2212yescomplemented
37C2×#381C2211yescharacteristic
38C2×#381C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C2×#379C2214yes—
41C2×#379C2214yes—
42C2×#379C2212yes—
43C2×#379C2212yes—
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46#1446C2212yescomplemented
47#1446C2212yescomplemented
48#1445C2212yescomplemented
49#1445C2212yescomplemented
50#1441C2214yescomplemented
51#1440C2212yes—
52#1440C2212yes—
53#1440C2214yes—
54#1440C2214yescomplemented
55#1440C2214yescomplemented
56#1439C2214yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2214yescomplemented
61#1439C2214yescomplemented
62#1439C2214yescomplemented
63#1439C2214yescomplemented
64#1438C2214yes—
65#1438C2212yes—
66#1438C2212yes—
67#1438C2212yes—
68#1438C2212yes—
69#1437C2214yes—
70#1437C2214yescomplemented
71#1437C2214yescomplemented
72#1436C2212yescomplemented
73#1436C2214yescomplemented
74#1436C2212yescomplemented
75#1434C2212yescomplemented
76C4×#95C2212yescomplemented
77C26C2312yes—
78C23×Q8C2311yescharacteristic
79C23×D4—24no—
80C23×D4—24no—
81C23×D4—24no—
82C23×D4—44no—
83C23×D4—44no—
84C23×D4C2312yes—
85C23×D4C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×#124—24no—
89C2×#124—24no—
90C2×#124—44no—
91C2×#124—44no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2314yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#397—48no—
129#397—48no—
130#396—28no—
131#396—28no—
132#394—216no—
133#390—24no—
134#390—24no—
135#390—44no—
136#390—44no—
137#389—28no—
138#389—28no—
139#388—48no—
140#388—48no—
141#387—28no—
142#387—28no—
143#381—24no—
144#381—24no—
145#380—24no—
146#379—216no—
147#379—48no—
148#379—44no—
149#379—44no—
150C4×C22⋊C4—24no—
151C4×C22⋊C4D418yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C25—24no—
173C25—24no—
174C25—24no—
175C25—24no—
176C25—24no—
177C25—42no—
178C25—42no—
179C25—44no—
180C25—44no—
181C25—44no—
182C25—44no—
183C25C2412yes—
184C25C2412yes—
185C25C2411yescharacteristic
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—24no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4—42no—
208C22×D4—416no—
209C22×D4—416no—
210C22×D4—432no—
211C22×D4—432no—
212C22×D4—416no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4C2×D414yes—
216C23×C4#4114yes—
217C23×C4#4114yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230#124—416no—
231#124—416no—
232C22⋊Q8—416no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
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⋊C4—44no—
308C2×C22⋊C4C2×D418yes—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42—44no—
314C2×C42—44no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—44no—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95C2×D418yes—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—22no—
345C24—216no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—24no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—48no—
367C24—48no—
368C24—48no—
369C24—48no—
370C24—48no—
371C24—416no—
372C24—416no—
373C24—48no—
374C24—88no—
375C24—84no—
376C24—84no—
377C24—84no—
378C24—84no—
379C24—88no—
380C24—88no—
381C24—88no—
382C24—88no—
383C24—88no—
384C24—88no—
385C24—88no—
386C24—88no—
387C24—88no—
388C24C22×D412yes—
389C24C2×#4112yes—
390C24C2×#4112yes—
391C24H(5,𝔽2)14yes—
392C24H(5,𝔽2)14yes—
393C24C2511yescharacteristic
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—84no—
398C2×Q8—84no—
399C2×D4—28no—
400C2×D4—28no—
401C2×D4—216no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—44no—
411C2×D4—44no—
412C2×D4—48no—
413C2×D4—4128no—
414C2×D4—48no—
415C2×D4—44no—
416C2×D4—44no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4#124116yes—
477C22×C4C22×D414yes—
478C22×C4C22×D414yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4H(5,𝔽2)14yes—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—84no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—416no—
498C22⋊C4—416no—
499C22⋊C4—416no—
500C22⋊C4—416no—
501C22⋊C4—48no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C22⋊C4—88no—
512C22⋊C4—88no—
513C22⋊C4—88no—
514C42—44no—
515C42—44no—
516C42—82no—
517C42—82no—
518C42—88no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—28no—
537C23—28no—
538C23—28no—
539C23—28no—
540C23—432no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23—44no—
548C23—44no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—44no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—44no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—44no—
566C23—44no—
567C23—42no—
568C23—42no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—816no—
582C23—816no—
583C23—816no—
584C23—816no—
585C23—832no—
586C23—832no—
587C23—832no—
588C23—832no—
589C23—816no—
590C23—816no—
591C23C2×#12414yes—
592C23C23×D411yescharacteristic
593C23C22×#4111yescharacteristic
594C23C22×#4111yescharacteristic
595C23C2×H(5,𝔽2)12yes—
596C23C2×H(5,𝔽2)12yes—
597C23C2×H(5,𝔽2)12yes—
598C23#58514yes—
599C23#58512yes—
600Q8—88no—
601D4—432no—
602D4—416no—
603D4—416no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—28no—
620C2×C4—416no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
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—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4C2×#12414yes—
668C2×C4C2×#12414yes—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—44no—
673C22—44no—
674C22—48no—
675C22—44no—
676C22—44no—
677C22—48no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—48no—
683C22—48no—
684C22—48no—
685C22—44no—
686C22—44no—
687C22—864no—
688C22—84no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—84no—
694C22—84no—
695C22—84no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22C22×#12411yescharacteristic
701C22#344912yes—
702C22#348411yescharacteristic
703C22#348411yescharacteristic
704C22#348812yes—
705C22#349012yes—
706C22#349012yes—
707C22#350612yes—
708C22#350811yescharacteristic
709C22#351012yes—
710C22#351012yes—
711C22#351011yescharacteristic
712C22#353414yes—
713C22#354214yes—
714C22#356014yes—
715C22#356014yes—
716C4—24no—
717C4—24no—
718C4—44no—
719C4—44no—
720C4—44no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4—84no—
725C4—82no—
726C4—84no—
727C4—82no—
728C2—44no—
729C2—84no—
730C2—84no—
731C2#3533811yescharacteristic
732C2#3536711yescharacteristic
733C2#3555611yescharacteristic
734C2#4060612yes—
735C2#4198612yes—
736C2#4645712yes—
737C2#4703312yes—
738C2#4800914yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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×#2608C2semidirect
3. #16888C2semidirect
4. #16558C2semidirect
5. #15639C2semidirect
6. #15348C2semidirect
7. #15297C2semidirect
8. #15297C2semidirect
9. #14982C2semidirect
10. #14932C2semidirect
11. #14660C2semidirect
13. #14292C2semidirect
15. #13780C2semidirect
22. C2×#397C22semidirect
23. C2×#397C22semidirect
31. C2×#389C22semidirect
32. C2×#389C22semidirect
33. C2×#388C22semidirect
34. C2×#388C22semidirect
35. C2×#387C22semidirect
36. C2×#387C22semidirect
46. #1446C22semidirect
47. #1446C22semidirect
48. #1445C22semidirect
49. #1445C22semidirect
50. #1441C22semidirect
54. #1440C22semidirect
55. #1440C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
63. #1439C22semidirect
70. #1437C22semidirect
71. #1437C22semidirect
72. #1436C22semidirect
73. #1436C22semidirect
74. #1436C22semidirect
75. #1434C22semidirect
76. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26082
C2×#24352
#168882
#165582
#156392
#153482
#152972
#149822
#149322
#146602
#143352
#142922
#139022
#137802