Finite Group Databasebeta

← #4431148

Subgroups — #4431148

Order 512.

3164 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2482C211yesmaximal, characteristic, complemented
2C2×#2624C211yesmaximal, characteristic, complemented
3C2×#2617C211yesmaximal, characteristic
4C2×#2347C211yesmaximal, characteristic, complemented
5#16679C212yesmaximal, complemented
6#16662C212yesmaximal, complemented
7#16549C214yesmaximal, complemented
8#15301C214yesmaximal, complemented
9#15281C214yesmaximal, complemented
10#14815C212yesmaximal, complemented
11#14808C212yesmaximal, complemented
12#14778C214yesmaximal, complemented
13#14675C211yesmaximal, characteristic, complemented
14#14290C211yesmaximal, characteristic, complemented
15#13903C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×#120C2212yes—
18C22×#117C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20#2624—28no—
21#2617—28no—
22#2482—28no—
23C2×#399C2212yes—
24C2×#399C2212yescomplemented
25C2×#397C2212yes—
26C2×#397C2212yes—
27C2×#397C2212yescomplemented
28C2×#396C2214yescomplemented
29C2×#394C2214yes—
30C2×#394C2214yes—
31C2×#394C2214yes—
32C2×#392C2212yes—
33#2347—28no—
34C2×#390C2214yes—
35C2×#390C2212yescomplemented
36C2×#389C2212yescomplemented
37C2×#389C2212yescomplemented
38C2×#388C2212yes—
39C2×#388C2212yes—
40C2×#388C2214yes—
41C2×#388C2214yes—
42C2×#386C2212yes—
43C2×#385C2212yes—
44C2×#383C2212yescomplemented
45C2×#381C2211yescharacteristic
46C2×#381C2212yes—
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#379C2212yes—
50C2×#379C2212yes—
51C2×C4×C4⋊C4C2212yescomplemented
52C2×C4×C4⋊C4C2212yescomplemented
53C2×C4×C22⋊C4C2211yescharacteristic
54#1445C2211yescharacteristic, complemented
55#1445C2211yescharacteristic, complemented
56#1444C2212yescomplemented
57#1442C2212yescomplemented
58#1442C2214yescomplemented
59#1441C2214yescomplemented
60#1441C2214yescomplemented
61#1441C2214yescomplemented
62#1440C2214yescomplemented
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yescomplemented
70#1438C2214yes—
71#1438C2214yes—
72#1438C2214yes—
73#1438C2211yescharacteristic, complemented
74#1438C2211yescharacteristic, complemented
75#1437C2214yes—
76#1437C2214yescomplemented
77#1437C2214yescomplemented
78#1436C2214yescomplemented
79C26C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C23×D4C2312yes—
82C23×D4C2314yes—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×#121—24no—
86C2×#121—24no—
87C2×#121—24no—
88C2×#120—28no—
89C2×#120—48no—
90C2×#117—24no—
91C2×#117—24no—
92C2×#117D418yescomplemented
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—24no—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C42C2312yes—
127C22×C42C2314yes—
128C22×C42C2312yes—
129#399—28no—
130#399—28no—
131#397—28no—
132#397—28no—
133#397—28no—
134#396—48no—
135#394—216no—
136#394—216no—
137#394—48no—
138#392—28no—
139#390—28no—
140#390—48no—
141#389—28no—
142#389—28no—
143#388—216no—
144#388—28no—
145#388—28no—
146#388—48no—
147#386—28no—
148#385—28no—
149#383—28no—
150#381—24no—
151#381—44no—
152#380—24no—
153#380D418yes—
154#379—28no—
155#379—44no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—28no—
158C4×C22⋊C4—24no—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171C2×#95C2311yescharacteristic
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C25—22no—
180C25—24no—
181C25—22no—
182C25—22no—
183C25—42no—
184C25—44no—
185C25—42no—
186C25—42no—
187C25C2412yes—
188C25C2412yes—
189C25C2411yescharacteristic
190C22×D4—24no—
191C22×D4—28no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—22no—
195C22×D4—28no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—44no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—44no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—44no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4#4114yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227#121—416no—
228#120—432no—
229#117—232no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
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×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×D418yes—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
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—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—416no—
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—42no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4C2×D418yes—
312C2×C22⋊C4C2×D418yes—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—28no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42C2×D418yes—
321#95—22no—
322#95—22no—
323#95—28no—
324#95—24no—
325#95—22no—
326#95—22no—
327#95—24no—
328#95—28no—
329#95—24no—
330#95—24no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95C2×D418yes—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—22no—
345C24—22no—
346C24—28no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—24no—
362C24—24no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—48no—
367C24—48no—
368C24—44no—
369C24—48no—
370C24—48no—
371C24—44no—
372C24—84no—
373C24—84no—
374C24—84no—
375C24—82no—
376C24—82no—
377C24—88no—
378C24—88no—
379C24—88no—
380C24—88no—
381C24C22×D412yes—
382C24C22×D412yes—
383C24C2×#4112yes—
384C24H(5,𝔽2)14yes—
385C24C2511yescharacteristic
386C2×D4—416no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—416no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—44no—
393C2×D4—84no—
394C2×D4—84no—
395C2×D4—88no—
396C2×D4—88no—
397C2×D4—88no—
398C2×D4—88no—
399C2×D4—88no—
400C2×D4—88no—
401C2×D4—88no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—44no—
461C22×C4—84no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4#121116yes—
465C22×C4C22×D418yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4H(5,𝔽2)14yes—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—48no—
478C4⋊C4—416no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—88no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C4⋊C4—88no—
485C22⋊C4—216no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—416no—
491C22⋊C4—416no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—88no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C22⋊C4—88no—
502C22⋊C4—84no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C42—216no—
506C42—48no—
507C42—88no—
508C23—24no—
509C23—24no—
510C23—22no—
511C23—24no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—416no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—42no—
535C23—44no—
536C23—42no—
537C23—44no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—42no—
551C23—44no—
552C23—42no—
553C23—44no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—42no—
559C23—42no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—88no—
569C23—88no—
570C23—816no—
571C23—832no—
572C23—816no—
573C23—816no—
574C23—816no—
575C23C2×#12114yes—
576C23C23×D411yescharacteristic
577C23C23×D411yescharacteristic
578C23C22×#4111yescharacteristic
579C23C2×H(5,𝔽2)12yes—
580C23C2×H(5,𝔽2)12yes—
581C23#58514yes—
582C23#58512yes—
583C23#58512yes—
584D4—88no—
585D4—832no—
586D4—816no—
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—28no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—416no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
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—88no—
651C2×C4C2×#12114yes—
652C2×C4C2×#12114yes—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—44no—
657C22—48no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—48no—
662C22—48no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—832no—
667C22—82no—
668C22—84no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—84no—
673C22—82no—
674C22—84no—
675C22—84no—
676C22—84no—
677C22—82no—
678C22—84no—
679C22—84no—
680C22—82no—
681C22—84no—
682C22C22×#12111yescharacteristic
683C22#344912yes—
684C22#346012yes—
685C22#348411yescharacteristic
686C22#348512yes—
687C22#348711yescharacteristic
688C22#348812yes—
689C22#348812yes—
690C22#349112yes—
691C22#349412yes—
692C22#349811yescharacteristic
693C22#350011yescharacteristic
694C22#353314yes—
695C22#353414yes—
696C22#356214yes—
697C22#356814yes—
698C4—24no—
699C4—24no—
700C4—44no—
701C4—48no—
702C4—44no—
703C4—44no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—84no—
708C4—82no—
709C4—82no—
710C2—44no—
711C2—84no—
712C2—162no—
713C2#3508911yescharacteristic
714C2#3525611yescharacteristic
715C2#3556311yescharacteristic
716C2#4161914yes—
717C2#4194612yes—
718C2#4195012yes—
719C2#4877612yes—
720C2#5017212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2617C2
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×#2482C2semidirect
2. C2×#2624C2semidirect
4. C2×#2347C2semidirect
5. #16679C2semidirect
6. #16662C2semidirect
7. #16549C2semidirect
8. #15301C2semidirect
9. #15281C2semidirect
10. #14815C2semidirect
11. #14808C2semidirect
12. #14778C2semidirect
13. #14675C2semidirect
14. #14290C2semidirect
15. #13903C2semidirect
24. C2×#399C22semidirect
27. C2×#397C22semidirect
28. C2×#396C22semidirect
35. C2×#390C22semidirect
36. C2×#389C22semidirect
37. C2×#389C22semidirect
44. C2×#383C22semidirect
51. C2×C4×C4⋊C4C22semidirect
52. C2×C4×C4⋊C4C22semidirect
54. #1445C22semidirect
55. #1445C22semidirect
56. #1444C22semidirect
57. #1442C22semidirect
58. #1442C22semidirect
59. #1441C22semidirect
60. #1441C22semidirect
61. #1441C22semidirect
62. #1440C22semidirect
69. #1439C22semidirect
73. #1438C22semidirect
74. #1438C22semidirect
76. #1437C22semidirect
77. #1437C22semidirect
78. #1436C22semidirect
92. C2×#117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24822
C2×#26242
C2×#26172
C2×#23472
#166792
#166622
#165492
#153012
#152812
#148152
#148082
#147782
#146752
#142902
#139032