Finite Group Databasebeta

← #5956185

Subgroups — #5956185

Order 512.

4372 conjugacy classes of subgroups.

The subgroup lattice has 809 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2328C211yesmaximal, characteristic, complemented
2C2×#2324C212yesmaximal, complemented
3C2×C4×#121C212yesmaximal, complemented
4C2×#2290C211yesmaximal, characteristic, complemented
5#15223C211yesmaximal, characteristic, complemented
6#15223C211yesmaximal, characteristic, complemented
7#15148C212yesmaximal, complemented
8#15124C211yesmaximal, characteristic, complemented
9#15124C211yesmaximal, characteristic, complemented
10#15123C212yesmaximal, complemented
11#14985C214yesmaximal, complemented
12#14977C214yesmaximal, complemented
13#14811C214yesmaximal, complemented
14#14796C214yesmaximal, complemented
15#13797C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#121C2212yes—
18C22×#121C2212yes—
19C22×C4×D4C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2214yes—
22C22×#117C2211yescharacteristic
23C23×C42C2211yescharacteristic
24C2×#399C2214yes—
25C2×#392C2214yes—
26C2×#392C2214yes—
27C2×#392C2214yes—
28#2328—28no—
29#2324—216no—
30C4×#121—216no—
31C2×#390C2212yes—
32C2×#390C2214yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37#2290—28no—
38C2×#388C2214yes—
39C2×#388C2212yes—
40C2×#388C2214yes—
41C2×#387C2212yes—
42C2×#386C2214yes—
43C2×#386C2214yes—
44C2×#386C2214yes—
45C2×#386C2214yes—
46C2×#386C2212yes—
47C2×#385C2214yes—
48C2×#385C2212yes—
49C2×#382C2211yescharacteristic
50C2×#380C2212yes—
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×C4×C22⋊C4C2214yes—
57C2×#376C2211yescharacteristic
58#1445C2214yes—
59#1445C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1438C2214yes—
67#1438C2214yes—
68#1438C2214yes—
69#1438C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2211yescharacteristic
73#1434C2214yes—
74#1434C2214yes—
75C23×D4C2312yes—
76C23×D4C2311yescharacteristic
77C23×D4C2314yes—
78C23×D4C2314yes—
79C23×D4C2314yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2314yes—
84C2×#127—28no—
85C2×#127—24no—
86C2×#121—28no—
87C2×#121—28no—
88C2×#121—28no—
89C2×#121—28no—
90C2×#121—28no—
91C2×#121—28no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4—28no—
96C2×C4×D4—28no—
97C2×C4×D4—216no—
98C2×C4×D4—28no—
99C2×C4×D4—48no—
100C2×C4×D4—48no—
101C2×#117—28no—
102C2×#117—24no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C42—24no—
128C22×C42—28no—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2314yes—
135C22×C42C2311yescharacteristic
136#399—48no—
137#392—48no—
138#392—48no—
139#392—48no—
140#390—28no—
141#390—28no—
142#390—216no—
143#390—28no—
144#389—28no—
145#389—28no—
146#388—216no—
147#388—28no—
148#388—216no—
149#387—28no—
150#386—216no—
151#386—216no—
152#386—216no—
153#386—28no—
154#386—216no—
155#385—216no—
156#385—28no—
157#382—24no—
158#380—28no—
159#380—28no—
160C4×C4⋊C4—24no—
161C4×C4⋊C4D4116yescomplemented
162C4×C22⋊C4—24no—
163C4×C22⋊C4—24no—
164C4×C22⋊C4—216no—
165#376D418yescomplemented
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C25C2411yescharacteristic
182C25C2414yes—
183C25C2411yescharacteristic
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—28no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—28no—
190C22×D4—28no—
191C22×D4—28no—
192C22×D4—28no—
193C22×D4—24no—
194C22×D4—28no—
195C22×D4—28no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—22no—
200C22×D4—28no—
201C22×D4—28no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—44no—
206C22×D4—42no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4—44no—
214C22×D4—44no—
215C22×D4—44no—
216C22×D4#4118yes—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—28no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—28no—
232C23×C4—28no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4#4114yes—
240C23×C4#4114yes—
241C23×C4C2412yes—
242C23×C4C2414yes—
243C23×C4C2411yescharacteristic
244C23×C4C2414yes—
245C23×C4C2414yes—
246C23×C4C2411yescharacteristic
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253#127—232no—
254#121—264no—
255#121—264no—
256C4×D4—232no—
257C4×D4—464no—
258C4×D4—432no—
259#117—416no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4C2×D418yes—
283C2×C4⋊C4C2×D418yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—44no—
343C2×C42—44no—
344C2×C42—416no—
345C2×C42C2×D414yes—
346C2×C42C2×D414yes—
347C2×C42C2×D4116yes—
348C2×C42C2×D414yes—
349C2×C42C2×D418yes—
350#95—28no—
351#95—24no—
352#95—28no—
353#95—24no—
354#95—24no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95C2×D418yes—
363#95C2×D418yes—
364#95C2×D4116yes—
365C24—21no—
366C24—22no—
367C24—21no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—21no—
373C24—24no—
374C24—24no—
375C24—22no—
376C24—24no—
377C24—21no—
378C24—24no—
379C24—24no—
380C24—22no—
381C24—24no—
382C24—24no—
383C24—24no—
384C24—22no—
385C24—24no—
386C24—22no—
387C24—24no—
388C24—22no—
389C24—24no—
390C24—24no—
391C24—24no—
392C24—24no—
393C24—22no—
394C24C2×#4118yes—
395C24C2×#4112yes—
396C24C2×#4112yes—
397C24H(5,𝔽2)14yes—
398C24C2511yescharacteristic
399C2×D4—28no—
400C2×D4—232no—
401C2×D4—232no—
402C2×D4—232no—
403C2×D4—216no—
404C2×D4—216no—
405C2×D4—216no—
406C2×D4—416no—
407C2×D4—48no—
408C2×D4—416no—
409C2×D4—48no—
410C2×D4—416no—
411C2×D4—416no—
412C2×D4—44no—
413C2×D4—416no—
414C2×D4—44no—
415C2×D4—416no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—88no—
421C2×D4—88no—
422C2×D4—88no—
423C2×D4—88no—
424C2×D4—88no—
425C2×D4—88no—
426C2×D4—88no—
427C2×D4—88no—
428C2×D4—88no—
429C2×D4—88no—
430C2×D4—88no—
431C2×D4—88no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—216no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—28no—
478C22×C4—28no—
479C22×C4—216no—
480C22×C4—216no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—24no—
491C22×C4—22no—
492C22×C4—24no—
493C22×C4—22no—
494C22×C4—24no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—416no—
501C22×C4—416no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—48no—
505C22×C4—48no—
506C22×C4—48no—
507C22×C4—44no—
508C22×C4—44no—
509C22×C4—416no—
510C22×C4—88no—
511C22×C4—88no—
512C22×C4—88no—
513C22×C4—88no—
514C22×C4C22×D414yes—
515C22×C4C22×D412yes—
516C22×C4C22×D418yes—
517C22×C4C22×D418yes—
518C22×C4C22×D412yes—
519C22×C4C22×D414yes—
520C22×C4C22×D414yes—
521C22×C4C22×D414yes—
522C22×C4C22×D412yes—
523C22×C4C22×D414yes—
524C22×C4C2×#4112yes—
525C22×C4C2×#4112yes—
526C22×C4C2×#4114yes—
527C22×C4C2×#4112yes—
528C22×C4C2×#4112yes—
529C22×C4H(5,𝔽2)14yes—
530C4⋊C4—28no—
531C4⋊C4—216no—
532C4⋊C4—216no—
533C4⋊C4—48no—
534C4⋊C4—416no—
535C4⋊C4—48no—
536C4⋊C4—48no—
537C4⋊C4—88no—
538C22⋊C4—28no—
539C22⋊C4—28no—
540C22⋊C4—232no—
541C22⋊C4—232no—
542C22⋊C4—416no—
543C22⋊C4—48no—
544C22⋊C4—416no—
545C22⋊C4—48no—
546C22⋊C4—416no—
547C22⋊C4—48no—
548C22⋊C4—416no—
549C22⋊C4—88no—
550C22⋊C4—88no—
551C22⋊C4—88no—
552C22⋊C4—88no—
553C22⋊C4—88no—
554C42—28no—
555C42—28no—
556C42—416no—
557C42—48no—
558C42—44no—
559C42—44no—
560C42#127116yes—
561C23—22no—
562C23—24no—
563C23—22no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—22no—
572C23—28no—
573C23—28no—
574C23—24no—
575C23—28no—
576C23—24no—
577C23—22no—
578C23—22no—
579C23—28no—
580C23—28no—
581C23—28no—
582C23—28no—
583C23—41no—
584C23—42no—
585C23—41no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—41no—
590C23—41no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—41no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—41no—
599C23—44no—
600C23—44no—
601C23—41no—
602C23—41no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23—42no—
607C23—44no—
608C23—44no—
609C23—42no—
610C23—44no—
611C23—42no—
612C23—44no—
613C23—44no—
614C23—42no—
615C23—42no—
616C23—44no—
617C23—44no—
618C23—44no—
619C23—44no—
620C23—44no—
621C23—44no—
622C23—44no—
623C23—42no—
624C23—42no—
625C23—44no—
626C23—44no—
627C23—44no—
628C23—44no—
629C23—44no—
630C23—42no—
631C23—42no—
632C23—44no—
633C23—42no—
634C23—44no—
635C23—44no—
636C23—42no—
637C23#54014yes—
638C23C23×D412yes—
639C23C23×D411yescharacteristic
640C23C22×#4111yescharacteristic
641C23C22×#4111yescharacteristic
642C23C22×#4112yes—
643C23C2×H(5,𝔽2)12yes—
644C23C2×H(5,𝔽2)12yes—
645C23#58512yes—
646C23#58512yes—
647D4—464no—
648D4—416no—
649D4—432no—
650D4—832no—
651D4—832no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—24no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—24no—
661C2×C4—24no—
662C2×C4—24no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—28no—
666C2×C4—28no—
667C2×C4—28no—
668C2×C4—28no—
669C2×C4—24no—
670C2×C4—24no—
671C2×C4—24no—
672C2×C4—28no—
673C2×C4—24no—
674C2×C4—24no—
675C2×C4—24no—
676C2×C4—28no—
677C2×C4—24no—
678C2×C4—432no—
679C2×C4—42no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—41no—
685C2×C4—41no—
686C2×C4—44no—
687C2×C4—41no—
688C2×C4—41no—
689C2×C4—42no—
690C2×C4—44no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—44no—
696C2×C4—41no—
697C2×C4—44no—
698C2×C4—41no—
699C2×C4—44no—
700C2×C4—42no—
701C2×C4—44no—
702C2×C4—42no—
703C2×C4—44no—
704C2×C4—44no—
705C2×C4—42no—
706C2×C4—42no—
707C2×C4—416no—
708C2×C4—416no—
709C2×C4—42no—
710C2×C4—44no—
711C2×C4—44no—
712C2×C4—42no—
713C2×C4—42no—
714C2×C4—44no—
715C2×C4—42no—
716C2×C4—42no—
717C2×C4—44no—
718C2×C4—44no—
719C2×C4—44no—
720C2×C4—42no—
721C2×C4—42no—
722C2×C4—44no—
723C2×C4—832no—
724C2×C4#52514yes—
725C2×C4#52514yes—
726C2×C4#52518yes—
727C2×C4#52518yes—
728C2×C4C2×#12718yes—
729C2×C4C2×#12714yes—
730C22—28no—
731C22—24no—
732C22—44no—
733C22—42no—
734C22—44no—
735C22—44no—
736C22—44no—
737C22—44no—
738C22—42no—
739C22—48no—
740C22—48no—
741C22—42no—
742C22—42no—
743C22—48no—
744C22—48no—
745C22—44no—
746C22—48no—
747C22—48no—
748C22—44no—
749C22—42no—
750C22—48no—
751C22—82no—
752C22—84no—
753C22—82no—
754C22—84no—
755C22—84no—
756C22—84no—
757C22—82no—
758C22—84no—
759C22—84no—
760C22—84no—
761C22—82no—
762C22—84no—
763C22C2×#52512yes—
764C22C2×#52511yescharacteristic
765C22C2×#52511yescharacteristic
766C22C2×#52512yes—
767C22C22×#12711yescharacteristic
768C22C2×#54011yescharacteristic
769C22#345511yescharacteristic
770C22#346012yes—
771C22#348412yes—
772C22#348712yes—
773C22#348812yes—
774C22#348812yes—
775C22#348812yes—
776C22#348812yes—
777C22#349412yes—
778C22#350811yescharacteristic
779C22#351211yescharacteristic
780C22#351312yes—
781C22#351412yes—
782C22#351912yes—
783C22#352112yes—
784C4—28no—
785C4—48no—
786C4—42no—
787C4—42no—
788C4—44no—
789C4—48no—
790C4—44no—
791C4—42no—
792C4—42no—
793C4—82no—
794C4—84no—
795C4#239718yes—
796C2—44no—
797C2—88no—
798C2—82no—
799C2C2×#239711yescharacteristic
800C2#3519512yes—
801C2#3522612yes—
802C2#3525911yescharacteristic
803C2#3526111yescharacteristic
804C2#3552212yes—
805C2#3552412yes—
806C2#3552512yes—
807C2#3556512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
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×#2328C2semidirect
2. C2×#2324C2semidirect
3. C2×C4×#121C2semidirect
4. C2×#2290C2semidirect
5. #15223C2semidirect
6. #15223C2semidirect
7. #15148C2semidirect
8. #15124C2semidirect
9. #15124C2semidirect
10. #15123C2semidirect
11. #14985C2semidirect
12. #14977C2semidirect
13. #14811C2semidirect
14. #14796C2semidirect
15. #13797C2semidirect
161. C4×C4⋊C4D4semidirect
165. #376D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23282
C2×#23242
C2×C4×#1212
C2×#22902
#152232
#151482
#151242
#151232
#149852
#149772
#148112
#147962
#137972