Finite Group Databasebeta

← #770061

Subgroups — #770061

Order 512.

3930 conjugacy classes of subgroups.

The subgroup lattice has 786 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2534C212yesmaximal, complemented
2C2×#2487C212yesmaximal, complemented
3C22×#397C211yesmaximal, characteristic, complemented
4C2×#2325C211yesmaximal, characteristic, complemented
5#15808C214yesmaximal, complemented
6#15696C214yesmaximal, complemented
7#15526C214yesmaximal, complemented
8#15493C214yesmaximal, complemented
9#14936C211yesmaximal, characteristic, complemented
10#14926C211yesmaximal, characteristic, complemented
11#14734C211yesmaximal, characteristic, complemented
12#14716C211yesmaximal, characteristic
13#13996C212yesmaximal, complemented
14#13995C212yesmaximal, complemented
15#13778C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#121C2211yescharacteristic
18C22×#120C2212yes—
19C22×C4×D4C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24#2534—216no—
25#2487—216no—
26C2×#400C2214yescomplemented
27C2×#399C2212yes—
28C2×#399C2214yescomplemented
29C2×#397—28no—
30C2×#397C2214yes—
31C2×#397C2214yes—
32C2×#396C2214yes—
33C2×#394C2214yes—
34C2×#392C2212yes—
35#2325—28no—
36C2×#390C2214yes—
37C2×#389C2214yescomplemented
38C2×#388C2214yes—
39C2×#388C2212yescomplemented
40C2×#387C2212yescomplemented
41C2×#386C2214yes—
42C2×#386C2212yes—
43C2×#386C2212yes—
44C2×#386C2212yes—
45C2×#386C2212yescomplemented
46C2×#385C2212yes—
47C2×#385C2212yes—
48C2×#385C2212yescomplemented
49C2×#381C2212yes—
50C2×#380C2211yescharacteristic
51C2×#380C2212yes—
52C2×#380C2214yescomplemented
53C2×#379C2212yes—
54C2×C4×C22⋊C4C2212yes—
55C22×#95C2211yescharacteristic
56C22×#95C2211yescharacteristic
57C22×#95C2211yescharacteristic
58#1445C2214yescomplemented
59#1445C2212yescomplemented
60#1445C2214yescomplemented
61#1444C2212yescomplemented
62#1444C2212yescomplemented
63#1439C2214yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2212yes—
67#1439C2214yescomplemented
68#1439C2214yescomplemented
69#1438C2214yes—
70#1438C2214yes—
71#1438C2212yescomplemented
72#1437C2214yes—
73#1437C2214yescomplemented
74#1436C2214yescomplemented
75#1436C2214yescomplemented
76C4⋊#95C2214yescomplemented
77C22⋊#95C2211yescharacteristic
78C22⋊#95C2211yescharacteristic
79C26C2311yescharacteristic
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C23×D4C2314yes—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×#123—28no—
91C2×#123—28no—
92C2×#123—44no—
93C2×#121—28no—
94C2×#121—24no—
95C2×#120—28no—
96C2×#120—28no—
97C2×#120—44no—
98C2×C4×D4—28no—
99C2×C4×D4—28no—
100C2×C4×D4—28no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—24no—
114C22×C22⋊C4—24no—
115C22×C22⋊C4—28no—
116C22×C22⋊C4—44no—
117C22×C22⋊C4—44no—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C42C2312yes—
136C22×C42C2314yes—
137#400—216no—
138#399—216no—
139#399—44no—
140#397—216no—
141#397—216no—
142#396—216no—
143#394—216no—
144#392—44no—
145#390—48no—
146#389—216no—
147#388—28no—
148#388—216no—
149#387—28no—
150#386—28no—
151#386—28no—
152#386—28no—
153#386—28no—
154#386—216no—
155#385—28no—
156#385—28no—
157#385—28no—
158#381—28no—
159#380—24no—
160#380—216no—
161#380D4116yescomplemented
162#379—28no—
163C4×C22⋊C4—44no—
164C2×#95—24no—
165C2×#95—24no—
166C2×#95D418yescomplemented
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C25—22no—
188C25—24no—
189C25—24no—
190C25—24no—
191C25—22no—
192C25—22no—
193C25—42no—
194C25—42no—
195C25—42no—
196C25C2412yes—
197C25C2412yes—
198C25C2411yescharacteristic
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—28no—
207C22×D4—28no—
208C22×D4—22no—
209C22×D4—28no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4—42no—
214C22×D4—44no—
215C22×D4—42no—
216C22×D4#4118yes—
217C23×C4—24no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—22no—
224C23×C4—24no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4—44no—
239C23×C4—44no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4C2×D414yes—
245C23×C4C2×D414yes—
246C23×C4C2×D414yes—
247C23×C4C2×D418yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2414yes—
254C23×C4C2411yescharacteristic
255C23×C4C2412yes—
256C23×C4C2414yes—
257C23×C4C2411yescharacteristic
258C23×C4C2412yes—
259C23×C4C2412yes—
260#123—432no—
261#121—232no—
262#120—432no—
263C4×D4—432no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—416no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4C2×D4116yes—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—416no—
329C2×C22⋊C4—416no—
330C2×C22⋊C4—416no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4C2×D418yes—
340C2×C22⋊C4C2×D418yes—
341C2×C42—24no—
342C2×C42—24no—
343C2×C42—24no—
344C2×C42—44no—
345C2×C42—44no—
346C2×C42—44no—
347#95—28no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—24no—
353#95—28no—
354#95—28no—
355#95—24no—
356#95—24no—
357#95—28no—
358#95—28no—
359#95—42no—
360#95—42no—
361#95—44no—
362#95—42no—
363#95—42no—
364#95C2×D418yes—
365#95C2×D418yes—
366#95C2×D4116yes—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—28no—
372C24—22no—
373C24—22no—
374C24—24no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—24no—
379C24—22no—
380C24—24no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—24no—
386C24—22no—
387C24—24no—
388C24—24no—
389C24—24no—
390C24—24no—
391C24—22no—
392C24—416no—
393C24—416no—
394C24—48no—
395C24—48no—
396C24—44no—
397C24—44no—
398C24—44no—
399C24—44no—
400C24—44no—
401C24—44no—
402C24—48no—
403C24—48no—
404C24—88no—
405C24—88no—
406C24—84no—
407C24—84no—
408C24C22×D412yes—
409C24C22×D414yes—
410C24C2×#4114yes—
411C24C2×#4114yes—
412C24H(5,𝔽2)14yes—
413C24H(5,𝔽2)12yes—
414C242−1+412yes—
415C24C2511yescharacteristic
416C2×D4—216no—
417C2×D4—216no—
418C2×D4—216no—
419C2×D4—28no—
420C2×D4—416no—
421C2×D4—416no—
422C2×D4—48no—
423C2×D4—48no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—416no—
427C2×D4—82no—
428C2×D4—88no—
429C2×D4—82no—
430C2×D4—88no—
431C2×D4—88no—
432C2×D4—88no—
433C2×D4—88no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—22no—
491C22×C4—416no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—416no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4—48no—
509C22×C4—48no—
510C22×C4—48no—
511C22×C4—84no—
512C22×C4—88no—
513C22×C4—84no—
514C22×C4—88no—
515C22×C4—84no—
516C22×C4—84no—
517C22×C4—88no—
518C22×C4—88no—
519C22×C4#127116yes—
520C22×C4C22×D414yes—
521C22×C4C22×D412yes—
522C22×C4C22×D418yes—
523C22×C4C22×D412yes—
524C22×C4C22×D418yes—
525C22×C4C22×D414yes—
526C22×C4C22×D414yes—
527C22×C4C22×D414yes—
528C22×C4C2×#4114yes—
529C4⋊C4—28no—
530C4⋊C4—416no—
531C4⋊C4—416no—
532C4⋊C4—416no—
533C4⋊C4—48no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4—88no—
537C4⋊C4—88no—
538C22⋊C4—216no—
539C22⋊C4—216no—
540C22⋊C4—216no—
541C22⋊C4—416no—
542C22⋊C4—416no—
543C22⋊C4—416no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C22⋊C4—416no—
547C22⋊C4—48no—
548C22⋊C4—416no—
549C22⋊C4—48no—
550C22⋊C4—84no—
551C22⋊C4—88no—
552C22⋊C4—84no—
553C22⋊C4—88no—
554C22⋊C4—88no—
555C42—48no—
556C42—88no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—28no—
561C23—22no—
562C23—28no—
563C23—28no—
564C23—28no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—28no—
574C23—28no—
575C23—28no—
576C23—28no—
577C23—24no—
578C23—24no—
579C23—416no—
580C23—416no—
581C23—416no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—44no—
588C23—42no—
589C23—42no—
590C23—44no—
591C23—42no—
592C23—42no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—42no—
597C23—44no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—44no—
610C23—44no—
611C23—42no—
612C23—44no—
613C23—42no—
614C23—44no—
615C23—44no—
616C23—44no—
617C23—44no—
618C23—832no—
619C23—832no—
620C23—88no—
621C23—88no—
622C23—88no—
623C23—88no—
624C23#52514yes—
625C23C2×#12714yes—
626C23#53714yes—
627C23#54618yes—
628C23#54818yes—
629C23C23×D411yescharacteristic
630C23C23×D412yes—
631C23C22×#4112yes—
632C23C2×H(5,𝔽2)11yescharacteristic
633C23C2×H(5,𝔽2)12yes—
634C23C2×2−1+411yescharacteristic
635C23#58512yes—
636C23#58512yes—
637C23#58512yes—
638D4—432no—
639D4—88no—
640D4—832no—
641C2×C4—28no—
642C2×C4—28no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—22no—
646C2×C4—24no—
647C2×C4—22no—
648C2×C4—24no—
649C2×C4—28no—
650C2×C4—28no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—28no—
657C2×C4—24no—
658C2×C4—28no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—28no—
662C2×C4—28no—
663C2×C4—416no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—44no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—44no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—44no—
700C2×C4—44no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—44no—
704C2×C4—44no—
705C2×C4—816no—
706C2×C4—88no—
707C2×C4—88no—
708C2×C4—88no—
709C2×C4—88no—
710C2×C4—88no—
711C2×C4C2×#12718yes—
712C22—216no—
713C22—216no—
714C22—24no—
715C22—24no—
716C22—24no—
717C22—24no—
718C22—24no—
719C22—24no—
720C22—48no—
721C22—48no—
722C22—48no—
723C22—48no—
724C22—44no—
725C22—44no—
726C22—44no—
727C22—44no—
728C22—44no—
729C22—44no—
730C22—832no—
731C22—84no—
732C22—84no—
733C22—84no—
734C22—84no—
735C22—82no—
736C22—82no—
737C22—82no—
738C22—82no—
739C22—82no—
740C22—82no—
741C22—82no—
742C22—82no—
743C22#254018yes—
744C22C2×#52511yescharacteristic
745C22C22×#12711yescharacteristic
746C22C2×#53711yescharacteristic
747C22C2×#54612yes—
748C22C2×#54812yes—
749C22#348411yescharacteristic
750C22#348514yes—
751C22#348511yescharacteristic
752C22#348511yescharacteristic
753C22#348711yescharacteristic
754C22#348814yes—
755C22#351314yes—
756C22#352114yes—
757C22#353412yes—
758C22#353512yes—
759C22#356012yes—
760C22#356412yes—
761C4—28no—
762C4—48no—
763C4—48no—
764C4—42no—
765C4—42no—
766C4—44no—
767C4—44no—
768C4—82no—
769C4—82no—
770C4—82no—
771C4—82no—
772C4—82no—
773C2—28no—
774C2—84no—
775C2—84no—
776C2C2×#254011yescharacteristic
777C2#3535411yescharacteristic
778C2#3545411yescharacteristic
779C2#3546311yescharacteristic
780C2#3556511yescharacteristic
781C2#3670911yescharacteristic
782C2#3697111yescharacteristic
783C2#4353014yes—
784C2#4353314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#14716C2
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×#2534C2semidirect
2. C2×#2487C2semidirect
3. C22×#397C2semidirect
4. C2×#2325C2semidirect
5. #15808C2semidirect
6. #15696C2semidirect
7. #15526C2semidirect
8. #15493C2semidirect
9. #14936C2semidirect
10. #14926C2semidirect
11. #14734C2semidirect
13. #13996C2semidirect
14. #13995C2semidirect
15. #13778C2semidirect
26. C2×#400C22semidirect
28. C2×#399C22semidirect
37. C2×#389C22semidirect
39. C2×#388C22semidirect
40. C2×#387C22semidirect
45. C2×#386C22semidirect
48. C2×#385C22semidirect
52. C2×#380C22semidirect
58. #1445C22semidirect
59. #1445C22semidirect
60. #1445C22semidirect
61. #1444C22semidirect
62. #1444C22semidirect
67. #1439C22semidirect
68. #1439C22semidirect
71. #1438C22semidirect
73. #1437C22semidirect
74. #1436C22semidirect
75. #1436C22semidirect
76. C4⋊#95C22semidirect
161. #380D4semidirect
166. C2×#95D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25342
C2×#24872
C22×#3972
C2×#23252
#158082
#156962
#155262
#154932
#149362
#149262
#147342
#147162
#139962
#139952
#137782