Finite Group Databasebeta

← #559848

Subgroups — #559848

Order 512.

3926 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#529C211yesmaximal, characteristic
2C2×#2379C212yesmaximal, complemented
3#14682C211yesmaximal, characteristic, complemented
4#14682C211yesmaximal, characteristic, complemented
5#14664C212yesmaximal, complemented
6#14475C214yesmaximal, complemented
7#14449C212yesmaximal, complemented
8#14417C212yesmaximal
9#14359C214yesmaximal, complemented
10#13962C212yesmaximal, complemented
11#13943C212yesmaximal, complemented
12#13931C212yesmaximal, complemented
13#13912C212yesmaximal, complemented
14#13899C212yesmaximal, complemented
15#13894C212yesmaximal, complemented
16C2×#529—216no—
17C2×#529C4116yescomplemented
18C22×#126C2212yes—
19C22×#125C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×#123C2212yes—
22C22×C22⋊Q8C2211yescharacteristic
23C22×#121C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C22×C4×D4C2211yescharacteristic
26C22×C4×D4C2212yes—
27C22×#117C2212yes—
28C23×C42C2211yescharacteristic
29#2379—216no—
30C2×#391C2214yes—
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#385C2214yes—
36C2×#385C2214yes—
37C2×#382C2212yes—
38C2×#382C2212yes—
39C2×#381C2212yes—
40C2×#380C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C22×#95C2212yes—
44#1445C2214yes—
45#1445C2214yes—
46#1445C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1439C2214yes—
55#1438C2214yes—
56#1437C2214yes—
57#1436C2214yes—
58C4⋊#95C2214yes—
59C4⋊#95C2212yes—
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2212yes—
63#1434C2212yes—
64#1434C2212yes—
65#1434C2212yes—
66#1434C2212yes—
67#1434C2212yes—
68#1434C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C4×#95C2212yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2314yes—
86#529—2128no—
87C2×#126—216no—
88C2×#126C2×C4116yescomplemented
89C2×#125—28no—
90C2×#125C2×C418yescomplemented
91C2×#124C2×C418yes—
92C2×#124D4116yes—
93C2×#123—216no—
94C2×#123C2×C4116yescomplemented
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8C2×C418yescomplemented
97C2×#121—28no—
98C2×#121C2×C418yes—
99C2×C4×Q8—28no—
100C2×C4×Q8C2×C418yescomplemented
101C2×C4×D4—28no—
102C2×C4×D4—216no—
103C2×C4×D4C2×C418yes—
104C2×C4×D4C2×C4116yescomplemented
105C2×#117—216no—
106C2×#117C2×C4116yescomplemented
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C42—28no—
130C22×C42C2×C418yescomplemented
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135C22×C42C2314yes—
136C22×C42C2311yescharacteristic
137#391—216no—
138#388—216no—
139#388—216no—
140#386—216no—
141#386—48no—
142#385—216no—
143#385—216no—
144#382—28no—
145#382—28no—
146#381—44no—
147#380—42no—
148#380—42no—
149#380—44no—
150C2×#95—44no—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C25C2412yes—
175C25C2411yescharacteristic
176C22×Q8—41no—
177C22×Q8—42no—
178C22×Q8—41no—
179C22×Q8C22×C414yes—
180C22×Q8C2×D418yes—
181C22×D4—28no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—41no—
185C22×D4—42no—
186C22×D4—41no—
187C22×D4—44no—
188C22×D4C22×C418yes—
189C22×D4C22×C414yes—
190C22×D4C2×D418yes—
191C23×C4—28no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—28no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—44no—
205C23×C4—42no—
206C23×C4—41no—
207C23×C4—41no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4C22×C418yes—
215C23×C4C22×C414yes—
216C23×C4C22×C414yes—
217C23×C4C22×C414yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231#126—264no—
232#125—232no—
233#124C22⋊C4164yescomplemented
234#123—264no—
235C22⋊Q8—232no—
236#121—232no—
237C4×Q8—232no—
238C4×D4—232no—
239C4×D4—264no—
240#117—264no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4C22×C418yes—
263C2×C4⋊C4C22×C418yes—
264C2×C4⋊C4C22×C414yes—
265C2×C4⋊C4C22×C418yes—
266C2×C4⋊C4C22×C414yes—
267C2×C4⋊C4C22×C418yes—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4C22×C418yes—
290C2×C22⋊C4C22×C418yes—
291C2×C22⋊C4C22×C418yes—
292C2×C22⋊C4C22×C418yes—
293C2×C22⋊C4C22×C418yes—
294C2×C22⋊C4C2×D4116yes—
295C2×C22⋊C4C2×D4116yes—
296C2×C42—24no—
297C2×C42—28no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—232no—
301C2×C42—28no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42C22×C414yes—
305C2×C42C22×C418yes—
306C2×C42C22×C414yes—
307C2×C42C22×C414yes—
308C2×C42C22×C414yes—
309C2×C42C2×D418yes—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—42no—
325#95—44no—
326#95—42no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333C24—22no—
334C24—21no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—21no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—21no—
343C24—21no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—22no—
349C24—22no—
350C24C23×C414yes—
351C24C23×C412yes—
352C24C22×D418yes—
353C24H(5,𝔽2)12yes—
354C242−1+412yes—
355C24C2511yescharacteristic
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8C2×C22⋊C4116yes—
361C2×D4—216no—
362C2×D4—216no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—416no—
369C2×D4—416no—
370C2×D4C2×C22⋊C4116yes—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—216no—
386C22×C4—28no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—216no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
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—44no—
437C22×C4—48no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—416no—
450C22×C4—88no—
451C22×C4—88no—
452C22×C4—88no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4—88no—
456C22×C4C23×C412yes—
457C22×C4C23×C412yes—
458C22×C4C23×C414yes—
459C22×C4C23×C414yes—
460C22×C4C23×C412yes—
461C22×C4C23×C412yes—
462C22×C4C23×C412yes—
463C22×C4C23×C414yes—
464C22×C4C23×C412yes—
465C22×C4C22×D414yes—
466C22×C4C22×D418yes—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4H(5,𝔽2)12yes—
470C22×C42−1+412yes—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—28no—
474C4⋊C4—216no—
475C4⋊C4—28no—
476C4⋊C4—216no—
477C4⋊C4—88no—
478C4⋊C4—84no—
479C4⋊C4—84no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C22⋊C4—216no—
483C22⋊C4—216no—
484C22⋊C4—216no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4—84no—
490C22⋊C4—84no—
491C22⋊C4C2×C22⋊C4132yes—
492C22⋊C4C2×C22⋊C4132yes—
493C42—28no—
494C42—28no—
495C42—28no—
496C42—216no—
497C42—88no—
498C42C2×C22⋊C4116yes—
499C23—22no—
500C23—22no—
501C23—24no—
502C23—24no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—24no—
516C23—24no—
517C23—28no—
518C23—28no—
519C23—24no—
520C23—24no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—42no—
543C23C22×C22⋊C418yes—
544C23#52014yes—
545C23C24×C411yescharacteristic
546C23C23×D412yes—
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×H(5,𝔽2)11yescharacteristic
549C23C2×2−1+411yescharacteristic
550C23C2×2−1+411yescharacteristic
551C23#58512yes—
552C23#58511yescharacteristic
553C23#58511yescharacteristic
554C23#58512yes—
555C23#58512yes—
556Q8—416no—
557D4—416no—
558D4—432no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—416no—
592C2×C4—432no—
593C2×C4—416no—
594C2×C4—416no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
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—832no—
632C2×C4C22×C22⋊C414yes—
633C2×C4C22×C22⋊C414yes—
634C2×C4C22×C22⋊C414yes—
635C2×C4C22×C22⋊C418yes—
636C2×C4#52014yes—
637C22—24no—
638C22—24no—
639C22—28no—
640C22—24no—
641C22—28no—
642C22—28no—
643C22—42no—
644C22—44no—
645C22—48no—
646C22—44no—
647C22—48no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—42no—
652C22—44no—
653C22—42no—
654C22—42no—
655C22—48no—
656C22—44no—
657C22—48no—
658C22—44no—
659C22C23×C22⋊C411yescharacteristic
660C22C2×#52011yescharacteristic
661C22C2×#52011yescharacteristic
662C22#342811yescharacteristic
663C22#342811yescharacteristic
664C22#342811yescharacteristic
665C22#342811yescharacteristic
666C22#348512yes—
667C22#348512yes—
668C22#348812yes—
669C22#348812yes—
670C22#348912yes—
671C22#348912yes—
672C22#352912yes—
673C22#353112yes—
674C22#354512yes—
675C22#355312yes—
676C22#355811yescharacteristic
677C22#355911yescharacteristic
678C22#356111yescharacteristic
679C22#356311yescharacteristic
680C22#356712yes—
681C22#356812yes—
682C4—24no—
683C4—28no—
684C4—24no—
685C4—24no—
686C4—24no—
687C4—42no—
688C4—44no—
689C4—42no—
690C4—44no—
691C4—84no—
692C4—84no—
693C4—84no—
694C4—84no—
695C2—44no—
696C2—48no—
697C2#3417211yescharacteristic
698C2#3417511yescharacteristic
699C2#3417511yescharacteristic
700C2#3435011yescharacteristic
701C2#3438211yescharacteristic
702C2#3439011yescharacteristic
703C2#3440511yescharacteristic
704C2#4597212yes—
705C2#4597312yes—
706C2#4613212yes—
707C2#4613312yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#121C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2379C2semidirect
3. #14682C2semidirect
4. #14682C2semidirect
5. #14664C2semidirect
6. #14475C2semidirect
7. #14449C2semidirect
9. #14359C2semidirect
10. #13962C2semidirect
11. #13943C2semidirect
12. #13931C2semidirect
13. #13912C2semidirect
14. #13899C2semidirect
15. #13894C2semidirect
17. C2×#529C4semidirect
88. C2×#126C2×C4semidirect
90. C2×#125C2×C4semidirect
94. C2×#123C2×C4semidirect
96. C2×C22⋊Q8C2×C4semidirect
100. C2×C4×Q8C2×C4semidirect
104. C2×C4×D4C2×C4semidirect
106. C2×#117C2×C4semidirect
130. C22×C42C2×C4semidirect
233. #124C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5292
C2×#23792
#146822
#146642
#144752
#144492
#144172
#143592
#139622
#139432
#139312
#139122
#138992
#138942