Finite Group Databasebeta

← #517795

Subgroups — #517795

Order 512.

5938 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#518C211yesmaximal, characteristic
2C2×#2342C212yesmaximal, complemented
3#14344C212yesmaximal, complemented
4#14341C211yesmaximal, characteristic, complemented
5#13943C214yesmaximal, complemented
6#13940C212yesmaximal, complemented
7#13899C214yesmaximal, complemented
8#13873C214yesmaximal, complemented
9#13871C212yesmaximal, complemented
10#13867C212yesmaximal
11#13839C212yesmaximal, complemented
12C2×#1440C211yesmaximal, characteristic, complemented
13#13813C214yesmaximal, complemented
14C24×D4C2211yescharacteristic
15C2×#518—216no—
16C2×#518C4116yescomplemented
17C22×C4×D4C2212yes—
18C22×C4×D4C2212yes—
19C22×C4×D4C2212yes—
20C22×C4×D4C2212yes—
21C22×#117C2212yes—
22C22×#117C2212yescomplemented
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2212yes—
25C23×C22⋊C4C2212yes—
26C23×C22⋊C4C2211yescharacteristic
27#2342—216no—
28C2×#388C2214yes—
29C2×#388C2214yes—
30C2×#386C2214yes—
31C2×#386C2214yes—
32C2×#385C2214yescomplemented
33C2×#381C2212yescomplemented
34C2×#380C2212yescomplemented
35C2×#380C2214yescomplemented
36C2×#379C2212yes—
37C2×#379C2214yes—
38C2×C4×C22⋊C4C2212yescomplemented
39C2×C4×C22⋊C4C2214yescomplemented
40C2×C4×C22⋊C4C2214yescomplemented
41C22×#95C2214yescomplemented
42#1441C2214yescomplemented
43#1441C2214yescomplemented
44#1440C2214yes—
45#1440C2214yes—
46#1440C2212yescomplemented
47#1440C2211yescharacteristic, complemented
48#1440C2211yescharacteristic, complemented
49#1440C2211yescharacteristic, complemented
50#1440C2211yescharacteristic, complemented
51#1440C2212yescomplemented
52#1439C2214yescomplemented
53#1439C2214yescomplemented
54#1438C2212yes—
55#1438C2212yes—
56#1438C2212yes—
57#1438C2212yes—
58#1437C2214yescomplemented
59#1437C2214yescomplemented
60#1436C2214yescomplemented
61#1436C2214yescomplemented
62C4⋊#95C2214yescomplemented
63#1434C2214yescomplemented
64#1434C2214yescomplemented
65C22⋊#95C2214yes—
66C22⋊#95C2214yescomplemented
67C22⋊#95C2214yescomplemented
68C22⋊#95C2214yescomplemented
69C4×#95C2214yescomplemented
70C26C2311yescharacteristic
71C26C2311yescharacteristic
72C23×D4—28no—
73C23×D4—216no—
74C23×D4C2×C418yes—
75C23×D4C2312yes—
76C23×D4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2314yes—
79C24×C4C2312yes—
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2312yes—
83#518—2128no—
84C2×C4×D4—28no—
85C2×C4×D4—28no—
86C2×C4×D4—28no—
87C2×C4×D4—28no—
88C2×C4×D4—28no—
89C2×C4×D4—28no—
90C2×C4×D4—28no—
91C2×C4×D4—28no—
92C2×C4×D4C2×C4116yes—
93C2×C4×D4C2×C4116yes—
94C2×C4×D4C2×C4116yescomplemented
95C2×C4×D4C2×C4116yescomplemented
96C2×#117—28no—
97C2×#117—28no—
98C2×#117—28no—
99C2×#117—48no—
100C2×#117C2×C4116yescomplemented
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C22⋊C4—28no—
106C22×C22⋊C4—28no—
107C22×C22⋊C4—28no—
108C22×C22⋊C4—28no—
109C22×C22⋊C4—48no—
110C22×C22⋊C4C2×C418yes—
111C22×C22⋊C4C2×C418yes—
112C22×C22⋊C4C2×C4116yescomplemented
113C22×C22⋊C4D4116yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2311yescharacteristic
133C22×C42C2314yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2314yes—
137#388—216no—
138#388—216no—
139#386—216no—
140#386—216no—
141#385—216no—
142#381—28no—
143#380—28no—
144#380—48no—
145#379—28no—
146#379—48no—
147C4×C22⋊C4—28no—
148C4×C22⋊C4—216no—
149C4×C22⋊C4—48no—
150C2×#95—48no—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C25—24no—
168C25—28no—
169C25—28no—
170C25—44no—
171C25—44no—
172C25—44no—
173C25—44no—
174C25C22×C414yes—
175C25C22×C414yes—
176C25C2×D418yes—
177C25C2411yescharacteristic
178C25C2411yescharacteristic
179C25C2412yes—
180C25C2411yescharacteristic
181C22×D4—2128no—
182C22×D4—232no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—264no—
190C22×D4—28no—
191C22×D4C22×C418yes—
192C22×D4C22×C418yes—
193C22×D4#41116yes—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—28no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4C22×C414yes—
218C23×C4C22×C418yes—
219C23×C4C22×C418yes—
220C23×C4C22×C418yes—
221C23×C4C22×C418yes—
222C23×C4C2×D418yes—
223C23×C4C2×D418yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C4×D4—264no—
234C4×D4—264no—
235C4×D4—264no—
236C4×D4—264no—
237#117—264no—
238#117—432no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C22×C418yes—
250C2×C4⋊C4C22×C418yes—
251C2×C22⋊C4—232no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—264no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
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—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—432no—
279C2×C22⋊C4—44no—
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⋊C4C22⋊C4164yescomplemented
292C2×C22⋊C4C22×C418yes—
293C2×C22⋊C4C22×C414yes—
294C2×C22⋊C4C22×C414yes—
295C2×C22⋊C4C22×C418yes—
296C2×C22⋊C4C22×C418yes—
297C2×C22⋊C4C22×C418yes—
298C2×C22⋊C4C22×C414yes—
299C2×C22⋊C4C22×C414yes—
300C2×C22⋊C4C2×D418yes—
301C2×C22⋊C4C2×D418yes—
302C2×C22⋊C4C2×D418yes—
303C2×C22⋊C4C2×D418yes—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—28no—
309C2×C42—28no—
310C2×C42—28no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42—44no—
314C2×C42C22×C418yes—
315C2×C42C22×C418yes—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332C24—24no—
333C24—24no—
334C24—232no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—24no—
341C24—24no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—28no—
349C24—216no—
350C24—216no—
351C24—232no—
352C24—48no—
353C24—416no—
354C24—48no—
355C24—416no—
356C24—416no—
357C24—88no—
358C24—88no—
359C24—88no—
360C24—88no—
361C24—84no—
362C24—84no—
363C24—88no—
364C24—88no—
365C24—88no—
366C24—88no—
367C24—88no—
368C24C2×C22⋊C4116yes—
369C24C23×C412yes—
370C24C23×C412yes—
371C24C23×C414yes—
372C24C23×C412yes—
373C24C22×D414yes—
374C24C22×D414yes—
375C24C22×D414yes—
376C24C2×#4118yes—
377C24C2×#4118yes—
378C24C2511yescharacteristic
379C2×D4—2256no—
380C2×D4—232no—
381C2×D4—232no—
382C2×D4—216no—
383C2×D4—216no—
384C2×D4—216no—
385C2×D4—216no—
386C2×D4—232no—
387C2×D4—232no—
388C2×D4—216no—
389C2×D4#117132yescomplemented
390C22×C4—232no—
391C22×C4—28no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—216no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—216no—
415C22×C4—216no—
416C22×C4—216no—
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—24no—
437C22×C4—24no—
438C22×C4—48no—
439C22×C4—416no—
440C22×C4—48no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4—84no—
450C22×C4—84no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4—88no—
458C22×C4—88no—
459C22×C4—88no—
460C22×C4—88no—
461C22×C4C2×C22⋊C4132yes—
462C22×C4C23×C414yes—
463C22×C4C23×C414yes—
464C22×C4C23×C414yes—
465C22×C4C23×C414yes—
466C22×C4C23×C414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C22×C4C2×#4118yes—
472C22×C4H(5,𝔽2)18yes—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C4⋊C4—416no—
476C4⋊C4—88no—
477C22⋊C4—216no—
478C22⋊C4—216no—
479C22⋊C4—28no—
480C22⋊C4—28no—
481C22⋊C4—216no—
482C22⋊C4—216no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—416no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C22⋊C4—88no—
493C22⋊C4C2×C22⋊C4132yes—
494C22⋊C4C2×C22⋊C4116yes—
495C22⋊C4C2×C22⋊C4116yes—
496C42—216no—
497C42—216no—
498C42—416no—
499C42—88no—
500C23—28no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—28no—
506C23—28no—
507C23—28no—
508C23—232no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—28no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—28no—
531C23—232no—
532C23—432no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—832no—
552C23—832no—
553C23—832no—
554C23—816no—
555C23—832no—
556C23—832no—
557C23—832no—
558C23C22×C22⋊C414yes—
559C23C22×C22⋊C414yes—
560C23C22×C22⋊C414yes—
561C23C2×#11718yes—
562C23C2×#11718yes—
563C23#525116yes—
564C23C24×C411yescharacteristic
565C23C23×D412yes—
566C23C22×#4114yes—
567C23C2×H(5,𝔽2)14yes—
568C23#58514yes—
569D4—264no—
570D4—264no—
571C2×C4—232no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—432no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
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—816no—
624C2×C4—816no—
625C2×C4—816no—
626C2×C4—832no—
627C2×C4C22×C22⋊C418yes—
628C2×C4C22×C22⋊C418yes—
629C2×C4C2×#11718yes—
630C22—216no—
631C22—28no—
632C22—216no—
633C22—216no—
634C22—28no—
635C22—28no—
636C22—216no—
637C22—24no—
638C22—24no—
639C22—28no—
640C22—48no—
641C22—48no—
642C22—48no—
643C22—48no—
644C22—44no—
645C22—44no—
646C22—48no—
647C22—48no—
648C22—48no—
649C22—48no—
650C22—48no—
651C22—48no—
652C22—48no—
653C22—832no—
654C22—832no—
655C22#228018yes—
656C22C23×C22⋊C411yescharacteristic
657C22C22×#11712yes—
658C22C4×H(5,𝔽2)14yes—
659C22C2×#52514yes—
660C22#348412yes—
661C22#348512yes—
662C22#349614yes—
663C22#351314yes—
664C22#351414yes—
665C22#351914yes—
666C22#352114yes—
667C4—28no—
668C4—28no—
669C4—28no—
670C4—48no—
671C4—44no—
672C4—44no—
673C4—84no—
674C4—84no—
675C4—84no—
676C2—28no—
677C2—44no—
678C2—48no—
679C2—44no—
680C2C2×#228011yescharacteristic
681C2#3425612yes—
682C2#3426114yes—
683C2#3571414yes—
684C2#3579214yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC24×D4C22
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×#2342C2semidirect
3. #14344C2semidirect
4. #14341C2semidirect
5. #13943C2semidirect
6. #13940C2semidirect
7. #13899C2semidirect
8. #13873C2semidirect
9. #13871C2semidirect
11. #13839C2semidirect
12. C2×#1440C2semidirect
13. #13813C2semidirect
16. C2×#518C4semidirect
22. C22×#117C22semidirect
32. C2×#385C22semidirect
33. C2×#381C22semidirect
34. C2×#380C22semidirect
35. C2×#380C22semidirect
38. C2×C4×C22⋊C4C22semidirect
39. C2×C4×C22⋊C4C22semidirect
40. C2×C4×C22⋊C4C22semidirect
41. C22×#95C22semidirect
42. #1441C22semidirect
43. #1441C22semidirect
46. #1440C22semidirect
47. #1440C22semidirect
48. #1440C22semidirect
49. #1440C22semidirect
50. #1440C22semidirect
51. #1440C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
58. #1437C22semidirect
59. #1437C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. C4⋊#95C22semidirect
63. #1434C22semidirect
64. #1434C22semidirect
66. C22⋊#95C22semidirect
67. C22⋊#95C22semidirect
68. C22⋊#95C22semidirect
69. C4×#95C22semidirect
94. C2×C4×D4C2×C4semidirect
95. C2×C4×D4C2×C4semidirect
100. C2×#117C2×C4semidirect
112. C22×C22⋊C4C2×C4semidirect
291. C2×C22⋊C4C22⋊C4semidirect
389. C2×D4#117semidirect

Maximal subgroups

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

subgroupindex
C22×#5182
C2×#23422
#143442
#143412
#139432
#139402
#138992
#138732
#138712
#138672
#138392
C2×#14402
#138132