Finite Group Databasebeta

← #4308047

Subgroups — #4308047

Order 512.

5182 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#117C211yesmaximal, characteristic
2C2×#2855C211yesmaximal, characteristic, complemented
3C2×#2627C211yesmaximal, characteristic, complemented
4C2×#2626C211yesmaximal, characteristic
5#14893C214yesmaximal, complemented
6#14891C218yesmaximal, complemented
7#14887C214yesmaximal, complemented
8#14705C212yesmaximal, complemented
9#14685C211yesmaximal, characteristic, complemented
10#14681C212yesmaximal
11#14339C211yesmaximal, characteristic, complemented
12#14338C211yesmaximal, characteristic
13#14263C212yesmaximal, complemented
14#14262C212yesmaximal
15C25×C4C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×#117—28no—
18C22×#117—28no—
19C22×#117—28no—
20C22×#117C2214yes—
21C22×#117C2214yes—
22C23×C4⋊C4C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25C23×C22⋊C4C2211yescharacteristic
26C23×C42C2211yescharacteristic
27C23×C42C2211yescharacteristic
28#2855—28no—
29#2627—28no—
30#2626—28no—
31C2×#400C2212yescomplemented
32C2×#399C2212yes—
33C2×#398C2214yescomplemented
34C2×#396C2214yes—
35C2×#395C2212yescomplemented
36C2×#393C2212yes—
37C2×#391C2212yescomplemented
38C2×#389C2212yescomplemented
39C2×#387C2214yescomplemented
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#381C2212yes—
44C2×#381C2212yes—
45C2×#381C2212yes—
46C2×#381C2212yes—
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#379C2212yes—
51C2×#379C2212yes—
52#1444C2218yescomplemented
53#1442C2218yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1439C2214yes—
57#1439C2214yes—
58#1439C2218yes—
59#1439C2218yes—
60#1436C2218yescomplemented
61#1436C2218yescomplemented
62C4⋊#95C2214yescomplemented
63#1434C2214yescomplemented
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2218yes—
67C22⋊#95C2218yes—
68C22⋊#95C2212yes—
69C26C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C24×C4—24no—
72C24×C4—24no—
73C24×C4—24no—
74C24×C4—24no—
75C24×C4—24no—
76C24×C4—24no—
77C24×C4Q818yescomplemented
78C24×C4C2314yes—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×#117—2128no—
89C2×#117—216no—
90C2×#117—216no—
91C2×#117—216no—
92C2×#117—216no—
93C2×#117—216no—
94C2×#117—216no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4D418yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—24no—
114C22×C22⋊C4—24no—
115C22×C22⋊C4—24no—
116C22×C22⋊C4—24no—
117C22×C22⋊C4D418yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2318yes—
128C22×C42—24no—
129C22×C42—24no—
130C22×C42—24no—
131C22×C42—24no—
132C22×C42—24no—
133C22×C42D418yes—
134C22×C42C2314yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137#400—28no—
138#399—28no—
139#398—216no—
140#396—216no—
141#395—28no—
142#393—28no—
143#391—28no—
144#389—28no—
145#387—216no—
146#385—28no—
147#385—28no—
148#385—28no—
149#381—28no—
150#381—28no—
151#381—28no—
152#381—28no—
153#380—24no—
154#380—24no—
155#380—24no—
156#379—28no—
157#379—28no—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2318yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2318yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2318yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C25—22no—
174C25—22no—
175C25—22no—
176C25—22no—
177C25—44no—
178C25—44no—
179C25C2×D414yes—
180C25C2×Q814yes—
181C25#4114yes—
182C25C2412yes—
183C25C2411yescharacteristic
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—44no—
188C23×C4—232no—
189C23×C4—232no—
190C23×C4—232no—
191C23×C4—232no—
192C23×C4—232no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—28no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—28no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—232no—
204C23×C4—232no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—28no—
208C23×C4—28no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×Q8116yes—
228C23×C4C2×Q814yes—
229C23×C4C2×Q814yes—
230C23×C4#4114yes—
231C23×C4#4114yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2414yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2412yes—
243C23×C4C2412yes—
244C22⋊Q8—416no—
245#117—2128no—
246#117—2128no—
247C2×C4⋊C4—232no—
248C2×C4⋊C4—232no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—48no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—48no—
273C2×C4⋊C4—48no—
274C2×C4⋊C4C2×D418yes—
275C2×C4⋊C4C2×D418yes—
276C2×C22⋊C4—232no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—232no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
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—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—216no—
300C2×C22⋊C4—48no—
301C2×C22⋊C4—48no—
302C2×C22⋊C4—48no—
303C2×C22⋊C4—48no—
304C2×C22⋊C4C2×D418yes—
305C2×C22⋊C4C2×D418yes—
306C2×C42—232no—
307C2×C42—232no—
308C2×C42—28no—
309C2×C42—28no—
310C2×C42—28no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42C2×D418yes—
316C2×C42C2×D418yes—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—216no—
328#95—216no—
329#95—48no—
330#95—48no—
331#95—48no—
332C24—24no—
333C24—24no—
334C24—28no—
335C24—28no—
336C24—28no—
337C24—28no—
338C24—28no—
339C24—28no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—28no—
345C24—28no—
346C24—48no—
347C24—48no—
348C24—416no—
349C24—416no—
350C24—48no—
351C24—48no—
352C24—416no—
353C24—48no—
354C24—48no—
355C24—416no—
356C24C22⋊Q8116yes—
357C24C22×D414yes—
358C24C22×D412yes—
359C24C22×D412yes—
360C24C22×D412yes—
361C24C22×Q814yes—
362C24C22×Q812yes—
363C24C2×#4114yes—
364C24C2×#4112yes—
365C24C2×#4112yes—
366C24C2×#4112yes—
367C24C2511yescharacteristic
368C2×Q8—48no—
369C2×Q8—44no—
370C2×Q8—84no—
371C2×Q8—84no—
372C22×C4—232no—
373C22×C4—2256no—
374C22×C4—232no—
375C22×C4—232no—
376C22×C4—232no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—232no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—28no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—232no—
401C22×C4—232no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4#120116yes—
451C22×C4C22⋊Q8116yes—
452C22×C4C22⋊Q8116yes—
453C22×C4C22×D414yes—
454C22×C4C22×D414yes—
455C22×C4C22×D414yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D414yes—
463C22×C4C22×D414yes—
464C22×C4C22×Q814yes—
465C22×C4C22×Q814yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4114yes—
468C4⋊C4—232no—
469C4⋊C4—216no—
470C4⋊C4—216no—
471C4⋊C4—48no—
472C4⋊C4—416no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—84no—
476C4⋊C4—84no—
477C4⋊C4—88no—
478C4⋊C4—88no—
479C22⋊C4—232no—
480C22⋊C4—216no—
481C22⋊C4—216no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—816no—
489C42—232no—
490C42—216no—
491C42—216no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—28no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—28no—
502C23—232no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—216no—
510C23—28no—
511C23—28no—
512C23—216no—
513C23—432no—
514C23—416no—
515C23—432no—
516C23—432no—
517C23—432no—
518C23—432no—
519C23—416no—
520C23—432no—
521C23—44no—
522C23—44no—
523C23—48no—
524C23—44no—
525C23—44no—
526C23—48no—
527C23C2×#12014yes—
528C23C2×C22⋊Q818yes—
529C23C2×C22⋊Q814yes—
530C23C2×C22⋊Q814yes—
531C23C2×C22⋊Q814yes—
532C23#52514yes—
533C23#52514yes—
534C23#52514yes—
535C23C23×D411yescharacteristic
536C23C23×D411yescharacteristic
537C23C23×D411yescharacteristic
538C23C23×Q811yescharacteristic
539C23C22×#4111yescharacteristic
540C23C22×#4111yescharacteristic
541C23C22×#4111yescharacteristic
542C23#58514yes—
543C23#58514yes—
544Q8—88no—
545C2×C4—2128no—
546C2×C4—232no—
547C2×C4—232no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—416no—
574C2×C4—416no—
575C2×C4—416no—
576C2×C4—416no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—48no—
593C2×C4—48no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4C2×#12014yes—
607C2×C4C2×#12014yes—
608C2×C4C2×C22⋊Q818yes—
609C2×C4C2×C22⋊Q818yes—
610C22—24no—
611C22—28no—
612C22—28no—
613C22—28no—
614C22—28no—
615C22—28no—
616C22—24no—
617C22—28no—
618C22—28no—
619C22—216no—
620C22—464no—
621C22—44no—
622C22—48no—
623C22—48no—
624C22—48no—
625C22—48no—
626C22—44no—
627C22—48no—
628C22#284818yes—
629C22C22×#12011yescharacteristic
630C22C22×C22⋊Q811yescharacteristic
631C22C22×C22⋊Q811yescharacteristic
632C22C22×C22⋊Q811yescharacteristic
633C22C2×#52511yescharacteristic
634C22C2×#52511yescharacteristic
635C22C2×#52511yescharacteristic
636C22#344912yes—
637C22#344912yes—
638C22#345514yes—
639C22#345814yes—
640C22#348512yes—
641C22#348512yes—
642C22#348512yes—
643C22#348712yes—
644C22#349114yes—
645C22#349314yes—
646C4—28no—
647C4—28no—
648C4—24no—
649C4—24no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—82no—
654C4—84no—
655C4—84no—
656C4—84no—
657C4—82no—
658C2—28no—
659C2—48no—
660C2#3463412yes—
661C2#3508811yescharacteristic
662C2#3509012yes—
663C2#3509312yes—
664C2#3509412yes—
665C2#3511812yes—
666C2#3514911yescharacteristic
667C2#3515112yes—
668C2C2×#284811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2855C2semidirect
3. C2×#2627C2semidirect
5. #14893C2semidirect
6. #14891C2semidirect
7. #14887C2semidirect
8. #14705C2semidirect
9. #14685C2semidirect
11. #14339C2semidirect
13. #14263C2semidirect
31. C2×#400C22semidirect
33. C2×#398C22semidirect
35. C2×#395C22semidirect
37. C2×#391C22semidirect
38. C2×#389C22semidirect
39. C2×#387C22semidirect
52. #1444C22semidirect
53. #1442C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. C4⋊#95C22semidirect
63. #1434C22semidirect
77. C24×C4Q8semidirect

Maximal subgroups

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

subgroupindex
C23×#1172
C2×#28552
C2×#26272
C2×#26262
#148932
#148912
#148872
#147052
#146852
#146812
#143392
#143382
#142632
#142622