Finite Group Databasebeta

← #618704

Subgroups — #618704

Order 512.

5434 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#124C211yesmaximal, characteristic
2C2×#2561C211yesmaximal, characteristic, complemented
3C2×#2434C211yesmaximal, characteristic, complemented
4C2×#2626C211yesmaximal, characteristic, complemented
5#15785C214yesmaximal, complemented
6#15761C214yesmaximal, complemented
7#15321C218yesmaximal, complemented
8#14918C212yesmaximal, complemented
9#14724C212yesmaximal
10#14705C212yesmaximal, complemented
11#14262C212yesmaximal, complemented
12#13922C211yesmaximal, characteristic, complemented
13#13903C211yesmaximal, characteristic, complemented
14#13775C211yesmaximal, characteristic, complemented
15C24×Q8C2211yescharacteristic
16C24×D4C2211yescharacteristic
17C22×C4⋊Q8C2212yescomplemented
18C22×#124—28no—
19C22×#124—28no—
20C22×#124—28no—
21C22×#124C2214yes—
22C22×#124C2214yes—
23C22×C22⋊Q8C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25C23×C22⋊C4C2211yescharacteristic
26C23×C22⋊C4C2211yescharacteristic
27C23×C22⋊C4C2211yescharacteristic
28C23×C42C2211yescharacteristic
29#2626—28no—
30#2561—28no—
31#2434—28no—
32C2×#399C2212yescomplemented
33C2×#397C2212yescomplemented
34C2×#396C2214yescomplemented
35C2×#395C2214yescomplemented
36C2×#394C2218yes—
37C2×#393C2212yescomplemented
38C2×#393C2212yescomplemented
39C2×#391C2212yescomplemented
40C2×#391C2214yescomplemented
41C2×#388C2214yescomplemented
42C2×#385C2212yes—
43C2×#385C2212yes—
44C2×#385C2212yes—
45C2×#381C2212yes—
46C2×#381C2212yes—
47C2×#380C2211yescharacteristic
48C2×#379C2212yes—
49C2×#379C2212yes—
50C2×#379C2212yes—
51C2×#379C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54#1446C2214yescomplemented
55#1440C2218yescomplemented
56#1439C2218yescomplemented
57#1439C2214yescomplemented
58#1439C2218yescomplemented
59#1439C2218yescomplemented
60#1438C2218yes—
61#1437C2214yescomplemented
62#1437C2214yescomplemented
63#1437C2214yescomplemented
64#1436C2218yescomplemented
65#1436C2218yescomplemented
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C26C2311yescharacteristic
70C26C2311yescharacteristic
71C23×Q8—24no—
72C23×Q8—24no—
73C23×Q8—48no—
74C23×Q8D418yes—
75C23×Q8C2312yes—
76C23×Q8C2312yes—
77C23×Q8C2311yescharacteristic
78C23×D4—24no—
79C23×D4—24no—
80C23×D4—24no—
81C23×D4—48no—
82C23×D4C2314yes—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—48no—
90C2×#124—2128no—
91C2×#124—216no—
92C2×#124—216no—
93C2×#124—216no—
94C2×#124—216no—
95C2×#124—216no—
96C2×#124—216no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—24no—
114C22×C22⋊C4—24no—
115C22×C22⋊C4—24no—
116C22×C22⋊C4D418yes—
117C22×C22⋊C4D418yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2318yes—
131C22×C22⋊C4C2318yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C42—24no—
135C22×C42—24no—
136C22×C42—24no—
137C22×C42C2314yes—
138C22×C42C2312yes—
139C22×C42C2312yes—
140#399—28no—
141#397—28no—
142#396—216no—
143#395—216no—
144#394—416no—
145#393—28no—
146#393—28no—
147#391—28no—
148#391—48no—
149#388—216no—
150#385—28no—
151#385—28no—
152#385—28no—
153#381—28no—
154#381—28no—
155#380—24no—
156#379—28no—
157#379—28no—
158#379—28no—
159#379—28no—
160C4×C22⋊C4—24no—
161C4×C22⋊C4—24no—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2318yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2318yes—
172C2×#95C2318yes—
173C2×#95C2318yes—
174C25—22no—
175C25—22no—
176C25—22no—
177C25—22no—
178C25—24no—
179C25—24no—
180C25—24no—
181C25—24no—
182C25—44no—
183C25—44no—
184C25—44no—
185C25—44no—
186C25C2×D414yes—
187C25C2×D414yes—
188C25C2412yes—
189C25C2412yes—
190C25C2411yescharacteristic
191C22×Q8—232no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—28no—
200C22×Q8—28no—
201C22×Q8—44no—
202C22×Q8—44no—
203C22×Q8—44no—
204C22×Q8—432no—
205C22×Q8—432no—
206C22×Q8—432no—
207C22×Q8C2×D418yes—
208C22×Q8C2×D418yes—
209C22×D4—232no—
210C22×D4—28no—
211C22×D4—28no—
212C22×D4—28no—
213C22×D4—216no—
214C22×D4—48no—
215C22×D4—432no—
216C22×D4—432no—
217C22×D4—432no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—28no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2414yes—
242C23×C4C2414yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2414yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250C4⋊Q8—432no—
251#124—2128no—
252#124—2128no—
253C22⋊Q8—416no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—216no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—48no—
266C2×C4⋊C4—48no—
267C2×C4⋊C4—44no—
268C2×C22⋊C4—232no—
269C2×C22⋊C4—232no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—232no—
275C2×C22⋊C4—232no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
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—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—216no—
302C2×C22⋊C4—216no—
303C2×C22⋊C4—48no—
304C2×C22⋊C4—48no—
305C2×C22⋊C4—48no—
306C2×C22⋊C4—48no—
307C2×C22⋊C4—48no—
308C2×C22⋊C4—48no—
309C2×C22⋊C4—48no—
310C2×C22⋊C4—48no—
311C2×C22⋊C4C2×D418yes—
312C2×C22⋊C4C2×D418yes—
313C2×C22⋊C4C2×D418yes—
314C2×C22⋊C4C2×D418yes—
315C2×C22⋊C4#4118yes—
316C2×C42—232no—
317C2×C42—28no—
318C2×C42—28no—
319C2×C42—28no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—44no—
323C2×C42—44no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—216no—
333#95—216no—
334#95—48no—
335#95—48no—
336C24—24no—
337C24—24no—
338C24—28no—
339C24—28no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—28no—
347C24—28no—
348C24—24no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—28no—
353C24—28no—
354C24—28no—
355C24—216no—
356C24—48no—
357C24—48no—
358C24—48no—
359C24—48no—
360C24—48no—
361C24—48no—
362C24—48no—
363C24—48no—
364C24—48no—
365C24—48no—
366C24—48no—
367C24—48no—
368C24—44no—
369C24—44no—
370C24—48no—
371C24—48no—
372C24—48no—
373C24—816no—
374C24—816no—
375C24—816no—
376C24—816no—
377C24C22×D414yes—
378C24C22×D414yes—
379C24C22×D412yes—
380C24C22×D412yes—
381C24C22×D412yes—
382C24C2×#4114yes—
383C242−1+414yes—
384C24C2511yescharacteristic
385C2×Q8—216no—
386C2×Q8—216no—
387C2×Q8—216no—
388C2×Q8—216no—
389C2×Q8—48no—
390C2×Q8—48no—
391C2×Q8—416no—
392C2×Q8—416no—
393C2×Q8—416no—
394C2×Q8—416no—
395C2×Q8—4128no—
396C2×Q8—48no—
397C2×Q8—44no—
398C2×Q8—84no—
399C2×Q8—84no—
400C2×D4—232no—
401C2×D4—432no—
402C2×D4—432no—
403C2×D4—4128no—
404C2×D4—416no—
405C2×D4—416no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—28no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—216no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—216no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—28no—
448C22×C4—24no—
449C22×C4—28no—
450C22×C4—28no—
451C22×C4—28no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—48no—
464C22×C4#120116yes—
465C22×C4C22×D414yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D414yes—
475C22×C4C2×#4118yes—
476C22×C4H(5,𝔽2)18yes—
477C4⋊C4—416no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—416no—
481C4⋊C4—84no—
482C4⋊C4—88no—
483C22⋊C4—232no—
484C22⋊C4—232no—
485C22⋊C4—216no—
486C22⋊C4—216no—
487C22⋊C4—216no—
488C22⋊C4—216no—
489C22⋊C4—216no—
490C22⋊C4—416no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—816no—
495C22⋊C4—816no—
496C22⋊C4—88no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C42—232no—
500C42—48no—
501C42—84no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—216no—
509C23—216no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—416no—
521C23—416no—
522C23—416no—
523C23—416no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—42no—
528C23—44no—
529C23—416no—
530C23—416no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—48no—
540C23—48no—
541C23—48no—
542C23—48no—
543C23—416no—
544C23—48no—
545C23—48no—
546C23—816no—
547C23—816no—
548C23—832no—
549C23—816no—
550C23—816no—
551C23—832no—
552C23—832no—
553C23—832no—
554C23C2×#12014yes—
555C23#53214yes—
556C23#53614yes—
557C23#54718yes—
558C23#54818yes—
559C23C23×D411yescharacteristic
560C23C23×D411yescharacteristic
561C23C23×D411yescharacteristic
562C23C22×#4112yes—
563C23C2×H(5,𝔽2)14yes—
564C23C2×2−1+412yes—
565C23#58514yes—
566Q8—432no—
567Q8—432no—
568Q8—88no—
569D4—464no—
570C2×C4—232no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—28no—
589C2×C4—216no—
590C2×C4—416no—
591C2×C4—416no—
592C2×C4—416no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—416no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—48no—
604C2×C4—42no—
605C2×C4—42no—
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—48no—
619C2×C4—48no—
620C2×C4—48no—
621C2×C4—48no—
622C2×C4C2×#12014yes—
623C2×C4C2×#12014yes—
624C2×C4#53418yes—
625C22—24no—
626C22—28no—
627C22—28no—
628C22—216no—
629C22—24no—
630C22—216no—
631C22—44no—
632C22—44no—
633C22—48no—
634C22—48no—
635C22—48no—
636C22—48no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—832no—
642C22—832no—
643C22—84no—
644C22—84no—
645C22—88no—
646C22—88no—
647C22—88no—
648C22#243218yes—
649C22C22×#12011yescharacteristic
650C22C2×#53211yescharacteristic
651C22C2×#53412yes—
652C22C2×#53611yescharacteristic
653C22#344912yes—
654C22C2×#54712yes—
655C22C2×#54812yes—
656C22#348412yes—
657C22#348412yes—
658C22#348512yes—
659C22#348712yes—
660C22#351418yes—
661C22#356118yes—
662C4—28no—
663C4—24no—
664C4—24no—
665C4—44no—
666C4—44no—
667C4—44no—
668C4—48no—
669C4—82no—
670C4—84no—
671C4—84no—
672C4—82no—
673C2—28no—
674C2—84no—
675C2—84no—
676C2C2×#243211yescharacteristic
677C2#3498011yescharacteristic
678C2#3514911yescharacteristic
679C2#3546612yes—
680C2#3696612yes—
681C2#3804414yes—
682C2#3912314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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×#2561C2semidirect
3. C2×#2434C2semidirect
4. C2×#2626C2semidirect
5. #15785C2semidirect
6. #15761C2semidirect
7. #15321C2semidirect
8. #14918C2semidirect
10. #14705C2semidirect
11. #14262C2semidirect
12. #13922C2semidirect
13. #13903C2semidirect
14. #13775C2semidirect
17. C22×C4⋊Q8C22semidirect
32. C2×#399C22semidirect
33. C2×#397C22semidirect
34. C2×#396C22semidirect
35. C2×#395C22semidirect
37. C2×#393C22semidirect
38. C2×#393C22semidirect
39. C2×#391C22semidirect
40. C2×#391C22semidirect
41. C2×#388C22semidirect
54. #1446C22semidirect
55. #1440C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
61. #1437C22semidirect
62. #1437C22semidirect
63. #1437C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C23×#1242
C2×#25612
C2×#24342
C2×#26262
#157852
#157612
#153212
#149182
#147242
#147052
#142622
#139222
#139032
#137752