Finite Group Databasebeta

← #4414798

Subgroups — #4414798

Order 512.

2974 conjugacy classes of subgroups.

The subgroup lattice has 746 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic, complemented
2C2×#2297C211yesmaximal, characteristic
3#17080C211yesmaximal, characteristic, complemented
4#17078C211yesmaximal, characteristic, complemented
5#17077C211yesmaximal, characteristic, complemented
6#17069C211yesmaximal, characteristic, complemented
7#16397C214yesmaximal, complemented
8#16395C212yesmaximal, complemented
9#16263C212yesmaximal, complemented
10#15114C212yesmaximal, complemented
11#15050C212yesmaximal, complemented
12#14978C214yesmaximal, complemented
13#14969C212yesmaximal, complemented
14#14889C212yesmaximal
15#14685C211yesmaximal, characteristic, complemented
16#14626C212yesmaximal, complemented
17#14624C212yesmaximal, complemented
18C23×#41C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2855—28no—
23C2×#397C2212yes—
24C2×#395C2212yes—
25C2×#394C2212yes—
26C2×#394C2214yes—
27C2×#393C2212yes—
28#2297—28no—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#387C2212yes—
38C2×#387C2212yes—
39C2×#387C2214yes—
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#385C2212yes—
43C2×#385C2214yes—
44C2×#381C2212yes—
45C2×#381C2212yes—
46C2×#380C2211yescharacteristic
47C2×#380C2212yes—
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50#1446C2212yes—
51#1446C2212yes—
52#1445C2212yes—
53#1445C2214yes—
54#1444C2212yes—
55#1444C2212yes—
56#1444C2212yes—
57#1443C2214yes—
58#1443C2214yes—
59#1442C2212yes—
60#1441C2212yes—
61#1441C2212yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2212yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2212yes—
75#1438C2214yes—
76#1438C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2211yescharacteristic
80#1434C2214yes—
81#1434C2212yes—
82#1434C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2214yes—
85C22×#41—24no—
86C22×#41—24no—
87C22×#41—24no—
88C22×#41—44no—
89C22×#41—44no—
90C23×Q8C2311yescharacteristic
91C23×Q8C2311yescharacteristic
92C23×D4C2312yes—
93C23×D4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C24×C4C2312yes—
96C24×C4C2311yescharacteristic
97C24×C4C2311yescharacteristic
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—44no—
100C2×#117—24no—
101C2×#117—44no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C42—44no—
127C22×C42D418yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#397—44no—
134#395—28no—
135#394—28no—
136#394—48no—
137#393—44no—
138#391—24no—
139#391—24no—
140#391—24no—
141#391—24no—
142#390—24no—
143#390—24no—
144#389—24no—
145#389—24no—
146#387—28no—
147#387—28no—
148#387—48no—
149#386—28no—
150#386—28no—
151#385—28no—
152#385—48no—
153#381—28no—
154#381—44no—
155#380—24no—
156#380—44no—
157C4×C22⋊C4—28no—
158C4×C22⋊C4—28no—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C25C2412yes—
188C25C2411yescharacteristic
189C2×#41—232no—
190C2×#41—432no—
191C2×#41—432no—
192C2×#41—416no—
193C2×#41—416no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—42no—
200C22×Q8—42no—
201C22×Q8—42no—
202C22×Q8—42no—
203C22×Q8#4114yes—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—28no—
207C22×D4—22no—
208C22×D4—24no—
209C22×D4—22no—
210C22×D4—42no—
211C22×D4—42no—
212C22×D4—44no—
213C22×D4#4114yes—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—22no—
224C23×C4—24no—
225C23×C4—22no—
226C23×C4—28no—
227C23×C4—41no—
228C23×C4—41no—
229C23×C4—42no—
230C23×C4—44no—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C22⋊Q8—416no—
250#117—416no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C42—24no—
314C2×C42—28no—
315C2×C42—416no—
316C2×C42—42no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42C2×D414yes—
321C2×C42C2×D414yes—
322C2×C42C2×D414yes—
323C2×C42C2×D414yes—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352C24—21no—
353C24—21no—
354C24—21no—
355C24—22no—
356C24—22no—
357C24—21no—
358C24—21no—
359C24—21no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—24no—
371C24C22×D412yes—
372C24C2×#4114yes—
373C24H(5,𝔽2)14yes—
374C24C2511yescharacteristic
375#41—4128no—
376C2×Q8—28no—
377C2×Q8—48no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—48no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—84no—
385C2×Q8—84no—
386C2×D4—216no—
387C2×D4—28no—
388C2×D4—48no—
389C2×D4—416no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—48no—
393C2×D4—416no—
394C2×D4—48no—
395C2×D4—48no—
396C22×C4—216no—
397C22×C4—216no—
398C22×C4—28no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—216no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—416no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—416no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C2×#4114yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4H(5,𝔽2)14yes—
494C22×C4H(5,𝔽2)12yes—
495C22×C42−1+414yes—
496C22×C42−1+412yes—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—416no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4—88no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—416no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—88no—
518C22⋊C4—88no—
519C22⋊C4—88no—
520C22⋊C4—88no—
521C42—44no—
522C42—44no—
523C42—44no—
524C42—44no—
525C42—48no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—28no—
540C23—24no—
541C23—24no—
542C23—28no—
543C23—41no—
544C23—41no—
545C23—42no—
546C23—41no—
547C23—41no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—42no—
561C23—42no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23#53414yes—
572C23C23×D411yescharacteristic
573C23C22×#4111yescharacteristic
574C23C22×#4111yescharacteristic
575C23C2×H(5,𝔽2)12yes—
576C23C2×H(5,𝔽2)12yes—
577C23C2×H(5,𝔽2)11yescharacteristic
578C23C2×2−1+412yes—
579C23C2×2−1+411yescharacteristic
580C23#58511yescharacteristic
581C23#58511yescharacteristic
582C23#58512yes—
583Q8—416no—
584Q8—88no—
585D4—416no—
586D4—432no—
587C2×C4—232no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—28no—
612C2×C4—416no—
613C2×C4—416no—
614C2×C4—432no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4#52514yes—
669C2×C4#53514yes—
670C2×C4#53614yes—
671C2×C4#53714yes—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—48no—
686C22—48no—
687C22—44no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—84no—
693C22—84no—
694C22C2×#52511yescharacteristic
695C22C2×#53411yescharacteristic
696C22C2×#53511yescharacteristic
697C22C2×#53611yescharacteristic
698C22C2×#53711yescharacteristic
699C22#344911yescharacteristic
700C22#344911yescharacteristic
701C22#349012yes—
702C22#349012yes—
703C22#351411yescharacteristic
704C22#351511yescharacteristic
705C22#351512yes—
706C22#351711yescharacteristic
707C22#351811yescharacteristic
708C22#351812yes—
709C22#352912yes—
710C22#352912yes—
711C22#353412yes—
712C22#353512yes—
713C22#353512yes—
714C22#353512yes—
715C22#354812yes—
716C22#355012yes—
717C4—24no—
718C4—24no—
719C4—24no—
720C4—24no—
721C4—42no—
722C4—42no—
723C4—44no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4—82no—
729C4—84no—
730C4—82no—
731C4—84no—
732C2—44no—
733C2—84no—
734C2#3505011yescharacteristic
735C2#3518911yescharacteristic
736C2#3520011yescharacteristic
737C2#3668711yescharacteristic
738C2#3744711yescharacteristic
739C2#3745211yescharacteristic
740C2#3764011yescharacteristic
741C2#4628812yes—
742C2#4629312yes—
743C2#4631512yes—
744C2#4631612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2855C2semidirect
3. #17080C2semidirect
4. #17078C2semidirect
5. #17077C2semidirect
6. #17069C2semidirect
7. #16397C2semidirect
8. #16395C2semidirect
9. #16263C2semidirect
10. #15114C2semidirect
11. #15050C2semidirect
12. #14978C2semidirect
13. #14969C2semidirect
15. #14685C2semidirect
16. #14626C2semidirect
17. #14624C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×#22972
#170802
#170782
#170772
#170692
#163972
#163952
#162632
#151142
#150502
#149782
#149692
#148892
#146852
#146262
#146242