Finite Group Databasebeta

← #545392

Subgroups — #545392

Order 512.

2578 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3C2×#2319C211yesmaximal, characteristic
4#14625C212yesmaximal, complemented
5#14409C214yesmaximal, complemented
6#14381C214yesmaximal, complemented
7#14323C212yesmaximal, complemented
8#14305C212yesmaximal, complemented
9#14277C211yesmaximal, characteristic, complemented
10#14231C212yesmaximal, complemented
11#14141C212yesmaximal
12#14133C212yesmaximal, complemented
13#14006C212yesmaximal, complemented
14#14002C212yesmaximal
15#13906C211yesmaximal, characteristic
16#13850C211yesmaximal, characteristic, complemented
17#13804C211yesmaximal, characteristic, complemented
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#2580C4116yescomplemented
21C2×#399C2212yes—
22C2×#393C2212yes—
23#2355—28no—
24#2319—28no—
25C2×#391C2212yes—
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#388C2212yes—
35C2×#388C2212yes—
36C2×#388C2214yes—
37C2×#387C2214yes—
38C2×#387C2214yes—
39C2×#385C2214yes—
40C2×#384C2211yescharacteristic
41C2×#382C2212yes—
42C2×#382C2211yescharacteristic
43C2×#381C2212yes—
44C2×#380C2212yes—
45C2×#380C2214yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×#376C2211yescharacteristic
52#1446C2211yescharacteristic
53#1446C2211yescharacteristic
54#1445C2211yescharacteristic
55#1445C2211yescharacteristic
56#1444C2212yes—
57#1444C2212yes—
58#1443C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2212yes—
62#1442C2212yes—
63#1441C2212yes—
64#1441C2212yes—
65#1441C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1439C2214yes—
70#1439C2212yes—
71#1439C2212yes—
72#1437C2214yes—
73#1437C2214yes—
74#1437C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C4×Q8—24no—
90C2×C4×Q8—24no—
91C2×C4×Q8C2×C418yes—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×C4×D4C2×C418yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2314yes—
123C22×C42C2314yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126#399C2×C4116yes—
127#393C2×C4116yes—
128#391—28no—
129#390—24no—
130#390—24no—
131#390—44no—
132#390C2×C4116yes—
133#389—24no—
134#389—24no—
135#389—44no—
136#389C2×C4116yes—
137#388—28no—
138#388—28no—
139#388—48no—
140#387—216no—
141#387—48no—
142#385C2×C4132yes—
143#384—24no—
144#382—28no—
145#382C2×C418yes—
146#381—28no—
147#380—28no—
148#380—48no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—44no—
153C4×C4⋊C4D418yes—
154#376—24no—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2311yescharacteristic
167C2×#95C2311yescharacteristic
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
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×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C25C2412yes—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8C22×C414yes—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4C22×C414yes—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C22×C418yes—
204C23×C4C2411yescharacteristic
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C4×Q8—232no—
221C4×D4—232no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4C22×C414yes—
254C2×C4⋊C4C22×C418yes—
255C2×C4⋊C4C22×C414yes—
256C2×C4⋊C4C22×C418yes—
257C2×C4⋊C4C22×C4116yes—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D414yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4C22×C418yes—
281C2×C22⋊C4C22×C4116yes—
282C2×C22⋊C4C22×C4116yes—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—28no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42C22×C418yes—
302C2×C42C22×C414yes—
303C2×C42C2×D418yes—
304C2×C42C2×D414yes—
305C2×C42#4114yes—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—22no—
316#95—22no—
317#95—28no—
318#95—24no—
319#95—28no—
320#95—24no—
321#95—42no—
322#95—42no—
323#95—42no—
324#95—42no—
325#95—44no—
326#95—42no—
327#95—44no—
328#95—42no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95C22×C418yes—
334#95C22×C418yes—
335#95C22×C4116yes—
336#95C2×D418yes—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—24no—
347C24C23×C414yes—
348C24C2511yescharacteristic
349C2×Q8—28no—
350C2×Q8—48no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×Q8—48no—
354C2×D4—28no—
355C2×D4—28no—
356C2×D4—28no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—216no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—416no—
437C22×C4—88no—
438C22×C4—84no—
439C22×C4—84no—
440C22×C4C23×C418yes—
441C22×C4C23×C414yes—
442C22×C4C23×C414yes—
443C22×C4C23×C414yes—
444C22×C4C23×C412yes—
445C22×C4C23×C412yes—
446C22×C4C23×C412yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4H(5,𝔽2)14yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C42−1+412yes—
458C22×C42−1+412yes—
459C22×C42−1+412yes—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—216no—
463C4⋊C4—216no—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—416no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—84no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C4⋊C4—88no—
474C22⋊C4—216no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—84no—
478C22⋊C4—84no—
479C22⋊C4—84no—
480C22⋊C4—84no—
481C22⋊C4—88no—
482C22⋊C4—88no—
483C42—216no—
484C42—48no—
485C42—44no—
486C42—44no—
487C42—48no—
488C42—88no—
489C42—88no—
490C42C4×D4116yes—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—28no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23C24×C411yescharacteristic
515C23C23×D411yescharacteristic
516C23C22×#4111yescharacteristic
517C23C2×H(5,𝔽2)11yescharacteristic
518C23C2×H(5,𝔽2)12yes—
519C23C2×2−1+411yescharacteristic
520C23C2×2−1+411yescharacteristic
521C23C2×2−1+411yescharacteristic
522C23#58512yes—
523C23#58512yes—
524C23#58512yes—
525Q8—416no—
526D4—416no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—816no—
613C2×C4C2×C4×D414yes—
614C2×C4C2×C4×D414yes—
615C2×C4C2×C4×D414yes—
616C2×C4#51914yes—
617C2×C4#52014yes—
618C2×C4#53614yes—
619C2×C4#53714yes—
620C2×C4#54114yes—
621C2×C4#54214yes—
622C22—44no—
623C22—44no—
624C22—48no—
625C22—48no—
626C22—44no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—84no—
632C22—84no—
633C22C22×C4×D411yescharacteristic
634C22C2×#51911yescharacteristic
635C22C2×#52011yescharacteristic
636C22#342812yes—
637C22C4×H(5,𝔽2)12yes—
638C22C2×#53611yescharacteristic
639C22C2×#53711yescharacteristic
640C22C2×#54111yescharacteristic
641C22C2×#54211yescharacteristic
642C22#348511yescharacteristic
643C22#348711yescharacteristic
644C22#349312yes—
645C22#349412yes—
646C22#349811yescharacteristic
647C22#350011yescharacteristic
648C22#355412yes—
649C22#355412yes—
650C22#355712yes—
651C22#355712yes—
652C22#355812yes—
653C22#355912yes—
654C22#356212yes—
655C22#356512yes—
656C4—24no—
657C4—24no—
658C4—44no—
659C4—44no—
660C4—44no—
661C4—48no—
662C4—44no—
663C4—44no—
664C4—44no—
665C4—81no—
666C4—81no—
667C4—84no—
668C4—81no—
669C4—81no—
670C4—82no—
671C4#232318yes—
672C2—84no—
673C2C2×#232311yescharacteristic
674C2#3420111yescharacteristic
675C2#3427511yescharacteristic
676C2#3448912yes—
677C2#3450012yes—
678C2#4264511yescharacteristic
679C2#4265111yescharacteristic
680C2#4268712yes—
681C2#4269312yes—
682C2#4287511yescharacteristic
683C2#4288111yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
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×#2355C2semidirect
4. #14625C2semidirect
5. #14409C2semidirect
6. #14381C2semidirect
7. #14323C2semidirect
8. #14305C2semidirect
9. #14277C2semidirect
10. #14231C2semidirect
12. #14133C2semidirect
13. #14006C2semidirect
16. #13850C2semidirect
17. #13804C2semidirect
20. #2580C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
C2×#23552
C2×#23192
#146252
#144092
#143812
#143232
#143052
#142772
#142312
#141412
#141332
#140062
#140022
#139062
#138502
#138042