Finite Group Databasebeta

← #4930476

Subgroups — #4930476

Order 512.

2642 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic
2C2×#2565C211yesmaximal, characteristic
3C2×#2584C211yesmaximal, characteristic
4C2×#2276C211yesmaximal, characteristic
5#15166C214yesmaximal
6#15133C214yesmaximal
7#14903C212yesmaximal
8#14901C212yesmaximal
9#14872C212yesmaximal
10#14818C212yesmaximal
11#14283C211yesmaximal, characteristic
12#14006C212yesmaximal
13#14005C212yesmaximal
14#13935C212yesmaximal
15#13935C212yesmaximal
16#13910C211yesmaximal, characteristic
17#13850C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22#2584—28no—
23#2582—28no—
24#2565—28no—
25C2×#398C2212yes—
26C2×#398C2214yes—
27C2×#398C2212yes—
28C2×#398C2212yes—
29C2×#395C2212yes—
30C2×#391C2212yes—
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#391C2212yes—
34C2×#389C2214yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2214yes—
42C2×#387C2212yes—
43C2×#387C2212yes—
44C2×#384C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#384C2212yes—
47C2×#382C2211yescharacteristic
48#2276—28no—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2214yes—
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×#376C2211yescharacteristic
55C2×#376C2212yes—
56#1446C2211yescharacteristic
57#1446C2211yescharacteristic
58#1445C2211yescharacteristic
59#1445C2211yescharacteristic
60#1444C2212yes—
61#1444C2212yes—
62#1444C2214yes—
63#1442C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2212yes—
67#1441C2212yes—
68#1441C2212yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1441C2212yes—
74#1441C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2214yes—
78C4⋊#95C2214yes—
79C4⋊#95C2214yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83#1434C2214yes—
84#1434C2214yes—
85C4×#95C2212yes—
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C2×#125—24no—
90C2×#125—24no—
91C2×#125—24no—
92C2×C4×Q8—28no—
93C2×C4×Q8—28no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8D418yes—
100C2×C4×Q8Q8116yescomplemented
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2314yes—
129C22×C42C2314yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132#398—28no—
133#398—28no—
134#398—216no—
135#398—28no—
136#395—28no—
137#391—24no—
138#391—24no—
139#391—28no—
140#391—28no—
141#389—28no—
142#389—28no—
143#389—28no—
144#389—28no—
145#389—28no—
146#389—28no—
147#389—216no—
148#389—48no—
149#387—28no—
150#387—28no—
151#384—24no—
152#384—24no—
153#384—44no—
154#382—24no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—216no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4D418yes—
160#376—24no—
161#376—28no—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2311yescharacteristic
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—24no—
199C22×Q8—22no—
200C22×Q8—24no—
201C22×Q8—42no—
202C22×Q8C2×D414yes—
203C22×Q8C2×Q818yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221#125—416no—
222C4×Q8—264no—
223C4×Q8—232no—
224C4×Q8—416no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—41no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×D418yes—
286C2×C4⋊C4C2×D418yes—
287C2×C4⋊C4C2×Q8116yes—
288C2×C4⋊C4C2×Q818yes—
289C2×C4⋊C4#4118yes—
290C2×C42—24no—
291C2×C42—28no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—28no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—28no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42—44no—
307C2×C42—44no—
308C2×C42C2×D418yes—
309C2×C42C2×D418yes—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312C2×C42C2×Q818yes—
313C2×C42C2×Q8116yes—
314C2×C42#4114yes—
315#95—22no—
316#95—24no—
317#95—28no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—24no—
322#95—24no—
323#95—24no—
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—28no—
333#95—28no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—42no—
339#95—42no—
340#95—42no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—42no—
345#95—44no—
346#95C2×D418yes—
347C24C2511yescharacteristic
348C2×Q8—28no—
349C2×Q8—28no—
350C2×Q8—216no—
351C2×Q8—28no—
352C2×Q8—216no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8C4⋊Q8116yes—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
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—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D418yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×Q814yes—
449C22×C4C22×Q814yes—
450C22×C4C22×Q818yes—
451C22×C4C22×Q818yes—
452C22×C4C22×Q814yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4118yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4114yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C42−1+412yes—
460C22×C42−1+414yes—
461C4⋊C4—216no—
462C4⋊C4—28no—
463C4⋊C4—232no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—416no—
467C4⋊C4—416no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—84no—
479C4⋊C4—88no—
480C4⋊C4—88no—
481C4⋊C4—88no—
482C4⋊C4C22⋊Q8132yes—
483C42—216no—
484C42—232no—
485C42—216no—
486C42—48no—
487C42—48no—
488C42—44no—
489C42—44no—
490C42—88no—
491C42#121116yes—
492C23C23×D411yescharacteristic
493C23C23×D411yescharacteristic
494C23C23×Q812yes—
495C23C22×#4111yescharacteristic
496C23C22×#4112yes—
497C23C2×H(5,𝔽2)11yescharacteristic
498C23C2×2−1+412yes—
499C23C2×2−1+411yescharacteristic
500C23#58512yes—
501C23#58511yescharacteristic
502C23#58511yescharacteristic
503Q8—232no—
504Q8—416no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—28no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—28no—
518C2×C4—22no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
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—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4C2×#12114yes—
600C2×C4C2×#12114yes—
601C2×C4C2×#12114yes—
602C2×C4C2×C22⋊Q8116yes—
603C2×C4C2×C22⋊Q818yes—
604C2×C4C2×C4⋊Q818yes—
605C2×C4C2×C4⋊Q814yes—
606C2×C4#53214yes—
607C2×C4#53318yes—
608C22C22×#12111yescharacteristic
609C22C22×C22⋊Q812yes—
610C22C22×C4⋊Q811yescharacteristic
611C22C2×#53211yescharacteristic
612C22C2×#53312yes—
613C22#348611yescharacteristic
614C22#348711yescharacteristic
615C22#348811yescharacteristic
616C22#348912yes—
617C22#348911yescharacteristic
618C22#349512yes—
619C22#349611yescharacteristic
620C22#349711yescharacteristic
621C22#350212yes—
622C22#350212yes—
623C22#350312yes—
624C22#350412yes—
625C22#351111yescharacteristic
626C22#351211yescharacteristic
627C22#351312yes—
628C22#351512yes—
629C22#352312yes—
630C22#352412yes—
631C4—28no—
632C4—24no—
633C4—216no—
634C4—24no—
635C4—42no—
636C4—44no—
637C4—42no—
638C4—44no—
639C4—44no—
640C4—44no—
641C4—81no—
642C4—84no—
643C4—82no—
644C4—81no—
645C4—82no—
646C4—82no—
647C4#285718yes—
648C2#3556411yescharacteristic
649C2#3558212yes—
650C2#3564212yes—
651C2#3566111yescharacteristic
652C2#3578612yes—
653C2#3580612yes—
654C2#3585311yescharacteristic
655C2#3593711yescharacteristic
656C2#3594911yescharacteristic
657C2#3595711yescharacteristic
658C2C2×#285711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
100. C2×C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
C2×#25652
C2×#25842
C2×#22762
#151662
#151332
#149032
#149012
#148722
#148182
#142832
#140062
#140052
#139352
#139102
#138502