Finite Group Databasebeta

← #4435292

Subgroups — #4435292

Order 512.

2481 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×#2821C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic, complemented
3C2×#2604C211yesmaximal, characteristic
4C2×#2642C211yesmaximal, characteristic, complemented
5C2×#2634C211yesmaximal, characteristic, complemented
6C2×#2664C211yesmaximal, characteristic
7#17064C212yesmaximal
8#16813C212yesmaximal, complemented
9#16813C212yesmaximal, complemented
10#16812C212yesmaximal
11#15808C214yesmaximal
12#15804C214yesmaximal, complemented
13#15173C212yesmaximal, complemented
14#15153C212yesmaximal, complemented
15#15065C212yesmaximal
16#15065C212yesmaximal
17#13806C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#126C2211yescharacteristic
20C22×#125C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22#2821—28no—
23#2664—28no—
24#2642—28no—
25#2634—28no—
26#2604—28no—
27#2567—28no—
28C2×#400C2211yescharacteristic
29C2×#400C2212yes—
30C2×#400C2212yes—
31C2×#400C2211yescharacteristic
32C2×#400C2212yes—
33C2×#400C2212yes—
34C2×#400C2212yes—
35C2×#400C2214yes—
36C2×#399C2211yescharacteristic
37C2×#399C2211yescharacteristic
38C2×#399C2212yes—
39C2×#398C2211yescharacteristic
40C2×#398C2212yes—
41C2×#398C2212yes—
42C2×#398C2212yes—
43C2×#397C2211yescharacteristic
44C2×#397C2212yes—
45C2×#396C2212yes—
46C2×#395C2212yes—
47C2×#395C2211yescharacteristic
48C2×#395C2212yes—
49C2×#395C2212yes—
50C2×#395C2212yes—
51C2×#393C2211yescharacteristic
52C2×#393C2212yes—
53C2×#391C2212yes—
54C2×#391C2212yes—
55C2×#391C2212yes—
56C2×#389C2212yes—
57C2×#389C2212yes—
58C2×#389C2214yes—
59C2×#388C2212yes—
60C2×#388C2212yes—
61C2×#388C2212yes—
62C2×#387C2214yes—
63C2×#387C2212yes—
64C2×#384C2212yes—
65C2×#383C2211yescharacteristic
66C2×C4×C4⋊C4C2211yescharacteristic
67C2×#376C2211yescharacteristic
68#1445C2214yes—
69#1445C2214yes—
70#1445C2214yes—
71#1445C2214yes—
72#1445C2214yes—
73#1445C2214yes—
74#1444C2214yes—
75#1444C2214yes—
76#1444C2214yes—
77#1439C2214yes—
78#1439C2214yes—
79#1439C2214yes—
80#1437C2214yes—
81#1436C2214yes—
82C4⋊#95C2214yes—
83C4⋊#95C2214yes—
84C4⋊#95C2214yes—
85#1434C2212yes—
86#1434C2212yes—
87C4×#95C2212yes—
88C4×#95C2212yes—
89C23×Q8C2311yescharacteristic
90C23×Q8C2312yes—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—24no—
94C2×#126—24no—
95C2×#126—24no—
96C2×#126—24no—
97C2×#125—24no—
98C2×#125—24no—
99C2×#125—24no—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C42C2312yes—
132C22×C42C2314yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136#400—28no—
137#400—24no—
138#400—28no—
139#400—28no—
140#400—216no—
141#400—28no—
142#400—28no—
143#400—24no—
144#399—28no—
145#399—24no—
146#399—24no—
147#398—28no—
148#398—28no—
149#398—28no—
150#398—24no—
151#397—28no—
152#397—24no—
153#396—28no—
154#395—28no—
155#395—28no—
156#395—28no—
157#395—24no—
158#395—28no—
159#393—28no—
160#393—24no—
161#391—28no—
162#391—28no—
163#391—28no—
164#389—216no—
165#389—28no—
166#389—28no—
167#388—28no—
168#388—28no—
169#388—28no—
170#387—28no—
171#387—216no—
172#384—28no—
173#383D418yes—
174C4×C4⋊C4D418yes—
175#376D418yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2312yes—
199C2×#95C2311yescharacteristic
200C2×#95C2311yescharacteristic
201C2×#95C2311yescharacteristic
202C2×#95C2312yes—
203C2×#95C2311yescharacteristic
204C2×#95C2312yes—
205C2×#95C2312yes—
206C25C2411yescharacteristic
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—24no—
211C22×Q8—24no—
212C22×Q8—24no—
213C22×Q8—24no—
214C22×Q8—42no—
215C22×Q8—44no—
216C22×Q8—44no—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C4⋊Q8—416no—
234#126—416no—
235#125—416no—
236C4×Q8—416no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
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—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4C2×D418yes—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—28no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42C2×D418yes—
326C2×C42C2×D418yes—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329C2×C42C2×D414yes—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—28no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—44no—
351#95—44no—
352#95C2×D418yes—
353#95C2×D414yes—
354#95C2×D414yes—
355#95C2×D414yes—
356#95C2×D414yes—
357#95C2×D418yes—
358#95C2×D418yes—
359#95C2×D418yes—
360C24—21no—
361C24—21no—
362C24—21no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—24no—
367C24—22no—
368C24C2511yescharacteristic
369C2×Q8—28no—
370C2×Q8—48no—
371C2×Q8—48no—
372C2×Q8—44no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—44no—
376C2×Q8—48no—
377C2×Q8—88no—
378C2×Q8—88no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4C22×D414yes—
449C22×C4C22×D418yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D414yes—
462C22×C42−1+414yes—
463C22×C42−1+414yes—
464C22×C42−1+414yes—
465C4⋊C4—216no—
466C4⋊C4—48no—
467C4⋊C4—416no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—84no—
482C4⋊C4—88no—
483C4⋊C4—88no—
484C4⋊C4—88no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—84no—
492C22⋊C4—88no—
493C42—48no—
494C42—48no—
495C42—48no—
496C42—88no—
497C42#127116yes—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23C23×D411yescharacteristic
515C23C23×D411yescharacteristic
516C23C23×D411yescharacteristic
517C23C2×2−1+412yes—
518C23C2×2−1+412yes—
519C23C2×2−1+412yes—
520C23#58514yes—
521C23#58512yes—
522Q8—416no—
523Q8—816no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4C2×#12714yes—
607C2×C4C2×#12714yes—
608C2×C4C2×#12714yes—
609C2×C4#53618yes—
610C22—81no—
611C22—82no—
612C22—81no—
613C22—82no—
614C22—82no—
615C22—84no—
616C22—82no—
617C22—81no—
618C22C22×#12711yescharacteristic
619C22C2×#53612yes—
620C22#344912yes—
621C22#348511yescharacteristic
622C22#348611yescharacteristic
623C22#348912yes—
624C22#348912yes—
625C22#349911yescharacteristic
626C22#349911yescharacteristic
627C22#349912yes—
628C22#350012yes—
629C22#350012yes—
630C22#355314yes—
631C22#355514yes—
632C22#355514yes—
633C22#356414yes—
634C4—24no—
635C4—24no—
636C4—28no—
637C4—24no—
638C4—44no—
639C4—44no—
640C4—44no—
641C4—48no—
642C4—44no—
643C4—82no—
644C4—81no—
645C4—84no—
646C4—81no—
647C4—82no—
648C4—81no—
649C4—82no—
650C2—161no—
651C2#3531711yescharacteristic
652C2#3532911yescharacteristic
653C2#3563411yescharacteristic
654C2#4137412yes—
655C2#4230512yes—
656C2#4231512yes—
657C2#4279312yes—
658C2#5133114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2567C2semidirect
4. C2×#2642C2semidirect
5. C2×#2634C2semidirect
8. #16813C2semidirect
9. #16813C2semidirect
12. #15804C2semidirect
13. #15173C2semidirect
14. #15153C2semidirect
17. #13806C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28212
C2×#25672
C2×#26042
C2×#26422
C2×#26342
C2×#26642
#170642
#168132
#168122
#158082
#158042
#151732
#151532
#150652
#138062