Finite Group Databasebeta

← #4389779

Subgroups — #4389779

Order 512.

2760 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2820C211yesmaximal, characteristic
2C2×#2545C211yesmaximal, characteristic
3C2×#2581C211yesmaximal, characteristic, complemented
4C2×#2584C211yesmaximal, characteristic, complemented
5C2×#2634C211yesmaximal, characteristic, complemented
6C2×#2632C211yesmaximal, characteristic
7#16813C212yesmaximal, complemented
8#16510C212yesmaximal
9#15825C214yesmaximal, complemented
10#15808C214yesmaximal, complemented
11#15765C214yesmaximal, complemented
12#15765C214yesmaximal, complemented
13#15170C212yesmaximal
14#15143C212yesmaximal, complemented
15#13781C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22#2820—28no—
23#2634—28no—
24#2632—28no—
25#2584—28no—
26#2581—28no—
27#2545—28no—
28C2×#400C2212yes—
29C2×#400C2212yes—
30C2×#400C2212yes—
31C2×#399C2212yes—
32C2×#399C2212yes—
33C2×#399C2214yes—
34C2×#398C2212yes—
35C2×#398C2212yes—
36C2×#397C2212yes—
37C2×#396C2214yes—
38C2×#396C2214yes—
39C2×#395C2212yes—
40C2×#395C2212yes—
41C2×#395C2212yes—
42C2×#395C2212yes—
43C2×#395C2212yes—
44C2×#394C2212yes—
45C2×#394C2212yes—
46C2×#393C2212yes—
47C2×#391C2212yes—
48C2×#391C2211yescharacteristic
49C2×#390C2211yescharacteristic
50C2×#389C2212yes—
51C2×#389C2212yes—
52C2×#389C2212yes—
53C2×#389C2212yes—
54C2×#389C2212yes—
55C2×#388C2214yes—
56C2×#388C2214yes—
57C2×#387C2214yes—
58C2×#387C2214yes—
59C2×#384C2214yes—
60C2×#383C2211yescharacteristic
61C2×#382C2211yescharacteristic
62C2×C4×C4⋊C4C2211yescharacteristic
63#1445C2214yes—
64#1445C2214yes—
65#1445C2214yes—
66#1444C2214yes—
67#1444C2214yes—
68#1444C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1437C2214yes—
75#1437C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2214yes—
79#1434C2212yes—
80C4×#95C2212yes—
81C23×Q8C2311yescharacteristic
82C23×Q8C2311yescharacteristic
83C23×Q8C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—44no—
87C2×#125—24no—
88C2×#125—24no—
89C2×#125—24no—
90C2×#124—24no—
91C2×#124—24no—
92C2×#124—24no—
93C2×#124—24no—
94C2×#124—44no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8—24no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#400—28no—
135#400—28no—
136#400—28no—
137#399—28no—
138#399—28no—
139#399—48no—
140#398—28no—
141#398—28no—
142#397—28no—
143#396—216no—
144#396—216no—
145#395—28no—
146#395—28no—
147#395—28no—
148#395—28no—
149#395—28no—
150#394—28no—
151#394—28no—
152#393—28no—
153#391—28no—
154#391—24no—
155#390—24no—
156#389—28no—
157#389—28no—
158#389—28no—
159#389—28no—
160#389—28no—
161#388—216no—
162#388—216no—
163#387—216no—
164#387—216no—
165#384—48no—
166#383D418yes—
167#382D418yes—
168C4×C4⋊C4D418yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C25C2412yes—
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—42no—
204C22×Q8—42no—
205C22×Q8—42no—
206C22×Q8—42no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—42no—
211C22×D4—42no—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C4⋊Q8—416no—
227#125—416no—
228#124—416no—
229#124—416no—
230C4×Q8—416no—
231C4×Q8—416no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4C2×D418yes—
284C2×C4⋊C4C2×D418yes—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—216no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
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×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—28no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42C2×D414yes—
320C2×C42C2×D414yes—
321C2×C42C2×D414yes—
322C2×C42C2×D414yes—
323C2×C42C2×D414yes—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—44no—
339#95—44no—
340#95C2×D418yes—
341#95C2×D418yes—
342#95C2×D418yes—
343#95C2×D418yes—
344#95C2×D418yes—
345#95C2×D418yes—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—24no—
350C24—28no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24C2511yescharacteristic
355C2×Q8—28no—
356C2×Q8—28no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—44no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—44no—
368C2×Q8—84no—
369C2×Q8—84no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—84no—
374C2×D4—84no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
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—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
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—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
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—24no—
429C22×C4—28no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4C22×D418yes—
438C22×C4C22×D418yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4H(5,𝔽2)14yes—
451C22×C42−1+414yes—
452C22×C42−1+418yes—
453C4⋊C4—216no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—416no—
457C4⋊C4—416no—
458C4⋊C4—48no—
459C4⋊C4—416no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—84no—
469C4⋊C4—84no—
470C4⋊C4—88no—
471C4⋊C4—88no—
472C22⋊C4—416no—
473C22⋊C4—416no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—416no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C42—44no—
481C42—44no—
482C42—416no—
483C42—48no—
484C42—48no—
485C42#127116yes—
486C23—42no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—48no—
499C23—48no—
500C23—48no—
501C23—48no—
502C23C23×D411yescharacteristic
503C23C23×D411yescharacteristic
504C23C23×D411yescharacteristic
505C23C2×H(5,𝔽2)12yes—
506C23C2×2−1+414yes—
507C23C2×2−1+412yes—
508C23#58512yes—
509C23#58512yes—
510Q8—416no—
511Q8—416no—
512Q8—88no—
513D4—88no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4C2×#12714yes—
589C2×C4C2×#12714yes—
590C2×C4C2×#12714yes—
591C2×C4#53618yes—
592C22—84no—
593C22—82no—
594C22—88no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22—82no—
599C22—82no—
600C22C22×#12711yescharacteristic
601C22C2×#53612yes—
602C22#348411yescharacteristic
603C22#348511yescharacteristic
604C22#348611yescharacteristic
605C22#348711yescharacteristic
606C22#348812yes—
607C22#348812yes—
608C22#348912yes—
609C22#348912yes—
610C22#349912yes—
611C22#349912yes—
612C22#354114yes—
613C22#355214yes—
614C22#355514yes—
615C22#356514yes—
616C4—24no—
617C4—28no—
618C4—24no—
619C4—24no—
620C4—44no—
621C4—48no—
622C4—48no—
623C4—44no—
624C4—44no—
625C4—82no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4—82no—
630C2—162no—
631C2#3531711yescharacteristic
632C2#3556511yescharacteristic
633C2#3563411yescharacteristic
634C2#4058912yes—
635C2#4289112yes—
636C2#4772214yes—
637C2#5069212yes—
638C2#5106912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2581C2semidirect
4. C2×#2584C2semidirect
5. C2×#2634C2semidirect
7. #16813C2semidirect
9. #15825C2semidirect
10. #15808C2semidirect
11. #15765C2semidirect
12. #15765C2semidirect
14. #15143C2semidirect
15. #13781C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28202
C2×#25452
C2×#25812
C2×#25842
C2×#26342
C2×#26322
#168132
#165102
#158252
#158082
#157652
#151702
#151432
#137812