Finite Group Databasebeta

← #4929828

Subgroups — #4929828

Order 512.

2864 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2591C211yesmaximal, characteristic
2C2×#2484C211yesmaximal, characteristic, complemented
3C2×#2500C211yesmaximal, characteristic
4C2×#2274C211yesmaximal, characteristic, complemented
5#15163C212yesmaximal, complemented
6#15152C212yesmaximal, complemented
7#15107C212yesmaximal, complemented
8#15085C212yesmaximal, complemented
9#14890C214yesmaximal, complemented
10#14838C214yesmaximal, complemented
11#14290C211yesmaximal, characteristic
12#14206C214yesmaximal, complemented
13#14119C214yesmaximal, complemented
14#13953C211yesmaximal, characteristic, complemented
15#13853C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20#2591—28no—
21#2500—28no—
22#2484—28no—
23C2×#400C2212yes—
24C2×#400C2212yes—
25C2×#398C2212yes—
26C2×#396C2214yes—
27C2×#394C2212yes—
28C2×#394C2214yes—
29C2×#390C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#387C2214yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#385C2212yes—
38C2×#384C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#381C2211yescharacteristic
41C2×#381C2212yes—
42#2274—28no—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51#1445C2212yes—
52#1445C2212yes—
53#1445C2212yes—
54#1443C2214yes—
55#1442C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1438C2212yes—
64#1438C2214yes—
65#1437C2214yes—
66#1437C2214yes—
67#1437C2212yes—
68#1437C2214yes—
69#1437C2212yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2214yes—
75#1434C2212yes—
76#1434C2214yes—
77C4×#95C2212yes—
78C4×#95C2212yes—
79C23×D4C2312yes—
80C23×D4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×#121—24no—
85C2×#121—24no—
86C2×#121—24no—
87C2×#117—24no—
88C2×#117—24no—
89C2×#117—24no—
90C2×#117—28no—
91C2×#117—28no—
92C2×#117—28no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4D418yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128#400—28no—
129#400—28no—
130#398—28no—
131#396—216no—
132#394—28no—
133#394—216no—
134#390—24no—
135#390—24no—
136#388—216no—
137#388—216no—
138#388—216no—
139#387—216no—
140#386—28no—
141#386—28no—
142#385—28no—
143#384—24no—
144#382—24no—
145#381—24no—
146#381—28no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—28no—
151C4×C22⋊C4—24no—
152C4×C22⋊C4—24no—
153C4×C22⋊C4—28no—
154C4×C22⋊C4D418yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C25C2412yes—
178C25C2411yescharacteristic
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—42no—
186C22×D4—42no—
187C22×D4—44no—
188C22×D4—44no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4C2×D414yes—
200C23×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4#4114yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2412yes—
217#121—416no—
218#117—264no—
219#117—416no—
220C2×C4⋊C4—232no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4C2×D418yes—
249C2×C4⋊C4C2×D418yes—
250C2×C4⋊C4#4118yes—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4C2×D418yes—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42—44no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42—42no—
305C2×C42—42no—
306C2×C42—44no—
307C2×C42C2×D414yes—
308C2×C42C2×D414yes—
309C2×C42#4118yes—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—28no—
317#95—24no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95C2×D418yes—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—24no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—24no—
345C24—24no—
346C24C22×D412yes—
347C24C22×D412yes—
348C24C2×#4112yes—
349C24C2511yescharacteristic
350C2×D4—44no—
351C2×D4—44no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—44no—
356C2×D4—84no—
357C2×D4—84no—
358C2×D4—88no—
359C2×D4—88no—
360C22×C4—216no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—216no—
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—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
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—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—84no—
429C22×C4#121116yes—
430C22×C4C22×D414yes—
431C22×C4C22×D414yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4114yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4114yes—
446C4⋊C4—216no—
447C4⋊C4—216no—
448C4⋊C4—216no—
449C4⋊C4—216no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—416no—
454C4⋊C4—88no—
455C4⋊C4—88no—
456C22⋊C4—216no—
457C22⋊C4—216no—
458C22⋊C4—416no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—416no—
464C22⋊C4—416no—
465C22⋊C4—88no—
466C22⋊C4—88no—
467C22⋊C4—88no—
468C42—216no—
469C42—216no—
470C42—44no—
471C42—44no—
472C42—48no—
473C42—84no—
474C42—82no—
475C42—82no—
476C42—84no—
477C42—84no—
478C23—24no—
479C23—24no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—44no—
495C23—44no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23C2×#12114yes—
515C23C23×D411yescharacteristic
516C23C23×D411yescharacteristic
517C23C22×#4111yescharacteristic
518C23C22×#4112yes—
519C23C22×#4112yes—
520C23#58512yes—
521C23#58512yes—
522C23#58514yes—
523D4—88no—
524D4—816no—
525C2×C4—232no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—416no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—88no—
595C2×C4C2×#12114yes—
596C2×C4C2×#12114yes—
597C2×C4#52518yes—
598C2×C4#53218yes—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—44no—
603C22—44no—
604C22—48no—
605C22—44no—
606C22—44no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—84no—
611C22—84no—
612C22—84no—
613C22—84no—
614C22—84no—
615C22—84no—
616C22C22×#12111yescharacteristic
617C22C2×#52512yes—
618C22C2×#53212yes—
619C22#344411yescharacteristic
620C22#344411yescharacteristic
621C22#348511yescharacteristic
622C22#348512yes—
623C22#348511yescharacteristic
624C22#348712yes—
625C22#349112yes—
626C22#349314yes—
627C22#350012yes—
628C22#351012yes—
629C22#351214yes—
630C22#352114yes—
631C22#352414yes—
632C4—28no—
633C4—24no—
634C4—24no—
635C4—28no—
636C4—44no—
637C4—44no—
638C4—44no—
639C4—44no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—84no—
644C4—84no—
645C2—44no—
646C2—162no—
647C2#3544711yescharacteristic
648C2#3550112yes—
649C2#3550612yes—
650C2#3567711yescharacteristic
651C2#3601112yes—
652C2#3601514yes—
653C2#3603212yes—
654C2#3662411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2484C2semidirect
4. C2×#2274C2semidirect
5. #15163C2semidirect
6. #15152C2semidirect
7. #15107C2semidirect
8. #15085C2semidirect
9. #14890C2semidirect
10. #14838C2semidirect
12. #14206C2semidirect
13. #14119C2semidirect
14. #13953C2semidirect
15. #13853C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25912
C2×#24842
C2×#25002
C2×#22742
#151632
#151522
#151072
#150852
#148902
#148382
#142902
#142062
#141192
#139532
#138532