Finite Group Databasebeta

← #4866501

Subgroups — #4866501

Order 512.

2314 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×#2598C211yesmaximal, characteristic
2C2×#2628C211yesmaximal, characteristic, complemented
3#17144C212yesmaximal, complemented
4#17084C212yesmaximal, complemented
5#17046C212yesmaximal, complemented
6#16965C212yesmaximal, complemented
7#16824C212yesmaximal, complemented
8#16506C212yesmaximal, complemented
9#16414C214yesmaximal, complemented
10#16395C212yesmaximal, complemented
11#16338C212yesmaximal
12#15506C214yesmaximal, complemented
13#14868C212yesmaximal, complemented
14#14735C212yesmaximal, complemented
15#13958C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#2628—28no—
20#2598—28no—
21C2×#400C2212yes—
22C2×#400C2211yescharacteristic
23C2×#399C2211yescharacteristic
24C2×#399C2212yes—
25C2×#397C2212yes—
26C2×#397C2212yes—
27C2×#395C2212yes—
28C2×#394C2211yescharacteristic
29C2×#394C2214yes—
30C2×#393C2212yes—
31C2×#393C2214yes—
32C2×#392C2211yescharacteristic
33C2×#389C2212yes—
34C2×#388C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2214yes—
38C2×#385C2212yes—
39C2×#383C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×#381C2211yescharacteristic
42C2×#380C2212yes—
43C22×#95C2211yescharacteristic
44#1446C2214yes—
45#1445C2212yes—
46#1445C2212yes—
47#1445C2214yes—
48#1445C2212yes—
49#1445C2212yes—
50#1444C2212yes—
51#1443C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1443C2214yes—
55#1442C2212yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2214yes—
70#1434C2212yes—
71#1434C2212yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C23×Q8C2312yes—
78C23×D4C2312yes—
79C23×D4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—42no—
85C2×C4⋊Q8—42no—
86C2×C4×D4—24no—
87C2×C4×D4—42no—
88C2×C4×D4—42no—
89C2×#117—42no—
90C2×#117—42no—
91C2×#117D418yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#400—24no—
117#400—28no—
118#399—24no—
119#399—28no—
120#397—44no—
121#397—44no—
122#395—28no—
123#394—24no—
124#394—48no—
125#393—28no—
126#393—48no—
127#392—24no—
128#389—28no—
129#388—28no—
130#386—28no—
131#386—28no—
132#386—48no—
133#385—28no—
134#383—24no—
135#382—24no—
136#381—42no—
137#380—28no—
138C2×#95—42no—
139C2×#95C2311yescharacteristic
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2311yescharacteristic
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C25C2412yes—
169C25C2411yescharacteristic
170C22×Q8—24no—
171C22×Q8—42no—
172C22×Q8—42no—
173C22×Q8—44no—
174C22×Q8—44no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—28no—
179C22×D4—22no—
180C22×D4—41no—
181C22×D4—41no—
182C22×D4—42no—
183C22×D4—42no—
184C22×D4—44no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—44no—
194C23×C4—42no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—42no—
198C23×C4—44no—
199C23×C4C2×D414yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C4⋊Q8—416no—
214C4×D4—416no—
215#117—416no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4C2×D414yes—
240C2×C4⋊C4C2×D414yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4C2×D414yes—
276C2×C22⋊C4C2×D414yes—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42C2×D414yes—
282C2×C42C2×D414yes—
283#95—28no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—22no—
289#95—22no—
290#95—24no—
291#95—24no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—42no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—42no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312C24—21no—
313C24—21no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—24no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—22no—
326C24—24no—
327C24—22no—
328C24C22×D412yes—
329C24H(5,𝔽2)14yes—
330C242−1+414yes—
331C24C2511yescharacteristic
332C2×Q8—48no—
333C2×Q8—88no—
334C2×Q8—84no—
335C2×Q8—84no—
336C2×Q8—88no—
337C2×D4—48no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—48no—
341C2×D4—44no—
342C2×D4—48no—
343C2×D4—416no—
344C2×D4—48no—
345C2×D4—84no—
346C2×D4—88no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—28no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—28no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—44no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4—84no—
420C22×C4—88no—
421C22×C4—88no—
422C22×C4—88no—
423C22×C4—84no—
424C22×C4—88no—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C4H(5,𝔽2)12yes—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—88no—
440C4⋊C4—84no—
441C4⋊C4—88no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—84no—
448C22⋊C4—88no—
449C22⋊C4—84no—
450C22⋊C4—84no—
451C22⋊C4—88no—
452C22⋊C4—88no—
453C22⋊C4—88no—
454C22⋊C4—88no—
455C42—28no—
456C42—28no—
457C23—22no—
458C23—22no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—28no—
464C23—28no—
465C23—24no—
466C23—28no—
467C23—24no—
468C23—24no—
469C23—28no—
470C23—41no—
471C23—42no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23—44no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23#53614yes—
496C23C23×D411yescharacteristic
497C23C2×H(5,𝔽2)11yescharacteristic
498C23C2×H(5,𝔽2)11yescharacteristic
499C23C2×H(5,𝔽2)12yes—
500C23C2×2−1+412yes—
501C23#58512yes—
502C23#58512yes—
503C23#58514yes—
504Q8—816no—
505D4—416no—
506D4—816no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—416no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
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—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—816no—
579C2×C4—816no—
580C2×C4#53414yes—
581C22—24no—
582C22—28no—
583C22—44no—
584C22—42no—
585C22—44no—
586C22—42no—
587C22—44no—
588C22—48no—
589C22—48no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—48no—
594C22—84no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—84no—
600C22C2×#53411yescharacteristic
601C22C2×#53611yescharacteristic
602C22#344911yescharacteristic
603C22#348812yes—
604C22#350012yes—
605C22#353512yes—
606C22#353514yes—
607C22#353512yes—
608C22#353711yescharacteristic
609C22#354011yescharacteristic
610C22#354012yes—
611C22#354212yes—
612C22#355112yes—
613C22#356014yes—
614C22#356414yes—
615C22#356614yes—
616C4—24no—
617C4—42no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—42no—
622C4—82no—
623C4—84no—
624C4—84no—
625C4—82no—
626C4—84no—
627C4—82no—
628C4—84no—
629C2—44no—
630C2—84no—
631C2#3726911yescharacteristic
632C2#3741911yescharacteristic
633C2#3744711yescharacteristic
634C2#4193812yes—
635C2#4239312yes—
636C2#5433312yes—
637C2#5456912yes—
638C2#5525914yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2628C2semidirect
3. #17144C2semidirect
4. #17084C2semidirect
5. #17046C2semidirect
6. #16965C2semidirect
7. #16824C2semidirect
8. #16506C2semidirect
9. #16414C2semidirect
10. #16395C2semidirect
12. #15506C2semidirect
13. #14868C2semidirect
14. #14735C2semidirect
15. #13958C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25982
C2×#26282
#171442
#170842
#170462
#169652
#168242
#165062
#164142
#163952
#163382
#155062
#148682
#147352
#139582