Finite Group Databasebeta

← #4457185

Subgroups — #4457185

Order 512.

2225 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2581C211yesmaximal, characteristic, complemented
2C2×#2604C211yesmaximal, characteristic
3C2×#2604C211yesmaximal, characteristic
4C2×#2320C211yesmaximal, characteristic, complemented
5#17074C212yesmaximal, complemented
6#17039C212yesmaximal
7#16993C212yesmaximal, complemented
8#16901C212yesmaximal
9#16676C212yesmaximal
10#16663C212yesmaximal, complemented
11#16567C212yesmaximal, complemented
12#16566C212yesmaximal
13#15005C214yesmaximal, complemented
14#14995C214yesmaximal
15#14291C211yesmaximal, characteristic
16#13963C211yesmaximal, characteristic, complemented
17#13850C211yesmaximal, characteristic, complemented
18C22×#126C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#2604—28no—
22#2604—28no—
23#2581—28no—
24C2×#400C2211yescharacteristic
25C2×#400C2211yescharacteristic
26C2×#400C2212yes—
27C2×#399C2211yescharacteristic
28C2×#399C2211yescharacteristic
29C2×#395C2212yes—
30C2×#395C2211yescharacteristic
31C2×#395C2211yescharacteristic
32C2×#393C2211yescharacteristic
33C2×#393C2211yescharacteristic
34#2320—28no—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#388C2212yes—
42C2×#388C2212yes—
43C2×#388C2212yes—
44C2×#388C2212yes—
45C2×#387C2214yes—
46C2×#387C2212yes—
47C2×#387C2212yes—
48C2×#387C2214yes—
49C2×#383C2212yes—
50C2×#382C2212yes—
51C2×#382C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×#376C2211yescharacteristic
54C2×#376C2211yescharacteristic
55#1446C2212yes—
56#1445C2214yes—
57#1445C2212yes—
58#1444C2214yes—
59#1444C2212yes—
60#1443C2214yes—
61#1442C2212yes—
62#1442C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1442C2214yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2212yes—
70#1441C2212yes—
71#1441C2212yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2214yes—
75#1440C2214yes—
76#1439C2214yes—
77#1439C2212yes—
78#1439C2214yes—
79#1439C2212yes—
80#1439C2214yes—
81#1437C2212yes—
82#1436C2214yes—
83#1436C2212yes—
84C4⋊#95C2212yes—
85#1434C2212yes—
86C4×#95C2214yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C2×#126—24no—
90C2×#126—24no—
91C2×#126—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8D418yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128#400—28no—
129#400—24no—
130#400—24no—
131#399—24no—
132#399—24no—
133#395—28no—
134#395—24no—
135#395—24no—
136#393—24no—
137#393—24no—
138#391—28no—
139#391—28no—
140#389—28no—
141#389—28no—
142#389—28no—
143#389—28no—
144#388—28no—
145#388—28no—
146#388—28no—
147#388—28no—
148#387—216no—
149#387—28no—
150#387—216no—
151#387—28no—
152#383—28no—
153#382—24no—
154#382—28no—
155C4×C4⋊C4—24no—
156#376—24no—
157#376D418yes—
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2311yescharacteristic
189C2×#95C2311yescharacteristic
190C2×#95C2312yes—
191C25C2411yescharacteristic
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—24no—
198C22×Q8—44no—
199C22×Q8—42no—
200C22×Q8C2×D414yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217#126—416no—
218C4×Q8—232no—
219C4×Q8—416no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C2×D418yes—
263C2×C4⋊C4C2×D414yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
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—44no—
295C2×C42—42no—
296C2×C42—44no—
297C2×C42C2×D418yes—
298C2×C42C2×D418yes—
299C2×C42C2×D414yes—
300C2×C42C2×D414yes—
301C2×C42#4114yes—
302#95—24no—
303#95—22no—
304#95—22no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—24no—
309#95—28no—
310#95—28no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95C2×D414yes—
333#95C2×D414yes—
334#95C2×D418yes—
335C24—21no—
336C24—21no—
337C24—21no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—22no—
343C24C2511yescharacteristic
344C2×Q8—28no—
345C2×Q8—28no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—44no—
352C2×Q8—44no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
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—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4C22×D414yes—
425C22×C4C22×D414yes—
426C22×C4C22×D414yes—
427C22×C4C22×D414yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C42−1+414yes—
438C22×C42−1+414yes—
439C4⋊C4—28no—
440C4⋊C4—216no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—416no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—88no—
454C4⋊C4—88no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—44no—
462C22⋊C4—88no—
463C42—216no—
464C42—48no—
465C42—48no—
466C42—48no—
467C42—84no—
468C42#121116yes—
469C23—41no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—41no—
474C23—41no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23C23×D411yescharacteristic
486C23C23×D411yescharacteristic
487C23C22×#4111yescharacteristic
488C23C2×2−1+412yes—
489C23C2×2−1+412yes—
490C23#58514yes—
491C23#58512yes—
492C23#58512yes—
493Q8—416no—
494Q8—416no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—42no—
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×C4C2×#12114yes—
579C2×C4C2×#12114yes—
580C2×C4C2×#12114yes—
581C22—81no—
582C22—81no—
583C22—82no—
584C22—81no—
585C22—84no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22C22×#12111yescharacteristic
590C22#344912yes—
591C22#344912yes—
592C22#345512yes—
593C22#348611yescharacteristic
594C22#348711yescharacteristic
595C22#348912yes—
596C22#348912yes—
597C22#349512yes—
598C22#349911yescharacteristic
599C22#350011yescharacteristic
600C22#350712yes—
601C22#355714yes—
602C22#356314yes—
603C22#356514yes—
604C22#356814yes—
605C4—24no—
606C4—24no—
607C4—24no—
608C4—44no—
609C4—44no—
610C4—44no—
611C4—44no—
612C4—44no—
613C4—44no—
614C4—81no—
615C4—82no—
616C4—81no—
617C4—81no—
618C4—84no—
619C4—82no—
620C4—84no—
621C2—161no—
622C2#3463311yescharacteristic
623C2#3562111yescharacteristic
624C2#3566111yescharacteristic
625C2#3700614yes—
626C2#4215612yes—
627C2#4217012yes—
628C2#4217112yes—
629C2#4231612yes—
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
1. C2×#2581C2semidirect
4. C2×#2320C2semidirect
5. #17074C2semidirect
7. #16993C2semidirect
10. #16663C2semidirect
11. #16567C2semidirect
13. #15005C2semidirect
16. #13963C2semidirect
17. #13850C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25812
C2×#26042
C2×#23202
#170742
#170392
#169932
#169012
#166762
#166632
#165672
#165662
#150052
#149952
#142912
#139632
#138502