Finite Group Databasebeta

← #4419845

Subgroups — #4419845

Order 512.

2396 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2611C211yesmaximal, characteristic
2C2×#2621C211yesmaximal, characteristic
3C2×#2660C211yesmaximal, characteristic, complemented
4C2×#2346C211yesmaximal, characteristic
5#17060C212yesmaximal
6#16701C212yesmaximal
7#16565C212yesmaximal, complemented
8#16264C212yesmaximal
9#15751C214yesmaximal
10#15618C214yesmaximal, complemented
11#15151C212yesmaximal, complemented
12#15085C212yesmaximal, complemented
13#14901C212yesmaximal, complemented
14#14763C212yesmaximal
15#14664C211yesmaximal, characteristic, complemented
16#14282C211yesmaximal, characteristic, complemented
17#13963C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C22⋊Q8C2212yes—
21C22×#117C2211yescharacteristic
22#2660—28no—
23#2621—28no—
24#2611—28no—
25C2×#400C2211yescharacteristic
26C2×#399C2211yescharacteristic
27C2×#398C2212yes—
28C2×#398C2214yes—
29C2×#398C2211yescharacteristic
30C2×#398C2212yes—
31C2×#398C2214yes—
32C2×#397C2211yescharacteristic
33C2×#397C2212yes—
34C2×#397C2212yes—
35C2×#395C2212yes—
36C2×#395C2212yes—
37#2346—28no—
38C2×#391C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2214yes—
42C2×#388C2212yes—
43C2×#388C2212yes—
44C2×#388C2212yes—
45C2×#387C2212yes—
46C2×#385C2212yes—
47C2×#384C2212yes—
48C2×#382C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#382C2212yes—
51C2×#382C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×#376C2211yescharacteristic
56C2×#376C2211yescharacteristic
57C2×#376C2211yescharacteristic
58#1446C2212yes—
59#1446C2212yes—
60#1445C2212yes—
61#1445C2212yes—
62#1445C2214yes—
63#1445C2212yes—
64#1445C2212yes—
65#1444C2212yes—
66#1443C2214yes—
67#1442C2214yes—
68#1442C2212yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2212yes—
75#1441C2212yes—
76#1440C2214yes—
77#1439C2214yes—
78#1439C2214yes—
79#1439C2214yes—
80#1436C2214yes—
81#1436C2214yes—
82C4⋊#95C2214yes—
83C4⋊#95C2214yes—
84C4⋊#95C2212yes—
85#1434C2214yes—
86#1434C2212yes—
87C23×Q8C2312yes—
88C24×C4C2311yescharacteristic
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—24no—
92C2×#125—24no—
93C2×#125—24no—
94C2×#125—24no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—48no—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117D418yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131#400—24no—
132#399—24no—
133#398—24no—
134#398—28no—
135#398—28no—
136#398—216no—
137#398—48no—
138#397—24no—
139#397—28no—
140#397—28no—
141#395—28no—
142#395—28no—
143#391—28no—
144#389—28no—
145#389—28no—
146#389—48no—
147#388—28no—
148#388—28no—
149#388—28no—
150#387—28no—
151#385—28no—
152#384—28no—
153#382—24no—
154#382—24no—
155#382—28no—
156#382—24no—
157#380—24no—
158C4×C4⋊C4—28no—
159C4×C22⋊C4—24no—
160#376—24no—
161#376—24no—
162#376D418yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2311yescharacteristic
193C2×#95C2311yescharacteristic
194C2×#95C2312yes—
195C25C2411yescharacteristic
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—44no—
200C22×Q8—44no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—42no—
205C23×C4C2×D414yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C4⋊Q8—416no—
223#125—416no—
224C22⋊Q8—432no—
225#117—232no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D414yes—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4C2×D414yes—
293C2×C22⋊C4C2×D414yes—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—28no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—28no—
303C2×C42—42no—
304C2×C42—42no—
305C2×C42C2×D414yes—
306C2×C42C2×D418yes—
307C2×C42C2×D414yes—
308C2×C42C2×D414yes—
309C2×C42#4114yes—
310#95—22no—
311#95—24no—
312#95—24no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—28no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95C2×D414yes—
340#95C2×D414yes—
341#95C2×D418yes—
342C24—21no—
343C24—21no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—22no—
348C24—22no—
349C24C22×D412yes—
350C24C2511yescharacteristic
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—88no—
355C2×Q8—88no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
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—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—44no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D414yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D414yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4H(5,𝔽2)14yes—
443C22×C42−1+414yes—
444C22×C42−1+418yes—
445C4⋊C4—28no—
446C4⋊C4—28no—
447C4⋊C4—48no—
448C4⋊C4—416no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—416no—
459C4⋊C4—88no—
460C4⋊C4—84no—
461C4⋊C4—88no—
462C4⋊C4—84no—
463C22⋊C4—28no—
464C22⋊C4—28no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—416no—
468C22⋊C4—48no—
469C22⋊C4—84no—
470C42—28no—
471C42—216no—
472C42—48no—
473C42—48no—
474C42—48no—
475C42#121116yes—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—24no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—41no—
485C23—41no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23C23×D411yescharacteristic
493C23C23×D411yescharacteristic
494C23C22×#4111yescharacteristic
495C23C2×H(5,𝔽2)12yes—
496C23C2×2−1+414yes—
497C23C2×2−1+412yes—
498C23#58512yes—
499C23#58512yes—
500Q8—816no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—416no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4C2×#12114yes—
582C2×C4C2×#12114yes—
583C2×C4C2×#12114yes—
584C2×C4#53514yes—
585C2×C4#53618yes—
586C22—42no—
587C22—44no—
588C22—42no—
589C22—44no—
590C22—42no—
591C22—84no—
592C22—82no—
593C22—82no—
594C22C22×#12111yescharacteristic
595C22C2×#53511yescharacteristic
596C22C2×#53612yes—
597C22#344911yescharacteristic
598C22#348812yes—
599C22#348912yes—
600C22#349412yes—
601C22#349911yescharacteristic
602C22#349912yes—
603C22#350011yescharacteristic
604C22#350712yes—
605C22#350912yes—
606C22#353814yes—
607C22#354714yes—
608C22#356314yes—
609C22#356514yes—
610C4—24no—
611C4—24no—
612C4—24no—
613C4—28no—
614C4—24no—
615C4—42no—
616C4—44no—
617C4—44no—
618C4—42no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—84no—
623C4—82no—
624C4—82no—
625C4—84no—
626C2—82no—
627C2#3525211yescharacteristic
628C2#3531611yescharacteristic
629C2#3562111yescharacteristic
630C2#4193912yes—
631C2#4231612yes—
632C2#4296512yes—
633C2#4299412yes—
634C2#5162714yes—
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
3. C2×#2660C2semidirect
7. #16565C2semidirect
10. #15618C2semidirect
11. #15151C2semidirect
12. #15085C2semidirect
13. #14901C2semidirect
15. #14664C2semidirect
16. #14282C2semidirect
17. #13963C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26112
C2×#26212
C2×#26602
C2×#23462
#170602
#167012
#165652
#162642
#157512
#156182
#151512
#150852
#149012
#147632
#146642
#142822
#139632