Finite Group Databasebeta

← #410813

Subgroups — #410813

Order 512.

1522 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C2×#1873C211yesmaximal, characteristic
3C2×#1872C211yesmaximal, characteristic
4#14258C211yesmaximal, characteristic
5#11268C212yesmaximal, complemented
6#11267C212yesmaximal, complemented
7#11052C212yesmaximal, complemented
8#11050C212yesmaximal, complemented
9#10731C211yesmaximal, characteristic
10#10730C211yesmaximal, characteristic
11#10713C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21#2317—24no—
22#2317—24no—
23C2×#391C2212yes—
24C2×#389—21no—
25C2×#389—21no—
26C2×#385—22no—
27C2×#384C2212yes—
28C2×#382C2212yes—
29C2×#380C2211yescharacteristic
30#1873—24no—
31#1872—24no—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1441—21no—
42#1441—21no—
43#1440—22no—
44C4⋊#95C2212yes—
45C23×Q8C2311yescharacteristic
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8Q814yes—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—24no—
61C2×#117—24no—
62C2×#117—42no—
63C2×#117—42no—
64C2×#117D414yes—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—42no—
71C22×C4⋊C4—42no—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D414yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4D412yes—
84C22×C42—21no—
85C22×C42C2312yes—
86C2×#103—24no—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—42no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#103—42no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99#391—24no—
100#391—24no—
101#389—28no—
102#389—28no—
103#385—216no—
104#384—24no—
105#384—24no—
106#382—24no—
107#382—24no—
108#380—22no—
109#380—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117#340—24no—
118#340—22no—
119#340—22no—
120#340—42no—
121#340—42no—
122#340—42no—
123#340—42no—
124#336—24no—
125#336—24no—
126C25C2×D411yescharacteristic
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8C2×Q812yes—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—24no—
147C23×C4—22no—
148C23×C4—44no—
149C23×C4—81no—
150C23×C4—82no—
151C23×C4—81no—
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2×Q812yes—
158C23×C4#4112yes—
159C23×C4#4112yes—
160C23×C4#4112yes—
161C23×C4C2411yescharacteristic
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22⋊Q8—28no—
180C22⋊Q8—28no—
181C22⋊Q8—44no—
182C22⋊Q8—44no—
183#117—28no—
184#117—28no—
185#117—216no—
186#117—44no—
187#117—44no—
188#117—48no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—48no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4C2×D414yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×Q812yes—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—81no—
243C2×C22⋊C4—81no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—44no—
250C2×C42#4114yes—
251C2×C42#4114yes—
252#103—48no—
253#103—44no—
254#103—44no—
255C22⋊C8—44no—
256C22⋊C8—44no—
257#95—28no—
258#95—28no—
259#95—44no—
260#95—44no—
261#95—42no—
262#95—82no—
263#95—82no—
264C24—22no—
265C24—22no—
266C24—21no—
267C24—21no—
268C24—22no—
269C24—21no—
270C24—22no—
271C24—21no—
272C24#12012yes—
273C24C22⋊Q812yes—
274C24#12412yes—
275C24C22×D411yescharacteristic
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—48no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—84no—
360C22×C4—84no—
361C22×C4—88no—
362C22×C4—88no—
363C22×C4#12012yes—
364C22×C4C22⋊Q814yes—
365C22×C4C22⋊Q814yes—
366C22×C4C22⋊Q812yes—
367C22×C4#12314yes—
368C22×C4#12314yes—
369C22×C4#12414yes—
370C22×C4#12414yes—
371C22×C4#12412yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×Q811yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C4C2×#4111yescharacteristic
380C22×C4H(5,𝔽2)12yes—
381C22×C4H(5,𝔽2)12yes—
382C2×C8—42no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—42no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—42no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C4⋊C4—28no—
400C4⋊C4—28no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—48no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—48no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—88no—
422C4⋊C4—88no—
423C22⋊C4—216no—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C22⋊C4—84no—
427C42—24no—
428C42—24no—
429C42—24no—
430C42—24no—
431C42—48no—
432C23—24no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—22no—
440C23—24no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—24no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23#39314yes—
455C23C2×#12011yescharacteristic
456C23C2×C22⋊Q811yescharacteristic
457C23C2×#12312yes—
458C23C2×#12411yescharacteristic
459C23#53512yes—
460C23#57611yescharacteristic
461C23#57712yes—
462C23#57711yescharacteristic
463C23#57711yescharacteristic
464C23#57812yes—
465C23#57811yescharacteristic
466Q8—44no—
467Q8—42no—
468Q8—42no—
469Q8—82no—
470Q8—81no—
471Q8—81no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—28no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—816no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4#39314yes—
534C2×C4C2×C22⋊Q811yescharacteristic
535C2×C4C2×C22⋊Q811yescharacteristic
536C2×C4#52511yescharacteristic
537C2×C4#52511yescharacteristic
538C2×C4#52512yes—
539C2×C4C2×#12411yescharacteristic
540C2×C4C2×#12411yescharacteristic
541C2×C4#53412yes—
542C2×C4#53411yescharacteristic
543C2×C4#53411yescharacteristic
544C2×C4#55312yes—
545C2×C4#55312yes—
546C2×C4#55312yes—
547C2×C4#55312yes—
548C2×C4#55312yes—
549C2×C4#55312yes—
550C2×C4#55412yes—
551C2×C4#55412yes—
552C8—82no—
553C8—81no—
554C8—81no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22C2×#39311yescharacteristic
567C22#248712yes—
568C22#250412yes—
569C22#261311yescharacteristic
570C22#284911yescharacteristic
571C22#300911yescharacteristic
572C22#301111yescharacteristic
573C22#301211yescharacteristic
574C22#301311yescharacteristic
575C22#308212yes—
576C22#308311yescharacteristic
577C22#308411yescharacteristic
578C22#309712yes—
579C22#309712yes—
580C22#309812yes—
581C22#309912yes—
582C22#313611yescharacteristic
583C22#313712yes—
584C22#313711yescharacteristic
585C22#322212yes—
586C22#322312yes—
587C22#322412yes—
588C22#322512yes—
589C4—24no—
590C4—44no—
591C4—44no—
592C4—82no—
593C4—82no—
594C4—84no—
595C4—82no—
596C4—82no—
597C4—162no—
598C4#257912yes—
599C4#257911yescharacteristic
600C4#257911yescharacteristic
601C4#259112yes—
602C4#259111yescharacteristic
603C4#259111yescharacteristic
604C2—44no—
605C2#1509111yescharacteristic
606C2#2345711yescharacteristic
607C2#2345811yescharacteristic
608C2#2768912yes—
609C2#2769012yes—
610C2#2798412yes—
611C2#2798512yes—
612C2#2963211yescharacteristic
613C2#2963311yescharacteristic
614C2#3298811yescharacteristic
615C2#3298911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11268C2semidirect
6. #11267C2semidirect
7. #11052C2semidirect
8. #11050C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#18732
C2×#18722
#142582
#112682
#112672
#110522
#110502
#107312
#107302
#107132