Finite Group Databasebeta

← #416949

Subgroups — #416949

Order 512.

1432 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C211yesmaximal, characteristic
2C2×#1984C211yesmaximal, characteristic, complemented
3C2×#1835C211yesmaximal, characteristic, complemented
4C2×#1804C211yesmaximal, characteristic
5C2×C4×#102C211yesmaximal, characteristic
6#13925C211yesmaximal, characteristic, complemented
7#11293C212yesmaximal, complemented
8#11292C212yesmaximal, complemented
9#11189C212yesmaximal
10#11189C212yesmaximal
11#10850C211yesmaximal, characteristic, complemented
12C22×C4×Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2580—24no—
20#2580—24no—
21C2×#399—21no—
22C2×#393—21no—
23C2×#390C2212yes—
24C2×#389C2212yes—
25C2×#385—22no—
26C2×#382C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1984—24no—
30#1835—24no—
31#1804—24no—
32C4×#102—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1446—21no—
43#1445—21no—
44#1436—22no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8D411yescharacteristic
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4D414yes—
65C22×C4⋊C4—22no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D412yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C42—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×#102—22no—
83C2×#102—22no—
84C2×#102—22no—
85C2×#102—22no—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×C4×C8—22no—
92C2×C4×C8—22no—
93C2×C4×C8D414yes—
94#399—28no—
95#393—28no—
96#390—24no—
97#390—24no—
98#389—24no—
99#389—24no—
100#385—216no—
101#382—22no—
102#382—22no—
103#382—22no—
104#382D414yes—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117#340—22no—
118#340—22no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#336—24no—
125#336—24no—
126C25—21no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—24no—
131C22×Q8—42no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4C2×D412yes—
144C22×D4#4112yes—
145C22×D4#4112yes—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—24no—
151C23×C4—22no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—82no—
155C23×C4—82no—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8C2×D412yes—
181C22×C8C2×D412yes—
182C4×Q8—44no—
183C4×Q8—44no—
184C4×Q8—44no—
185C4×Q8—44no—
186C4×D4—28no—
187C4×D4—28no—
188C4×D4—44no—
189C4×D4—44no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4#4112yes—
228C2×C4⋊C4#4112yes—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—82no—
237C2×C22⋊C4—82no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—42no—
245C2×C42—82no—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42#4112yes—
250C2×C42#4112yes—
251C2×C42#4112yes—
252C4⋊C8—44no—
253#102—28no—
254#102—28no—
255#102—48no—
256C4×C8—28no—
257#95—28no—
258#95—24no—
259#95—24no—
260#95—44no—
261#95—42no—
262#95—42no—
263#95—42no—
264#95—44no—
265#95—82no—
266#95—82no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—41no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24C22×D411yescharacteristic
279C2×Q8—28no—
280C2×Q8—48no—
281C2×Q8—48no—
282C2×Q8—44no—
283C2×Q8—44no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4#12414yes—
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—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—48no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—88no—
360C22×C4—88no—
361C22×C4—84no—
362C22×C4—84no—
363C22×C4#12012yes—
364C22×C4#12012yes—
365C22×C4#12112yes—
366C22×C4#12112yes—
367C22×C4#12312yes—
368C22×C4#12312yes—
369C22×C4#13514yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C42−1+411yescharacteristic
381C22×C42−1+411yescharacteristic
382C22×C42−1+411yescharacteristic
383C22×C42−1+411yescharacteristic
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—48no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—88no—
428C22⋊C4—416no—
429C22⋊C4—48no—
430C22⋊C4—88no—
431C42—42no—
432C42—42no—
433C42—42no—
434C42—42no—
435C42—84no—
436C42#12114yes—
437C42#12114yes—
438C42#12314yes—
439C42#12414yes—
440C23—24no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—44no—
450C23—44no—
451C23—42no—
452C23—42no—
453C23—82no—
454C23—82no—
455C23—82no—
456C23—82no—
457C23—82no—
458C23—82no—
459C23—81no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23C2×#12011yescharacteristic
464C23C2×#12111yescharacteristic
465C23C2×#12311yescharacteristic
466C23#54811yescharacteristic
467C23#54811yescharacteristic
468C23#54811yescharacteristic
469C23#54811yescharacteristic
470C23C2×#13512yes—
471C23#57512yes—
472C23#57612yes—
473C23#57712yes—
474Q8—416no—
475D4—24no—
476D4—24no—
477D4—42no—
478D4—42no—
479D4—42no—
480D4—42no—
481D4—42no—
482D4—42no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—816no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4#39414yes—
536C2×C4#39414yes—
537C2×C4C2×#12111yescharacteristic
538C2×C4C2×#12111yescharacteristic
539C2×C4C2×#12311yescharacteristic
540C2×C4#52511yescharacteristic
541C2×C4#52511yescharacteristic
542C2×C4C2×#12411yescharacteristic
543C2×C4C2×#12411yescharacteristic
544C2×C4#52911yescharacteristic
545C2×C4#52911yescharacteristic
546C2×C4#53211yescharacteristic
547C2×C4#53611yescharacteristic
548C2×C4#53611yescharacteristic
549C2×C4#53611yescharacteristic
550C2×C4#53611yescharacteristic
551C2×C4#54111yescharacteristic
552C2×C4#54111yescharacteristic
553C2×C4#54711yescharacteristic
554C2×C4#54711yescharacteristic
555C2×C4#54811yescharacteristic
556C2×C4#54811yescharacteristic
557C2×C4#55211yescharacteristic
558C2×C4#55211yescharacteristic
559C2×C4#55211yescharacteristic
560C2×C4#55211yescharacteristic
561C2×C4#55211yescharacteristic
562C2×C4#55211yescharacteristic
563C2×C4#55611yescharacteristic
564C2×C4#55611yescharacteristic
565C8—42no—
566C8—42no—
567C8—82no—
568C22—48no—
569C22—44no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—84no—
575C22—84no—
576C22—84no—
577C22—82no—
578C22—82no—
579C22C2×#39411yescharacteristic
580C22#242111yescharacteristic
581C22#242111yescharacteristic
582C22#243211yescharacteristic
583C22#243611yescharacteristic
584C22#253611yescharacteristic
585C22#253611yescharacteristic
586C22#300212yes—
587C22#301212yes—
588C22#305011yescharacteristic
589C22#305411yescharacteristic
590C22#305711yescharacteristic
591C22#305811yescharacteristic
592C22#309112yes—
593C22#309612yes—
594C22#336012yes—
595C22#336112yes—
596C22#336112yes—
597C22#336512yes—
598C22#337012yes—
599C22#337012yes—
600C22#337112yes—
601C22#337612yes—
602C4—24no—
603C4—24no—
604C4—44no—
605C4—82no—
606C4—82no—
607C4—84no—
608C4—82no—
609C4—82no—
610C4—162no—
611C4#241511yescharacteristic
612C4#241511yescharacteristic
613C4#248911yescharacteristic
614C4#248911yescharacteristic
615C4#249911yescharacteristic
616C4#250311yescharacteristic
617C4#262111yescharacteristic
618C4#285011yescharacteristic
619C2—84no—
620C2#1523611yescharacteristic
621C2#2347911yescharacteristic
622C2#2348611yescharacteristic
623C2#2499812yes—
624C2#2499912yes—
625C2#2565312yes—
626C2#2593812yes—
627C2#2818811yescharacteristic
628C2#2818911yescharacteristic
629C2#2819111yescharacteristic
630C2#2819211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1984C2semidirect
3. C2×#1835C2semidirect
6. #13925C2semidirect
7. #11293C2semidirect
8. #11292C2semidirect
11. #10850C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
C2×#19842
C2×#18352
C2×#18042
C2×C4×#1022
#139252
#112932
#112922
#111892
#108502