Finite Group Databasebeta

← #401809

Subgroups — #401809

Order 512.

1542 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2360C211yesmaximal, characteristic
2C2×#2083C211yesmaximal, characteristic, complemented
3C2×#2036C212yesmaximal
4C2×#1803C211yesmaximal, characteristic, complemented
5#15195C211yesmaximal, characteristic
6#11228C211yesmaximal, characteristic, complemented
7#11190C212yesmaximal, complemented
8#11042C211yesmaximal, characteristic
9#10907C212yesmaximal, complemented
10#10789C211yesmaximal, characteristic, complemented
11#10730C212yesmaximal
12C22×C22⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2212yes—
20#2360—24no—
21#2360—24no—
22C2×#391C2211yescharacteristic
23C2×#389—21no—
24C2×#389C2211yescharacteristic
25C2×#387—21no—
26C2×#385—21no—
27C2×#384C2211yescharacteristic
28C2×#380C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4—21no—
31C2×C4×C22⋊C4C2211yescharacteristic
32#2083—24no—
33#2036—28no—
34#1803—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1442—21no—
43#1442—21no—
44#1439—21no—
45#1439—21no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2212yes—
48C23×Q8C2311yescharacteristic
49C24×C4—21no—
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8D414yes—
59C2×C4×Q8—22no—
60C2×C4×Q8—24no—
61C2×C4×Q8—42no—
62C2×C4×Q8D414yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—42no—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4D411yescharacteristic
81C22×C22⋊C4D411yescharacteristic
82C22×C42—21no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#107—24no—
87C2×#107—24no—
88C2×#107—24no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—42no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—42no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—42no—
100#391—22no—
101#391—22no—
102#389—28no—
103#389—22no—
104#389D414yes—
105#387—28no—
106#385—28no—
107#384—22no—
108#384—22no—
109#380—24no—
110#380—24no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C22⋊C4—28no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95D411yescharacteristic
126C2×#95D411yescharacteristic
127#340—24no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340—42no—
132#336—24no—
133#336—22no—
134C25C2×D411yescharacteristic
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8C2×D412yes—
143C22×Q8C2×D412yes—
144C22×Q8#4112yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—44no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—82no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4#4112yes—
171C23×C4C2411yescharacteristic
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8#4114yes—
186C22⋊Q8—28no—
187C22⋊Q8—28no—
188C22⋊Q8—28no—
189C22⋊Q8—28no—
190C4×Q8—28no—
191C4×Q8—44no—
192C4×Q8—48no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
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—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—48no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4#4112yes—
234C2×C4⋊C4#4114yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—82no—
245C2×C22⋊C4—82no—
246C2×C22⋊C4Q1614yes—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—24no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—24no—
255C2×C42—44no—
256C2×C42—82no—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259#107—48no—
260C4⋊C8—44no—
261#103—48no—
262#103—48no—
263C22⋊C8—48no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—22no—
269#95—22no—
270#95—42no—
271#95—44no—
272#95—42no—
273#95—82no—
274#95—82no—
275C24—21no—
276C24—21no—
277C24—22no—
278C24—22no—
279C24—21no—
280C24—21no—
281C24—22no—
282C24—22no—
283C24#12112yes—
284C24C2×Q1612yes—
285C24#13712yes—
286C24C22×D411yescharacteristic
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8#12114yes—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—48no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—48no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—84no—
379C22×C4—88no—
380C22×C4—88no—
381C22×C4#12012yes—
382C22×C4#12012yes—
383C22×C4#12112yes—
384C22×C4C2×Q1612yes—
385C22×C4C2×Q1612yes—
386C22×C4#13712yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4112yes—
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4112yes—
395C22×C4H(5,𝔽2)12yes—
396C22×C42−1+412yes—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C4⋊C4—216no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4#12114yes—
434C4⋊C4#12314yes—
435C22⋊C4—216no—
436C22⋊C4—28no—
437C22⋊C4—44no—
438C22⋊C4—88no—
439C42—22no—
440C42—22no—
441C42—24no—
442C42—48no—
443C42—41no—
444C42—41no—
445C42—42no—
446C42—84no—
447C23—24no—
448C23—24no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—41no—
463C23—41no—
464C23—42no—
465C23—41no—
466C23—41no—
467C23—42no—
468C23—42no—
469C23#46214yes—
470C23C2×#12011yescharacteristic
471C23C2×#12111yescharacteristic
472C23#52511yescharacteristic
473C23#54612yes—
474C23#54712yes—
475C23C22×Q1611yescharacteristic
476C23C2×#13711yescharacteristic
477C23C2×#13711yescharacteristic
478C23#57511yescharacteristic
479C23#57612yes—
480C23#57712yes—
481Q8—24no—
482Q8—24no—
483Q8—42no—
484Q8—42no—
485Q8—44no—
486Q8—44no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—44no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—816no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4#45114yes—
553C2×C4C2×#12111yescharacteristic
554C2×C4C2×#12111yescharacteristic
555C2×C4C2×#12311yescharacteristic
556C2×C4#52511yescharacteristic
557C2×C4#52511yescharacteristic
558C2×C4#52912yes—
559C2×C4#52912yes—
560C2×C4#53211yescharacteristic
561C2×C4#53411yescharacteristic
562C2×C4#53611yescharacteristic
563C2×C4#53912yes—
564C2×C4#54212yes—
565C2×C4#54612yes—
566C2×C4#54612yes—
567C2×C4#54712yes—
568C2×C4#54812yes—
569C2×C4#55212yes—
570C2×C4#55312yes—
571C8—44no—
572C8—82no—
573C22—28no—
574C22—24no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22—42no—
579C22—44no—
580C22—42no—
581C22—42no—
582C22—44no—
583C22—42no—
584C22#240511yescharacteristic
585C22#242512yes—
586C22#243211yescharacteristic
587C22#246812yes—
588C22#284711yescharacteristic
589C22C2×#45111yescharacteristic
590C22#300711yescharacteristic
591C22#300811yescharacteristic
592C22#301011yescharacteristic
593C22C2×#46211yescharacteristic
594C22#303711yescharacteristic
595C22#304512yes—
596C22#310411yescharacteristic
597C22#311311yescharacteristic
598C22#311912yes—
599C22#329912yes—
600C22#331412yes—
601C22#332012yes—
602C22#332212yes—
603C22#333812yes—
604C22#334412yes—
605C22#334812yes—
606C22#335712yes—
607C4—24no—
608C4—42no—
609C4—44no—
610C4—42no—
611C4—44no—
612C4—42no—
613C4—44no—
614C4—81no—
615C4—82no—
616C4—162no—
617C4—162no—
618C4#239511yescharacteristic
619C4#242712yes—
620C4#246012yes—
621C4#248611yescharacteristic
622C4#254311yescharacteristic
623C4#263811yescharacteristic
624C2—44no—
625C2#1482411yescharacteristic
626C2#2404211yescharacteristic
627C2#2404911yescharacteristic
628C2#2520612yes—
629C2#2522212yes—
630C2#2570611yescharacteristic
631C2#2571511yescharacteristic
632C2#2710012yes—
633C2#2711712yes—
634C2#3290611yescharacteristic
635C2#3291311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2083C2semidirect
4. C2×#1803C2semidirect
6. #11228C2semidirect
7. #11190C2semidirect
9. #10907C2semidirect
10. #10789C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23602
C2×#20832
C2×#20362
C2×#18032
#151952
#112282
#111902
#110422
#109072
#107892
#107302