Finite Group Databasebeta

← #642002

Subgroups — #642002

Order 512.

2528 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic, complemented
2C2×#2290C211yesmaximal, characteristic
3#16894C214yesmaximal, complemented
4#16893C214yesmaximal
5#15235C211yesmaximal, characteristic, complemented
6#15223C211yesmaximal, characteristic, complemented
7#15218C211yesmaximal, characteristic, complemented
8#15216C211yesmaximal, characteristic, complemented
9#15059C214yesmaximal, complemented
10#15055C214yesmaximal, complemented
11#14266C212yesmaximal, complemented
12#14264C212yesmaximal, complemented
13#13951C212yesmaximal, complemented
14#13871C212yesmaximal
15#13756C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×#117C2212yes—
19C22×#117C2211yescharacteristic
20C23×C42C2211yescharacteristic
21#2851—28no—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25#2290—28no—
26C2×#388C2212yes—
27C2×#387C2214yes—
28C2×#387C2212yes—
29C2×#386C2212yes—
30C2×#385C2214yes—
31C2×#385C2212yes—
32C2×#382C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×#376C2211yescharacteristic
40#1445C2214yes—
41#1445C2214yes—
42#1444C2214yes—
43#1444C2214yes—
44#1443C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1440C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2214yes—
60#1437C2214yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2211yescharacteristic
67C4⋊#95C2212yes—
68C4⋊#95C2214yes—
69C4⋊#95C2212yes—
70#1434C2214yes—
71C22⋊#95C2214yes—
72C4×#95C2214yes—
73C23×D4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C2×#125—24no—
78C2×#125—44no—
79C2×C4×D4—24no—
80C2×C4×D4—24no—
81C2×C4×D4Q818yes—
82C2×#117—28no—
83C2×#117—28no—
84C2×#117—24no—
85C2×#117—44no—
86C2×#117—44no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C42—44no—
108C22×C42D418yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2314yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116#389—28no—
117#389—28no—
118#389—44no—
119#388—28no—
120#387—216no—
121#387—28no—
122#386—28no—
123#385—216no—
124#385—28no—
125#382—24no—
126#380—28no—
127#380—28no—
128#380—44no—
129C4×C4⋊C4—24no—
130C4×C22⋊C4—24no—
131C4×C22⋊C4—24no—
132#376—24no—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C25C2411yescharacteristic
158C25C2411yescharacteristic
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—42no—
162C22×D4—42no—
163C22×D4C2×Q814yes—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4C2×D414yes—
177C23×C4C2×D418yes—
178C23×C4C2×Q814yes—
179C23×C4C2×Q814yes—
180C23×C4#4114yes—
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2414yes—
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194#125—416no—
195C4×D4—232no—
196#117—416no—
197#117—432no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4C2×Q814yes—
230C2×C4⋊C4#4118yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4C2×Q814yes—
257C2×C22⋊C4C2×Q814yes—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—28no—
265C2×C42—28no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—416no—
269C2×C42—44no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42C2×D414yes—
273C2×C42C2×D414yes—
274C2×C42C2×D418yes—
275C2×C42C2×Q814yes—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—21no—
307C24—22no—
308C24—21no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24C22×D412yes—
315C24C22×Q812yes—
316C24C22×Q812yes—
317C24C2×#4112yes—
318C24C2511yescharacteristic
319C2×D4—28no—
320C2×D4—48no—
321C2×D4—48no—
322C2×D4—44no—
323C2×D4—44no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—216no—
329C22×C4—216no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—24no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—28no—
361C22×C4—216no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—48no—
407C22×C4C22⋊Q8116yes—
408C22×C4C22×D414yes—
409C22×C4C22×D414yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×Q812yes—
413C22×C4C22×Q812yes—
414C22×C4C22×Q812yes—
415C22×C4C22×Q812yes—
416C22×C4C22×Q812yes—
417C22×C4C2×#4118yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4114yes—
421C22×C4H(5,𝔽2)14yes—
422C4⋊C4—28no—
423C4⋊C4—216no—
424C4⋊C4—48no—
425C4⋊C4—416no—
426C4⋊C4—48no—
427C4⋊C4—416no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—88no—
432C4⋊C4—88no—
433C22⋊C4—28no—
434C22⋊C4—28no—
435C22⋊C4—416no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—416no—
439C22⋊C4—48no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C42—28no—
443C42—28no—
444C42—28no—
445C42—48no—
446C42—416no—
447C42—44no—
448C42—44no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—42no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23C2×C22⋊Q814yes—
491C23C23×D411yescharacteristic
492C23C23×Q811yescharacteristic
493C23C22×#4111yescharacteristic
494C23C22×#4112yes—
495C23C2×H(5,𝔽2)12yes—
496C23#58512yes—
497C23#58512yes—
498C23#58512yes—
499C23#58512yes—
500D4—416no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—232no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—432no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—416no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4C2×C22⋊Q814yes—
580C2×C4C2×C22⋊Q814yes—
581C2×C4#53214yes—
582C2×C4#53414yes—
583C2×C4#55118yes—
584C2×C4#55418yes—
585C22—24no—
586C22—24no—
587C22—24no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—44no—
598C22—44no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22C22×C22⋊Q811yescharacteristic
604C22C2×#53211yescharacteristic
605C22C2×#53411yescharacteristic
606C22#344912yes—
607C22#344912yes—
608C22#345511yescharacteristic
609C22#345511yescharacteristic
610C22#345812yes—
611C22C2×#55112yes—
612C22C2×#55412yes—
613C22#350411yescharacteristic
614C22#350411yescharacteristic
615C22#351212yes—
616C22#352112yes—
617C22#352112yes—
618C22#352412yes—
619C22#352412yes—
620C22#353912yes—
621C22#353912yes—
622C22#354012yes—
623C22#354012yes—
624C4—24no—
625C4—42no—
626C4—42no—
627C4—44no—
628C4—44no—
629C4—42no—
630C4—48no—
631C4—82no—
632C4—84no—
633C4—82no—
634C4—84no—
635C4—84no—
636C4#256418yes—
637C2—44no—
638C2—82no—
639C2#3463411yescharacteristic
640C2C2×#256411yescharacteristic
641C2#3521212yes—
642C2#3533311yescharacteristic
643C2#3744512yes—
644C2#3810612yes—
645C2#3810712yes—
646C2#3811812yes—
647C2#3811912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2851C2semidirect
3. #16894C2semidirect
5. #15235C2semidirect
6. #15223C2semidirect
7. #15218C2semidirect
8. #15216C2semidirect
9. #15059C2semidirect
10. #15055C2semidirect
11. #14266C2semidirect
12. #14264C2semidirect
13. #13951C2semidirect
15. #13756C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#22902
#168942
#168932
#152352
#152232
#152182
#152162
#150592
#150552
#142662
#142642
#139512
#138712
#137562