Finite Group Databasebeta

← #482498

Subgroups — #482498

Order 512.

927 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34971C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic, complemented
3C2×#1467C211yesmaximal, characteristic, complemented
4#11850C212yesmaximal, complemented
5#11848C212yesmaximal
6#11831C212yesmaximal, complemented
7#11829C212yesmaximal, complemented
8#11792C212yesmaximal, complemented
9#11778C212yesmaximal, complemented
10C2×C4×#41C411yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#2920—21no—
13C2×#432—21no—
14#2599—21no—
15#2599—21no—
16#2581C411yescharacteristic
17#2581C2211yescharacteristic
18#2536C411yescharacteristic
19#2536C2211yescharacteristic
20#2486—21no—
21#2484C411yescharacteristic
22#2484C2211yescharacteristic
23#2409—21no—
24#2406C411yescharacteristic
25#2406C2211yescharacteristic
26#2399C411yescharacteristic
27#2399C2211yescharacteristic
28#2398C411yescharacteristic
29#2398C2211yescharacteristic
30C4×C22⋊Q8—21no—
31C4×C22⋊Q8—21no—
32C4×#123—21no—
33C4×#123—21no—
34#2283C411yescharacteristic
35#1821C2212yes—
36#1801—21no—
37#1801C2212yes—
38#1784—21no—
39C4×#101—21no—
40C4×#101C2212yes—
41#1467—21no—
42#1467—21no—
43#1467C2212yes—
44#1467C2212yes—
45C2×#324—21no—
46C2×#324C2212yes—
47#1354C2214yes—
48#1353C2214yes—
49#1346C2214yes—
50#1345C2214yes—
51C22×#41C2×C411yescharacteristic
52C23×Q8D411yescharacteristic
53C2×#123—21no—
54C2×#123—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C4×#41—41no—
58C4×#41—41no—
59C4×#41—41no—
60C4×#41—41no—
61C2×C4×Q8—41no—
62C2×C4×Q8—41no—
63C2×C4×Q8—41no—
64C2×C4×Q8—41no—
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4C2×C411yescharacteristic
79C2×#117—21no—
80C2×#117C2×C411yescharacteristic
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C42—21no—
85C22×C42C2×C411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88#432—22no—
89#432—22no—
90#432—41no—
91#432—41no—
92#432—41no—
93#432—41no—
94#431—41no—
95#431—41no—
96#423—22no—
97C2×C4⋊C8—41no—
98C2×#101—22no—
99C2×#101—22no—
100#407—22no—
101C2×C22⋊C8—22no—
102C4×M16—22no—
103C2×#97—41no—
104#400C2×C411yescharacteristic
105#400C2×C411yescharacteristic
106#400C2×C411yescharacteristic
107#400C2311yescharacteristic
108#398—21no—
109#398—21no—
110#398C2×C411yescharacteristic
111#398C2×C411yescharacteristic
112#398C2×C411yescharacteristic
113#398C2311yescharacteristic
114#397—21no—
115#397—21no—
116#395—21no—
117#395—21no—
118#395C2×C411yescharacteristic
119#395C2×C411yescharacteristic
120#395C2×C411yescharacteristic
121#395C2311yescharacteristic
122#394C2×C411yescharacteristic
123#394C2×C411yescharacteristic
124#394C2×C411yescharacteristic
125#394C2311yescharacteristic
126#393—21no—
127#393—21no—
128#391—21no—
129#391—21no—
130#389—21no—
131#389—21no—
132#388—21no—
133#388—21no—
134#388—21no—
135#388—21no—
136#387—21no—
137#387—21no—
138#387—21no—
139#387—21no—
140#387—21no—
141#387—21no—
142#387—21no—
143#387—21no—
144#386—21no—
145#386—21no—
146#385—21no—
147#385—21no—
148#385—21no—
149#385—21no—
150#385—21no—
151#385—21no—
152#383—21no—
153#382C2×C411yescharacteristic
154#382C2×C411yescharacteristic
155#381—21no—
156#381—21no—
157#381C2×C411yescharacteristic
158#381C2×C411yescharacteristic
159#380—21no—
160#380—21no—
161#380—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4—21no—
167C4×C22⋊C4—21no—
168C4×C22⋊C4—21no—
169C4×C22⋊C4—21no—
170C4×C22⋊C4—21no—
171C4×C22⋊C4—21no—
172#376C2×C411yescharacteristic
173#376C2×C411yescharacteristic
174#343—22no—
175#343C2314yes—
176#324—22no—
177#324—22no—
178#324—22no—
179#324C2314yes—
180C2×#41—41no—
181C2×#41—41no—
182C2×#41—44no—
183C22×Q8—41no—
184C22×Q8—41no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×Q8C22⋊C411yescharacteristic
188C22×Q8C22⋊C411yescharacteristic
189C22×Q8C22×C411yescharacteristic
190C22×Q8C2×D411yescharacteristic
191C22×Q8C2×D411yescharacteristic
192C22×D4—21no—
193C22×D4C22×C411yescharacteristic
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4C22⋊C411yescharacteristic
198C23×C4C22⋊C411yescharacteristic
199C23×C4C22⋊C411yescharacteristic
200C23×C4C22⋊C411yescharacteristic
201C23×C4C22×C411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C2×M16—24no—
205C22×C8—42no—
206#123—44no—
207#123—44no—
208C22⋊Q8—44no—
209C22⋊Q8—44no—
210C4×Q8—41no—
211C4×Q8—41no—
212C4×Q8—41no—
213C4×Q8—41no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×Q8—41no—
217C4×Q8—41no—
218C4×Q8—41no—
219C4×Q8—41no—
220C4×Q8—41no—
221C4×Q8—41no—
222C4×Q8—41no—
223C4×Q8—41no—
224C4×Q8—41no—
225C4×Q8—41no—
226C4×Q8—41no—
227C4×Q8—41no—
228C4×Q8—41no—
229C4×Q8—41no—
230C4×D4—41no—
231C4×D4—41no—
232C4×D4—41no—
233C4×D4—41no—
234C4×D4—42no—
235C4×D4—42no—
236C4×D4—41no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4—41no—
240#117—42no—
241#117—42no—
242#117—41no—
243#117—41no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4C22⋊C411yescharacteristic
263C2×C4⋊C4C22⋊C411yescharacteristic
264C2×C4⋊C4C22⋊C411yescharacteristic
265C2×C4⋊C4C22⋊C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C4⋊C4C2×D411yescharacteristic
272C2×C4⋊C4C2×D411yescharacteristic
273C2×C4⋊C4C2×D411yescharacteristic
274C2×C4⋊C4C2×D411yescharacteristic
275C2×C4⋊C4C2411yescharacteristic
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—42no—
307C2×C42—42no—
308C2×C42C22⋊C411yescharacteristic
309C2×C42C22⋊C411yescharacteristic
310C2×C42C22⋊C411yescharacteristic
311C2×C42C22⋊C411yescharacteristic
312C2×C42C22⋊C411yescharacteristic
313C2×C42C22⋊C411yescharacteristic
314C2×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C4⋊C8—42no—
321C4⋊C8—42no—
322C4⋊C8—42no—
323C4⋊C8—42no—
324#101—44no—
325#101—44no—
326C22⋊C8—24no—
327C22⋊C8—42no—
328C22⋊C8—42no—
329C22⋊C8—42no—
330#97—42no—
331#97—42no—
332#97—42no—
333C4×C8—42no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95C22×C411yescharacteristic
357#95C22×C411yescharacteristic
358#95C22×C411yescharacteristic
359#95C22×C411yescharacteristic
360C24—21no—
361C24C2×C22⋊C411yescharacteristic
362#41—44no—
363#41—44no—
364#41—44no—
365#41—44no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—41no—
371C2×Q8—41no—
372C2×Q8—41no—
373C2×Q8—41no—
374C2×Q8—41no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8—42no—
379C2×Q8—41no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×D4—24no—
383C2×D4—41no—
384C2×D4—42no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4—42no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—42no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—42no—
427C22×C4#9911yescharacteristic
428C22×C4#9911yescharacteristic
429C22×C4#9911yescharacteristic
430C22×C4#9911yescharacteristic
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C2×C22⋊C411yescharacteristic
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C23×C411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445M16—44no—
446M16—44no—
447C2×C8—44no—
448C2×C8—44no—
449C2×C8—84no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—42no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42—42no—
520C42—41no—
521C42—41no—
522C23—22no—
523C23—21no—
524C23—21no—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23C2×#9911yescharacteristic
534C23C2×#9911yescharacteristic
535C23C22×C22⋊C411yescharacteristic
536Q8—44no—
537Q8—44no—
538Q8—44no—
539Q8—42no—
540Q8—44no—
541Q8—42no—
542D4—44no—
543D4—42no—
544D4—42no—
545D4—44no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—48no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—82no—
613C2×C4—84no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—82no—
617C2×C4C2×#9911yescharacteristic
618C2×C4C2×#9911yescharacteristic
619C2×C4C2×#9911yescharacteristic
620C2×C4C2×#9911yescharacteristic
621C2×C4#42112yes—
622C2×C4#57512yes—
623C8—84no—
624C8—84no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22#212012yes—
642C22C22×#9911yescharacteristic
643C22#289611yescharacteristic
644C22C2×#42111yescharacteristic
645C4—21no—
646C4—21no—
647C4—21no—
648C4—22no—
649C4—22no—
650C4—41no—
651C4—41no—
652C4—42no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4—81no—
667C4#213312yes—
668C2—21no—
669C2—21no—
670C2—22no—
671C2—41no—
672C2—82no—
673C2#1758211yescharacteristic
674C2C2×#212011yescharacteristic
675C2C2×#213311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20727C2semidirect
3. C2×#1467C2semidirect
4. #11850C2semidirect
6. #11831C2semidirect
7. #11829C2semidirect
8. #11792C2semidirect
9. #11778C2semidirect

Maximal subgroups

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

subgroupindex
#349712
#207272
C2×#14672
#118502
#118482
#118312
#118292
#117922
#117782