Finite Group Databasebeta

← #482553

Subgroups — #482553

Order 512.

1018 conjugacy classes of subgroups.

The subgroup lattice has 725 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34875C211yesmaximal, characteristic
2C2×#1820C211yesmaximal, characteristic, complemented
3#17357C211yesmaximal, characteristic, complemented
4#11842C212yesmaximal, complemented
5#11840C212yesmaximal, complemented
6#11810C212yesmaximal, complemented
7#11798C212yesmaximal, complemented
8#11755C212yesmaximal, complemented
9#11753C212yesmaximal
10C2×C4×#41C411yescharacteristic, complemented
11C22×#117C2211yescharacteristic
12#2920—21no—
13#2609—21no—
14#2609—21no—
15#2599C411yescharacteristic, complemented
16#2599C2211yescharacteristic
17#2598C411yescharacteristic, complemented
18#2598C2211yescharacteristic
19#2579—21no—
20#2534—21no—
21#2484C411yescharacteristic, complemented
22#2482C2211yescharacteristic
23#2406C411yescharacteristic, complemented
24#2404C2211yescharacteristic
25#2396C411yescharacteristic, complemented
26#2396C2211yescharacteristic
27#2392C411yescharacteristic, complemented
28#2392C2211yescharacteristic
29C4×#123—21no—
30C4×#120—21no—
31#2290—21no—
32#2290—21no—
33#2283C411yescharacteristic, complemented
34C2×#381—21no—
35#1820—21no—
36#1820—21no—
37#1820C2212yes—
38#1820C2212yes—
39C2×#342—21no—
40C2×#342C2212yes—
41#1464—21no—
42#1464C2212yes—
43#1463—21no—
44#1462C2212yes—
45#1458—21no—
46#1458C2212yes—
47#1352C2214yes—
48#1351C2214yes—
49#1344C2214yes—
50#1343C2214yes—
51C22×#41C2×C411yescharacteristic
52C24×C4D411yescharacteristic
53C2×#123—21no—
54C2×#120—21no—
55C4×#41—41no—
56C4×#41—41no—
57C4×#41—41no—
58C4×#41—41no—
59C2×C4×Q8C2×C411yescharacteristic, complemented
60C2×C4×Q8C2×C411yescharacteristic, complemented
61C2×C4×D4—21no—
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4C2×C411yescharacteristic, complemented
66C2×C4×D4C2×C411yescharacteristic, complemented
67C2×#117—21no—
68C2×#117—41no—
69C2×#117—41no—
70C2×#117—41no—
71C2×#117—41no—
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117C2×C411yescharacteristic
75C2×#117D411yescharacteristic
76C2×#117D411yescharacteristic
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4D411yescharacteristic
83C22×C22⋊C4—22no—
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4C2311yescharacteristic
86C22×C42—21no—
87C22×C42C2×C411yescharacteristic
88C22×C42D411yescharacteristic
89#432—22no—
90#432—41no—
91#431—22no—
92#431—41no—
93#423—41no—
94#407—22no—
95C4×M16—41no—
96#400C2×C411yescharacteristic, complemented
97#400C2×C411yescharacteristic, complemented
98#399C2×C411yescharacteristic
99#399C2311yescharacteristic
100#398—21no—
101#398—21no—
102#398C2×C411yescharacteristic, complemented
103#398C2×C411yescharacteristic, complemented
104#397C2×C411yescharacteristic
105#397C2311yescharacteristic
106#396—21no—
107#396—21no—
108#396—21no—
109#396—21no—
110#395C2×C411yescharacteristic, complemented
111#395C2×C411yescharacteristic, complemented
112#394—21no—
113#394—21no—
114#394C2×C411yescharacteristic, complemented
115#394C2×C411yescharacteristic, complemented
116#393C2×C411yescharacteristic
117#393C2311yescharacteristic
118#392C2×C411yescharacteristic
119#392C2311yescharacteristic
120#388—21no—
121#388—21no—
122#388—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#387—21no—
127#387—21no—
128#387—21no—
129#387—21no—
130#387—21no—
131#387—21no—
132#386—21no—
133#386—21no—
134#386—21no—
135#386—21no—
136#386—21no—
137#386—21no—
138#386—21no—
139#386—21no—
140#385—21no—
141#385—21no—
142#385—21no—
143#385—21no—
144#385—21no—
145#385—21no—
146#385—21no—
147#385—21no—
148#383C2×C411yescharacteristic, complemented
149#383C2×C411yescharacteristic, complemented
150#382—21no—
151#381—21no—
152#381—22no—
153#381—22no—
154#381—41no—
155#381—41no—
156#381C2×C411yescharacteristic
157#381C2×C411yescharacteristic
158#380—21no—
159#379—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4C2×C411yescharacteristic, complemented
163C4×C4⋊C4C2×C411yescharacteristic, complemented
164C4×C22⋊C4—21no—
165C4×C22⋊C4—21no—
166C4×C22⋊C4—21no—
167C4×C22⋊C4—21no—
168C4×C22⋊C4—21no—
169#376—21no—
170C2×#95—41no—
171C2×#95—41no—
172#342—22no—
173#342—22no—
174#342—22no—
175#342C2314yes—
176#324—22no—
177#324—22no—
178#323—22no—
179#323C2314yes—
180C25C2×D411yescharacteristic
181C2×#41—41no—
182C2×#41—41no—
183C2×#41—44no—
184C22×Q8C22×C411yescharacteristic
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4C22×C411yescharacteristic
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4C22⋊C411yescharacteristic
200C23×C4C22⋊C411yescharacteristic
201C23×C4C22⋊C411yescharacteristic
202C23×C4C22⋊C411yescharacteristic
203C23×C4C22⋊C411yescharacteristic
204C23×C4C22⋊C411yescharacteristic
205C23×C4C22×C411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C2×M16—42no—
210#123—44no—
211#120—44no—
212C4×Q8—41no—
213C4×Q8—41no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×D4—41no—
217C4×D4—41no—
218C4×D4—41no—
219C4×D4—41no—
220C4×D4—44no—
221C4×D4—44no—
222C4×D4—42no—
223C4×D4—42no—
224C4×D4—41no—
225C4×D4—41no—
226C4×D4—41no—
227C4×D4—41no—
228#117—41no—
229#117—41no—
230#117—41no—
231#117—41no—
232#117—41no—
233#117—41no—
234#117—41no—
235#117—41no—
236#117—41no—
237#117—41no—
238#117—41no—
239#117—41no—
240#117—41no—
241#117—41no—
242#117—41no—
243#117—41no—
244#117—42no—
245#117—42no—
246#117—41no—
247#117—41no—
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—21no—
257C2×C4⋊C4—21no—
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⋊C4C2×D411yescharacteristic
271C2×C4⋊C4C2×D411yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
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—24no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—41no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4C22⋊C411yescharacteristic
298C2×C22⋊C4C22⋊C411yescharacteristic
299C2×C22⋊C4C22×C411yescharacteristic
300C2×C22⋊C4C2×D411yescharacteristic
301C2×C22⋊C4C2×D411yescharacteristic
302C2×C22⋊C4C2×D411yescharacteristic
303C2×C22⋊C4C2×D411yescharacteristic
304C2×C22⋊C4C2411yescharacteristic
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42C22⋊C411yescharacteristic
321C2×C42C22⋊C411yescharacteristic
322C2×C42C22⋊C411yescharacteristic
323C2×C42C22⋊C411yescharacteristic
324C2×C42C22×C411yescharacteristic
325C2×C42C22×C411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C4⋊C8—42no—
329C4⋊C8—42no—
330C22⋊C8—24no—
331C22⋊C8—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—42no—
353#95—42no—
354#95—42no—
355#95—42no—
356#95C22×C411yescharacteristic
357#95C22×C411yescharacteristic
358#95C22×C411yescharacteristic
359#95C22×C411yescharacteristic
360C24—21no—
361C24—22no—
362C24—21no—
363C24—22no—
364C24—21no—
365C24—42no—
366C24—81no—
367C24C2×C22⋊C411yescharacteristic
368C24C2×C22⋊C411yescharacteristic
369C24C22×D411yescharacteristic
370#41—44no—
371#41—44no—
372#41—44no—
373#41—44no—
374C2×Q8—42no—
375C2×Q8—41no—
376C2×D4—24no—
377C2×D4—41no—
378C2×D4—42no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—81no—
387C2×D4—81no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—44no—
440C22×C4—81no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—81no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4#9911yescharacteristic
448C22×C4#9911yescharacteristic
449C22×C4#9911yescharacteristic
450C22×C4#9911yescharacteristic
451C22×C4C2×C22⋊C411yescharacteristic
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C2×C22⋊C411yescharacteristic
455C22×C4C2×C22⋊C411yescharacteristic
456C22×C4C2×C22⋊C411yescharacteristic
457C22×C4C2×C22⋊C411yescharacteristic
458C22×C4C2×C22⋊C411yescharacteristic
459C22×C4C2×C22⋊C411yescharacteristic
460C22×C4C2×C22⋊C411yescharacteristic
461C22×C4C23×C411yescharacteristic
462C22×C4C22×D411yescharacteristic
463M16—84no—
464C2×C8—44no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—42no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—81no—
512C22⋊C4—81no—
513C22⋊C4—81no—
514C22⋊C4—81no—
515C22⋊C4—84no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—42no—
533C42—42no—
534C42—42no—
535C42—42no—
536C42—41no—
537C42—41no—
538C23—24no—
539C23—24no—
540C23—21no—
541C23—21no—
542C23—22no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—22no—
548C23—41no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—42no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—81no—
562C23—81no—
563C23—84no—
564C23—84no—
565C23C2×#9911yescharacteristic
566C23C2×#9911yescharacteristic
567C23#42112yes—
568C23C22×C22⋊C411yescharacteristic
569C23#57512yes—
570Q8—42no—
571Q8—42no—
572D4—44no—
573D4—42no—
574D4—42no—
575D4—44no—
576D4—84no—
577D4—84no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—48no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—82no—
645C2×C4—82no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4—81no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4—168no—
655C2×C4C2×#9911yescharacteristic
656C2×C4C2×#9911yescharacteristic
657C2×C4C2×#9911yescharacteristic
658C2×C4C2×#9911yescharacteristic
659C8—84no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—22no—
667C22—21no—
668C22—22no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—48no—
674C22—42no—
675C22—42no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—42no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—84no—
686C22—81no—
687C22—81no—
688C22#212012yes—
689C22C22×#9911yescharacteristic
690C22#289611yescharacteristic
691C22C2×#42111yescharacteristic
692C4—21no—
693C4—21no—
694C4—21no—
695C4—21no—
696C4—41no—
697C4—42no—
698C4—41no—
699C4—41no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—41no—
706C4—41no—
707C4—81no—
708C4—81no—
709C4—81no—
710C4—81no—
711C4—81no—
712C4—81no—
713C4—164no—
714C2—21no—
715C2—21no—
716C2—22no—
717C2—42no—
718C2—41no—
719C2—42no—
720C2—82no—
721C2#1758711yescharacteristic
722C2C2×#212011yescharacteristic
723C2#3335811yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#2404C22
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1820C2semidirect
3. #17357C2semidirect
4. #11842C2semidirect
5. #11840C2semidirect
6. #11810C2semidirect
7. #11798C2semidirect
8. #11755C2semidirect
10. C2×C4×#41C4semidirect
15. #2599C4semidirect
17. #2598C4semidirect
21. #2484C4semidirect
23. #2406C4semidirect
25. #2396C4semidirect
27. #2392C4semidirect
33. #2283C4semidirect
59. C2×C4×Q8C2×C4semidirect
60. C2×C4×Q8C2×C4semidirect
65. C2×C4×D4C2×C4semidirect
66. C2×C4×D4C2×C4semidirect
96. #400C2×C4semidirect
97. #400C2×C4semidirect
102. #398C2×C4semidirect
103. #398C2×C4semidirect
110. #395C2×C4semidirect
111. #395C2×C4semidirect
114. #394C2×C4semidirect
115. #394C2×C4semidirect
148. #383C2×C4semidirect
149. #383C2×C4semidirect
162. C4×C4⋊C4C2×C4semidirect
163. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#348752
C2×#18202
#173572
#118422
#118402
#118102
#117982
#117552
#117532