Finite Group Databasebeta

← #214730

Subgroups — #214730

Order 512.

1116 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35703C211yesmaximal, characteristic
2#20633C211yesmaximal, characteristic, complemented
3#17350C211yesmaximal, characteristic, complemented
4#9467C212yesmaximal, complemented
5#9384C212yesmaximal, complemented
6#9203C212yesmaximal, complemented
7#9013C212yesmaximal, complemented
8#8762C212yesmaximal, complemented
9#8377C212yesmaximal, complemented
10C2×#548C2211yescharacteristic
11C2×#518C2211yescharacteristic
12#2972C2212yes—
13C22×#99—21no—
14#2638C411yescharacteristic, complemented
15#2631C411yescharacteristic, complemented
16#2617C2211yescharacteristic
17#2610C411yescharacteristic, complemented
18#2599C411yescharacteristic, complemented
19#2561C411yescharacteristic, complemented
20#2543C2211yescharacteristic
21#2477—21no—
22#2472—21no—
23#2463—21no—
24#2459—21no—
25#2444—21no—
26#2410—21no—
27#2409C2211yescharacteristic
28#2404C411yescharacteristic, complemented
29#2398C411yescharacteristic, complemented
30#2388C411yescharacteristic, complemented
31C2×#393C2211yescharacteristic
32#2364—21no—
33#2358—21no—
34#2353C2211yescharacteristic
35#2315C2212yes—
36#1799—21no—
37#1799—21no—
38#1799C2212yes—
39#1799C2212yes—
40#1787C2214yescomplemented
41C2×#342—21no—
42#1519C2214yescomplemented
43#1460—21no—
44#1459—21no—
45#1459C2212yes—
46#1458—21no—
47C2×#323—21no—
48C2×#323C2212yes—
49#1294C2214yescomplemented
50#1289C2214yescomplemented
51C22×#41C2×C411yescharacteristic, complemented
52C23×Q8C2311yescharacteristic
53C23×D4D411yescharacteristic
54#548—44no—
55C2×#124—21no—
56C2×#124C2×C411yescharacteristic, complemented
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2×C411yescharacteristic, complemented
60C2×#121—21no—
61C2×#121—21no—
62C2×#121C2311yescharacteristic
63C2×#120—21no—
64C2×#120—21no—
65#518—44no—
66C2×C4×Q8C2×C411yescharacteristic, complemented
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×#117C2×C411yescharacteristic
78C2×#117D411yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4C2×C411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4C2311yescharacteristic
85C8×D4—44no—
86#431—22no—
87C2×#103—44no—
88C2×#99—22no—
89C2×#99—22no—
90C2×#99—44no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8C2314yes—
93#400—21no—
94#398C2×C411yescharacteristic, complemented
95#397—21no—
96#397—21no—
97#397C2×C411yescharacteristic
98#397C2×C411yescharacteristic
99#397C2×C411yescharacteristic, complemented
100#395C2×C411yescharacteristic, complemented
101#395C2×C411yescharacteristic, complemented
102#394—21no—
103#394—21no—
104#394—21no—
105#394—21no—
106#394—21no—
107#394C2×C411yescharacteristic, complemented
108#393—21no—
109#393—21no—
110#393C2×C411yescharacteristic
111#393C2×C411yescharacteristic
112#393C2×C411yescharacteristic
113#393C2×C411yescharacteristic
114#393C2×C411yescharacteristic, complemented
115#393C2×C411yescharacteristic, complemented
116#392—21no—
117#392—21no—
118#392C2×C411yescharacteristic
119#392C2×C411yescharacteristic
120#392C2×C411yescharacteristic, complemented
121#391—21no—
122#390—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#388—21no—
129#388—21no—
130#387—21no—
131#387—21no—
132#387—21no—
133#387—21no—
134#386—21no—
135#386—21no—
136#386—21no—
137#386—21no—
138#386—21no—
139#386—21no—
140#386—21no—
141#386—21no—
142#386—21no—
143#386—21no—
144#386—21no—
145#385—21no—
146#385—21no—
147#385—21no—
148#385—44no—
149#383C2×C411yescharacteristic, complemented
150#381—21no—
151#381C2×C411yescharacteristic, complemented
152#380—22no—
153#380C2314yes—
154#380C2311yescharacteristic
155#379—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4C2×C411yescharacteristic, complemented
161#376C2×C411yescharacteristic, complemented
162C2×#95—21no—
163C2×#95C2311yescharacteristic
164#342—22no—
165#342—22no—
166#327—44no—
167#324—22no—
168#323—22no—
169#323—22no—
170#323—22no—
171C25C22⋊C411yescharacteristic
172C25C2×D411yescharacteristic
173C2×#41—42no—
174C2×#41—82no—
175C22×Q8—21no—
176C22×Q8—24no—
177C22×Q8—82no—
178C22×Q8C22×C411yescharacteristic
179C22×Q8C22×C411yescharacteristic
180C22×D4—24no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—21no—
184C22×D4—21no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×D4—42no—
189C22×D4C22⋊C411yescharacteristic
190C22×D4C22⋊C411yescharacteristic
191C22×D4C22×C411yescharacteristic
192C22×D4C2×D411yescharacteristic
193C22×D4C2×D411yescharacteristic
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—42no—
198C23×C4C22⋊C411yescharacteristic
199C23×C4C22⋊C411yescharacteristic
200C23×C4C22⋊C411yescharacteristic
201C23×C4C22⋊C411yescharacteristic
202C23×C4C22⋊C411yescharacteristic
203C23×C4C22×C411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2411yescharacteristic
207C22×C8—24no—
208C22×C8—42no—
209#124—42no—
210#124—44no—
211C22⋊Q8—42no—
212C22⋊Q8—44no—
213C22⋊Q8—82no—
214#121—24no—
215#121—44no—
216#121—42no—
217#121—42no—
218#120—44no—
219#120—82no—
220C4×Q8—42no—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—42no—
224C4×D4—42no—
225C4×D4—42no—
226C4×D4—42no—
227C4×D4—42no—
228C4×D4—42no—
229C4×D4—44no—
230C4×D4—82no—
231#117—42no—
232#117—42no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C22⋊C4—24no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—82no—
276C2×C22⋊C4C22⋊C411yescharacteristic
277C2×C22⋊C4C22⋊C411yescharacteristic
278C2×C22⋊C4C22⋊C411yescharacteristic
279C2×C22⋊C4C22⋊C411yescharacteristic
280C2×C22⋊C4C22×C411yescharacteristic
281C2×C22⋊C4C22×C411yescharacteristic
282C2×C22⋊C4C22×C411yescharacteristic
283C2×C22⋊C4C2×D411yescharacteristic
284C2×C22⋊C4C2×D411yescharacteristic
285C2×C22⋊C4C2×D411yescharacteristic
286C2×C22⋊C4C2×D411yescharacteristic
287C2×C22⋊C4C2×D411yescharacteristic
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42C22⋊C411yescharacteristic
292C2×C42C22⋊C411yescharacteristic
293C2×C42C22×C411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C4⋊C8—42no—
296#103—88no—
297#99—44no—
298#99—44no—
299C22⋊C8—24no—
300C22⋊C8—24no—
301C22⋊C8—42no—
302C4×C8—42no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—24no—
321#95—42no—
322#95C22×C411yescharacteristic
323#95C22×C411yescharacteristic
324#95C22×C411yescharacteristic
325#95C22×C411yescharacteristic
326C24—22no—
327C24—21no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—21no—
332C24—41no—
333C24—81no—
334C24—81no—
335C24—81no—
336C24—81no—
337C24C2×C22⋊C411yescharacteristic
338C24C2×C22⋊C411yescharacteristic
339C24C2×C22⋊C411yescharacteristic
340C24C2×C22⋊C411yescharacteristic
341C24#13514yes—
342C24C22×D411yescharacteristic
343#41—84no—
344#41—84no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—42no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—48no—
351C2×Q8—81no—
352C2×Q8—81no—
353C2×Q8—84no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—42no—
368C2×D4—48no—
369C2×D4—48no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—81no—
379C2×D4—81no—
380C2×D4—81no—
381C2×D4—81no—
382C2×D4—81no—
383C2×D4—81no—
384C2×D4—81no—
385C2×D4—81no—
386C2×D4AGL(1,ℤ/8)12yes—
387C2×D4#13712yes—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—44no—
412C22×C4—81no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—84no—
425C22×C4—162no—
426C22×C4—162no—
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C23×C411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4H(5,𝔽2)14yes—
438C2×C8—44no—
439C2×C8—84no—
440C2×C8—162no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C4⋊C4—84no—
453C4⋊C4—162no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—81no—
481C22⋊C4—81no—
482C22⋊C4—81no—
483C22⋊C4—81no—
484C22⋊C4—81no—
485C22⋊C4—81no—
486C22⋊C4—81no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C22⋊C4—84no—
490C42—22no—
491C42—22no—
492C42—41no—
493C42—41no—
494C42—42no—
495C42—81no—
496C42—81no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—24no—
504C23—24no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—41no—
509C23—41no—
510C23—44no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—82no—
527C23—82no—
528C23—84no—
529C23—161no—
530C23—161no—
531C23—161no—
532C23—161no—
533C23—162no—
534C23—162no—
535C23#41612yes—
536C23#41612yes—
537C23#41712yes—
538C23C22×C22⋊C411yescharacteristic
539C23#53412yes—
540C23C2×#13512yes—
541C23C2×AGL(1,ℤ/8)11yescharacteristic
542C23C2×#13711yescharacteristic
543Q8—24no—
544Q8—42no—
545Q8—44no—
546Q8—84no—
547D4—24no—
548D4—24no—
549D4—44no—
550D4—44no—
551D4—42no—
552D4—42no—
553D4—82no—
554D4—84no—
555D4—84no—
556D4—82no—
557D4—82no—
558D4—162no—
559D4—162no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—22no—
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—44no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
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—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—161no—
611C2×C4—161no—
612C2×C4—161no—
613C2×C4—161no—
614C2×C4—161no—
615C2×C4—161no—
616C2×C4—168no—
617C2×C4#41712yes—
618C2×C4#51812yes—
619C2×C4#53412yes—
620C8—84no—
621C22—21no—
622C22—22no—
623C22—21no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—84no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—84no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—164no—
658C22#230811yescharacteristic
659C22C2×#41611yescharacteristic
660C22C2×#41711yescharacteristic
661C22#318812yes—
662C22#319012yes—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—41no—
667C4—42no—
668C4—41no—
669C4—41no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—41no—
674C4—81no—
675C4—81no—
676C4—81no—
677C4—81no—
678C4—81no—
679C4—81no—
680C4—164no—
681C2—21no—
682C2—21no—
683C2—22no—
684C2—22no—
685C2—42no—
686C2—42no—
687C2—42no—
688C2—81no—
689C2—161no—
690C2#2176311yescharacteristic
691C2#2177611yescharacteristic
692C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35703C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20633C2semidirect
3. #17350C2semidirect
4. #9467C2semidirect
5. #9384C2semidirect
6. #9203C2semidirect
7. #9013C2semidirect
8. #8762C2semidirect
9. #8377C2semidirect
14. #2638C4semidirect
15. #2631C4semidirect
17. #2610C4semidirect
18. #2599C4semidirect
19. #2561C4semidirect
28. #2404C4semidirect
29. #2398C4semidirect
30. #2388C4semidirect
40. #1787C22semidirect
42. #1519C22semidirect
49. #1294C22semidirect
50. #1289C22semidirect
51. C22×#41C2×C4semidirect
56. C2×#124C2×C4semidirect
59. C2×C22⋊Q8C2×C4semidirect
66. C2×C4×Q8C2×C4semidirect
94. #398C2×C4semidirect
99. #397C2×C4semidirect
100. #395C2×C4semidirect
101. #395C2×C4semidirect
107. #394C2×C4semidirect
114. #393C2×C4semidirect
115. #393C2×C4semidirect
120. #392C2×C4semidirect
149. #383C2×C4semidirect
151. #381C2×C4semidirect
160. C4×C22⋊C4C2×C4semidirect
161. #376C2×C4semidirect

Maximal subgroups

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

subgroupindex
#357032
#206332
#173502
#94672
#93842
#92032
#90132
#87622
#83772