Finite Group Databasebeta

← #481660

Subgroups — #481660

Order 512.

1047 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35836C211yesmaximal, characteristic
2#20392C211yesmaximal, characteristic
3#17367C211yesmaximal, characteristic
4#11789C212yesmaximal, complemented
5#11788C212yesmaximal, complemented
6#11786C212yesmaximal, complemented
7#11784C212yesmaximal, complemented
8#11716C212yesmaximal, complemented
9C2×#1352C212yesmaximal, complemented
10C2×#554C411yescharacteristic, complemented
11C2×#518C2211yescharacteristic
12#2973—21no—
13C22×#99—21no—
14#2628C411yescharacteristic, complemented
15#2626C411yescharacteristic, complemented
16#2626C2211yescharacteristic
17#2580C2211yescharacteristic
18#2564C411yescharacteristic, complemented
19#2564C2211yescharacteristic
20#2542C2211yescharacteristic
21#2487C411yescharacteristic, complemented
22#2487C2211yescharacteristic
23#2454—21no—
24#2449—21no—
25#2449—21no—
26#2441—21no—
27#2437—21no—
28#2437—21no—
29#2396C411yescharacteristic, complemented
30C2×#399C411yescharacteristic, complemented
31C2×#399C2211yescharacteristic
32#2349C411yescharacteristic, complemented
33D4×C4⋊C4—21no—
34D4×C22⋊C4—21no—
35#2273—21no—
36C4×#99—21no—
37C4×#99—21no—
38C4×#99C2212yes—
39C4×#99C2212yes—
40C2×#342C2212yes—
41#1466C2212yes—
42#1465C2212yes—
43#1464—21no—
44#1462—21no—
45#1459—21no—
46C2×#323C2212yes—
47#1352C2214yescomplemented
48#1352C2214yescomplemented
49#1352C2214yescomplemented
50#1351C2214yescomplemented
51C23×D4D411yescharacteristic
52#554—44no—
53C2×C4⋊Q8C2×C411yescharacteristic, complemented
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2×C411yescharacteristic
60C2×C22⋊Q8C2×C411yescharacteristic, complemented
61#518—44no—
62C2×C4×Q8C2×C411yescharacteristic, complemented
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117D411yescharacteristic
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic, complemented
79C22×C22⋊C4—22no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4C2×C411yescharacteristic
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4C2311yescharacteristic
86#435—44no—
87#432—22no—
88C2×#99—28no—
89C2×#99—22no—
90C2×#99—22no—
91#407—22no—
92C2×C22⋊C8—22no—
93#400C2×C411yescharacteristic, complemented
94#400C2×C411yescharacteristic, complemented
95#400C2×C411yescharacteristic, complemented
96#399—21no—
97#399—21no—
98#399—21no—
99#399—21no—
100#399C2×C411yescharacteristic
101#399C2×C411yescharacteristic
102#399C2×C411yescharacteristic
103#399C2×C411yescharacteristic, complemented
104#399C2×C411yescharacteristic, complemented
105#399C2×C411yescharacteristic, complemented
106#399C2311yescharacteristic
107#399C2311yescharacteristic
108#399C2311yescharacteristic
109#398—21no—
110#396—21no—
111#396—21no—
112#396—21no—
113#396—21no—
114#396—21no—
115#395—21no—
116#395C2×C411yescharacteristic, complemented
117#393—21no—
118#393C2×C411yescharacteristic
119#393C2×C411yescharacteristic, complemented
120#393C2311yescharacteristic
121#390—21no—
122#390—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#387—21no—
129#387—21no—
130#386—21no—
131#386—21no—
132#386—21no—
133#386—21no—
134#385—21no—
135#385—21no—
136#385—21no—
137#385—21no—
138#385—21no—
139#385—21no—
140#385—21no—
141#385—21no—
142#385—21no—
143#385—21no—
144#384—21no—
145#382C2×C411yescharacteristic, complemented
146#380—21no—
147#380—21no—
148#380—21no—
149#380—21no—
150#380—21no—
151#380C2×C411yescharacteristic
152#380C2×C411yescharacteristic, complemented
153#379—21no—
154#379—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4C2×C411yescharacteristic, complemented
157C4×C22⋊C4—21no—
158C4×C22⋊C4—22no—
159C4×C22⋊C4—22no—
160C4×C22⋊C4—44no—
161C2×#95C2×C411yescharacteristic
162C2×#95C2×C411yescharacteristic, complemented
163#342—22no—
164#342C2314yes—
165#324—22no—
166#323—22no—
167#323—22no—
168#323C2314yes—
169C25C22⋊C411yescharacteristic
170C25C2×D411yescharacteristic
171C22×Q8—21no—
172C22×Q8C22×C411yescharacteristic
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—42no—
176C22×D4C22⋊C411yescharacteristic
177C22×D4C22⋊C411yescharacteristic
178C22×D4C2×D411yescharacteristic
179C22×D4C2×D411yescharacteristic
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—42no—
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C2×M16—42no—
195C22×C8—42no—
196C4⋊Q8—42no—
197#125—44no—
198C22⋊Q8—44no—
199C22⋊Q8—42no—
200C22⋊Q8—44no—
201C22⋊Q8—42no—
202C22⋊Q8—82no—
203C22⋊Q8—82no—
204C4×Q8—42no—
205C4×D4—24no—
206C4×D4—24no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—44no—
214C4×D4—44no—
215#117—42no—
216#117—42no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4—82no—
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
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—21no—
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—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—82no—
273C2×C22⋊C4—82no—
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C22⋊C4C22⋊C411yescharacteristic
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⋊C4C2×D411yescharacteristic
283C2×C22⋊C4C2×D411yescharacteristic
284C2×C22⋊C4C2×D411yescharacteristic
285C2×C22⋊C4C2411yescharacteristic
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42C22⋊C411yescharacteristic
291C2×C42C22⋊C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C4⋊C8—42no—
295#99—28no—
296#99—28no—
297C22⋊C8—24no—
298C22⋊C8—42no—
299C22⋊C8—42no—
300#97—42no—
301#95—21no—
302#95—21no—
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—42no—
315#95—42no—
316#95C22×C411yescharacteristic
317#95C22×C411yescharacteristic
318#95C22×C411yescharacteristic
319#95C22×C411yescharacteristic
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—41no—
325C24—81no—
326C24—81no—
327C24—81no—
328C24—81no—
329C24C2×C22⋊C411yescharacteristic
330C24C2×C22⋊C411yescharacteristic
331C24C2×C22⋊C411yescharacteristic
332C24C2×C22⋊C411yescharacteristic
333C24#13714yes—
334C24C22×D411yescharacteristic
335C2×Q8—41no—
336C2×Q8—42no—
337C2×Q8—81no—
338C2×Q8—81no—
339C2×Q8—81no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4#10414yes—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—48no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—162no—
396C22×C4—162no—
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C23×C411yescharacteristic
406C22×C4C22×D411yescharacteristic
407M16—162no—
408C2×C8—44no—
409C2×C8—162no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C4⋊C4—81no—
433C4⋊C4—81no—
434C4⋊C4—81no—
435C4⋊C4—81no—
436C4⋊C4—81no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—28no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—48no—
453C22⋊C4—42no—
454C22⋊C4—44no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—81no—
459C22⋊C4—81no—
460C22⋊C4—81no—
461C22⋊C4—81no—
462C22⋊C4—81no—
463C22⋊C4—81no—
464C22⋊C4—81no—
465C22⋊C4—81no—
466C22⋊C4—81no—
467C22⋊C4#9914yes—
468C42—22no—
469C42—22no—
470C42—41no—
471C42—41no—
472C42—42no—
473C42—42no—
474C42—162no—
475C42—162no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—24no—
483C23—24no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—41no—
488C23—41no—
489C23—44no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—82no—
499C23—82no—
500C23—84no—
501C23—84no—
502C23—161no—
503C23—161no—
504C23—162no—
505C23—162no—
506C23C2×#9912yes—
507C23#41912yes—
508C23#41912yes—
509C23C2×#10412yes—
510C23C2×#10412yes—
511C23C22×C22⋊C411yescharacteristic
512C23C2×#13712yes—
513Q8—82no—
514Q8—162no—
515D4—28no—
516D4—44no—
517D4—44no—
518D4—44no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—44no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—48no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—82no—
570C2×C4—84no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4—161no—
575C2×C4—161no—
576C2×C4—162no—
577C2×C4—162no—
578C2×C4—168no—
579C2×C4C2×#9912yes—
580C2×C4C2×#9912yes—
581C2×C4C2×#10412yes—
582C8—84no—
583C22—21no—
584C22—22no—
585C22—21no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—84no—
607C22—81no—
608C22—81no—
609C22—84no—
610C22—81no—
611C22—164no—
612C22#150814yes—
613C22C22×#9911yescharacteristic
614C22C2×#41911yescharacteristic
615C22C22×#10411yescharacteristic
616C4—22no—
617C4—22no—
618C4—22no—
619C4—42no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4—88no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—81no—
635C2—21no—
636C2—21no—
637C2—24no—
638C2—42no—
639C2—42no—
640C2—42no—
641C2—81no—
642C2C2×#150811yescharacteristic
643C2#3335011yescharacteristic
644C2#3335911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11789C2semidirect
5. #11788C2semidirect
6. #11786C2semidirect
7. #11784C2semidirect
8. #11716C2semidirect
9. C2×#1352C2semidirect
10. C2×#554C4semidirect
14. #2628C4semidirect
15. #2626C4semidirect
18. #2564C4semidirect
21. #2487C4semidirect
29. #2396C4semidirect
30. C2×#399C4semidirect
32. #2349C4semidirect
47. #1352C22semidirect
48. #1352C22semidirect
49. #1352C22semidirect
50. #1351C22semidirect
53. C2×C4⋊Q8C2×C4semidirect
60. C2×C22⋊Q8C2×C4semidirect
62. C2×C4×Q8C2×C4semidirect
78. C22×C4⋊C4C2×C4semidirect
93. #400C2×C4semidirect
94. #400C2×C4semidirect
95. #400C2×C4semidirect
103. #399C2×C4semidirect
104. #399C2×C4semidirect
105. #399C2×C4semidirect
116. #395C2×C4semidirect
119. #393C2×C4semidirect
145. #382C2×C4semidirect
152. #380C2×C4semidirect
156. C4×C4⋊C4C2×C4semidirect
162. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#358362
#203922
#173672
#117892
#117882
#117862
#117842
#117162
C2×#13522