Finite Group Databasebeta

← #495410

Subgroups — #495410

Order 512.

1151 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35349C211yesmaximal, characteristic
2#20638C211yesmaximal, characteristic
3#20392C211yesmaximal, characteristic
4#13297C212yesmaximal, complemented
5#13297C212yesmaximal, complemented
6#13296C212yesmaximal, complemented
7#13295C212yesmaximal, complemented
8#12574C212yesmaximal
9C2×#1390C212yesmaximal
10C22×C22⋊Q8C411yescharacteristic, complemented
11C2×#518C2211yescharacteristic
12#2905—21no—
13C22×#100C2212yes—
14C22×#99C2212yes—
15#2579C411yescharacteristic, complemented
16#2579C411yescharacteristic, complemented
17#2579C2211yescharacteristic
18#2579C2211yescharacteristic
19#2487C411yescharacteristic, complemented
20#2487C2211yescharacteristic
21#2450—21no—
22#2450—21no—
23#2449—21no—
24#2449—21no—
25#2438—21no—
26#2438—21no—
27#2437—21no—
28#2437—21no—
29#2409C411yescharacteristic, complemented
30#2409C2211yescharacteristic
31#2315C411yescharacteristic, complemented
32#2315C411yescharacteristic, complemented
33#2308C2211yescharacteristic
34C2×#386C2211yescharacteristic
35C2×#385C411yescharacteristic, complemented
36#2273—21no—
37#1800—21no—
38#1800—21no—
39#1800C2212yes—
40#1800C2212yes—
41C4×#99—21no—
42C4×#99—21no—
43C4×#99C2212yes—
44C4×#99C2212yes—
45#1742—21no—
46C2×#342—21no—
47#1390C2214yes—
48#1390C2214yes—
49#1390C2214yes—
50#1390C2214yes—
51C23×Q8C2×C411yescharacteristic, complemented
52C23×D4C2311yescharacteristic
53C24×C4C2×C411yescharacteristic
54C22×M16—22no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—44no—
62#518—44no—
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×#117D411yescharacteristic
72C2×#117D411yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2×C411yescharacteristic, complemented
75C22×C4⋊C4C2×C411yescharacteristic, complemented
76C22×C4⋊C4C2×C411yescharacteristic, complemented
77C22×C22⋊C4—22no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C22⋊C4C2×C411yescharacteristic
82C22×C22⋊C4C2×C411yescharacteristic
83C22×C22⋊C4C2×C411yescharacteristic
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86#424—44no—
87#423—22no—
88#423—22no—
89C2×#100—28no—
90C2×#100—22no—
91C2×#100C2314yes—
92C2×#99—22no—
93C2×#99—24no—
94C2×#99Q818yescomplemented
95C2×#99C2314yes—
96#400—21no—
97#400—21no—
98#399—21no—
99#399—21no—
100#398—21no—
101#398—21no—
102#398—21no—
103#398—21no—
104#396—21no—
105#396—21no—
106#396—21no—
107#396—21no—
108#395—21no—
109#395—21no—
110#393—21no—
111#393—21no—
112#391—21no—
113#391—21no—
114#389—21no—
115#389—21no—
116#387—21no—
117#387—21no—
118#387—21no—
119#387—21no—
120#386—21no—
121#386—21no—
122#386—21no—
123#386—21no—
124#386C2×C411yescharacteristic
125#386C2×C411yescharacteristic
126#386C2×C411yescharacteristic
127#386C2×C411yescharacteristic
128#386C2311yescharacteristic
129#386C2311yescharacteristic
130#386C2311yescharacteristic
131#386C2311yescharacteristic
132#385—21no—
133#385—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#385C2×C411yescharacteristic, complemented
145#385C2×C411yescharacteristic, complemented
146#385C2×C411yescharacteristic, complemented
147#385C2×C411yescharacteristic, complemented
148#385C2×C411yescharacteristic, complemented
149#385C2×C411yescharacteristic, complemented
150#385C2×C411yescharacteristic, complemented
151#385C2×C411yescharacteristic, complemented
152#381—21no—
153#380—21no—
154#380—21no—
155#380—21no—
156#380—21no—
157#380C2×C411yescharacteristic, complemented
158#380C2×C411yescharacteristic, complemented
159#380C2×C411yescharacteristic, complemented
160#380C2×C411yescharacteristic, complemented
161#379—21no—
162C4×C22⋊C4—22no—
163C4×C22⋊C4—22no—
164C4×C22⋊C4—44no—
165C2×#95—21no—
166C2×#95—21no—
167#343—22no—
168#343—22no—
169#342—22no—
170#342—22no—
171C25C22×C411yescharacteristic
172C25C2×D411yescharacteristic
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—42no—
176C22×Q8—82no—
177C22×D4—42no—
178C22×D4—42no—
179C22×D4C22×C411yescharacteristic
180C22×D4C2×D411yescharacteristic
181C22×D4C2×D411yescharacteristic
182C22×D4C2×Q814yes—
183C22×D4#4114yes—
184C22×D4C2411yescharacteristic
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—82no—
192C23×C4C22⋊C411yescharacteristic
193C23×C4C22⋊C411yescharacteristic
194C23×C4C22⋊C411yescharacteristic
195C23×C4C22⋊C411yescharacteristic
196C23×C4C22×C411yescharacteristic
197C23×C4C22×C411yescharacteristic
198C23×C4C22×C411yescharacteristic
199C23×C4C22×C411yescharacteristic
200C23×C4C22×C411yescharacteristic
201C23×C4C22×C411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C2×M16—24no—
206C2×M16—42no—
207C2×M16—44no—
208C2×M16—44no—
209C22×C8—42no—
210C22⋊Q8—44no—
211C22⋊Q8—44no—
212C22⋊Q8—44no—
213C22⋊Q8—44no—
214C22⋊Q8—82no—
215C22⋊Q8—82no—
216C4×D4—42no—
217C4×D4—42no—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—42no—
223C4×D4—42no—
224#117—24no—
225#117—42no—
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—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—82no—
248C2×C4⋊C4C22⋊C411yescharacteristic
249C2×C4⋊C4C22⋊C411yescharacteristic
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
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—21no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
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⋊C4C22⋊C411yescharacteristic
284C2×C22⋊C4C22×C411yescharacteristic
285C2×C22⋊C4C22×C411yescharacteristic
286C2×C22⋊C4C22×C411yescharacteristic
287C2×C22⋊C4C22×C411yescharacteristic
288C2×C22⋊C4C22×C411yescharacteristic
289C2×C22⋊C4C22×C411yescharacteristic
290C2×C22⋊C4C2×D411yescharacteristic
291C2×C22⋊C4C2×D411yescharacteristic
292C2×C42—42no—
293C2×C42—42no—
294C2×C42C22⋊C411yescharacteristic
295C2×C42C22⋊C411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C4⋊C8—44no—
299C4⋊C8—44no—
300#100—28no—
301#100—44no—
302#99—28no—
303#99—24no—
304#99C2×Q818yes—
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—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—42no—
326#95—42no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—81no—
332C24—81no—
333C24—81no—
334C24—81no—
335C24C2×C22⋊C411yescharacteristic
336C24C2×C22⋊C411yescharacteristic
337C24C2×C22⋊C411yescharacteristic
338C24C4×Q814yes—
339C24C22⋊Q814yes—
340C24C23×C411yescharacteristic
341C24C22×D411yescharacteristic
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—81no—
347C2×Q8—81no—
348C2×Q8—84no—
349C2×Q8—84no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—41no—
362C2×D4C4×Q814yes—
363C2×D4C22⋊Q814yes—
364C2×D4#12412yes—
365C2×D4C4⋊Q812yes—
366C2×D4C22×Q812yes—
367C2×D4C2×#4112yes—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—48no—
395C22×C4—42no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—81no—
400C22×C4—81no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4—81no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4—162no—
419C22×C4—162no—
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4#12412yes—
430C22×C4C4⋊Q812yes—
431C22×C4C22×D411yescharacteristic
432M16—44no—
433M16—44no—
434M16—84no—
435C2×C8—44no—
436C2×C8—84no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—81no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—28no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—44no—
474C22⋊C4—48no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—81no—
484C22⋊C4—81no—
485C22⋊C4—81no—
486C22⋊C4—81no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C22⋊C4#9914yes—
490C42—22no—
491C42—22no—
492C42—41no—
493C42—41no—
494C42—162no—
495C42—162no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—82no—
519C23—82no—
520C23—82no—
521C23—82no—
522C23—84no—
523C23—84no—
524C23—84no—
525C23—162no—
526C23#39112yes—
527C23#39112yes—
528C23#39112yes—
529C23C2×#9912yes—
530C23#41012yes—
531C23C22×C22⋊C411yescharacteristic
532C23C2×C4×Q812yes—
533C23C2×C22⋊Q812yes—
534Q8—84no—
535Q8—84no—
536D4—44no—
537D4—44no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—42no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—48no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
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—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—161no—
602C2×C4—161no—
603C2×C4—161no—
604C2×C4—161no—
605C2×C4—162no—
606C2×C4—168no—
607C2×C4#39112yes—
608C2×C4C2×#9912yes—
609C2×C4C2×#9912yes—
610C2×C4C2×#12411yescharacteristic
611C2×C4C2×C4⋊Q811yescharacteristic
612C8—84no—
613C22—21no—
614C22—22no—
615C22—21no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—84no—
638C22—81no—
639C22—81no—
640C22—84no—
641C22—84no—
642C22—84no—
643C22#199614yes—
644C22C2×#39111yescharacteristic
645C22C22×#9911yescharacteristic
646C22C2×#41011yescharacteristic
647C4—22no—
648C4—22no—
649C4—21no—
650C4—21no—
651C4—22no—
652C4—42no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—81no—
660C4—81no—
661C4—81no—
662C4—81no—
663C4—88no—
664C4—81no—
665C4—81no—
666C4—81no—
667C2—21no—
668C2—21no—
669C2—24no—
670C2—42no—
671C2—42no—
672C2—42no—
673C2—42no—
674C2C2×#199611yescharacteristic
675C2#3334811yescharacteristic
676C2#3335011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13297C2semidirect
5. #13297C2semidirect
6. #13296C2semidirect
7. #13295C2semidirect
10. C22×C22⋊Q8C4semidirect
15. #2579C4semidirect
16. #2579C4semidirect
19. #2487C4semidirect
29. #2409C4semidirect
31. #2315C4semidirect
32. #2315C4semidirect
35. C2×#385C4semidirect
51. C23×Q8C2×C4semidirect
74. C22×C4⋊C4C2×C4semidirect
75. C22×C4⋊C4C2×C4semidirect
76. C22×C4⋊C4C2×C4semidirect
94. C2×#99Q8semidirect
144. #385C2×C4semidirect
145. #385C2×C4semidirect
146. #385C2×C4semidirect
147. #385C2×C4semidirect
148. #385C2×C4semidirect
149. #385C2×C4semidirect
150. #385C2×C4semidirect
151. #385C2×C4semidirect
157. #380C2×C4semidirect
158. #380C2×C4semidirect
159. #380C2×C4semidirect
160. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
#353492
#206382
#203922
#132972
#132962
#132952
#125742
C2×#13902