Finite Group Databasebeta

← #495420

Subgroups — #495420

Order 512.

1137 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35529C211yesmaximal, characteristic
2#20633C211yesmaximal, characteristic
3#20392C211yesmaximal, characteristic
4#13300C212yesmaximal, complemented
5#13300C212yesmaximal, complemented
6#13299C212yesmaximal, complemented
7#13298C212yesmaximal, complemented
8#12582C212yesmaximal
9C2×#1393C212yesmaximal
10C2×#544C411yescharacteristic, complemented
11C2×#518C2211yescharacteristic
12C22×#99C2212yes—
13C22×#99C2212yes—
14#2566C411yescharacteristic, complemented
15#2566C411yescharacteristic, complemented
16#2566C2211yescharacteristic
17#2566C2211yescharacteristic
18#2563C411yescharacteristic, complemented
19#2563C2211yescharacteristic
20#2542C411yescharacteristic, complemented
21#2542C2211yescharacteristic
22#2454—21no—
23#2454—21no—
24#2452—21no—
25#2452—21no—
26#2437—21no—
27#2437—21no—
28#2437—21no—
29#2437—21no—
30#2317C411yescharacteristic, complemented
31#2315—21no—
32#2309C2211yescharacteristic
33#2296C411yescharacteristic, complemented
34C2×#390C2211yescharacteristic
35#2285C411yescharacteristic, complemented
36#2273—21no—
37#1799—21no—
38#1799—21no—
39#1799C2212yes—
40#1799C2212yes—
41C4×#99—21no—
42C4×#99—21no—
43C4×#99C2212yes—
44C4×#99C2212yes—
45C2×#342—21no—
46C2×#342—21no—
47#1393C2214yes—
48#1393C2214yes—
49#1393C2214yes—
50#1393C2214yes—
51C23×D4C2311yescharacteristic
52#544—44no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C411yescharacteristic, complemented
55C2×C4⋊Q8C2×C411yescharacteristic, complemented
56C2×#125—21no—
57C2×#125C2×C411yescharacteristic, complemented
58C2×#125C2×C411yescharacteristic, complemented
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65#518—44no—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C2×#117C2×C411yescharacteristic
75C2×#117C2×C411yescharacteristic
76C2×#117C2×C411yescharacteristic
77C2×#117C2×C411yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C22⋊C4—22no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C22⋊C4D411yescharacteristic
87C22×C42C2×C411yescharacteristic
88C2×#99—22no—
89C2×#99—24no—
90C2×#99—28no—
91C2×#99—22no—
92C2×#99Q818yescomplemented
93C2×#99C2314yes—
94C2×#99C2314yes—
95#399—21no—
96#399—21no—
97#399—21no—
98#399—21no—
99#396—21no—
100#396—21no—
101#396—21no—
102#396—21no—
103#396—21no—
104#396—21no—
105#396—21no—
106#396—21no—
107#393—21no—
108#393—21no—
109#393—21no—
110#393—21no—
111#390—21no—
112#390—21no—
113#390—21no—
114#390—21no—
115#390C2×C411yescharacteristic
116#390C2×C411yescharacteristic
117#390C2×C411yescharacteristic
118#390C2×C411yescharacteristic
119#390C2311yescharacteristic
120#390C2311yescharacteristic
121#390C2311yescharacteristic
122#390C2311yescharacteristic
123#389—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#385—21no—
128#385—21no—
129#385—21no—
130#385—21no—
131#385—21no—
132#385—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#385—21no—
137#385—21no—
138#385—21no—
139#385—44no—
140#384C2×C411yescharacteristic, complemented
141#384C2×C411yescharacteristic, complemented
142#384C2×C411yescharacteristic, complemented
143#384C2×C411yescharacteristic, complemented
144#382C2×C411yescharacteristic, complemented
145#382C2×C411yescharacteristic, complemented
146#382C2×C411yescharacteristic, complemented
147#382C2×C411yescharacteristic, complemented
148#380—21no—
149#380—21no—
150#380—21no—
151#380—21no—
152#380—21no—
153#380—21no—
154#380—22no—
155#380—22no—
156#379—21no—
157#379—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4C2×C411yescharacteristic, complemented
161C4×C4⋊C4C2×C411yescharacteristic, complemented
162C4×C4⋊C4C2×C411yescharacteristic, complemented
163C4×C4⋊C4C2×C411yescharacteristic, complemented
164C4×C22⋊C4—22no—
165C4×C22⋊C4—22no—
166C4×C22⋊C4—44no—
167#342—22no—
168#342—22no—
169#342—22no—
170#342—22no—
171C25C2×D411yescharacteristic
172C25C2×D411yescharacteristic
173C22×Q8—21no—
174C22×Q8—21no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4C22×C411yescharacteristic
178C22×D4C2×D411yescharacteristic
179C22×D4C2×D411yescharacteristic
180C22×D4C2×Q814yes—
181C22×D4#4114yes—
182C22×D4C2411yescharacteristic
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C4⋊Q8—42no—
197C4⋊Q8—42no—
198C4⋊Q8—44no—
199#125—42no—
200#125—42no—
201#125—44no—
202C22⋊Q8—44no—
203C22⋊Q8—44no—
204C22⋊Q8—82no—
205C22⋊Q8—82no—
206C22⋊Q8—82no—
207C22⋊Q8—82no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—42no—
214C4×D4—42no—
215C4×D4—42no—
216#117—24no—
217#117—42no—
218#117—42no—
219#117—42no—
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—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4C22⋊C411yescharacteristic
244C2×C4⋊C4C22⋊C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
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—24no—
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—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—82no—
282C2×C22⋊C4—82no—
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⋊C4C22⋊C411yescharacteristic
291C2×C22⋊C4C2×D411yescharacteristic
292C2×C22⋊C4C2×D411yescharacteristic
293C2×C22⋊C4C2×D411yescharacteristic
294C2×C22⋊C4C2×D411yescharacteristic
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42C22⋊C411yescharacteristic
301C2×C42C22⋊C411yescharacteristic
302C2×C42C22×C411yescharacteristic
303C2×C42C22×C411yescharacteristic
304C2×C42C22×C411yescharacteristic
305C2×C42C22×C411yescharacteristic
306C2×C42C22×C411yescharacteristic
307C2×C42C22×C411yescharacteristic
308#99—28no—
309#99—24no—
310#99—28no—
311#99—44no—
312#99C2×Q818yes—
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—
327#95—42no—
328#95—42no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—81no—
334C24—81no—
335C24—81no—
336C24—81no—
337C24C2×C22⋊C411yescharacteristic
338C24C2×C22⋊C411yescharacteristic
339C24C2×C22⋊C411yescharacteristic
340C24C22⋊Q814yes—
341C24#12412yes—
342C24C4⋊Q812yes—
343C24C22×D411yescharacteristic
344C24C22×D411yescharacteristic
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—81no—
348C2×Q8—81no—
349C2×Q8—81no—
350C2×Q8—81no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—41no—
363C2×D4C4×Q814yes—
364C2×D4C22⋊Q814yes—
365C2×D4#12412yes—
366C2×D4C4⋊Q812yes—
367C2×D4C22×Q812yes—
368C2×D4C2×#4112yes—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—48no—
391C22×C4—42no—
392C22×C4—44no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4—81no—
398C22×C4—81no—
399C22×C4—81no—
400C22×C4—81no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4—81no—
404C22×C4—84no—
405C22×C4—162no—
406C22×C4—162no—
407C22×C4—162no—
408C22×C4—162no—
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C4×Q814yes—
419C22×C4C23×C411yescharacteristic
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—81no—
441C4⋊C4—81no—
442C4⋊C4—81no—
443C4⋊C4—81no—
444C4⋊C4—81no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—162no—
449C4⋊C4—162no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—28no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—44no—
467C22⋊C4—48no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—81no—
473C22⋊C4—81no—
474C22⋊C4—81no—
475C22⋊C4—81no—
476C22⋊C4—81no—
477C22⋊C4—81no—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C22⋊C4—81no—
481C22⋊C4—81no—
482C22⋊C4—81no—
483C22⋊C4—81no—
484C22⋊C4—84no—
485C22⋊C4#9914yes—
486C42—22no—
487C42—22no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—41no—
494C42—42no—
495C42—42no—
496C42—162no—
497C42—162no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—82no—
521C23—82no—
522C23—82no—
523C23—82no—
524C23—84no—
525C23—82no—
526C23—82no—
527C23—162no—
528C23—162no—
529C23#39112yes—
530C23#39112yes—
531C23#39112yes—
532C23C2×#9912yes—
533C23#41012yes—
534C23C22×C22⋊C411yescharacteristic
535C23C2×C22⋊Q812yes—
536C23C2×#12411yescharacteristic
537C23C2×C4⋊Q811yescharacteristic
538Q8—162no—
539Q8—162no—
540D4—44no—
541D4—44no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—42no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—48no—
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—41no—
590C2×C4—41no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—84no—
601C2×C4—161no—
602C2×C4—161no—
603C2×C4—161no—
604C2×C4—161no—
605C2×C4—162no—
606C2×C4—168no—
607C2×C4—168no—
608C2×C4#39112yes—
609C2×C4C2×#9912yes—
610C2×C4C2×#9912yes—
611C2×C4C2×C4×Q812yes—
612C22—21no—
613C22—22no—
614C22—21no—
615C22—22no—
616C22—22no—
617C22—21no—
618C22—21no—
619C22—22no—
620C22—22no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—41no—
625C22—41no—
626C22—42no—
627C22—42no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—84no—
637C22—84no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22—84no—
643C22—84no—
644C22#199614yes—
645C22C2×#39111yescharacteristic
646C22C22×#9911yescharacteristic
647C22C2×#41011yescharacteristic
648C4—22no—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—42no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—41no—
659C4—81no—
660C4—81no—
661C4—81no—
662C4—88no—
663C4—81no—
664C4—81no—
665C4—81no—
666C4—81no—
667C4—164no—
668C2—21no—
669C2—21no—
670C2—24no—
671C2—42no—
672C2—41no—
673C2—41no—
674C2—42no—
675C2—42no—
676C2C2×#199611yescharacteristic
677C2#3335911yescharacteristic
678C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×D4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ 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. #13300C2semidirect
5. #13300C2semidirect
6. #13299C2semidirect
7. #13298C2semidirect
10. C2×#544C4semidirect
14. #2566C4semidirect
15. #2566C4semidirect
18. #2563C4semidirect
20. #2542C4semidirect
30. #2317C4semidirect
33. #2296C4semidirect
35. #2285C4semidirect
54. C2×C4⋊Q8C2×C4semidirect
55. C2×C4⋊Q8C2×C4semidirect
57. C2×#125C2×C4semidirect
58. C2×#125C2×C4semidirect
92. C2×#99Q8semidirect
140. #384C2×C4semidirect
141. #384C2×C4semidirect
142. #384C2×C4semidirect
143. #384C2×C4semidirect
144. #382C2×C4semidirect
145. #382C2×C4semidirect
146. #382C2×C4semidirect
147. #382C2×C4semidirect
160. C4×C4⋊C4C2×C4semidirect
161. C4×C4⋊C4C2×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
#355292
#206332
#203922
#133002
#132992
#132982
#125822
C2×#13932