Finite Group Databasebeta

← #214008

Subgroups — #214008

Order 512.

891 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35532C211yesmaximal, characteristic
2#17372C211yesmaximal, characteristic
3#17368C211yesmaximal, characteristic
4#9455C212yesmaximal, complemented
5#9454C212yesmaximal, complemented
6#9450C212yesmaximal, complemented
7#9449C212yesmaximal, complemented
8#8742C212yesmaximal, complemented
9#8525C212yesmaximal, complemented
10C2×#544C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2983C2212yes—
13#2973C2212yes—
14#2567C411yescharacteristic
15#2567C2211yescharacteristic
16#2566C411yescharacteristic
17#2566C2211yescharacteristic
18#2565C411yescharacteristic
19#2565C2211yescharacteristic
20#2542C411yescharacteristic
21#2542C2211yescharacteristic
22#2449—21no—
23#2449—21no—
24#2449—21no—
25#2449—21no—
26#2441—21no—
27#2441—21no—
28#2440—21no—
29#2440—21no—
30#2317C411yescharacteristic
31#2311C411yescharacteristic
32#2297C411yescharacteristic
33#2296C2211yescharacteristic
34C2×#384C411yescharacteristic
35#1556C2214yes—
36#1534C2214yes—
37#1467C2212yes—
38#1466—21no—
39#1466C2212yes—
40#1465—21no—
41#1464—21no—
42#1464C2212yes—
43#1464C2212yes—
44#1463—21no—
45#1460—21no—
46#1459—21no—
47#1458—21no—
48C2×#324—21no—
49#1285C2214yes—
50#1285C2214yes—
51C22×#41C2×C411yescharacteristic
52#544—44no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2311yescharacteristic
56C2×#125—21no—
57C2×#125C2×C411yescharacteristic
58C2×#125C2311yescharacteristic
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#520—44no—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2311yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C2×#117C2×C411yescharacteristic
75C2×#117C2×C411yescharacteristic
76C2×#117D411yescharacteristic
77C2×#117D411yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4D411yescharacteristic
85C22×C42C2×C411yescharacteristic
86#446—44no—
87#435—44no—
88#432—22no—
89#432C2314yes—
90#432C2314yes—
91#431—22no—
92#407—22no—
93C2×C22⋊C8—22no—
94#399—21no—
95#399—21no—
96#399—21no—
97#399—21no—
98#398—21no—
99#398—21no—
100#398—21no—
101#398—21no—
102#396—21no—
103#396—21no—
104#396—21no—
105#396—21no—
106#395—21no—
107#395—21no—
108#395—21no—
109#395—21no—
110#391C2×C411yescharacteristic
111#391C2×C411yescharacteristic
112#391C2×C411yescharacteristic
113#391C2×C411yescharacteristic
114#390C2×C411yescharacteristic
115#390C2×C411yescharacteristic
116#390C2×C411yescharacteristic
117#390C2×C411yescharacteristic
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#387—21no—
127#387—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#384—21no—
143#384—21no—
144#384C2×C411yescharacteristic
145#384C2×C411yescharacteristic
146#384C2×C411yescharacteristic
147#384C2×C411yescharacteristic
148#384C2×C411yescharacteristic
149#384C2×C411yescharacteristic
150#382C2×C411yescharacteristic
151#382C2×C411yescharacteristic
152#380—21no—
153#380—21no—
154C4×C4⋊C4C2×C411yescharacteristic
155C4×C4⋊C4C2×C411yescharacteristic
156C4×C4⋊C4C2311yescharacteristic
157C4×C4⋊C4C2311yescharacteristic
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160#332—44no—
161#330—44no—
162#324—22no—
163#324—22no—
164#324—22no—
165#324—22no—
166#323—22no—
167#323—22no—
168#323—22no—
169#323—22no—
170C2×#41—24no—
171C2×#41—42no—
172C2×#41—42no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8C22×C411yescharacteristic
176C22×D4C22⋊C411yescharacteristic
177C22×D4C22⋊C411yescharacteristic
178C22×D4C22×C411yescharacteristic
179C23×C4—21no—
180C23×C4—21no—
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C2×M16—42no—
191C22×C8—42no—
192C4⋊Q8—24no—
193C4⋊Q8—42no—
194C4⋊Q8—44no—
195#125—24no—
196#125—42no—
197#125—44no—
198C22⋊Q8—44no—
199C22⋊Q8—44no—
200C22⋊Q8—82no—
201C22⋊Q8—82no—
202C22⋊Q8—82no—
203C22⋊Q8—82no—
204C4×Q8—24no—
205C4×Q8—42no—
206C4×D4—24no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212#117—24no—
213#117—42no—
214#117—42no—
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—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—82no—
246C2×C4⋊C4—82no—
247C2×C4⋊C4C22⋊C411yescharacteristic
248C2×C4⋊C4C22⋊C411yescharacteristic
249C2×C4⋊C4C22⋊C411yescharacteristic
250C2×C4⋊C4C22⋊C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D411yescharacteristic
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—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4C22⋊C411yescharacteristic
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C2×D411yescharacteristic
274C2×C22⋊C4C2×D411yescharacteristic
275C2×C22⋊C4C2×D411yescharacteristic
276C2×C22⋊C4C2×D411yescharacteristic
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—42no—
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22⋊C411yescharacteristic
282C2×C42C22×C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C22×C411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2411yescharacteristic
290C4⋊C8—24no—
291C4⋊C8—24no—
292C4⋊C8—42no—
293C4⋊C8—42no—
294C22⋊C8—24no—
295C22⋊C8—24no—
296C22⋊C8—42no—
297C22⋊C8—42no—
298#97—24no—
299#97—24no—
300#97—42no—
301C4×C8—42no—
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—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318C24C2×C22⋊C411yescharacteristic
319C24C2×C22⋊C411yescharacteristic
320C24C2×C22⋊C411yescharacteristic
321#41—416no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—41no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—81no—
328C2×Q8—81no—
329C2×Q8—81no—
330C2×Q8—81no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—81no—
367C22×C4—81no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4—81no—
371C22×C4—81no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4#10114yes—
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C23×C411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387M16—162no—
388C2×C8—44no—
389C2×C8—44no—
390C2×C8—162no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—81no—
422C4⋊C4—81no—
423C4⋊C4—81no—
424C4⋊C4—81no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—81no—
442C22⋊C4—81no—
443C22⋊C4—81no—
444C22⋊C4—81no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—41no—
449C42—41no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42—42no—
455C42—42no—
456C42H(5,𝔽2)14yes—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23C2×#10112yes—
471C23#41312yes—
472C23C22×C22⋊C411yescharacteristic
473Q8—44no—
474Q8—162no—
475Q8—162no—
476D4—44no—
477D4—44no—
478D4—44no—
479C2×C4—28no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—48no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—161no—
533C2×C4—161no—
534C2×C4—161no—
535C2×C4—161no—
536C2×C4—162no—
537C2×C4C2×#10112yes—
538C2×C4C2×#10112yes—
539C2×C4#51812yes—
540C2×C4#53412yes—
541C2×C4#53412yes—
542C8—84no—
543C8—84no—
544C22—21no—
545C22—21no—
546C22—22no—
547C22—22no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—81no—
560C22—81no—
561C22#230811yescharacteristic
562C22C22×#10111yescharacteristic
563C22C2×#41311yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—22no—
568C4—22no—
569C4—42no—
570C4—42no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—81no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—81no—
584C2—21no—
585C2—21no—
586C2—42no—
587C2—42no—
588C2—81no—
589C2#2175311yescharacteristic
590C2#2178211yescharacteristic
591C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9455C2semidirect
5. #9454C2semidirect
6. #9450C2semidirect
7. #9449C2semidirect
8. #8742C2semidirect
9. #8525C2semidirect

Maximal subgroups

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

subgroupindex
#355322
#173722
#173682
#94552
#94542
#94502
#94492
#87422
#85252