Finite Group Databasebeta

← #213947

Subgroups — #213947

Order 512.

863 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35639C211yesmaximal, characteristic
2#17370C211yesmaximal, characteristic
3#17368C211yesmaximal, characteristic
4#9518C212yesmaximal, complemented
5#9517C212yesmaximal, complemented
6#9510C212yesmaximal, complemented
7#9509C212yesmaximal, complemented
8#8757C212yesmaximal, complemented
9#8554C212yesmaximal, complemented
10C22×#125C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2975C2212yes—
13#2973C2212yes—
14#2579C411yescharacteristic
15#2579C411yescharacteristic
16#2579C2211yescharacteristic
17#2579C2211yescharacteristic
18#2487C411yescharacteristic
19#2487C411yescharacteristic
20#2487C2211yescharacteristic
21#2487C2211yescharacteristic
22#2454—21no—
23#2454—21no—
24#2453—21no—
25#2453—21no—
26#2442—21no—
27#2442—21no—
28#2441—21no—
29#2441—21no—
30#2317C411yescharacteristic
31#2316C2211yescharacteristic
32#2311C411yescharacteristic
33#2297C411yescharacteristic
34#2296C411yescharacteristic
35#1821C2214yes—
36#1467—21no—
37#1465—21no—
38#1465C2212yes—
39#1465C2212yes—
40#1464—21no—
41#1463—21no—
42#1463C2212yes—
43#1463C2212yes—
44#1463C2214yes—
45#1461—21no—
46#1459—21no—
47#1458—21no—
48C2×#324—21no—
49#1305C2214yes—
50#1305C2214yes—
51C22×#41C2×C411yescharacteristic
52C2×#125—21no—
53C2×#125—21no—
54C2×#125—21no—
55C2×#125—21no—
56C2×#125—44no—
57#520—44no—
58C2×C4×Q8D411yescharacteristic
59C2×C4×Q8D411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4C2311yescharacteristic
65C2×C4×D4C2311yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117C2×C411yescharacteristic
68C2×#117C2×C411yescharacteristic
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C22×C4⋊C4C2×C411yescharacteristic
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C42C2×C411yescharacteristic
80#436—44no—
81#435—44no—
82#432—22no—
83#431—22no—
84#407—22no—
85#407C2314yes—
86#407C2314yes—
87C2×C22⋊C8—22no—
88#400—21no—
89#400—21no—
90#400—21no—
91#400—21no—
92#399—21no—
93#399—21no—
94#399—21no—
95#399—21no—
96#398—21no—
97#398—21no—
98#398—21no—
99#398—21no—
100#396—21no—
101#396—21no—
102#396—21no—
103#396—21no—
104#391—21no—
105#391—21no—
106#390—21no—
107#390—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#387—21no—
121#387—21no—
122#387—21no—
123#387—21no—
124#386C2×C411yescharacteristic
125#386C2×C411yescharacteristic
126#386C2×C411yescharacteristic
127#386C2×C411yescharacteristic
128#386C2×C411yescharacteristic
129#386C2×C411yescharacteristic
130#386C2×C411yescharacteristic
131#386C2×C411yescharacteristic
132#385—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#385C2×C411yescharacteristic
137#385C2×C411yescharacteristic
138#385C2×C411yescharacteristic
139#385C2×C411yescharacteristic
140#385C2×C411yescharacteristic
141#385C2×C411yescharacteristic
142#385C2×C411yescharacteristic
143#385C2×C411yescharacteristic
144#384—21no—
145#382—21no—
146#380—21no—
147#380—21no—
148#380—21no—
149#380—21no—
150#380—21no—
151#380—21no—
152#380C2×C411yescharacteristic
153#380C2×C411yescharacteristic
154#380C2311yescharacteristic
155#380C2311yescharacteristic
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160#343—44no—
161#324—22no—
162#324—22no—
163#324—22no—
164#324—22no—
165#324—22no—
166#324—22no—
167#324—44no—
168#323—22no—
169#323—22no—
170C2×#41—24no—
171C2×#41—42no—
172C2×#41—42no—
173C22×Q8C22⋊C411yescharacteristic
174C22×D4C22⋊C411yescharacteristic
175C22×D4C22×C411yescharacteristic
176C22×D4C22×C411yescharacteristic
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2411yescharacteristic
190C2×M16—24no—
191C2×M16—24no—
192C2×M16—42no—
193C22×C8—42no—
194#125—44no—
195#125—44no—
196#125—44no—
197#125—44no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×D4—24no—
201C4×D4—24no—
202C4×D4—42no—
203C4×D4—42no—
204C4×D4—42no—
205C4×D4—42no—
206#117—42no—
207#117—42no—
208#117—42no—
209#117—42no—
210#117—42no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
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—24no—
225C2×C4⋊C4—24no—
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—42no—
239C2×C4⋊C4—42no—
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⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2×D411yescharacteristic
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⋊C4C22⋊C411yescharacteristic
262C2×C22⋊C4C22⋊C411yescharacteristic
263C2×C22⋊C4C22×C411yescharacteristic
264C2×C22⋊C4C22×C411yescharacteristic
265C2×C22⋊C4C22×C411yescharacteristic
266C2×C22⋊C4C22×C411yescharacteristic
267C2×C22⋊C4C22×C411yescharacteristic
268C2×C22⋊C4C22×C411yescharacteristic
269C2×C22⋊C4C2×D411yescharacteristic
270C2×C22⋊C4C2×D411yescharacteristic
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—42no—
276C2×C42C22⋊C411yescharacteristic
277C2×C42C22⋊C411yescharacteristic
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22⋊C411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C4⋊C8—42no—
283C4⋊C8—42no—
284C22⋊C8—24no—
285C22⋊C8—24no—
286C22⋊C8—24no—
287C22⋊C8—24no—
288C22⋊C8—42no—
289C22⋊C8—42no—
290#97—42no—
291C4×C8—42no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
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—
312C24C2×C22⋊C411yescharacteristic
313C24C2×C22⋊C411yescharacteristic
314C24C23×C411yescharacteristic
315#41—416no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—41no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—81no—
354C22×C4—81no—
355C22×C4—81no—
356C22×C4—81no—
357C22×C4—81no—
358C22×C4—81no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4#10114yes—
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4H(5,𝔽2)14yes—
377M16—84no—
378M16—84no—
379M16—162no—
380C2×C8—44no—
381C2×C8—44no—
382C2×C8—162no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—81no—
406C4⋊C4—81no—
407C4⋊C4—81no—
408C4⋊C4—81no—
409C4⋊C4—81no—
410C4⋊C4—81no—
411C4⋊C4—81no—
412C4⋊C4—81no—
413C4⋊C4—81no—
414C4⋊C4—81no—
415C4⋊C4—81no—
416C4⋊C4—81no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—81no—
428C22⋊C4—81no—
429C22⋊C4—81no—
430C22⋊C4—81no—
431C22⋊C4—81no—
432C22⋊C4—81no—
433C22⋊C4—81no—
434C22⋊C4—81no—
435C42—22no—
436C42—22no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—42no—
442C42—42no—
443C42—42no—
444C42—42no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23C2×#10112yes—
459C23C22×C22⋊C411yescharacteristic
460C23#51812yes—
461Q8—44no—
462D4—44no—
463D4—44no—
464D4—44no—
465C2×C4—28no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—44no—
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—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—161no—
524C2×C4—161no—
525C2×C4C2×#10112yes—
526C2×C4C2×#10112yes—
527C2×C4#41312yes—
528C2×C4#53412yes—
529C2×C4#53412yes—
530C8—84no—
531C8—84no—
532C22—21no—
533C22—21no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—81no—
548C22—81no—
549C22#230811yescharacteristic
550C22C22×#10111yescharacteristic
551C22C2×#41311yescharacteristic
552C4—22no—
553C4—22no—
554C4—22no—
555C4—22no—
556C4—22no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—81no—
571C4—81no—
572C2—21no—
573C2—21no—
574C2—42no—
575C2—42no—
576C2—81no—
577C2#2175311yescharacteristic
578C2#2177711yescharacteristic
579C2#2178111yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ 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
4. #9518C2semidirect
5. #9517C2semidirect
6. #9510C2semidirect
7. #9509C2semidirect
8. #8757C2semidirect
9. #8554C2semidirect

Maximal subgroups

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

subgroupindex
#356392
#173702
#173682
#95182
#95172
#95102
#95092
#87572
#85542