Finite Group Databasebeta

← #481881

Subgroups — #481881

Order 512.

864 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36031C211yesmaximal, characteristic
2#20695C211yesmaximal, characteristic
3#17368C211yesmaximal, characteristic
4#11818C212yesmaximal, complemented
5#11817C212yesmaximal, complemented
6#11816C212yesmaximal, complemented
7#11814C212yesmaximal, complemented
8#11722C212yesmaximal, complemented
9C2×#1353C212yesmaximal, complemented
10C2×#554C411yescharacteristic
11C2×#520C2211yescharacteristic
12#2973—21no—
13#2879—21no—
14#2628C2211yescharacteristic
15#2627C2211yescharacteristic
16#2627C2211yescharacteristic
17#2581C411yescharacteristic
18#2568C411yescharacteristic
19#2564C411yescharacteristic
20#2564C411yescharacteristic
21#2487C411yescharacteristic
22#2487C411yescharacteristic
23#2453—21no—
24#2450—21no—
25#2450—21no—
26#2442—21no—
27#2438—21no—
28#2438—21no—
29#2396C2211yescharacteristic
30C2×#400C2211yescharacteristic
31C2×#400C2211yescharacteristic
32#2349—21no—
33#2349C411yescharacteristic
34#2339—21no—
35#1822—21no—
36#1822C2212yes—
37#1821—21no—
38#1821C2212yes—
39#1742—21no—
40C2×#343C2212yes—
41#1467—21no—
42#1465—21no—
43#1464C2212yes—
44#1463C2212yes—
45#1459—21no—
46C2×#324C2212yes—
47#1354C2214yes—
48#1353C2214yes—
49#1353C2214yes—
50#1353C2214yes—
51C22×#41D411yescharacteristic
52C22×M16—22no—
53#554—44no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#125—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8C2×C411yescharacteristic
59C2×C22⋊Q8C2×C411yescharacteristic
60#520—44no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×#117D411yescharacteristic
73C2×#117D411yescharacteristic
74C2×#117C2311yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80#435—44no—
81#432—22no—
82#408—44no—
83#407—22no—
84#407—22no—
85#407—22no—
86C2×C22⋊C8—22no—
87#400—21no—
88#400—21no—
89#400—21no—
90#400—21no—
91#400C2×C411yescharacteristic
92#400C2×C411yescharacteristic
93#400C2×C411yescharacteristic
94#400C2×C411yescharacteristic
95#400C2×C411yescharacteristic
96#400C2×C411yescharacteristic
97#400C2311yescharacteristic
98#400C2311yescharacteristic
99#400C2311yescharacteristic
100#399C2×C411yescharacteristic
101#399C2×C411yescharacteristic
102#399C2×C411yescharacteristic
103#398—21no—
104#398—21no—
105#398—21no—
106#398—21no—
107#398—21no—
108#396—21no—
109#395—21no—
110#395C2×C411yescharacteristic
111#395C2×C411yescharacteristic
112#395C2311yescharacteristic
113#393—21no—
114#393C2×C411yescharacteristic
115#391—21no—
116#391—21no—
117#391—21no—
118#391—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#387—21no—
125#387—21no—
126#387—21no—
127#387—21no—
128#386—21no—
129#386—21no—
130#386—21no—
131#386—21no—
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#382—21no—
141#382C2×C411yescharacteristic
142#381—21no—
143#381—21no—
144#380—21no—
145#380—21no—
146#380—21no—
147#380—21no—
148#380—21no—
149#380C2×C411yescharacteristic
150#380C2×C411yescharacteristic
151C4×C4⋊C4—21no—
152C4×C4⋊C4C2×C411yescharacteristic
153C4×C22⋊C4—21no—
154C2×#95—21no—
155C2×#95—21no—
156C2×#95—21no—
157C2×#95—21no—
158C2×#95C2×C411yescharacteristic
159C2×#95C2×C411yescharacteristic
160#343—22no—
161#343—22no—
162#343—22no—
163#343C2314yes—
164#324—22no—
165#324—22no—
166#324C2314yes—
167#323—22no—
168C2×#41—24no—
169C2×#41—24no—
170C2×#41—42no—
171C22×Q8—21no—
172C22×Q8C22⋊C411yescharacteristic
173C22×Q8C22×C411yescharacteristic
174C22×D4C22⋊C411yescharacteristic
175C22×D4C22⋊C411yescharacteristic
176C22×D4C22⋊C411yescharacteristic
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C2×M16—24no—
191C2×M16—42no—
192C2×M16—42no—
193C2×M16—44no—
194C2×M16—44no—
195C22×C8—42no—
196C22×C8—42no—
197C4⋊Q8—42no—
198#125—44no—
199C22⋊Q8—44no—
200C22⋊Q8—44no—
201C22⋊Q8—42no—
202C22⋊Q8—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×Q8—42no—
206C4×D4—24no—
207C4×D4—24no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—44no—
213C4×D4—44no—
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—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4C22⋊C411yescharacteristic
239C2×C4⋊C4C22⋊C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2411yescharacteristic
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⋊C4C22⋊C411yescharacteristic
260C2×C22⋊C4C22⋊C411yescharacteristic
261C2×C22⋊C4C22⋊C411yescharacteristic
262C2×C22⋊C4C22⋊C411yescharacteristic
263C2×C22⋊C4C22×C411yescharacteristic
264C2×C22⋊C4C2×D411yescharacteristic
265C2×C42—21no—
266C2×C42—21no—
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C4⋊C8—42no—
274C22⋊C8—24no—
275C22⋊C8—42no—
276C22⋊C8—44no—
277C22⋊C8—44no—
278C22⋊C8—42no—
279#97—42no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
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#95C22×C411yescharacteristic
306#95C22×C411yescharacteristic
307#95C22×C411yescharacteristic
308#95C22×C411yescharacteristic
309C24C2×C22⋊C411yescharacteristic
310C24C2×C22⋊C411yescharacteristic
311C24C2×C22⋊C411yescharacteristic
312#41—48no—
313#41—48no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—41no—
317C2×Q8—42no—
318C2×Q8—41no—
319C2×Q8—81no—
320C2×Q8—81no—
321C2×Q8—81no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—41no—
330C2×D4#9914yes—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—81no—
358C22×C4—81no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4—81no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4—81no—
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4#13714yes—
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384M16—44no—
385M16—44no—
386M16—84no—
387M16—162no—
388C2×C8—44no—
389C2×C8—44no—
390C2×C8—84no—
391C2×C8—162no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
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—81no—
413C4⋊C4—81no—
414C4⋊C4—81no—
415C4⋊C4—81no—
416C4⋊C4—81no—
417C4⋊C4—81no—
418C4⋊C4—81no—
419C4⋊C4—81no—
420C4⋊C4—81no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
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—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23C2×#9912yes—
455C23C2×#9912yes—
456C23#41912yes—
457C23#42112yes—
458C23C22×C22⋊C411yescharacteristic
459Q8—44no—
460Q8—82no—
461Q8—162no—
462D4—28no—
463D4—44no—
464D4—44no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—24no—
472C2×C4—24no—
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—21no—
482C2×C4—21no—
483C2×C4—21no—
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—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—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—161no—
523C2×C4—161no—
524C2×C4—162no—
525C2×C4—162no—
526C2×C4—162no—
527C2×C4—162no—
528C2×C4C2×#9912yes—
529C2×C4#41912yes—
530C2×C4C2×#13712yes—
531C8—84no—
532C8—84no—
533C22—21no—
534C22—22no—
535C22—21no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—81no—
549C22#151114yes—
550C22C22×#9911yescharacteristic
551C22C2×#41911yescharacteristic
552C22C2×#42111yescharacteristic
553C4—22no—
554C4—22no—
555C4—22no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4—81no—
573C2—21no—
574C2—21no—
575C2—24no—
576C2—42no—
577C2—81no—
578C2C2×#151111yescharacteristic
579C2#3335011yescharacteristic
580C2#3337011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11818C2semidirect
5. #11817C2semidirect
6. #11816C2semidirect
7. #11814C2semidirect
8. #11722C2semidirect
9. C2×#1353C2semidirect

Maximal subgroups

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

subgroupindex
#360312
#206952
#173682
#118182
#118172
#118162
#118142
#117222
C2×#13532