Finite Group Databasebeta

← #482465

Subgroups — #482465

Order 512.

922 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35739C211yesmaximal, characteristic
2#20731C211yesmaximal, characteristic
3#17370C211yesmaximal, characteristic
4#11908C212yesmaximal, complemented
5#11905C212yesmaximal, complemented
6#11817C212yesmaximal, complemented
7#11815C212yesmaximal, complemented
8#11793C212yesmaximal, complemented
9#11791C212yesmaximal, complemented
10C2×D4×Q8C411yescharacteristic
11C2×#520C2211yescharacteristic
12#2975—21no—
13#2921—21no—
14#2628C411yescharacteristic
15#2627C411yescharacteristic
16#2627C2211yescharacteristic
17#2581C2211yescharacteristic
18#2567C2211yescharacteristic
19#2564C411yescharacteristic
20#2564C2211yescharacteristic
21#2451—21no—
22#2449—21no—
23#2449—21no—
24#2440—21no—
25#2438—21no—
26#2438—21no—
27#2409C411yescharacteristic
28#2409C2211yescharacteristic
29#2396C411yescharacteristic
30C2×#395C411yescharacteristic
31C2×#395C2211yescharacteristic
32#2349C411yescharacteristic
33D4×C4⋊C4—21no—
34#2339—21no—
35#1821C2212yes—
36#1801C2212yes—
37#1800—21no—
38#1785—21no—
39C4×#101—21no—
40C4×#100C2212yes—
41#1465—21no—
42#1465C2212yes—
43#1463—21no—
44#1463C2212yes—
45#1461C2212yes—
46#1458—21no—
47#1354C2214yes—
48#1354C2214yes—
49#1354C2214yes—
50#1353C2214yes—
51C22×#41D411yescharacteristic
52C23×Q8C2×C411yescharacteristic
53C23×Q8C2×C411yescharacteristic
54D4×Q8—44no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8C2×C411yescharacteristic
62C2×C22⋊Q8C2×C411yescharacteristic
63#520—44no—
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8C2311yescharacteristic
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×#117C2×C411yescharacteristic
76C2×#117D411yescharacteristic
77C2×#117D411yescharacteristic
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2×C411yescharacteristic
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83#436—44no—
84#433—44no—
85#431—22no—
86#423—22no—
87C2×#101—22no—
88C2×#100—22no—
89#407—22no—
90#407—22no—
91#407—22no—
92C4×M16—22no—
93#400—21no—
94#400C2×C411yescharacteristic
95#400C2×C411yescharacteristic
96#400C2311yescharacteristic
97#399—21no—
98#399C2×C411yescharacteristic
99#398—21no—
100#398—21no—
101#398—21no—
102#398—21no—
103#398—21no—
104#396—21no—
105#395—21no—
106#395—21no—
107#395—21no—
108#395—21no—
109#395C2×C411yescharacteristic
110#395C2×C411yescharacteristic
111#395C2×C411yescharacteristic
112#395C2×C411yescharacteristic
113#395C2×C411yescharacteristic
114#395C2×C411yescharacteristic
115#395C2311yescharacteristic
116#395C2311yescharacteristic
117#395C2311yescharacteristic
118#393C2×C411yescharacteristic
119#393C2×C411yescharacteristic
120#393C2×C411yescharacteristic
121#391—21no—
122#391—21no—
123#391—21no—
124#391—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#387—21no—
131#387—21no—
132#387—21no—
133#387—21no—
134#386—21no—
135#386—21no—
136#386—21no—
137#386—21no—
138#385—21no—
139#385—21no—
140#385—21no—
141#385—21no—
142#385—21no—
143#385—21no—
144#385—21no—
145#385—21no—
146#384—21no—
147#382C2×C411yescharacteristic
148#381—21no—
149#381—21no—
150#380—21no—
151#380—21no—
152#380—21no—
153#380C2×C411yescharacteristic
154#380C2×C411yescharacteristic
155C4×C4⋊C4—21no—
156C4×C4⋊C4C2×C411yescharacteristic
157C4×C22⋊C4—21no—
158C2×#95—21no—
159C2×#95—21no—
160C2×#95C2×C411yescharacteristic
161C2×#95C2×C411yescharacteristic
162#343—22no—
163#343C2314yes—
164#324—22no—
165#324—22no—
166#324C2314yes—
167#323—22no—
168C2×#41—24no—
169C2×#41—42no—
170C2×#41—42no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8—42no—
174C22×Q8—42no—
175C22×Q8—82no—
176C22×Q8—82no—
177C22×Q8C22×C411yescharacteristic
178C22×Q8C22×C411yescharacteristic
179C22×Q8C22×C411yescharacteristic
180C22×Q8C2×D411yescharacteristic
181C22×D4C22⋊C411yescharacteristic
182C22×D4C22⋊C411yescharacteristic
183C22×D4C2×D411yescharacteristic
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22⋊C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C2×M16—24no—
198C2×M16—42no—
199C2×M16—42no—
200C4⋊Q8—42no—
201C4⋊Q8—44no—
202C22⋊Q8—44no—
203C22⋊Q8—42no—
204C22⋊Q8—44no—
205C22⋊Q8—42no—
206C22⋊Q8—82no—
207C22⋊Q8—82no—
208C4×Q8—42no—
209C4×Q8—42no—
210C4×Q8—42no—
211C4×D4—24no—
212C4×D4—24no—
213C4×D4—42no—
214C4×D4—42no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—44no—
218C4×D4—44no—
219#117—24no—
220#117—42no—
221#117—42no—
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—82no—
245C2×C4⋊C4—82no—
246C2×C4⋊C4C22⋊C411yescharacteristic
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⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C4⋊C4C2×D411yescharacteristic
256C2×C4⋊C4C2411yescharacteristic
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⋊C4C22⋊C411yescharacteristic
266C2×C22⋊C4C22⋊C411yescharacteristic
267C2×C22⋊C4C22⋊C411yescharacteristic
268C2×C22⋊C4C22⋊C411yescharacteristic
269C2×C22⋊C4C22×C411yescharacteristic
270C2×C22⋊C4C2×D411yescharacteristic
271C2×C42—21no—
272C2×C42—21no—
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22⋊C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C4⋊C8—44no—
280C4⋊C8—42no—
281#101—44no—
282#100—44no—
283C22⋊C8—24no—
284C22⋊C8—44no—
285C22⋊C8—42no—
286C22⋊C8—42no—
287#97—42no—
288C4×C8—42no—
289C4×C8—42no—
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#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95C22×C411yescharacteristic
312#95C22×C411yescharacteristic
313#95C22×C411yescharacteristic
314#95C22×C411yescharacteristic
315C24C2×C22⋊C411yescharacteristic
316C24C2×C22⋊C411yescharacteristic
317C24C2×C22⋊C411yescharacteristic
318#41—416no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—42no—
329C2×Q8—81no—
330C2×Q8—81no—
331C2×Q8—81no—
332C2×Q8—81no—
333C2×Q8—84no—
334C2×Q8—84no—
335C2×Q8—84no—
336C2×Q8—84no—
337C2×Q8#13714yes—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—41no—
346C2×D4—41no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C23×C411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400M16—44no—
401M16—44no—
402M16—162no—
403M16—162no—
404C2×C8—44no—
405C2×C8—44no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C4⋊C4—81no—
433C4⋊C4—81no—
434C4⋊C4—81no—
435C4⋊C4—81no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—81no—
448C22⋊C4—81no—
449C22⋊C4#9914yes—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—41no—
455C42—41no—
456C42—42no—
457C42—42no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23C2×#9912yes—
472C23#42112yes—
473C23C22×C22⋊C411yescharacteristic
474Q8—44no—
475Q8—42no—
476Q8—42no—
477Q8—84no—
478Q8—84no—
479Q8—82no—
480Q8—82no—
481D4—28no—
482D4—44no—
483D4—44no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
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—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4—161no—
543C2×C4—161no—
544C2×C4C2×#9912yes—
545C2×C4C2×#9912yes—
546C2×C4#41912yes—
547C2×C4#41912yes—
548C2×C4C2×#13712yes—
549C8—84no—
550C8—84no—
551C22—21no—
552C22—21no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—81no—
568C22C22×#9911yescharacteristic
569C22C2×#41911yescharacteristic
570C22C2×#42111yescharacteristic
571C4—22no—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—42no—
577C4—41no—
578C4—42no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—81no—
591C2—21no—
592C2—21no—
593C2—42no—
594C2—42no—
595C2—81no—
596C2#1757811yescharacteristic
597C2#3334811yescharacteristic
598C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ 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. #11908C2semidirect
5. #11905C2semidirect
6. #11817C2semidirect
7. #11815C2semidirect
8. #11793C2semidirect
9. #11791C2semidirect

Maximal subgroups

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

subgroupindex
#357392
#207312
#173702
#119082
#119052
#118172
#118152
#117932
#117912