Finite Group Databasebeta

← #480772

Subgroups — #480772

Order 512.

903 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35574C211yesmaximal, characteristic
2#17372C211yesmaximal, characteristic
3#17368C211yesmaximal, characteristic
4#11580C212yesmaximal, complemented
5#11578C212yesmaximal, complemented
6#11577C212yesmaximal, complemented
7#11575C212yesmaximal, complemented
8#11520C212yesmaximal, complemented
9#11518C212yesmaximal, complemented
10C22×C4⋊Q8C411yescharacteristic
11C2×#520C2211yescharacteristic
12#2983—21no—
13#2973—21no—
14#2579C411yescharacteristic
15#2579C411yescharacteristic
16#2579C2211yescharacteristic
17#2579C2211yescharacteristic
18#2452—21no—
19#2452—21no—
20#2451—21no—
21#2451—21no—
22#2440—21no—
23#2440—21no—
24#2439—21no—
25#2439—21no—
26#2409C411yescharacteristic
27#2409C411yescharacteristic
28#2409C2211yescharacteristic
29#2409C2211yescharacteristic
30#2317C2211yescharacteristic
31#2316C411yescharacteristic
32#2311C411yescharacteristic
33#2297C411yescharacteristic
34#2296C2211yescharacteristic
35#1467—21no—
36#1466—21no—
37#1466C2212yes—
38#1465C2212yes—
39#1464—21no—
40#1464C2212yes—
41#1464C2212yes—
42#1463—21no—
43#1460C2212yes—
44#1459C2212yes—
45#1458—21no—
46C2×#324—21no—
47#1326C2214yes—
48#1326C2214yes—
49#1326C2214yes—
50#1326C2214yes—
51C22×#41C2×C411yescharacteristic
52C23×Q8C2×C411yescharacteristic
53C23×Q8C2×C411yescharacteristic
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—44no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63#520—44no—
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117D411yescharacteristic
75C2×#117D411yescharacteristic
76C2×#117C2311yescharacteristic
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C4⋊C4C2×C411yescharacteristic
80C22×C4⋊C4C2×C411yescharacteristic
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4D411yescharacteristic
83C22×C42C2×C411yescharacteristic
84#446—44no—
85#435—44no—
86#432—22no—
87#432—22no—
88#432—22no—
89#431—22no—
90#407—22no—
91C2×C22⋊C8—22no—
92#398—21no—
93#398—21no—
94#398—21no—
95#398—21no—
96#396—21no—
97#396—21no—
98#396—21no—
99#396—21no—
100#395—21no—
101#395—21no—
102#395—21no—
103#395—21no—
104#393—21no—
105#393—21no—
106#393—21no—
107#393—21no—
108#391—21no—
109#391—21no—
110#391—21no—
111#391—21no—
112#391—21no—
113#391—21no—
114#390—21no—
115#390—21no—
116#389—21no—
117#389—21no—
118#389—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#386C2×C411yescharacteristic
129#386C2×C411yescharacteristic
130#386C2×C411yescharacteristic
131#386C2×C411yescharacteristic
132#386C2×C411yescharacteristic
133#386C2×C411yescharacteristic
134#386C2×C411yescharacteristic
135#386C2×C411yescharacteristic
136#385—21no—
137#385—21no—
138#385—21no—
139#385—21no—
140#385C2×C411yescharacteristic
141#385C2×C411yescharacteristic
142#385C2×C411yescharacteristic
143#385C2×C411yescharacteristic
144#385C2311yescharacteristic
145#385C2311yescharacteristic
146#385C2311yescharacteristic
147#385C2311yescharacteristic
148#384—21no—
149#382—21no—
150#380—21no—
151#380—21no—
152#380C2×C411yescharacteristic
153#380C2×C411yescharacteristic
154#380C2×C411yescharacteristic
155#380C2×C411yescharacteristic
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160#324—22no—
161#324—22no—
162#324—22no—
163#324—22no—
164#323—22no—
165#323—22no—
166#323C2314yes—
167#323C2314yes—
168C2×#41—24no—
169C2×#41—42no—
170C2×#41—42no—
171C22×Q8—21no—
172C22×Q8—21no—
173C22×Q8—21no—
174C22×Q8—21no—
175C22×Q8—42no—
176C22×Q8—42no—
177C22×Q8—82no—
178C22×Q8—82no—
179C22×Q8C22⋊C411yescharacteristic
180C22×D4C22⋊C411yescharacteristic
181C22×D4C22×C411yescharacteristic
182C22×D4C22×C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C2×M16—42no—
197C22×C8—42no—
198C4⋊Q8—44no—
199C4⋊Q8—44no—
200C4⋊Q8—44no—
201C4⋊Q8—44no—
202C22⋊Q8—82no—
203C22⋊Q8—82no—
204C22⋊Q8—82no—
205C22⋊Q8—82no—
206C4×Q8—24no—
207C4×Q8—42no—
208C4×D4—24no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—42no—
214#117—42no—
215#117—42no—
216#117—42no—
217#117—42no—
218#117—42no—
219C2×C4⋊C4—24no—
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⋊C4—42no—
244C2×C4⋊C4—42no—
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×C4⋊C4C2×D411yescharacteristic
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C4⋊C4C2×D411yescharacteristic
254C2×C4⋊C4C2×D411yescharacteristic
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—21no—
262C2×C22⋊C4—21no—
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⋊C4C22×C411yescharacteristic
270C2×C22⋊C4C2×D411yescharacteristic
271C2×C22⋊C4C2×D411yescharacteristic
272C2×C22⋊C4C2411yescharacteristic
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—42no—
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22⋊C411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C4⋊C8—42no—
285C4⋊C8—42no—
286C4⋊C8—42no—
287C4⋊C8—42no—
288C22⋊C8—24no—
289C22⋊C8—24no—
290C22⋊C8—42no—
291C22⋊C8—42no—
292#97—42no—
293#97—42no—
294#97—42no—
295C4×C8—42no—
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—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316C24C2×C22⋊C411yescharacteristic
317C24C2×C22⋊C411yescharacteristic
318C24C23×C411yescharacteristic
319#41—416no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—41no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—81no—
332C2×Q8—81no—
333C2×Q8—81no—
334C2×Q8—81no—
335C2×Q8—84no—
336C2×Q8—84no—
337C2×Q8—84no—
338C2×Q8—84no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
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—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4#10114yes—
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
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×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397M16—162no—
398C2×C8—44no—
399C2×C8—44no—
400C2×C8—162no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
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—42no—
422C4⋊C4—81no—
423C4⋊C4—81no—
424C4⋊C4—81no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4#9914yes—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—81no—
446C22⋊C4—81no—
447C22⋊C4—81no—
448C22⋊C4—81no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—41no—
454C42—41no—
455C42—42no—
456C42—42no—
457C42—42no—
458C42—42no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23C2×#10112yes—
473C23C22×C22⋊C411yescharacteristic
474C23#56812yes—
475Q8—44no—
476Q8—84no—
477Q8—84no—
478Q8—84no—
479Q8—84no—
480D4—44no—
481D4—44no—
482D4—44no—
483C2×C4—28no—
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—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
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—42no—
518C2×C4—42no—
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—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—161no—
544C2×C4—161no—
545C2×C4C2×#9912yes—
546C2×C4C2×#9912yes—
547C2×C4C2×#9912yes—
548C2×C4C2×#10112yes—
549C2×C4C2×#10112yes—
550C8—84no—
551C8—84no—
552C22—21no—
553C22—21no—
554C22—22no—
555C22—22no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—81no—
568C22—81no—
569C22#287911yescharacteristic
570C22C22×#9911yescharacteristic
571C22C22×#10111yescharacteristic
572C4—22no—
573C4—22no—
574C4—22no—
575C4—22no—
576C4—22no—
577C4—41no—
578C4—42no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—42no—
585C4—41no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—81no—
592C2—21no—
593C2—21no—
594C2—42no—
595C2—42no—
596C2—81no—
597C2#1739711yescharacteristic
598C2#3334811yescharacteristic
599C2#3334911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#41C2×C4
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11580C2semidirect
5. #11578C2semidirect
6. #11577C2semidirect
7. #11575C2semidirect
8. #11520C2semidirect
9. #11518C2semidirect

Maximal subgroups

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

subgroupindex
#355742
#173722
#173682
#115802
#115782
#115772
#115752
#115202
#115182