Finite Group Databasebeta

← #482608

Subgroups — #482608

Order 512.

867 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36041C211yesmaximal, characteristic
2#20691C211yesmaximal, characteristic, complemented
3#17371C211yesmaximal, characteristic, complemented
4#11844C212yesmaximal, complemented
5#11840C212yesmaximal, complemented
6#11810C212yesmaximal, complemented
7#11800C212yesmaximal
8#11758C212yesmaximal, complemented
9#11754C212yesmaximal, complemented
10C2×#559C411yescharacteristic, complemented
11C2×#519C2211yescharacteristic
12#2983—21no—
13#2632C411yescharacteristic, complemented
14#2632C411yescharacteristic, complemented
15#2619C2211yescharacteristic
16#2610C2211yescharacteristic
17#2610C2211yescharacteristic
18#2598C2211yescharacteristic
19#2598C2211yescharacteristic
20#2581C411yescharacteristic, complemented
21#2548C411yescharacteristic, complemented
22#2471—21no—
23#2471—21no—
24#2458—21no—
25#2458—21no—
26#2448—21no—
27#2436—21no—
28#2398C411yescharacteristic, complemented
29#2398C411yescharacteristic, complemented
30#2396C2211yescharacteristic
31#2367—21no—
32#2363—21no—
33#2356C411yescharacteristic, complemented
34#2310—21no—
35#1820—21no—
36#1820—21no—
37#1820C2212yes—
38#1820C2212yes—
39#1740—21no—
40#1740C2212yes—
41#1467—21no—
42#1467—21no—
43#1464C2212yes—
44#1464C2212yes—
45#1461—21no—
46#1458C2212yes—
47#1352C2214yes—
48#1351C2214yes—
49#1343C2214yes—
50#1343C2214yes—
51C23×Q8D411yescharacteristic
52#559—44no—
53C2×#127C2×C411yescharacteristic
54C2×#124—21no—
55C2×#124—21no—
56C2×#124C2×C411yescharacteristic
57C2×#124C2×C411yescharacteristic
58C2×#123—21no—
59C2×#123—21no—
60C2×#121—21no—
61#519—44no—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×#117—22no—
75C2×#117D411yescharacteristic
76C2×#117D411yescharacteristic
77C2×#117D411yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81#446—44no—
82#432—22no—
83#432—22no—
84#431—22no—
85#400C2×C411yescharacteristic
86#400C2×C411yescharacteristic
87#400C2×C411yescharacteristic
88#399—21no—
89#399—21no—
90#399C2311yescharacteristic
91#398—21no—
92#397—21no—
93#397—21no—
94#396—21no—
95#395C2×C411yescharacteristic
96#395C2×C411yescharacteristic
97#395C2×C411yescharacteristic
98#394—21no—
99#394C2×C411yescharacteristic
100#394C2×C411yescharacteristic
101#394C2×C411yescharacteristic
102#394C2×C411yescharacteristic
103#394C2×C411yescharacteristic
104#394C2×C411yescharacteristic
105#393—21no—
106#393—21no—
107#393—21no—
108#393—21no—
109#393C2311yescharacteristic
110#392—21no—
111#392C2311yescharacteristic
112#392C2311yescharacteristic
113#391—21no—
114#391—21no—
115#391—44no—
116#390—21no—
117#390—21no—
118#390—21no—
119#390—21no—
120#390—21no—
121#390—21no—
122#389—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#388—21no—
129#388—21no—
130#388—21no—
131#388—21no—
132#388—21no—
133#387—21no—
134#387—21no—
135#387—21no—
136#387—21no—
137#386—21no—
138#386—21no—
139#386—21no—
140#386—21no—
141#386—21no—
142#386—21no—
143#385—21no—
144#385—21no—
145#383C2×C411yescharacteristic
146#383C2×C411yescharacteristic
147#382C2×C411yescharacteristic
148#381—22no—
149#381—22no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4C2×C411yescharacteristic
153C4×C22⋊C4—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158#376C2×C411yescharacteristic
159#376C2×C411yescharacteristic
160#342—22no—
161#342—22no—
162#342—22no—
163#342C2314yes—
164#324—22no—
165#324—22no—
166#324—22no—
167#323C2314yes—
168C22×Q8—24no—
169C22×Q8—42no—
170C22×Q8—42no—
171C22×Q8C22⋊C411yescharacteristic
172C22×Q8C22⋊C411yescharacteristic
173C22×Q8C22⋊C411yescharacteristic
174C22×Q8C22⋊C411yescharacteristic
175C22×Q8C22×C411yescharacteristic
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4C22×C411yescharacteristic
181C22×D4C22×C411yescharacteristic
182C23×C4—21no—
183C23×C4—21no—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187#127—42no—
188#124—42no—
189#124—42no—
190#124—44no—
191#124—44no—
192#123—44no—
193#123—44no—
194#121—44no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×D4—42no—
205C4×D4—42no—
206C4×D4—44no—
207#117—42no—
208#117—42no—
209#117—42no—
210#117—42no—
211#117—42no—
212#117—42no—
213#117—44no—
214#117—44no—
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—42no—
223C2×C4⋊C4C22⋊C411yescharacteristic
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
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⋊C4C2×D411yescharacteristic
260C2×C22⋊C4C2×D411yescharacteristic
261C2×C22⋊C4C2×D411yescharacteristic
262C2×C22⋊C4C2411yescharacteristic
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—42no—
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22⋊C411yescharacteristic
273C2×C42C22⋊C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C4⋊C8—42no—
281C4⋊C8—42no—
282C4⋊C8—42no—
283C22⋊C8—24no—
284#97—42no—
285#97—42no—
286C4×C8—42no—
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—44no—
305#95—44no—
306#95C22×C411yescharacteristic
307#95C22×C411yescharacteristic
308#95C22×C411yescharacteristic
309#95C22×C411yescharacteristic
310C24—21no—
311C24—21no—
312C24C22×D411yescharacteristic
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—41no—
327C2×Q8—81no—
328C2×Q8#9914yes—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—81no—
338C2×D4—81no—
339C2×D4—81no—
340C2×D4—81no—
341C2×D4—81no—
342C2×D4—81no—
343C2×D4—81no—
344C2×D4—81no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—44no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4—81no—
366C22×C4—81no—
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×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C2×C8—44no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—84no—
398C4⋊C4—81no—
399C4⋊C4—81no—
400C4⋊C4—81no—
401C4⋊C4—81no—
402C4⋊C4—81no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—81no—
422C22⋊C4—81no—
423C22⋊C4—81no—
424C22⋊C4—81no—
425C22⋊C4—81no—
426C22⋊C4—81no—
427C22⋊C4—81no—
428C22⋊C4—81no—
429C22⋊C4—81no—
430C22⋊C4—81no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—42no—
444C42—42no—
445C42—84no—
446C42—81no—
447C42—81no—
448C23—21no—
449C23—21no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23C22×C22⋊C411yescharacteristic
467C23#57512yes—
468Q8—44no—
469Q8—44no—
470Q8—44no—
471Q8—44no—
472Q8—82no—
473D4—84no—
474D4—82no—
475D4—82no—
476D4—162no—
477D4—162no—
478D4—162no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
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—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—44no—
504C2×C4—44no—
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—41no—
531C2×C4—84no—
532C2×C4—84no—
533C2×C4—84no—
534C2×C4—162no—
535C2×C4—162no—
536C2×C4C2×#9912yes—
537C2×C4C2×#9912yes—
538C2×C4C2×#9912yes—
539C2×C4#42112yes—
540C8—84no—
541C22—21no—
542C22—21no—
543C22—22no—
544C22—42no—
545C22—41no—
546C22—41no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22C22×#9911yescharacteristic
563C22#289611yescharacteristic
564C22C2×#42111yescharacteristic
565C4—22no—
566C4—22no—
567C4—22no—
568C4—22no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—81no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—164no—
586C2—21no—
587C2—21no—
588C2—42no—
589C2—161no—
590C2—161no—
591C2#1758711yescharacteristic
592C2#3335111yescharacteristic
593C2#3336211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#2619C22
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#519 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20691C2semidirect
3. #17371C2semidirect
4. #11844C2semidirect
5. #11840C2semidirect
6. #11810C2semidirect
8. #11758C2semidirect
9. #11754C2semidirect
10. C2×#559C4semidirect
13. #2632C4semidirect
14. #2632C4semidirect
20. #2581C4semidirect
21. #2548C4semidirect
28. #2398C4semidirect
29. #2398C4semidirect
33. #2356C4semidirect

Maximal subgroups

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

subgroupindex
#360412
#206912
#173712
#118442
#118402
#118102
#118002
#117582
#117542