Finite Group Databasebeta

← #495626

Subgroups — #495626

Order 512.

997 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35454C211yesmaximal, characteristic
2#20725C211yesmaximal, characteristic, complemented
3#20397C211yesmaximal, characteristic, complemented
4#13292C212yesmaximal, complemented
5#13219C212yesmaximal, complemented
6#13083C212yesmaximal
7#13081C212yesmaximal, complemented
8#12622C212yesmaximal, complemented
9C4×#352C212yesmaximal, complemented
10C22×#126C411yescharacteristic, complemented
11C2×#520C2211yescharacteristic
12C2×#413C2212yes—
13#2873—21no—
14#2534C411yescharacteristic, complemented
15#2534C2211yescharacteristic
16#2487C411yescharacteristic, complemented
17#2487C2211yescharacteristic
18#2484C411yescharacteristic, complemented
19#2482C411yescharacteristic, complemented
20#2481—21no—
21#2481—21no—
22#2479—21no—
23#2479—21no—
24#2469—21no—
25#2469—21no—
26#2467—21no—
27#2467—21no—
28#2406C2211yescharacteristic
29#2404C2211yescharacteristic
30#2321C411yescharacteristic, complemented
31#2321C2211yescharacteristic
32#2313—21no—
33#2313C411yescharacteristic, complemented
34C4×#123C411yescharacteristic, complemented
35C4×#123C2211yescharacteristic
36C2×#352C2214yes—
37C2×AΣL(1,𝔽16)C2214yes—
38#1820—21no—
39#1799—21no—
40#1799C2212yes—
41#1782C2212yes—
42C4×#101—21no—
43C4×#101—21no—
44C4×#100C2212yes—
45C4×#100C2212yes—
46C4×#99—21no—
47C4×#99C2212yes—
48C2×#343—21no—
49#1390C2214yes—
50#1390C2214yes—
51C22×#41C2311yescharacteristic
52C22×M16—22no—
53C2×#126—21no—
54C2×#126—21no—
55C2×#126—21no—
56C2×#126—21no—
57C2×#126—44no—
58C2×#123C2×C411yescharacteristic
59C2×#123C2311yescharacteristic
60C2×#120C2×C411yescharacteristic
61C2×#120C2311yescharacteristic
62#520—44no—
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×#117C2×C411yescharacteristic
72C2×#117D411yescharacteristic
73C2×#117D411yescharacteristic
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4C2×C411yescharacteristic
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C22⋊C4C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83#413—28no—
84C2×#101—22no—
85C2×#100C2314yes—
86C2×#99—22no—
87C2×#99C2314yes—
88#405—44no—
89C4×M16—22no—
90C4×M16—22no—
91#400—21no—
92#400—21no—
93#400—21no—
94#399—21no—
95#399—21no—
96#399—21no—
97#398—21no—
98#397—21no—
99#396—21no—
100#396—21no—
101#395—21no—
102#394—21no—
103#394—21no—
104#394—21no—
105#393—21no—
106#392—21no—
107#390—21no—
108#390—21no—
109#389—21no—
110#389—21no—
111#388—21no—
112#388—21no—
113#388—21no—
114#388—21no—
115#388—21no—
116#388—21no—
117#388—21no—
118#388—21no—
119#388—21no—
120#388—21no—
121#387—21no—
122#387—21no—
123#387—21no—
124#387—21no—
125#387—21no—
126#387—21no—
127#387—21no—
128#387—21no—
129#387—44no—
130#386—21no—
131#386—21no—
132#386—21no—
133#386—21no—
134#386—21no—
135#386—21no—
136#386C2×C411yescharacteristic
137#386C2×C411yescharacteristic
138#386C2311yescharacteristic
139#386C2311yescharacteristic
140#385—21no—
141#385—21no—
142#385C2×C411yescharacteristic
143#385C2×C411yescharacteristic
144#385C2×C411yescharacteristic
145#385C2×C411yescharacteristic
146#381—22no—
147#381C2×C411yescharacteristic
148#381C2×C411yescharacteristic
149#381C2×C411yescharacteristic
150#381C2×C411yescharacteristic
151#380—21no—
152#380—21no—
153#380—22no—
154#380C2×C411yescharacteristic
155#380C2×C411yescharacteristic
156#379C2×C411yescharacteristic
157#379C2×C411yescharacteristic
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—22no—
163C4×C22⋊C4C2×C411yescharacteristic
164C4×C22⋊C4C2×C411yescharacteristic
165#352—28no—
166AΣL(1,𝔽16)—28no—
167#343—22no—
168#343—22no—
169#342—22no—
170#342—22no—
171C25C22×C411yescharacteristic
172C2×#41—24no—
173C2×#41—42no—
174C2×#41—42no—
175C22×Q8C2×D411yescharacteristic
176C22×D4—21no—
177C22×D4—21no—
178C22×D4C22×C411yescharacteristic
179C22×D4C2×D411yescharacteristic
180C22×D4C2411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C2×M16—28no—
194C2×M16—24no—
195C2×M16—44no—
196C2×M16—42no—
197C22×C8—42no—
198#126—44no—
199#126—44no—
200#126—44no—
201#126—44no—
202#123—24no—
203#123—42no—
204#120—24no—
205#120—42no—
206C4×Q8—42no—
207C4×Q8—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—42no—
214#117—24no—
215#117—42no—
216#117—42no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
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—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C22⋊C411yescharacteristic
238C2×C4⋊C4C22⋊C411yescharacteristic
239C2×C4⋊C4C22⋊C411yescharacteristic
240C2×C4⋊C4C22⋊C411yescharacteristic
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
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—24no—
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⋊C4—21no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4C22⋊C411yescharacteristic
268C2×C22⋊C4C22⋊C411yescharacteristic
269C2×C22⋊C4C22×C411yescharacteristic
270C2×C22⋊C4C22×C411yescharacteristic
271C2×C22⋊C4C22×C411yescharacteristic
272C2×C22⋊C4C22×C411yescharacteristic
273C2×C22⋊C4C22×C411yescharacteristic
274C2×C22⋊C4C22×C411yescharacteristic
275C2×C22⋊C4C2×D411yescharacteristic
276C2×C22⋊C4C2×D411yescharacteristic
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42C22⋊C411yescharacteristic
284C2×C42C22⋊C411yescharacteristic
285C2×C42C22⋊C411yescharacteristic
286C2×C42C22⋊C411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289#101—28no—
290#100—24no—
291#100#4118yes—
292#99—24no—
293#99—24no—
294#99—44no—
295#97—42no—
296#97—42no—
297C4×C8—42no—
298C4×C8—42no—
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—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—42no—
320#95—42no—
321#95—44no—
322C24—21no—
323C24—22no—
324C24—41no—
325C24—41no—
326C24—81no—
327C24—81no—
328C24C2×C22⋊C411yescharacteristic
329C24C23×C411yescharacteristic
330C24C22×D411yescharacteristic
331#41—48no—
332#41—48no—
333C2×Q8—22no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—81no—
347C2×D4—81no—
348C2×D4—81no—
349C2×D4—81no—
350C2×D4C2×#4114yes—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—44no—
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×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×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C22×D411yescharacteristic
399M16—28no—
400M16—48no—
401M16—44no—
402C2×C8—44no—
403C2×C8—48no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—81no—
420C4⋊C4—81no—
421C4⋊C4—81no—
422C4⋊C4—81no—
423C4⋊C4—81no—
424C4⋊C4—81no—
425C4⋊C4—162no—
426C4⋊C4#9914yes—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—81no—
450C22⋊C4—81no—
451C22⋊C4—81no—
452C22⋊C4—81no—
453C22⋊C4—81no—
454C22⋊C4—81no—
455C22⋊C4—81no—
456C22⋊C4—81no—
457C42—22no—
458C42—22no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—42no—
464C42—42no—
465C42—81no—
466C42—81no—
467C42—162no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—162no—
491C23—162no—
492C23#41012yes—
493C23C22×C22⋊C411yescharacteristic
494C23C4×#4112yes—
495C23#53212yes—
496Q8—44no—
497D4—44no—
498D4—44no—
499D4—44no—
500D4—162no—
501D4—162no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—44no—
523C2×C4—44no—
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—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—84no—
559C2×C4—84no—
560C2×C4—84no—
561C2×C4—161no—
562C2×C4—161no—
563C2×C4C2×#9912yes—
564C2×C4C2×#9912yes—
565C2×C4C2×#9912yes—
566C2×C4#52512yes—
567C8—48no—
568C22—21no—
569C22—21no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—164no—
593C22#229711yescharacteristic
594C22C22×#9911yescharacteristic
595C22C2×#41011yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—81no—
609C4—81no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4—164no—
616C2—21no—
617C2—21no—
618C2—42no—
619C2—42no—
620C2—42no—
621C2—161no—
622C2#2142011yescharacteristic
623C2#3335011yescharacteristic
624C2#3335111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radical#2404C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20725C2semidirect
3. #20397C2semidirect
4. #13292C2semidirect
5. #13219C2semidirect
7. #13081C2semidirect
8. #12622C2semidirect
9. C4×#352C2semidirect
10. C22×#126C4semidirect
14. #2534C4semidirect
16. #2487C4semidirect
18. #2484C4semidirect
19. #2482C4semidirect
30. #2321C4semidirect
33. #2313C4semidirect
34. C4×#123C4semidirect

Maximal subgroups

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

subgroupindex
#354542
#207252
#203972
#132922
#132192
#130832
#130812
#126222
C4×#3522