Finite Group Databasebeta

← #481734

Subgroups — #481734

Order 512.

1020 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34971C211yesmaximal, characteristic
2C4×#410C211yesmaximal, characteristic
3#17363C211yesmaximal, characteristic
4#11785C212yesmaximal, complemented
5#11773C212yesmaximal, complemented
6#11756C214yesmaximal, complemented
7#11728C214yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9C22×C4×Q8C411yescharacteristic, complemented
10C2×#432—21no—
11C2×#410—21no—
12#2599C412yescomplemented
13#2599C2212yes—
14#2581—21no—
15#2536—21no—
16#2486C411yescharacteristic, complemented
17#2486C2211yescharacteristic
18#2484—21no—
19#2409C411yescharacteristic, complemented
20#2409C2211yescharacteristic
21#2406—21no—
22#2399C412yescomplemented
23#2398C2212yes—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26C4×#123—21no—
27C4×#123—21no—
28#2283C411yescharacteristic, complemented
29C4×#117—21no—
30C4×#99—21no—
31C4×#99—21no—
32C4×#99C2212yes—
33C4×#99C2212yes—
34#1740C2212yes—
35#1467—21no—
36#1466C2212yes—
37#1464—21no—
38#1464C2212yes—
39#1459—21no—
40#1459C2212yes—
41#1351C2218yes—
42#1344C2218yes—
43C22×#41D411yescharacteristic
44C23×Q8C2×C411yescharacteristic
45C2×#123—21no—
46C2×#123—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C4×#41—41no—
50C4×#41—41no—
51C4×#41—41no—
52C4×#41—41no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—41no—
56C2×C4×Q8—41no—
57C2×C4×Q8—41no—
58C2×C4×Q8—41no—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8D411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×C4×D4C2311yescharacteristic
69C2×#117—22no—
70C2×#117C2×C411yescharacteristic
71C2×#117D411yescharacteristic
72C2×#117C2311yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C42—21no—
76C22×C42C2×C411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79#432—22no—
80#432—22no—
81#432—41no—
82#432—41no—
83C2×C4⋊C8—41no—
84#410—28no—
85C2×#99—22no—
86C2×#99—22no—
87C2×C22⋊C8—22no—
88C2×#97—41no—
89#400—22no—
90#398—22no—
91#398C2×C412yes—
92#398C2×C412yes—
93#397C2×C412yes—
94#397C2312yes—
95#395—22no—
96#395C2×C412yes—
97#395C2×C412yes—
98#394—22no—
99#393C2×C412yes—
100#393C2312yes—
101#391—21no—
102#391—21no—
103#389—21no—
104#389—21no—
105#388—22no—
106#388—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#386—21no—
112#386—21no—
113#385—22no—
114#385—22no—
115#385—21no—
116#385—21no—
117#383C2×C412yes—
118#382—21no—
119#381—21no—
120#381—21no—
121#381—21no—
122#380—21no—
123#380—21no—
124#380C2×C411yescharacteristic
125#380C2×C411yescharacteristic
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—41no—
130C4×C4⋊C4—41no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136#376—41no—
137#376C2×C412yes—
138C43—41no—
139#342—22no—
140#342C2314yes—
141#324—22no—
142#324—22no—
143#323—22no—
144#323C2314yes—
145C2×#41—24no—
146C2×#41—41no—
147C2×#41—41no—
148C2×#41—42no—
149C22×Q8—21no—
150C22×Q8—41no—
151C22×Q8—41no—
152C22×Q8—44no—
153C22×Q8C22⋊C411yescharacteristic
154C22×Q8C22×C412yes—
155C22×D4C22⋊C411yescharacteristic
156C22×D4C2×D411yescharacteristic
157C22×D4C2×D411yescharacteristic
158C23×C4—21no—
159C23×C4—21no—
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C22×C8—42no—
170#123—44no—
171#123—44no—
172C22⋊Q8—44no—
173C22⋊Q8—44no—
174C4×Q8—41no—
175C4×Q8—41no—
176C4×Q8—41no—
177C4×Q8—41no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×Q8—42no—
184C4×Q8—41no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—41no—
188C4×D4—41no—
189C4×D4—41no—
190C4×D4—41no—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—41no—
194C4×D4—41no—
195C4×D4—41no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—41no—
199C4×D4—41no—
200#117—28no—
201#117—41no—
202#117—41no—
203#117—41no—
204#117—41no—
205#117—41no—
206#117—41no—
207#117—44no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4C22⋊C411yescharacteristic
223C2×C4⋊C4C22⋊C411yescharacteristic
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4C22⋊C411yescharacteristic
237C2×C22⋊C4C22⋊C411yescharacteristic
238C2×C22⋊C4C22×C411yescharacteristic
239C2×C22⋊C4C2×D411yescharacteristic
240C2×C22⋊C4C2×D411yescharacteristic
241C2×C22⋊C4C2411yescharacteristic
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—42no—
258C2×C42—42no—
259C2×C42—42no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C4⋊C8—42no—
273C4⋊C8—42no—
274#99—28no—
275#99—28no—
276C22⋊C8—24no—
277C22⋊C8—42no—
278#97—42no—
279#97—42no—
280#95—22no—
281#95—22no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—42no—
292#95—42no—
293#95C22×C412yes—
294#95C22×C412yes—
295C24C2×C22⋊C411yescharacteristic
296C24C2×C22⋊C411yescharacteristic
297C24C22×D411yescharacteristic
298#41—48no—
299#41—48no—
300C2×Q8—24no—
301C2×Q8—22no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—44no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—44no—
308C2×Q8—48no—
309C2×Q8—44no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4#10414yes—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—44no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C23×C411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C2×C8—44no—
374C2×C8—84no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—48no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—44no—
394C22⋊C4—28no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—42no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—42no—
407C22⋊C4—44no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—44no—
427C42—44no—
428C42—48no—
429C42—42no—
430C42—42no—
431C42—84no—
432C42—84no—
433C42#9911yescharacteristic
434C42#9911yescharacteristic
435C42#9911yescharacteristic
436C42#9911yescharacteristic
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23C2×#10412yes—
450C23C2×#10412yes—
451C23C22×C22⋊C411yescharacteristic
452C23#57512yes—
453Q8—44no—
454Q8—44no—
455Q8—44no—
456Q8—44no—
457Q8—44no—
458D4—44no—
459D4—44no—
460D4—44no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
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—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—48no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—48no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4C2×#9911yescharacteristic
521C2×C4C2×#9911yescharacteristic
522C2×C4C2×#9911yescharacteristic
523C2×C4C2×#9911yescharacteristic
524C2×C4C2×#9911yescharacteristic
525C2×C4C2×#9911yescharacteristic
526C2×C4C2×#10412yes—
527C8—84no—
528C22—21no—
529C22—21no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—81no—
544C22—81no—
545C22C22×#9911yescharacteristic
546C22#289611yescharacteristic
547C22C22×#10411yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—22no—
553C4—44no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—82no—
563C4—82no—
564C4—88no—
565C4—82no—
566C4#212012yes—
567C4#212912yes—
568C2—21no—
569C2—21no—
570C2—42no—
571C2—42no—
572C2—42no—
573C2#1754211yescharacteristic
574C2C2×#212011yescharacteristic
575C2C2×#212911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
socleC22C22×#99
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11785C2semidirect
5. #11773C2semidirect
6. #11756C2semidirect
7. #11728C2semidirect
9. C22×C4×Q8C4semidirect
12. #2599C4semidirect
16. #2486C4semidirect
19. #2409C4semidirect
22. #2399C4semidirect
28. #2283C4semidirect

Maximal subgroups

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

subgroupindex
#349712
C4×#4102
#173632
#117852
#117732
#117562
#117282