Finite Group Databasebeta

← #481732

Subgroups — #481732

Order 512.

1180 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34963C211yesmaximal, characteristic
2C4×#410C211yesmaximal, characteristic, complemented
3#17363C211yesmaximal, characteristic, complemented
4#11786C212yesmaximal, complemented
5#11772C212yesmaximal, complemented
6#11754C214yesmaximal, complemented
7#11726C214yesmaximal, complemented
8C2×C4×#41C2211yescharacteristic
9C22×C4×D4C411yescharacteristic, complemented
10C2×#432—21no—
11C2×#410—21no—
12#2598C412yescomplemented
13#2598C2212yes—
14#2580—21no—
15#2535—21no—
16#2487C411yescharacteristic, complemented
17#2487C2211yescharacteristic
18#2482—21no—
19#2408C411yescharacteristic, complemented
20#2408C2211yescharacteristic
21#2404—21no—
22#2398C412yescomplemented
23#2397C2212yes—
24C4×#121—21no—
25C4×#121—21no—
26C4×#120—21no—
27C4×#120—21no—
28#2283C411yescharacteristic, complemented
29C4×#117—21no—
30C4×#99—21no—
31C4×#99—21no—
32C4×#99C2212yes—
33C4×#99C2212yescomplemented
34#1740C2212yescomplemented
35#1467—21no—
36#1466C2212yes—
37#1464—21no—
38#1464C2212yescomplemented
39#1459—21no—
40#1459C2212yescomplemented
41#1352C2218yescomplemented
42#1343C2218yes—
43C22×#41D411yescharacteristic
44C23×D4C2×C411yescharacteristic
45C24×C4—21no—
46C2×#121—21no—
47C2×#121—21no—
48C2×#120—21no—
49C2×#120—21no—
50C4×#41—41no—
51C4×#41—41no—
52C4×#41—41no—
53C4×#41—41no—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4—41no—
61C2×C4×D4—41no—
62C2×C4×D4—41no—
63C2×C4×D4—41no—
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4C2×C412yes—
66C2×C4×D4C2×C412yes—
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117—22no—
73C2×#117C2×C411yescharacteristic
74C2×#117D411yescharacteristic
75C2×#117C2311yescharacteristic
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C22⋊C4—21no—
78C22×C42C2×C411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42D411yescharacteristic
81#432—22no—
82#432—22no—
83#432—41no—
84#432—41no—
85C2×C4⋊C8—41no—
86#410—28no—
87C2×#99—22no—
88C2×#99—22no—
89C2×C22⋊C8—22no—
90C2×#97—41no—
91#400C2×C412yes—
92#400C2×C412yes—
93#399—22no—
94#399C2×C412yes—
95#399C2312yes—
96#397—22no—
97#394C2×C412yes—
98#394C2×C412yes—
99#393—22no—
100#392—22no—
101#392C2×C412yes—
102#392C2312yes—
103#390—21no—
104#390—21no—
105#389—21no—
106#389—21no—
107#388—22no—
108#388—22no—
109#386—21no—
110#386—21no—
111#386—22no—
112#386—22no—
113#386—22no—
114#386—22no—
115#385—22no—
116#385—22no—
117#385—21no—
118#385—21no—
119#383C2×C412yes—
120#382—21no—
121#381—21no—
122#380—21no—
123#380—21no—
124#380C2×C411yescharacteristic
125#380C2×C411yescharacteristic
126#379—21no—
127#379—21no—
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—
145C25—21no—
146C2×#41—24no—
147C2×#41—41no—
148C2×#41—41no—
149C2×#41—42no—
150C22×Q8C22⋊C411yescharacteristic
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—21no—
154C22×D4—41no—
155C22×D4—41no—
156C22×D4—44no—
157C22×D4C22⋊C411yescharacteristic
158C22×D4C22×C412yes—
159C22×D4C2×D411yescharacteristic
160C22×D4C2×D411yescharacteristic
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—84no—
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C22×C8—42no—
177#121—44no—
178#121—44no—
179#120—44no—
180#120—44no—
181C4×Q8—41no—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×Q8—41no—
185C4×D4—41no—
186C4×D4—41no—
187C4×D4—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—42no—
198C4×D4—42no—
199C4×D4—42no—
200C4×D4—42no—
201C4×D4—42no—
202C4×D4—42no—
203C4×D4—41no—
204C4×D4—41no—
205C4×D4—41no—
206C4×D4—41no—
207C4×D4—84no—
208C4×D4—84no—
209#117—28no—
210#117—41no—
211#117—41no—
212#117—41no—
213#117—41no—
214#117—41no—
215#117—41no—
216#117—44no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C22⋊C4—24no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4C22⋊C411yescharacteristic
248C2×C22⋊C4C22⋊C411yescharacteristic
249C2×C22⋊C4C22×C411yescharacteristic
250C2×C22⋊C4C2×D411yescharacteristic
251C2×C22⋊C4C2×D411yescharacteristic
252C2×C22⋊C4C2411yescharacteristic
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—22no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—42no—
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
279C4⋊C8—42no—
280C4⋊C8—42no—
281#99—28no—
282#99—28no—
283C22⋊C8—24no—
284C22⋊C8—42no—
285#97—42no—
286#97—42no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—42no—
295#95—42no—
296#95C22×C412yes—
297#95C22×C412yes—
298C24—22no—
299C24—42no—
300C24—44no—
301C24—42no—
302C24—42no—
303C24—82no—
304C24C2×C22⋊C411yescharacteristic
305C24C2×C22⋊C411yescharacteristic
306C24C22×D411yescharacteristic
307#41—48no—
308#41—48no—
309C2×Q8—22no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×D4—24no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—48no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—82no—
334C2×D4—82no—
335C2×D4#10414yes—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4—88no—
380C22×C4—88no—
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C2×C8—44no—
394C2×C8—84no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—48no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C22⋊C4—28no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—42no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—44no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C22⋊C4—82no—
429C22⋊C4—82no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—44no—
449C42—48no—
450C42—84no—
451C42—84no—
452C42#9911yescharacteristic
453C42#9911yescharacteristic
454C42#9911yescharacteristic
455C42#9911yescharacteristic
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—24no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—48no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—84no—
477C23—88no—
478C23—88no—
479C23—84no—
480C23C2×#10412yes—
481C23C2×#10412yes—
482C23C22×C22⋊C411yescharacteristic
483C23#57512yes—
484Q8—44no—
485D4—44no—
486D4—44no—
487D4—44no—
488D4—44no—
489D4—44no—
490D4—44no—
491D4—44no—
492D4—88no—
493D4—88no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
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—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—48no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—84no—
547C2×C4—88no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4C2×#9911yescharacteristic
553C2×C4C2×#9911yescharacteristic
554C2×C4C2×#9911yescharacteristic
555C2×C4C2×#9911yescharacteristic
556C2×C4C2×#9911yescharacteristic
557C2×C4C2×#9911yescharacteristic
558C2×C4C2×#10412yes—
559C8—84no—
560C22—21no—
561C22—21no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—44no—
570C22—44no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—44no—
578C22—81no—
579C22—81no—
580C22—82no—
581C22—82no—
582C22—816no—
583C22—82no—
584C22—82no—
585C22C22×#9911yescharacteristic
586C22#289611yescharacteristic
587C22C22×#10411yescharacteristic
588C4—21no—
589C4—21no—
590C4—21no—
591C4—21no—
592C4—22no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—82no—
602C4—82no—
603C4—88no—
604C4—82no—
605C4#212012yes—
606C4#212812yes—
607C2—21no—
608C2—21no—
609C2—42no—
610C2—42no—
611C2—42no—
612C2—84no—
613C2#1754211yescharacteristic
614C2C2×#212011yescharacteristic
615C2C2×#212811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
socleC22C22×#99
prime radical#34963C2
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
2. C4×#410C2semidirect
3. #17363C2semidirect
4. #11786C2semidirect
5. #11772C2semidirect
6. #11754C2semidirect
7. #11726C2semidirect
9. C22×C4×D4C4semidirect
12. #2598C4semidirect
16. #2487C4semidirect
19. #2408C4semidirect
22. #2398C4semidirect
28. #2283C4semidirect
33. C4×#99C22semidirect
34. #1740C22semidirect
38. #1464C22semidirect
40. #1459C22semidirect
41. #1352C22semidirect

Maximal subgroups

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

subgroupindex
#349632
C4×#4102
#173632
#117862
#117722
#117542
#117262