Finite Group Databasebeta

← #482044

Subgroups — #482044

Order 512.

1237 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34935C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic, complemented
3#17349C211yesmaximal, characteristic, complemented
4#11875C214yesmaximal, complemented
5#11874C212yesmaximal, complemented
6#11873C212yesmaximal, complemented
7#11691C214yesmaximal, complemented
8C2×#525—21no—
9C2×C4×#41C2211yescharacteristic
10C22×C4×D4C411yescharacteristic, complemented
11C2×#431—21no—
12C2×#410—21no—
13#2538—21no—
14#2537—21no—
15#2536C412yescomplemented
16#2535C2212yes—
17#2534—21no—
18#2534—21no—
19#2486C411yescharacteristic, complemented
20#2486C2211yescharacteristic
21#2408C411yescharacteristic, complemented
22#2408C2211yescharacteristic
23#2392C412yescomplemented
24#2392C2212yes—
25C4×#121—21no—
26C4×#121—21no—
27#2285—21no—
28#2285C411yescharacteristic, complemented
29#2280—21no—
30#1799—21no—
31#1799—21no—
32#1799C2212yes—
33#1799C2212yescomplemented
34#1740C2212yescomplemented
35#1461—21no—
36#1460C2212yes—
37#1459—21no—
38#1459C2212yescomplemented
39#1458—21no—
40#1458C2212yescomplemented
41#1344C2218yescomplemented
42#1343C2218yes—
43C22×#41D411yescharacteristic
44C23×D4C2×C411yescharacteristic
45C24×C4—21no—
46#525—44no—
47#525—44no—
48C2×#123—21no—
49C2×#123—21no—
50C2×C22⋊Q8—22no—
51C2×#121—22no—
52C2×#121—21no—
53C2×#121—21no—
54C2×#120—21no—
55C2×#120—21no—
56C4×#41—41no—
57C4×#41—41no—
58C4×#41—41no—
59C4×#41—41no—
60C2×C4×Q8C2×C411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—41no—
67C2×C4×D4—41no—
68C2×C4×D4—41no—
69C2×C4×D4—41no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C412yes—
72C2×C4×D4C2×C412yes—
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C2×#117—21no—
79C2×#117—22no—
80C2×#117C2×C411yescharacteristic
81C2×#117D411yescharacteristic
82C2×#117C2311yescharacteristic
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C22⋊C4—21no—
85C22×C42C2×C411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88#431—22no—
89#431—22no—
90#431—41no—
91#431—41no—
92C2×C4⋊C8—41no—
93#410—44no—
94C2×#99—22no—
95C2×#99—22no—
96C2×C22⋊C8—22no—
97C2×C4×C8—41no—
98#398C2×C412yes—
99#398C2×C412yes—
100#397C2×C412yes—
101#397C2312yes—
102#396—22no—
103#396—22no—
104#394—22no—
105#394—22no—
106#394C2×C412yes—
107#394C2×C412yes—
108#392C2×C412yes—
109#392C2312yes—
110#390—21no—
111#390—21no—
112#389—21no—
113#389—21no—
114#388—22no—
115#388—22no—
116#386—22no—
117#386—22no—
118#384—21no—
119#384—41no—
120#382—41no—
121#382C2×C412yes—
122#381—21no—
123#381—21no—
124#380—21no—
125#380—21no—
126#380—21no—
127#380—22no—
128#380—22no—
129#380—22no—
130#380—22no—
131#380C2×C411yescharacteristic
132#380C2×C411yescharacteristic
133#379—21no—
134#379—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—41no—
137C4×C4⋊C4—41no—
138C4×C4⋊C4C2×C412yes—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141#342—22no—
142#342C2314yes—
143#324—22no—
144#324—22no—
145#323—22no—
146#323C2314yes—
147C25—21no—
148C2×#41—41no—
149C2×#41—41no—
150C2×#41—42no—
151C2×#41D814yes—
152C2×#41SD1614yes—
153C22×Q8C22⋊C411yescharacteristic
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—21no—
157C22×D4—41no—
158C22×D4—41no—
159C22×D4—44no—
160C22×D4C22⋊C411yescharacteristic
161C22×D4C22×C412yes—
162C22×D4C2×D411yescharacteristic
163C22×D4C2×D411yescharacteristic
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22⋊C411yescharacteristic
174C23×C4C22⋊C411yescharacteristic
175C23×C4C22×C411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C22×C8—42no—
181#123—44no—
182#123—42no—
183#123—42no—
184C22⋊Q8—48no—
185#121—44no—
186#121—44no—
187#121—44no—
188#121—44no—
189#120—44no—
190#120—42no—
191#120—42no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—41no—
199C4×D4—41no—
200C4×D4—41no—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—41no—
205C4×D4—41no—
206C4×D4—41no—
207C4×D4—41no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—42no—
214C4×D4—48no—
215C4×D4—44no—
216C4×D4—44no—
217C4×D4—41no—
218C4×D4—41no—
219C4×D4—41no—
220C4×D4—41no—
221#117—41no—
222#117—41no—
223#117—41no—
224#117—41no—
225#117—41no—
226#117—41no—
227#117—44no—
228#117—44no—
229#117—44no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C22⋊C411yescharacteristic
244C2×C4⋊C4C22×C412yes—
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4C22⋊C411yescharacteristic
265C2×C22⋊C4C22⋊C411yescharacteristic
266C2×C22⋊C4C22×C411yescharacteristic
267C2×C22⋊C4C2×D411yescharacteristic
268C2×C22⋊C4C2×D411yescharacteristic
269C2×C22⋊C4C2411yescharacteristic
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—22no—
283C2×C42—42no—
284C2×C42C22⋊C411yescharacteristic
285C2×C42C22⋊C411yescharacteristic
286C2×C42C22⋊C411yescharacteristic
287C2×C42C22⋊C411yescharacteristic
288C2×C42C22⋊C411yescharacteristic
289C2×C42C22⋊C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C4⋊C8—42no—
295C4⋊C8—42no—
296#99—44no—
297#99—44no—
298C22⋊C8—24no—
299C22⋊C8—42no—
300C4×C8—42no—
301C4×C8—42no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—42no—
308#95—42no—
309#95C22×C412yes—
310#95C22×C412yes—
311C24—22no—
312C24—42no—
313C24—44no—
314C24—42no—
315C24—42no—
316C24—82no—
317C24C2×C22⋊C411yescharacteristic
318C24C2×C22⋊C411yescharacteristic
319C24C22×D411yescharacteristic
320#41—44no—
321#41—44no—
322#41—44no—
323#41—44no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8#10214yes—
327C2×D4—24no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—48no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—44no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—82no—
348C2×D4—82no—
349C2×D4#10214yes—
350C2×D4C2×D812yes—
351C2×D4C2×D812yes—
352C2×D4C2×SD1612yes—
353C2×D4C2×SD1612yes—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—88no—
393C22×C4—88no—
394C22×C4—81no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4#10214yes—
398C22×C4#10214yes—
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×D812yes—
410C22×C4C2×SD1612yes—
411C22×C4C23×C411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C2×C8—44no—
414C2×C8—84no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—84no—
430C4⋊C4—82no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—44no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—82no—
454C22⋊C4—82no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—44no—
474C42—84no—
475C42#9911yescharacteristic
476C42#9911yescharacteristic
477C42#9911yescharacteristic
478C42#9911yescharacteristic
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—24no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—48no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—84no—
500C23—88no—
501C23—88no—
502C23—84no—
503C23C2×#10212yes—
504C23C2×#10212yes—
505C23C22×C22⋊C411yescharacteristic
506C23C22×D811yescharacteristic
507C23C22×SD1611yescharacteristic
508Q8—44no—
509D4—44no—
510D4—42no—
511D4—42no—
512D4—42no—
513D4—42no—
514D4—44no—
515D4—44no—
516D4—44no—
517D4—44no—
518D4—84no—
519D4—84no—
520D4—88no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—88no—
571C2×C4—84no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4C2×#9911yescharacteristic
578C2×C4C2×#9911yescharacteristic
579C2×C4C2×#9911yescharacteristic
580C2×C4C2×#9911yescharacteristic
581C2×C4C2×#9911yescharacteristic
582C2×C4C2×#9911yescharacteristic
583C2×C4C2×#10212yes—
584C2×C4C2×#10212yes—
585C2×C4C2×#10212yes—
586C2×C4C2×#10212yes—
587C2×C4#42112yes—
588C8—84no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—22no—
593C22—21no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—44no—
601C22—44no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—44no—
609C22—81no—
610C22—81no—
611C22—82no—
612C22—82no—
613C22—816no—
614C22—82no—
615C22—82no—
616C22C22×#9911yescharacteristic
617C22C22×#10211yescharacteristic
618C22C2×#42111yescharacteristic
619C4—21no—
620C4—21no—
621C4—21no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—82no—
632C4—82no—
633C4—82no—
634C4—164no—
635C4#151912yes—
636C4#151912yes—
637C4#212012yes—
638C2—21no—
639C2—21no—
640C2—41no—
641C2—42no—
642C2—41no—
643C2—42no—
644C2—84no—
645C2C2×#151911yescharacteristic
646C2C2×#212011yescharacteristic
647C2#3335811yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
socleC22C22×#99
prime radical#34935C2
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. #20632C2semidirect
3. #17349C2semidirect
4. #11875C2semidirect
5. #11874C2semidirect
6. #11873C2semidirect
7. #11691C2semidirect
10. C22×C4×D4C4semidirect
15. #2536C4semidirect
19. #2486C4semidirect
21. #2408C4semidirect
23. #2392C4semidirect
28. #2285C4semidirect
33. #1799C22semidirect
34. #1740C22semidirect
38. #1459C22semidirect
40. #1458C22semidirect
41. #1344C22semidirect

Maximal subgroups

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

subgroupindex
#349352
#206322
#173492
#118752
#118742
#118732
#116912