Finite Group Databasebeta

← #482048

Subgroups — #482048

Order 512.

1171 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34937C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic, complemented
3#17349C211yesmaximal, characteristic, complemented
4#11878C212yesmaximal, complemented
5#11877C212yesmaximal, complemented
6#11876C214yesmaximal, complemented
7#11691C214yesmaximal, complemented
8C2×#532—21no—
9C2×#532C411yescharacteristic, complemented
10C2×C4×#41C2211yescharacteristic
11C2×#431—21no—
12C2×#410—21no—
13#2548C411yescharacteristic, complemented
14#2547C2211yescharacteristic
15#2546C411yescharacteristic, complemented
16#2545—21no—
17#2544—21no—
18#2543C412yescomplemented
19#2543C2212yes—
20#2541C2211yescharacteristic
21#2416—21no—
22#2416—21no—
23#2392C412yescomplemented
24#2392C2212yes—
25C4×#121—21no—
26C4×#121—21no—
27#2297—21no—
28#2285—21no—
29C2×C4×C4⋊C4C411yescharacteristic, complemented
30#1799—21no—
31#1799—21no—
32#1799C2212yes—
33#1799C2212yes—
34#1740C2212yes—
35#1461—21no—
36#1460C2212yes—
37#1459—21no—
38#1459C2212yes—
39#1458—21no—
40#1458C2212yes—
41#1344C2218yes—
42#1343C2218yes—
43C22×#41—21no—
44C22×#41D411yescharacteristic
45#532—41no—
46#532—41no—
47#532—41no—
48#532—41no—
49#532—44no—
50#532—44no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#127—21no—
54C2×#127C2×C411yescharacteristic
55C2×#124—21no—
56C2×#124—21no—
57C2×#124—21no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121C2×C412yes—
64C4×#41—41no—
65C4×#41—41no—
66C4×#41—41no—
67C4×#41—41no—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—21no—
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C412yes—
75C2×C4×D4D411yescharacteristic
76C2×C4×D4D411yescharacteristic
77C2×C4×D4D411yescharacteristic
78C2×C4×D4C2311yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C2×#117—22no—
81C2×#117C2×C411yescharacteristic
82C2×#117D411yescharacteristic
83C2×#117C2311yescharacteristic
84C22×C4⋊C4—21no—
85C22×C42—21no—
86C22×C42C2×C411yescharacteristic
87C22×C42D411yescharacteristic
88C22×C42D411yescharacteristic
89#431—22no—
90#431—22no—
91#431—41no—
92#431—41no—
93C2×C4⋊C8—41no—
94#410—44no—
95C2×#99—22no—
96C2×#99—22no—
97C2×C22⋊C8—22no—
98C2×C4×C8—41no—
99#398C2×C412yes—
100#398C2×C412yes—
101#397C2×C412yes—
102#397C2312yes—
103#396—22no—
104#396—22no—
105#394—22no—
106#394—22no—
107#394C2×C412yes—
108#394C2×C412yes—
109#392C2×C412yes—
110#392C2312yes—
111#391—21no—
112#391—21no—
113#390—21no—
114#390—21no—
115#390—21no—
116#390—21no—
117#389—21no—
118#389—21no—
119#388—22no—
120#388—22no—
121#386—22no—
122#386—22no—
123#386—22no—
124#386—22no—
125#384—41no—
126#382—41no—
127#380—21no—
128#380—21no—
129#380—22no—
130#380—22no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—41no—
134C4×C4⋊C4—41no—
135C4×C4⋊C4C2×C412yes—
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C22⋊C4—22no—
139C4×C22⋊C4—22no—
140C2×#95C2×C412yes—
141#342—22no—
142#342C2314yes—
143#324—22no—
144#324—22no—
145#323—22no—
146#323C2314yes—
147C2×#41—41no—
148C2×#41—41no—
149C2×#41—42no—
150C2×#41—42no—
151C2×#41—42no—
152C2×#41—44no—
153C2×#41—44no—
154C2×#41D814yes—
155C2×#41SD1614yes—
156C22×Q8—21no—
157C22×Q8C22⋊C411yescharacteristic
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4C22⋊C411yescharacteristic
164C22×D4C22×C412yes—
165C22×D4C2×D411yescharacteristic
166C22×D4C2×D411yescharacteristic
167C23×C4—22no—
168C23×C4—22no—
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22⋊C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C22×C8—42no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C4⋊Q8—44no—
182#127—41no—
183#127—41no—
184#127—42no—
185#127—42no—
186#124—42no—
187#124—42no—
188#124—44no—
189#124—42no—
190#124—42no—
191#121—44no—
192#121—44no—
193#121—42no—
194#121—42no—
195#121—48no—
196#121—44no—
197#121—44no—
198#121—42no—
199#121—42no—
200C4×Q8—41no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×D4—41no—
205C4×D4—41no—
206C4×D4—41no—
207C4×D4—41no—
208C4×D4—41no—
209C4×D4—41no—
210C4×D4—41no—
211C4×D4—41no—
212C4×D4—41no—
213C4×D4—41no—
214C4×D4—41no—
215C4×D4—41no—
216C4×D4—42no—
217C4×D4—42no—
218C4×D4—48no—
219C4×D4—44no—
220C4×D4—44no—
221C4×D4—41no—
222C4×D4—41no—
223C4×D4—41no—
224C4×D4—41no—
225#117—41no—
226#117—41no—
227#117—41no—
228#117—41no—
229#117—41no—
230#117—41no—
231#117—44no—
232#117—44no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4C22⋊C411yescharacteristic
244C2×C4⋊C4C22⋊C411yescharacteristic
245C2×C4⋊C4C22×C412yes—
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C22⋊C4—22no—
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—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4C22⋊C411yescharacteristic
262C2×C22⋊C4C22⋊C411yescharacteristic
263C2×C22⋊C4C22×C411yescharacteristic
264C2×C22⋊C4C2×D411yescharacteristic
265C2×C22⋊C4C2×D411yescharacteristic
266C2×C22⋊C4C2411yescharacteristic
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
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—22no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—44no—
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—
312C24C2×C22⋊C411yescharacteristic
313C24C2×C22⋊C411yescharacteristic
314C24C22×D411yescharacteristic
315#41—44no—
316#41—44no—
317#41—44no—
318#41—44no—
319#41—88no—
320#41—88no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8#10214yes—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—42no—
349C2×D4—82no—
350C2×D4—82no—
351C2×D4—82no—
352C2×D4#10214yes—
353C2×D4C2×D812yes—
354C2×D4C2×D812yes—
355C2×D4C2×SD1612yes—
356C2×D4C2×SD1612yes—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
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—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—82no—
394C22×C4—81no—
395C22×C4#10214yes—
396C22×C4#10214yes—
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
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×D812yes—
408C22×C4C2×SD1612yes—
409C22×C4C23×C411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C2×C8—44no—
412C2×C8—84no—
413C4⋊C4—24no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C4⋊C4—84no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—44no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—82no—
449C22⋊C4—82no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
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—44no—
469C42—42no—
470C42—42no—
471C42—84no—
472C42#9911yescharacteristic
473C42#9911yescharacteristic
474C42#9911yescharacteristic
475C42#9911yescharacteristic
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23C2×#10212yes—
496C23C2×#10212yes—
497C23C22×C22⋊C411yescharacteristic
498C23C22×D811yescharacteristic
499C23C22×SD1611yescharacteristic
500Q8—44no—
501Q8—84no—
502D4—44no—
503D4—42no—
504D4—42no—
505D4—42no—
506D4—42no—
507D4—82no—
508D4—84no—
509D4—84no—
510D4—84no—
511D4—88no—
512D4—84no—
513D4—88no—
514D4—82no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—88no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4—84no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4C2×#9911yescharacteristic
571C2×C4C2×#9911yescharacteristic
572C2×C4C2×#9911yescharacteristic
573C2×C4C2×#9911yescharacteristic
574C2×C4C2×#9911yescharacteristic
575C2×C4C2×#9911yescharacteristic
576C2×C4C2×#10212yes—
577C2×C4C2×#10212yes—
578C2×C4C2×#10212yes—
579C2×C4C2×#10212yes—
580C2×C4#42112yes—
581C8—84no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—22no—
586C22—21no—
587C22—22no—
588C22—42no—
589C22—41no—
590C22—41no—
591C22—42no—
592C22—42no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—81no—
600C22—81no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22C22×#9911yescharacteristic
609C22C22×#10211yescharacteristic
610C22C2×#42111yescharacteristic
611C4—21no—
612C4—21no—
613C4—21no—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—44no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—82no—
625C4—82no—
626C4—164no—
627C4#151912yes—
628C4#151912yes—
629C4#212612yes—
630C2—21no—
631C2—21no—
632C2—41no—
633C2—42no—
634C2—41no—
635C2—42no—
636C2—162no—
637C2C2×#151911yescharacteristic
638C2#3334711yescharacteristic
639C2C2×#212611yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
socleC22C22×#99
prime radical#34937C2
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. #11878C2semidirect
5. #11877C2semidirect
6. #11876C2semidirect
7. #11691C2semidirect
9. C2×#532C4semidirect
13. #2548C4semidirect
15. #2546C4semidirect
18. #2543C4semidirect
23. #2392C4semidirect
29. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#349372
#206322
#173492
#118782
#118772
#118762
#116912