Finite Group Databasebeta

← #190046

Subgroups — #190046

Order 512.

1013 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34026C211yesmaximal, characteristic
2#20728C211yesmaximal, characteristic
3#20723C211yesmaximal, characteristic
4#8741C212yesmaximal, complemented
5#8740C212yesmaximal, complemented
6#8739C212yesmaximal
7#8738C212yesmaximal
8#7941C212yesmaximal, complemented
9#7940C212yesmaximal, complemented
10C2×C4×#41C2211yescharacteristic
11#2920C2212yes—
12#2298C411yescharacteristic, complemented
13#2298C411yescharacteristic, complemented
14#2298C2211yescharacteristic
15#2298C2211yescharacteristic
16#2297C2211yescharacteristic
17#2297C2211yescharacteristic
18#2296C411yescharacteristic, complemented
19#2296C411yescharacteristic, complemented
20#2295—21no—
21#2295—21no—
22#2293—21no—
23#2293—21no—
24#2290—21no—
25#2290—21no—
26#2290—21no—
27#2290—21no—
28#2283C2212yes—
29C4×#117C411yescharacteristic, complemented
30C4×#117C411yescharacteristic, complemented
31C4×#117C411yescharacteristic, complemented
32C4×#117C2211yescharacteristic
33C4×#117C2211yescharacteristic
34C2×C43C411yescharacteristic, complemented
35#1822C2212yes—
36#1820C2212yes—
37#1801—21no—
38#1800—21no—
39#1799—21no—
40#1799—21no—
41#1785C2212yes—
42#1782C2212yes—
43C4×#101—21no—
44C4×#100—21no—
45C4×#99—21no—
46C4×#99—21no—
47#1284C2214yes—
48#1284C2214yes—
49#1284C2214yes—
50#1284C2214yes—
51C22×#41C2311yescharacteristic
52C4×#41—22no—
53C4×#41—22no—
54C4×#41—22no—
55C4×#41—22no—
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8D411yescharacteristic
58C2×C4×D4D411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×#117—21no—
65C2×#117—21no—
66C2×#117—21no—
67C2×#117—21no—
68C2×#117C2×C411yescharacteristic
69C2×#117C2×C411yescharacteristic
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C2×#117C2311yescharacteristic
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2311yescharacteristic
80#432—22no—
81#432—22no—
82#431—22no—
83#431—22no—
84#423—22no—
85C2×#101—22no—
86C2×#100—22no—
87C2×#99—22no—
88C2×#99—22no—
89#407C2314yes—
90C4×M16—22no—
91#391—21no—
92#391—21no—
93#390—21no—
94#390—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#388—21no—
100#388—21no—
101#388—21no—
102#388—21no—
103#388—21no—
104#388—21no—
105#388—21no—
106#388—21no—
107#387—21no—
108#387—21no—
109#387—21no—
110#387—21no—
111#387—21no—
112#387—21no—
113#387—21no—
114#387—21no—
115#386—21no—
116#386—21no—
117#386—21no—
118#386—21no—
119#385—21no—
120#385—21no—
121#385—21no—
122#385—21no—
123#383—22no—
124#382—22no—
125#381C2314yes—
126#380—22no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2×C411yescharacteristic
139C4×C4⋊C4C2×C411yescharacteristic
140C4×C4⋊C4C2×C411yescharacteristic
141C4×C4⋊C4C2×C411yescharacteristic
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—21no—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4—22no—
149C4×C22⋊C4C2×C411yescharacteristic
150C4×C22⋊C4C2×C411yescharacteristic
151C4×C22⋊C4C2×C411yescharacteristic
152C4×C22⋊C4C2×C411yescharacteristic
153C4×C22⋊C4C2311yescharacteristic
154C4×C22⋊C4C2311yescharacteristic
155C4×C22⋊C4C2311yescharacteristic
156C4×C22⋊C4C2311yescharacteristic
157#376—21no—
158#376—21no—
159#376—21no—
160#376—21no—
161#376—22no—
162#376C2×C411yescharacteristic
163#376C2×C411yescharacteristic
164C43—21no—
165C43—21no—
166C43C2×C411yescharacteristic
167C43C2×C411yescharacteristic
168C43C2×C411yescharacteristic
169C43C2×C411yescharacteristic
170C43C2×C411yescharacteristic
171C43C2×C411yescharacteristic
172#343—22no—
173#343—22no—
174#342—22no—
175#342—22no—
176C2×#41—22no—
177C2×#41—22no—
178C2×#41—42no—
179C2×#41—42no—
180C22×Q8C2×D411yescharacteristic
181C22×D4C2×D411yescharacteristic
182C22×D4C2×D411yescharacteristic
183C22×D4C2×D411yescharacteristic
184C23×C4—21no—
185C23×C4—21no—
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2411yescharacteristic
195C2×M16—24no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—22no—
212#117—22no—
213#117—22no—
214#117—22no—
215#117—22no—
216#117—22no—
217#117—22no—
218#117—44no—
219#117—44no—
220#117—44no—
221#117—44no—
222#117—42no—
223#117—42no—
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—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C22⋊C411yescharacteristic
238C2×C4⋊C4C22⋊C411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
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—24no—
252C2×C22⋊C4C22⋊C411yescharacteristic
253C2×C22⋊C4C22⋊C411yescharacteristic
254C2×C22⋊C4C22⋊C411yescharacteristic
255C2×C22⋊C4C22⋊C411yescharacteristic
256C2×C22⋊C4C2×D411yescharacteristic
257C2×C22⋊C4C2×D411yescharacteristic
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
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—21no—
280C2×C42—44no—
281C2×C42—44no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42C22⋊C411yescharacteristic
286C2×C42C22⋊C411yescharacteristic
287C2×C42C22⋊C411yescharacteristic
288C2×C42C22⋊C411yescharacteristic
289C2×C42C22×C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C22×C411yescharacteristic
297C2×C42C22×C411yescharacteristic
298C2×C42C22×C411yescharacteristic
299C2×C42C22×C411yescharacteristic
300C2×C42C22×C411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42#4112yes—
304C2×C42#4112yes—
305C2×C42#4112yes—
306C2×C42#4112yes—
307C4⋊C8—42no—
308C4⋊C8—42no—
309#101—44no—
310#100—44no—
311#99—44no—
312#99—44no—
313C22⋊C8—24no—
314C22⋊C8—24no—
315#97—42no—
316C4×C8—42no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—24no—
338#95—24no—
339#95—42no—
340#95—42no—
341C24C2×C22⋊C411yescharacteristic
342C24C2×C22⋊C411yescharacteristic
343C24C22×D411yescharacteristic
344#41—48no—
345#41—48no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—41no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
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—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—21no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—44no—
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×C4#12312yes—
408C22×C4#12312yes—
409C22×C4#12312yes—
410C22×C4#12312yes—
411C22×C4C23×C411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C2×#4112yes—
414C22×C4C2×#4112yes—
415M16—44no—
416M16—44no—
417C2×C8—44no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—84no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C42—24no—
454C42—24no—
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—41no—
468C42—41no—
469C42—41no—
470C42—41no—
471C42—41no—
472C42—41no—
473C42—41no—
474C42—41no—
475C42—42no—
476C42—42no—
477C42—42no—
478C42—42no—
479C42—42no—
480C42—42no—
481C42—42no—
482C42—42no—
483C42—42no—
484C42—42no—
485C42—42no—
486C42—42no—
487C42—84no—
488C42#11711yescharacteristic
489C42#11711yescharacteristic
490C42#11712yes—
491C42#11712yes—
492C42#11711yescharacteristic
493C42#11711yescharacteristic
494C42#12312yes—
495C42#12312yes—
496C42#12312yes—
497C42#12312yes—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23#41012yes—
512C23C22×C22⋊C411yescharacteristic
513C23C2×#12312yes—
514Q8—44no—
515D4—44no—
516D4—44no—
517D4—44no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—84no—
584C2×C4—84no—
585C2×C4—84no—
586C2×C4#38111yescharacteristic
587C2×C4#38111yescharacteristic
588C2×C4#38112yes—
589C2×C4#38112yes—
590C2×C4#38111yescharacteristic
591C2×C4#38111yescharacteristic
592C2×C4#41012yes—
593C2×C4C2×#11711yescharacteristic
594C2×C4C2×#11711yescharacteristic
595C2×C4C2×#12312yes—
596C8—84no—
597C22—21no—
598C22—21no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—42no—
604C22—42no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22C2×#38111yescharacteristic
615C22C2×#41011yescharacteristic
616C22C2×#41011yescharacteristic
617C4—22no—
618C4—21no—
619C4—21no—
620C4—22no—
621C4—21no—
622C4—21no—
623C4—22no—
624C4—22no—
625C4—42no—
626C4—42no—
627C4—42no—
628C4—42no—
629C4—41no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—41no—
634C4—81no—
635C4—81no—
636C4—81no—
637C4—81no—
638C4—164no—
639C4#182712yes—
640C4#182712yes—
641C2—21no—
642C2—21no—
643C2—42no—
644C2—42no—
645C2—81no—
646C2#2068811yescharacteristic
647C2C2×#182711yescharacteristic
648C2C2×#182711yescharacteristic
subgroupG/H
centerC2×C4#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#381
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8741C2semidirect
5. #8740C2semidirect
8. #7941C2semidirect
9. #7940C2semidirect
12. #2298C4semidirect
13. #2298C4semidirect
18. #2296C4semidirect
19. #2296C4semidirect
29. C4×#117C4semidirect
30. C4×#117C4semidirect
31. C4×#117C4semidirect
34. C2×C43C4semidirect

Maximal subgroups

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

subgroupindex
#340262
#207282
#207232
#87412
#87402
#87392
#87382
#79412
#79402