Finite Group Databasebeta

← #189232

Subgroups — #189232

Order 512.

978 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60210C211yesmaximal, characteristic
2#21899C211yesmaximal, characteristic
3#21856C211yesmaximal, characteristic
4C4×#428C211yesmaximal, characteristic, complemented
5#17796C211yesmaximal, characteristic, complemented
6#8669C212yesmaximal, complemented
7#8668C212yesmaximal, complemented
8#8667C212yesmaximal, complemented
9#8666C212yesmaximal, complemented
10#8092C211yesmaximal, characteristic, complemented
11C2×#1303C211yesmaximal, characteristic, complemented
12#3444C2211yescharacteristic
13#2969C412yescomplemented
14#2969C412yescomplemented
15#2968—22no—
16#2965—22no—
17#2965C2212yes—
18C8×#41C2212yes—
19#2962C411yescharacteristic, complemented
20#2962C411yescharacteristic, complemented
21#2935C411yescharacteristic, complemented
22#2935C411yescharacteristic, complemented
23#2933—21no—
24#2933—21no—
25#2922—21no—
26#2920C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#428—21no—
29C2×#423C2211yescharacteristic
30C4×#131—21no—
31C2×C4×M16C2211yescharacteristic
32C4×#117C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1994—21no—
35#1986—21no—
36#1980—21no—
37#1962—21no—
38#1896—22no—
39#1871—22no—
40#1806C2212yes—
41#1806C2212yes—
42#1806C2212yes—
43#1806C2212yes—
44#1776—22no—
45C4×#107C2212yes—
46C4×#106C2212yes—
47#1736—21no—
48#1569—22no—
49#1567C2212yes—
50#1566C2212yes—
51#1565—21no—
52#1565—21no—
53#1303C2212yes—
54#1303C2212yes—
55#1303C2211yescharacteristic
56#1303C2211yescharacteristic
57#1303C2211yescharacteristic
58#1303C2211yescharacteristic
59C2×#131—21no—
60C2×#131—21no—
61C22×M16—21no—
62C22×M16D411yescharacteristic
63C22×M16D411yescharacteristic
64#533C2×C411yescharacteristic
65#533D411yescharacteristic
66#533Q811yescharacteristic
67#532C2×C411yescharacteristic
68#532D411yescharacteristic
69#532Q811yescharacteristic
70#529—21no—
71#529—21no—
72#529—21no—
73#529—21no—
74#529C2×C412yes—
75#529Q812yes—
76#525—21no—
77#525—21no—
78#525D412yes—
79C4×#41—21no—
80C4×#41—21no—
81C4×#41C2312yes—
82C2×#117—21no—
83C22×C4⋊C4—21no—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#446—22no—
87#446—22no—
88#446C2×C412yescomplemented
89#446C2×C412yescomplemented
90C8×Q8C2×C412yescomplemented
91C8×Q8C2×C412yescomplemented
92#436—22no—
93#436—22no—
94#436C2×C412yescomplemented
95#436C2×C412yescomplemented
96#435—22no—
97#435—22no—
98#435—22no—
99#435—22no—
100#435—22no—
101#435—22no—
102C8×D4—22no—
103C8×D4—22no—
104C8×D4C2×C412yescomplemented
105C8×D4C2×C412yescomplemented
106#433—22no—
107#433—22no—
108#432—21no—
109#432—21no—
110#431—21no—
111#431—21no—
112#431—22no—
113#431—22no—
114#431—21no—
115#431—21no—
116#431C2×C411yescharacteristic
117#431C2×C411yescharacteristic
118#431C2311yescharacteristic
119#431C2311yescharacteristic
120#428—28no—
121C2×#107—22no—
122C2×#106—22no—
123#424—21no—
124#424—21no—
125#423—21no—
126#423—21no—
127#423—21no—
128#423C2311yescharacteristic
129#423C2312yes—
130#423C2311yescharacteristic
131C2×C4⋊C8—21no—
132C2×C4⋊C8—21no—
133C2×C4⋊C8C2×C411yescharacteristic
134C2×C4⋊C8C2×C411yescharacteristic
135C2×#104—22no—
136C2×#104—22no—
137C2×#104—22no—
138C2×#104—22no—
139#408—21no—
140#408—21no—
141#407—21no—
142C2×C22⋊C8—21no—
143#405—21no—
144#405—21no—
145#405—22no—
146#405—22no—
147C4×M16—22no—
148C4×M16—21no—
149C4×M16C2×C411yescharacteristic
150C4×M16C2×C411yescharacteristic
151C4×M16C2×C412yes—
152C4×M16D411yescharacteristic
153C4×M16D411yescharacteristic
154C4×M16D412yes—
155C2×#97D412yes—
156C2×C4×C8—21no—
157C2×C4×C8—22no—
158C2×C4×C8C2312yes—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—21no—
162C4×C4⋊C4—21no—
163C4×C4⋊C4C2312yes—
164C4×C4⋊C4C2311yescharacteristic
165C4×C4⋊C4C2311yescharacteristic
166C4×C22⋊C4—21no—
167C4×C22⋊C4—21no—
168#376—21no—
169C2×#95—21no—
170C43—21no—
171#340—22no—
172#340—22no—
173#339—22no—
174#339—22no—
175#339—22no—
176#339—22no—
177#335—22no—
178#335—22no—
179#334—22no—
180#334—22no—
181C2×#41—21no—
182C2×#41—21no—
183C2×#41C2×D412yes—
184C23×C4—41no—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4#4111yescharacteristic
188#131—44no—
189#131—44no—
190C2×M16—24no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—42no—
197C2×M16—41no—
198C2×M16—81no—
199C2×M16—81no—
200C2×M16C22⋊C411yescharacteristic
201C2×M16C22⋊C411yescharacteristic
202C2×M16C22⋊C411yescharacteristic
203C2×M16C22⋊C412yes—
204C2×M16C22⋊C412yes—
205C2×M16C22⋊C411yescharacteristic
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—41no—
211C22×C8C2×D412yes—
212C22×C8C2×D412yes—
213C4⋊Q8C4⋊C411yescharacteristic
214C4⋊Q8C4⋊C411yescharacteristic
215C4⋊Q8C4⋊C411yescharacteristic
216#127C4⋊C411yescharacteristic
217#126—22no—
218#126—22no—
219#125—21no—
220#125—21no—
221#125C4⋊C412yes—
222#124—21no—
223#124—21no—
224#124C4⋊C412yes—
225#123—22no—
226#123—22no—
227#123—22no—
228C22⋊Q8—21no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232C22⋊Q8C4⋊C412yes—
233C22⋊Q8C4⋊C412yes—
234#121—21no—
235#121—21no—
236#121—21no—
237#121—21no—
238#121C4⋊C412yes—
239#121C4⋊C412yes—
240#120—22no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8C22×C412yes—
246C4×Q8C2×Q812yes—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4C22×C412yes—
258C4×D4C2×Q812yes—
259#117—21no—
260#117—21no—
261#117—22no—
262#117—22no—
263#117—22no—
264#117—24no—
265#117—42no—
266#117C2×D412yes—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C42—22no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—22no—
278C2×C42—41no—
279C2×C42—41no—
280C2×C42—41no—
281C2×C42—81no—
282C2×C42—81no—
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×Q811yescharacteristic
290C2×C42C2×Q811yescharacteristic
291C2×C42#4111yescharacteristic
292C2×C42#4111yescharacteristic
293C2×C42#4111yescharacteristic
294C2×C42C2411yescharacteristic
295#107—28no—
296#106—28no—
297C4⋊C8—22no—
298C4⋊C8—22no—
299C4⋊C8—22no—
300C4⋊C8—22no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—22no—
305C4⋊C8—21no—
306C4⋊C8—22no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8—41no—
310C4⋊C8—41no—
311C4⋊C8C22×C411yescharacteristic
312C4⋊C8C22×C411yescharacteristic
313C4⋊C8C22×C411yescharacteristic
314C4⋊C8C22×C411yescharacteristic
315#104—82no—
316#104—82no—
317#104—82no—
318#104—82no—
319C22⋊C8—21no—
320C22⋊C8—21no—
321C22⋊C8—21no—
322C22⋊C8—21no—
323C22⋊C8—22no—
324C22⋊C8—22no—
325C22⋊C8—22no—
326C22⋊C8—22no—
327C22⋊C8—41no—
328C22⋊C8—41no—
329#97—22no—
330#97—22no—
331#97—41no—
332#97C22⋊C412yes—
333#97C22⋊C412yes—
334#97C22⋊C412yes—
335#97C22⋊C412yes—
336C4×C8—22no—
337C4×C8—22no—
338C4×C8—21no—
339C4×C8—21no—
340C4×C8—21no—
341C4×C8—21no—
342C4×C8—41no—
343C4×C8C22×C412yes—
344C4×C8C22×C412yes—
345C4×C8C2×D412yes—
346C4×C8C2×D412yes—
347#95—22no—
348#95—21no—
349#95—21no—
350#95—41no—
351#95—41no—
352C24#12111yescharacteristic
353#41—22no—
354#41—22no—
355#41—22no—
356#41—22no—
357#41—42no—
358#41—42no—
359#41—82no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8C2×C4⋊C412yes—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4C2×C4⋊C412yes—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—44no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—81no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C4⋊C411yescharacteristic
388C22×C4C2×C4⋊C411yescharacteristic
389C22×C4#11711yescharacteristic
390C22×C4#11711yescharacteristic
391C22×C4#12111yescharacteristic
392C22×C4#12111yescharacteristic
393C22×C4#12111yescharacteristic
394C22×C4#12111yescharacteristic
395C22×C4#12111yescharacteristic
396C22×C4C22⋊Q811yescharacteristic
397C22×C4C22⋊Q811yescharacteristic
398C22×C4C22⋊Q811yescharacteristic
399C22×C4C22⋊Q811yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C2×#4111yescharacteristic
403M16—22no—
404M16—22no—
405M16—22no—
406M16—24no—
407M16—42no—
408M16—44no—
409M16—44no—
410M16—41no—
411M16—42no—
412M16—41no—
413M16—42no—
414M16—41no—
415M16—41no—
416M16—82no—
417M16#10414yescomplemented
418C2×C8—24no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—42no—
424C2×C8—44no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—44no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—42no—
431C2×C8—82no—
432C2×C8C2×C22⋊C412yes—
433C2×C8C2×C22⋊C412yes—
434C2×C8C2×C22⋊C412yes—
435C2×C8C2×C22⋊C412yes—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4C2×C4⋊C412yes—
454C4⋊C4C2×C4⋊C412yes—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—44no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—44no—
469C42—41no—
470C42—42no—
471C42—41no—
472C42—42no—
473C42—41no—
474C42—41no—
475C42—82no—
476C42—82no—
477C42C2×C22⋊C411yescharacteristic
478C42C2×C22⋊C411yescharacteristic
479C42C2×C4⋊C411yescharacteristic
480C42C2×C4⋊C411yescharacteristic
481C42#11711yescharacteristic
482C42#11711yescharacteristic
483C42#12111yescharacteristic
484C42#12111yescharacteristic
485C42C22⋊Q811yescharacteristic
486C42C22⋊Q811yescharacteristic
487C42C22⋊Q811yescharacteristic
488C42C22⋊Q811yescharacteristic
489C42C23×C411yescharacteristic
490C42C22×D411yescharacteristic
491C42C22×Q811yescharacteristic
492C42C2×#4111yescharacteristic
493C23—22no—
494C23—21no—
495C23—21no—
496C23—22no—
497C23—41no—
498C23#38011yescharacteristic
499C23#38011yescharacteristic
500C23C2×#12111yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—42no—
504Q8—81no—
505D4—22no—
506D4—22no—
507D4—44no—
508D4—42no—
509D4—42no—
510D4—42no—
511D4—81no—
512D4—82no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—82no—
550C2×C4—88no—
551C2×C4—82no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4#38011yescharacteristic
555C2×C4#38011yescharacteristic
556C2×C4#38011yescharacteristic
557C2×C4#38011yescharacteristic
558C2×C4#38011yescharacteristic
559C2×C4#38011yescharacteristic
560C2×C4C2×#10412yes—
561C2×C4#42112yes—
562C2×C4C22×C22⋊C411yescharacteristic
563C2×C4C22×C4⋊C411yescharacteristic
564C2×C4C2×#11711yescharacteristic
565C2×C4C2×#12111yescharacteristic
566C2×C4C2×C22⋊Q811yescharacteristic
567C2×C4C2×C22⋊Q811yescharacteristic
568C2×C4#57611yescharacteristic
569C2×C4#57711yescharacteristic
570C2×C4#57711yescharacteristic
571C2×C4#57811yescharacteristic
572C8—22no—
573C8—22no—
574C8—22no—
575C8—41no—
576C8—41no—
577C8—41no—
578C8—41no—
579C8—82no—
580C8—81no—
581C8—81no—
582C8C2×#10414yes—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22#180612yes—
596C22C2×#38011yescharacteristic
597C22#306211yescharacteristic
598C22#306311yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—21no—
606C4—21no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—82no—
611C4—82no—
612C4—84no—
613C4#180612yes—
614C4C22×#10411yescharacteristic
615C4C2×#42111yescharacteristic
616C4#291211yescharacteristic
617C4#291311yescharacteristic
618C4#306211yescharacteristic
619C4#306311yescharacteristic
620C2—21no—
621C2—21no—
622C2—22no—
623C2—41no—
624C2—82no—
625C2C2×#180611yescharacteristic
626C2#2066411yescharacteristic
627C2#2067511yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#380
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C4×#428C2semidirect
5. #17796C2semidirect
6. #8669C2semidirect
7. #8668C2semidirect
8. #8667C2semidirect
9. #8666C2semidirect
10. #8092C2semidirect
11. C2×#1303C2semidirect
13. #2969C4semidirect
14. #2969C4semidirect
19. #2962C4semidirect
20. #2962C4semidirect
21. #2935C4semidirect
22. #2935C4semidirect
88. #446C2×C4semidirect
89. #446C2×C4semidirect
90. C8×Q8C2×C4semidirect
91. C8×Q8C2×C4semidirect
94. #436C2×C4semidirect
95. #436C2×C4semidirect
104. C8×D4C2×C4semidirect
105. C8×D4C2×C4semidirect
417. M16#104semidirect

Maximal subgroups

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

subgroupindex
#602102
#218992
#218562
C4×#4282
#177962
#86692
#86682
#86672
#86662
#80922
C2×#13032